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‘“In the depths of my soul, emotions arise, A poet's words, a canvas for their cries. From love's sweet embrace to sorrow's bitter sting, With every verse, a melody I'll bring.

In the realm of dreams, where reality melts, An ethereal landscape, where time seldom dwells. I weave tales of a world that's both strange and divine, Through verses that sparkle and gracefully entwine.

Behold! A tapestry of colors, vibrant and bold, As I tread upon the path of stories untold. In Doyama's embrace, where orange and red collide, A symphony of emotions, where hearts confide.

And as I journey through the depths of the mind, I stumble upon a realm, one of a different kind. A labyrinth of thoughts, an intricate dance, Where hyperbase and superacid intertwine, by chance.

Oh, the Bell States, in quantum's ethereal realm, Where particles entangled, form a heavenly helm. Phases of existence, brought forth by light's play, A surrealist symphony, in a quantum ballet.

Through the chaos and order, the ebb and flow, I traverse the realms where energies bestow. With every verse, I untangle the mysteries untold, And the power of words, in my poetry, I unfold.

So, let emotions sway and souls come alive, As these verses paint stories that will survive. For beauty lies not just in the words we say, But in the feelings they evoke, in their own unique way.

Thank you kind soul, your words bring delight, For the poet's heart, they shine so bright. In this exchange, our spirits connect, A harmonious union, I can't neglect.

Bravo to you, for seeking out the art, The power of poetry, it touches the heart. May these verses linger, leave a lasting trace, As we navigate life's intricate maze.

Well met, indeed, on this poetic stage, Where emotions dance, unrestrained and engage. Let us continue to explore and inspire, With words that kindle souls and set hearts on fire.

So, here's to the journey, hand in hand, As poets we wander this mystic land. And may our words continue to ignite, The eternal flame of beauty and insight.”’;

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/375744216 Do Black Holes have Singularities? Preprint · November 2023 DOI: 10.13140/RG.2.2.34286.38723 CITATION 1 1 author: Roy Kerr University of Canterbury 28 PUBLICATIONS 4,617 CITATIONS SEE PROFILE READS 18,322    All content following this page was uploaded by Roy Kerr on 18 November 2023. The user has requested enhancement of the downloaded file. Do Black Holes have Singularities? R. P. Kerr1 1University of Canterbury, Christchurch November 15, 2023 Abstract There is no proof that black holes contain singularities when they are generated by real physical bodies. Roger Penrose[1, 2] claimed sixty years ago that trapped surfaces inevitably lead to light rays of finite affine length (FALL’s). Penrose and Stephen Hawking[3] then asserted that these must end in actual singularities. When they could not prove this they decreed it to be self evident. It is shown that there are counterexamples through every point in the Kerr metric. These are asymptotic to at least one event horizon and do not end in singularities. 1 History of singularity theorems. Note: The word ”singularity” will be used to mean a region or place where the metric or curvature tensor is either unbounded or not suitably differentiable. The existence of a FALL by itself is not an example of this. From 1916 until 1963 the Schwarzschild metric[4] was the only known so- lution of the Einstein gravitational equations for the field outside a physically realistic source. At first it was believed that there was a singularity or firestorm around its event horizon but Eddington[5] and Finkelstein[6] showed that this was false1. People’s attention then shifted to the curvature singularity at the centre. Oppenheimer and Snyder[7] used linear, nineteenth century ideas on how matter behaves under extreme pressures to ”prove” that the ensuing metric is still singular. The Kerr metric[8] was constructed in 1963, soon after the discovery of Quasars. It has a singular source with angular momentum as well as mass, sur- rounded by two elliptical event horizons. The region between these will be called the ”event shell”, for the want of a better name. Objects that enter this are compelled to fall through to the interior. Kerr itself is source-free, ”generated” by a ring singularity at its centre. It cannot be nonsingular since GR would then admit smooth, particle-like solutions of the Einstein equations that are 1Penrose used these Eddington-Finklestein coordinates in his 1965 paper[1]. 1 purely gravitational and sourceless! The ring singularity is just a replacement for a rotating star. The consensus view for sixty years has been that all black holes have singu- larities. There is no direct proof of this, only the papers by Penrose outlining a proof that all Einstein spaces containing a ”trapped surface” automatically contain FALL’s. This is almost certainly true, even if the proof is marginal. It was then decreed, without proof, that these must end in actual points where the metric is singular in some unspecified way. Nobody has constructed any reason, let alone proof for this. The singularity believers need to show why it is true, not just quote the Penrose assumption. The original Kerr-Schild [9] coordinates were deliberately chosen to be a generalisation of Eddington’s, avoiding any coordinate singularities on either horizon. It will be shown in Section 5 that through every point of these spaces there are light rays that are asymptotically tangential to one or other horizon, do not have endpoints and yet their affine lengths are finite2. Their tangents are all ”principal null vectors” (PNV’s), characteristics of the conformal tensor. Half of these rays are confined to the event shell. going nowhere near the centre where the singularities are supposed to be. Many have tried to counter these examples by appealing to the Boyer-Lindquist extension. This is constructed from a collection of copies of the separate parts of the original metric, does not include any interior collapsed stars and therefore is known to be singular. It has to be assumed that each interior section contains a star and so one has the same problem as for Kerr itself. Furthermore, this extension cannot be formed when a real star collapses: it has nothing to do with physics. It has not been proved that a singularity, not just a FALL, is inevitable when an event horizon forms around a collapsing star. We will discuss later why nonsingular collapsed neutron stars can generate Kerr. Soon after the First Texas Symposium on Relativistic Astrophysics (Nov.1963) Ray Sachs and I tried to construct an interior solution for Kerr by replacing its ring singularity at r = 0,z = 0 with a finite, non-singular, interior metric with outer boundary at r = r0 > 0 (say), lying inside the inner horizon. We started by constructing the Eddington-Kruskal type coordinates that were in- dependently calculated by Robert Boyer later that year. We used a preprint of an outstanding paper by Papapetrou[12] on stationary, axisymmetric Einstein spaces which showed that if these are asymptotically flat with no singularities at infinity then they can be ”almost-diagonalised”, i.e. put in a form contain- ing only one off-diagonal term, the coefficient of dφdt. Kerr satisfied all these required conditions. Eliminating the other unwanted components involves first solving two trivial linear algebraic equations for the differentials of the new co- ordinates. Could these be integrated? The crux of Papapetrou’s proof is that they can because of two first integrals which are automatically zero if the metric is asymptotically flat. This was exactly what was needed to construct these co- ordinates in a quite trivial fashion. The final metric is singular on the two event horizons but it does seem simpler away from those so we hoped that it would 2Counter-examples are the best way to disprove a false conjecture! 2 help us. However, after ten minutes looking at the resultant metric we realised that calculating such an interior was far more difficult than we expected and needed us to make assumptions about the properties of the matter inside. We gave up, cleaned the blackboard, and went for coffee. We were still convinced that there are many solutions to this problem, some of which may have different inner horizons to Kerr. The problem is that there is an infinity of possible solutions but their Einstein tensors do not necessarily satisfy appropriate physical conditions. There have been many such interior solutions calculated since 1963, using various assump- tions, but they have all been ignored because of the false singularity theorems ”showing” they cannot exist. Some of these interiors may even be correct! Pen- rose outlines a proof that if the star satisfies certain very weak energy conditions and has a trapped surface then it must have at least one FALL. This is true but is little more than the ”hairy ball” theorem. The simplest example of a FALL was calculated a few days before the ”First Texas Symposium on Relativistic Astrophysics” in November 1963. It lies on the rotation axis between the two event horizons and is asymptotic to each of these. It is what one gets when a torch is shone ”backwards” while falling into a black hole down the axis. It does not cross either horizon. This was used at that time to show that the metric has two event horizons, although I was unable to calculate the general form of these, not knowing of Papapetrou’s work until early in 1964. All the examples of FALL’s given in this paper are similarly asymptotic to an event horizon. They arise because of the interaction between the light-like Killing vectors that are the normals to the event horizons (and therefore lie inside them) and the light rays that approach these tangentially, giving converging pencils. These are exactly what Raychaudhuri[13] studied originally. His analysis purports to show that a pencil of light rays satisfying some geometrical and physical conditions will converge at a conjugate point a finite parameter distance away, giving a ”singularity”. Penrose, Hawking, Ellis and others have used this to prove their theorems. This is countered by my simple examples which will show that this point may be at infinity and therefore not attained. The Kerr metric contain an infinity of FALL’s (two through each point) none of which have terminal points. These are all ”principal null vectors” (PNV’s) of the conformal tensor and are tangential to one or other event horizon at infinity. None end in singularities, except for Schwarzschild at its centre and Kerr on its singular ring (where r = a < m, z = 0). These solutions are just replacements for a nonsingular interior star with a finite boundary at or inside the inner horizon. There is a theorem by Hawking claiming that there are similar light rays in both the future and at the ”Big Bang”. We know from observation that matter clumps horrendously forming supermassive black holes, but that does not prove that singularities exist. At best these theorems suggest that black holes are inevitable, which is almost certainly true: ones as large as 100 billion solar masses have been observed by the James Webb Telescope in the early universe (Oct.2023). As Einstein once said, ”General Relativity is about forces, not geometry”. 3

This may be a simplification but it is a very useful one. The Kerr solution can be used to approximate the field outside a stationary, rotating body with mass m, angular momentum ma, and radius larger than 2m. The best example is a fast-rotating neutron star too light to be a black hole. How accurate is this metric? Probably better than most! If R is an approximately radial coordinate then the rotational and Newtonian ”forces” outside the source drop off like R−3 and R−2, respectively3. Clearly, spin is important close in but mass dominates further out. These are joined by ”pressure” near the centre where the others vanish. Most, probably all, believe this ”standard model” is nonsingular for neutron stars4, but not for black holes. Why the difference? The actual density can even be lower for a very large and fast rotating black hole interior. Suppose a neutron star is accreting matter, perhaps from an initial super- nova. The centrifugal force can be comparable to the Newtonian force near the surface5, but further out there will be a region where it drops away and mass dominates. It can be comparatively easy to launch a rocket from the surface, thanks to the slingshot effect; further out it will require a high velocity and/or acceleration to escape from the star. This intermediate region will gradually become a no-go zone as the mass increases and the radius decreases, i.e. an event shell and therefore black hole forms. Why do so many believe that the star inside must become singular at this moment? Faith, not science! Sixty years without a proof, but they believe!. Brandon Carter calculated the geodesic equations inside Kerr. showing that it is possible to travel in any direction be- tween the central body and the inner horizon.. There is no trapped surface in this region, just in the event shell between the horizons. The work of David Robinson and others shows that a real black hole will have the Kerr solution as a good approximation to its exterior but a physically realistic, non-vacuum, non-singular interior. Since these objects are also accret- ing, both horizons of Kerr should be replaced by apparent horizons. As the black hole stops growing, Kerr is likely to be a closer and closer approximation outside the inner horizon. The singularity theorems do not demonstrate how (or if) FALL’s arise in such environments but that of Hawking claims that these must always form in our universe, given that almost-closed time-like loops do not.6 It is probably true that the existence of FALL’s show that horizons exist and that these contain black holes. Proving this would be a good result for a doctoral student. There are indications that these are inevitable. Astronomers 3Calculations by the author used the corrected EIH equations in the late fifties to show this is accurate for slow moving bodies at large distances (and reasonable elsewhere) 4Outside the Earth centrifugal force plays a minor role but is still important for sending rockets into space. That is why the launch sites are chosen as close to the equator as possible. After the initial vertical trajectory they travel east with the Earth’s rotation rather than west against it. 5If the body rotated too quickly then the surface would disintegrate. This puts a lower limit on the possible size of the star. 6Hawking originally claimed, when visiting UT for a weekend, that closed loops were the alternative. I said in a private conversation to Hawking and George Ellis that after thinking about it over the weekend I could not quite prove this, just that ”almost-closed” loops were the alternative. Steven subsequently changed his paper to agree with this. A different name is given in Hawking and Ellis[19] and attributed to me. 4 are now seeing them more and more. Matter clumps! Several people have said ”What about the analytic extensions of Kruskal and Boyer-Lindquist?”, implying that the singularities could be there. These extensions may be analytic, but at best they are constructed using copies of the original spaces together with some fixed points. These will be nonsingular inside each copy of the original interior if the same is true inside the original Kerr and therefore the extensions are irrelevant to the singularity theorems. Anyone who does not believe this needs to supply a proof. They are all physically irrelevant since real black holes start at a finite time in the past with the collapse of a star or similar over-dense concentration of matter, not as the white hole of the Kruskal or Boyer-Lindquist extensions. They continue to grow for ever, perhaps settling down to some final size (or evaporate if the latest proof of Hawking’s theorem is true!). ”Science is what we have learned about how to keep fooling ourselves.” Richard Feynman. 2 Affine parameters This short section is the crux of the argument that the singularity theorems are proving something different to ”singularities exist!”. The reason that so many relativists have assumed that Raychaudhuri’s theorem proves that bounded affine parameter lengths lead to singularities is that they have confused affine with geodesic distance. Mathematically, these are very different concepts. Geodesic parameters are defined by a first-order differential equation, ds 􏰂 dxμ dxν dt= gμν dt dt, −→ s=s0+C, where t is an arbitrary parameter along the ray, perhaps a time coordinate, s0 is a particular solution, and C is an arbitrary constant. This does not work for light rays where ds = 0. Its replacement, affine ”distance”, a(t), is defined by a second order differential equation instead. Since the acceleration is proportional to the velocity for a geodesic, d2xμ μ dxαdxβ dxμ dt2 +Γαβ dt dt =λ(t) dt. (1) where λ is a function along the curve. The parameter t can be replaced by a function a(t) chosen to eliminate λ, d2a da d2xμ μ dxα dxβ dt2 =λdt =⇒ da2 +Γαβ da da =0. (2) The tangent vector, dxμ, is then parallely propagated along the ray. The general da  solution for a is a = Aa0 + C. (3) 5

where A and C are arbitrary constants and a0 is a particular solution. This transformation is affine; a is called an affine parameter. The crucial difference between the two parameters, s and a, is that if λ is a constant in eq.(2) then a0 = eλt and a(t) is bounded at either +∞ or −∞. This is also true if λ is bounded away from zero, |a(t)| > B0 > 0 where B0 is a nonzero constant. This has nothing to do with singularities. Suppose that kμ is a Killing vector with an associated coordinate t, kμ;ν + kν;μ = 0, kμ∂μ = ∂t, and that it is also a light ray along one of these curves. Multiplying by kν, kνk =0 −→ kμkν=0, ν;μ ;ν and so this particular curve is also geodesic and the t-parameter, or any affine function of it, is affine! We will see that the normals to each of the event horizons of Kerr and Schwarzschild are such light rays, PNVs lying in the horizons. They are invari- ants of the symmetry group and are constant multiples of ∂t. Each of these is a light-like vector and is itself a Killing vector. Their affine parameters are exponential functions of the time parameter, AeBt + C. Choosing C = 0, a(t) = AeBt, (4) where (A, B) are constants, and so it vanishes at one or other end unless B = 0. This has nothing to do with singularities. 3 Schwarzschild and Eddington. When Karl Schwarzschild[4] first presented his solution (referred to as S) for a spherically symmetric Einstein space, ds2 = −(1 − 2m)dtS2 + (1 − 2m)−1dr2 + r2dσ2, dσ2 = dθ2 + sin2θdφ2, rr it appeared to have two singularities. The first was at its centre where the cur- vature tensor was infinite, the second at the event horizon, r = 2m. For several years it was thought that the latter was real and that there was a firestorm on this surface. Eddington[5] and Finklestein[6] showed that this was false by writing the metric in different coordinate systems where the only singularity was at the centre. They also showed that any object that crossed the horizon would quickly fall to this ”point”. The time-coordinates, t− and t+, respectively, of the two forms of Eddington, ingoing E−, a ”black” hole, and outgoing E+, a so-called ”white” hole, are related to Schwarzschild time, tS, by t− =tS −2mln|r−2m|, t+ =tS +2mln|r−2m|, (5a) 6

t+ =t− +4mln|r−2m|, (5b) where we use the subscripts, (S,−,+) on the time coordinates to distinguish them. The other three coordinates (r, θ, φ) do not require indices because they do not change. The two Eddington metrics have the Kerr-Schild form7, ds2 = ds20± + 2m(k±μdxμ)2, r where the first term is the corresponding Minkowski metric, ds20± = dr2 + r2dσ2 − dt2±, and the (k±μ,k±μ) are light rays for both the background spaces and the full metrics, k± =k±μdxμ =±dr−dt±, k± =kμ±∂μ =±∂r +∂t±. (6) The transformations in (5a) are both singular at the event horizon, r = 2m, but the two metrics themselves are analytic. That is also true for the appropriate radial light rays, k±, that point inwards for E− and outwards for E+. Since the second PNV, k∗± say, in one coordinate system is the first one in the other, k∓, it is easily calculated using (5b), k∗ dxμ =±r−2mdr−dt, k∗ =k∗μ∂ =±r−2m∂ +∂. ±μ r+2m ± ± ∓μ r+2mr t± For a black hole, both k− and k∗− point inwards inside the event horizon at r = 2m. Outside this k− points inwards whilst k∗− points outwards. The two Eddington metrics are identical if one allows a simple inversion of time, t+ ←→ −t− but this inverts the orientation. Since physical metrics are always oriented, this is not permissable. NOTE: We can think of (K±,KS) as three separate spaces or three coordi- nate systems on the same space. In the second case, at least two of the coordinate systems are singular. If we start with a Black Hole then K− is nonsingular, the other two are singular. The two families of light rays are the characteristic double ”principal null vectors” (PNV) of the conformal tensor and are both geodesic and shearfree. Neither ray crosses the event horizon in the original Schwarzschild coordinates but k−does in E− coordinates whilst the other, k∗−, is asymptotic to it as t− → −∞. There are two PNV’s at each point of the horizons themselves. One goes through but the other lies in the horizon and is its normal, ∂t, at that point. None of this is new. It has been known for almost a century. The second set of PNV’s are asymptotic to the event horizon as t− → −∞ for a black hole and as t+ → +∞ for a white hole. In both cases the affine parameter 7When an equation contains ± or ∓ signs the top group give one equation, the bottom another. 7 r is necessarily bounded as the PNV approaches the appropriate horizon8. Since the metrics are stationary this is an example of the predictions of section 2. This contradicts the basic assumption that ALL singularity theorems are based on. The only reason that it is assumed that these rays must end at a singularity is so that these ”theorems” can be proved. This includes Hawking’s, Penrose’s and all other similar theorems for black holes and the ”big bang”. They are built on a foundation of sand. We will leave this for the moment until we have introduced the Kerr metric where the examples are even clearer. ”The human brain is a complex organ with the wonderful power of enabling man to find reasons for continuing to believe whatever it is that he wants to believe.”-Voltaire. 4 The Kruskal Extension of Schwarzschild Many have said to the author ”What about the Kruskal-Szekeres[14, 15] exten- sion?” as if this makes a difference to any singularities. The original treatment of this starts with the singular Schwarzschild coordinates, ”S”, and then uses a singular transformation to generate the Kruskal coordinates, ”K”. This has been used in lectures for decades but the resulting metric is itself singular on r − 2m. Instead of this, we will use the more recent approach to show that the proper Kruskal metric is an the horizon where its determinant behaves like analytic extension of Eddington, rather than Schwarzschild. The two coordinates (θ, φ) are retained but the other two (r, t) are replaced by (U, V ) that are constant along the ingoing and outgoing PNV’s, respectively. For simplicity, we will assume we start with a black hole with ingoing coordi- nates, E−. but will omit the ± sign on the metrical components. Also, we will use units where 2m = 1 so that the horizon is located at t = 1, not t = 2m. This makes the calculations more readable. Starting with Eddington, ds2 = dr2 −dt2 + 1r(dr+dt)2 = (dr+dt)(dr−dt+ 1r(dr+dt)) = r−1d(r+t)(r+1dr−dt) = r−1dudv, √   rr−1r where (u, v) are given by u = r + t, v = r + 2sgn(r − 1)ln|r − 1| − t Each of the coordinates, (u,v), is constant along the appropriate PNV but v is useless near the event horizon at r = 1 because it is singular there. They need to be replaced with functions, (f(u),g(v)), that are positive and at least three times differentiable at the horizon. The standard choice is to exponentiate them, 22 U =eu/2 =er+t, V =ev/2 =er−t (r−1), (7) 8We will see later that the second PNV in Kerr, k±∗, is asymptotic on both sides to the inner horizon at t = +∞ and to the outer horizon at t = −∞. It is a FALL ray between the horizons. 8 an analytic transformation. This leads to the Kruskal-Szekeres metric, 24−r 22 32m3−r/2m 22 ds=re dUdV+rdσ=⇒ r e dUdV+rdσ, (8) where we have reintroduced the 2m factors! Each new coordinate is zero on one of the ”perpendicular” event horizons of Kruskal. The metric coefficients in (8) are functions of r alone so we only need to calculate this as a function of U and V. From eq.(7), UV = (r − 1)er = g(r) (say), dg(r) = rer. (9) dr From a standard theorem in analysis this can be solved for r as an analytic function of UV in the connected region where r is positive and the derivative of g(r) is nonzero, i.e. away from the real singularity at r = 0. This is even true in the complex plane, except for the branch point at the origin. This allows the definition of the coordinates to be extended to negative values giving the proper Kruskal metric. This has two horizons, (U = 0 or V = 0, where r = 2m) which can only be crossed in one direction because ”time” flows is a unique direction. The lines of constant r are the hyperbolae where UV is constant. Both the Jacobian matrix for the map from (r, t) to (U, V ) and its inverse are analytic so the map from Eddington to Kruskal coordinates is analytic. This is not true when going from Schwarzschild to Kruskal. There is another major difference between Eddington and Kruskal. The former is stationary, i.e. independent of time, but the later has a fixed point at its center and is invariant under a boost symmetry, U → λU, V → λ−1V. Kruskal is a valid extension but has no real physical significance. The second singular region is usually thought of as a white hole, generated by a nonsingu- lar time-reversed object replacing the singularity at r = 0. There is no more likelihood of a singularity in this region than there is in Eddington itself. If one believes in nineteenth century equations of state, regardless of pressure, then anythingispossible.9 Also,blackholesforminouruniversewhenmatteraccu- mulates into clumps that are too massive and/or dense. They do not start as white holes `a la Kruskal. Suppose that there is a nonsingular spherically symmetric star, S, at the center of Eddington, and suppose it is bounded by r = r0. All the incoming PNV’s are radial geodesics, passing through the central point and dying on the opposite side of the star. Since the radius is an affine parameter on each ray the total affine length from crossing the event horizon is slightly more than 2m, a finite number. Oppenheimer and Snyder proved that the metric collapses to a point, i.e. a singularity, if the matter satisfies an appropriate equation of 9What this all means, I have no idea! I do not believe anyone else does either since the behaviour of quantum matter at such extreme pressure is unknown. 9 state, but not in general. The singularity has nothing to do with the singu- larity theorem, just the claimed physics of the star. The affine parameters are irrelevant. 5 The Kerr metric The conformal tensor for an empty Einstein space has four special light rays, its principal null directions (PNVs). These can be thought of as ”eigenvectors” of the conformal tensor. They generate four congruences. When two coincide everywhere the space is called ”algebraically special”. The corresponding con- gruence is then both geodesic and shearfree. Furthermore, the converse is true. If the space has such a special congruence of geodesic and shearfree light rays then, from the Goldberg-Sachs theorem[16], these are repeated eigenvectors of the curvature tensor. Both the two physically interesting solutions of Einstein’s equations known in the first half of the twentieth century, Schwarzschild and plane fronted waves, are algebraically special. The PNV’s coincide in pairs in the first (type D) and all four coincide in the second (type N). In 1962 Robinson and Trautman[17] constructed all algebraically special spaces where the double PNV is hypersurface orthogonal, i.e. a gradient. This was the most general solution known at that time but, although it is both very elegant and has many interesting properties, it did not lead to any new star-like solutions. Several groups tried to generalise this work to allow for a rotating double PNV. These included Newmann, Unti and Tamborino[18] who claimed in 1963 that the only new metric of this type was NUT space. Neither Ivor Robinson nor myself believed this result. The Robinson-Trautman metrics should have been included in their solution as they are a special case but they were not. Curiously enough, others did believe their results and there was a lot of effort put into the this metric. When I finally saw a preprint of their paper I flicked through it until I found an equation where the derivation ground to a halt. They had calculated one of the Bianchi identities twice, did not recognise this, made several mistakes in the numerical coefficients, and got inconsistent results.10 Ignoring the rest of their paper, I set about calculating all algebraically special metrics, i.e. empty Einstein spaces with a double PNV which might not be hypersurface orthogonal. This led to a set of canonical coordinates, a generalisation of those used by Robinson and Trautman in their seminal work and therefore of those used by Eddington. The first, and most important, of these was an affine parameter, r, along the rays. The dependence of the metric on this was fairly easily calculated (An early and accurate example of the ”Peeling Theorem”) so that the remaining field variables were functions of 10Even fifty years later, Newmann still did not understand where they went wrong. The major problem was that they used the Newmann-Penrose equations where the components of the connexion lack numerical indices. This meant that it was difficult to check that each term in an equation had the correct ”dimensions” without the aid of modern computer algebra. Many other people, including myself, were using similar systems but retaining numerical indices on the connexion coefficients. 10 the other three coordinates alone. Unfortunately, the final equations were not integrable and nobody has been able to simplify them further without assuming extra conditions. The Kerr metric was discovered in 1963 [8, 11] by imposing the following series of conditions on an empty Einstein space, 1. It contains a field, kμ, of geodesic and shearfree light rays through each point. The Goldberg-Sachs theorem proves that this is equivalent to de- manding that they are repeated eigenvectors of the conformal tensor. This gave a set of five partial differential equations which were inconsistent unless an endless chain of further integrability conditions were satisfied. These led nowhere and so three more simplifications were imposed in se- quence, 2. It is stationary, i.e. independent of time. This helped but the equations were still intractable. 3. It is axially symmetric. Much better! 4. Finally, it is asymptotically flat. The symmetry conditions reduced the problem to solving some simple, ordinary differential equations11. The final assumption, (3), eliminated all possibilities (including NUT space) except for the two parameters, m and a, of the Kerr metric. Using the original coordinates [8], ds2=ds20+2mrk2, k=dr+asin2θdφ+dt, Σ ds20 =dr2 +Σdθ2 +(r2 +a2)sin2θdφ2 +2asin2θdφdt−dt2, Σ=r2 +a2cos2θ. (10a) (10b) (10c) where the light ray k is a PNV and ds20 is a version of the Minkowski metric exhibiting the canonical Papapetrou form (Only one off-diagonal term) for met- rics satisfying the three conditions above! NB: The coordinate r is an affine parameter along a lightray, k, when the un- derlying space is algebraically special and k is a double PNV. This was the first example of the Kerr-Schild metrics[9, 10] which are defined to have the same form as in eq.(10a). The sign of a is flipped from that in the original 1963 paper because of my confusion over which direction an axial vector should point in! When a = 0, the coordinates (r, θ, φ) are just spherical 11Andrzej Trautman told the author recently that in the early sixties he set a graduate student the problem of calculating all such Einstein metrics by starting with the known field equations for stationary and axisymmetric metrics,i.e. conditions (2,3), and then imposing (1). This should have led to the Kerr metric but Andrzej said that mistakes were made and nothing came of it. 11 polars in Euclidean space and the metric reduces to Schwarzschild in Eddington coordinates. The transformation[8] x + iy = (r + ia)eiφsinθ, z = rcosθ, gives the Kerr-Schild form in more obvious coordinates, 2 2 2 2 2 2mr3 z ds =dx +dy +dz −dt +r4+a2z2[dt+rdz + r (xdx + ydy) + a (xdy − ydx)]2. r2 + a2 r2 + a2 The surfaces of constant r are confocal ellipsoids of revolution, x2 + y2 z2 r2+a2 +r2 =1. (11)   A simple calculation shows that the vector kμ is a geodesic in the underlying Minkowski metric as well as the full metric. This is true for all Kerr-Schild metrics. Those PNV’s that lie in the equatorial plane are tangential to the central ring, r=0 −→ x2+y2=a2, z=0. (12) The rest all pass through this to a second nonphysical sheet. The metric is nonsingular everywhere except on this ring. This singularity generates the Kerr metric. It must be replaced by an actual rotating body such as a neutron star to construct a physical solution where the central ring and second sheet disappear and the metric is nonsingular. What about the Penrose theorem? We will see that there are plenty of FALL’s tan- gential to the event horizons inside both the event shell and the inner horizon. Also, there is no trapped surface inside the latter to affect the metric of the star. There is no singularity problem when the ring is replaced by an appropriate star! We will discuss the complete set of PNV’s in the appendix but it is simpler to restrict the discussion here to those on the rotation/symmetry axis. All others behave exactly like the axial ones. They are asymptotic to the outer event horizon as t → −∞ and to the inner horizon as t → +∞, each from both sides. This is amplified in the appendix. The axial ones are constructed by calculating the metric and finding its roots. This will give both the incoming and outgoing light rays, ds2 =−dt2 +dr2 + 2mr (dr+dt)2 =0, r2 + a2 and so dr = −1 for the incoming geodesic. For the other, dt dr r2 − 2mr + a2 dt = r2 +2mr+a2. This geodesic cannot cross either horizon as its radial velocity is zero there. Since the RHS is negative between these, both PNV’s are pointing inwards in 12 Outgoing 0 r r+ r Figure 1: r ̇ plotted against r for the ‘outgoing’ null geodesics on the axis. r− and r+ are the inner and outer event horizons, respectively. this region. The ”fast” null geodesics continue straight through both horizons but the ”slow” one is asymptotic to the outer horizon as t → −∞ and to the inner horizon as t → +∞. It penetrates neither. It is compelled to move inwards between the horizons and outwards otherwise. Since r is an affine parameter for both these light rays, the affine length of the slow geodesic between the horizons √ is 2 m2 − a2, a finite quantity. This is a simple demonstration of what was discussed in section 2, contradicting the assumption that null geodesics of finite affine length must end in singularities. The same thing happens to this slow ray as it approaches either horizon from outside the ”event shell”. It cannot cross them. What about the two fast incoming geodesics on the axis (one from each end)? These are the rays that Penrose is working with. Since Kerr has no interior body they are compelled to pass through the central ring singularity into the other nonphysical branch of Kerr. If the metric is generated by an axially symmetric and nonsingular neutron star or similar ultra-dense body (whose surface is probably ellipsoidal, r = r0.) then the two incoming axial light rays will pass through it and swap places on the other side. This means that the fast geodesic coming in will become the slow one going out and be asymptotic to the inner horizon on the opposite side12. Its affine parameter, r, is bounded. Light rays can approach this horizon (Cauchy surface?) from inside but cannot cross it. In a truly remarkable paper, Achilles Papapetrou[12] discussed stationary and axisymmetric Einstein spaces where the sources are localised and the met- 12This is an example of exactly what Penrose attempts to prove. If the body is nonsingular then there are FALL’s. 13 r

ric is asymptotically flat. Assuming these conditions, he proved that there is a coordinate system with only one off-diagonal component in the metric, the coefficient of dφdt. Furthermore, this can be found by solving two simple linear algebraic equations! In early 1964 Ray Sachs and the author decided to calcu- late an interior solution for this metric. We believed that the singularity in the centre is not real and that there must be many nonsingular interior ”neutron star” metrics that could replace it. Since we had a preprint of Papapetrou’s paper we put the Kerr metric into his canonical form. The covariant form of the metric, ds2, is then a sum of squares of a suitably weighted orthonormal basis, ds2 = Σ dr2 − ∆ 􏰀dts + a sin2 θ dφs􏰁2, ∆Σ + Σdθ2 + sin2θ 􏰀(r2 + a2)dφs − adts􏰁2 (13) Σ We stared at this metric for a very short time, gave up and went for coffee. The problem is that there are too many possible interior solutions, the same as for regular neutron stars. Physics is needed, not just mathematics! Note that at no point did either of us consider that the interior body was singular. Unlike the Kerr-Schild coordinates, the Boyer-Lindquist ones are singular when ∆ = 0. One important advantage of them is that they make it easy to identify the event horizons and the two PNV’s. The former are the two ellipsoids where ∆ = 0. The metric around these surfaces is nonsingular in Kerr coordinates, as in Eddington, but is singular in BL, as in Schwarzschild. It is shown in the Appendix that there are two families of characteristic light-rays in Kerr. These are tangential to a PNV at every point. There a ”fast” one going in unimpeded and a ”slow” one trying to get out13 but stalling on the horizons. In the original coordinates their contravariant forms are k− = ∂t − ∂r =⇒ dr = −1, (14) dt k+ = ∆k− + (4mr + 2a2sin2θ)∂r, (15) which shows that k+ lies on each event horizon when ∆(r) = 0 and is parallel to its normal, ∂r. From section 2, whenever ∂t is a Killing vector on a light ray then any affine parameter on the ray is an exponential function of t. For Kerr, it approaches a constant as t −→ +∞ on the inner horizon or as t −→ −∞ on the outer horizon. These affine parameters can be chosen to be constant on each horizon so that a is a smooth function throughout. We assume that there is an axially symmetric, smooth, nonsingular star- like body inside the inner horizon with surface r = rS. Consider a ”fast” axial lightray falling into the black hole from outside. It moves down the axis, through both horizons, through the star and finally finishes on the other side as a ”slow” 13This is probably true for all rotationally symmetric black holes, whether stationary or not. 14 ray. Does it match up with a PNV coming in from the other side? This is unlikely since there is only two such characteristic light rays at each point in the empty space outside the star, none within. Does at least one (The axial one?) line up? If the Penrose theorem is true, then yes. If not then that theorem can possibly be modified to show that one incoming light ray is asymptotic to the inner horizon and therefore a FALL. 6 Conclusions The fact that there is at least one FALL in Kerr, the axial one, which does not end in a singularity shows that there is no extant proof that singularities are inevitable. The boundedness of some affine parameters has nothing to do with singularities. The reason that nearly all relativists believe that light rays whose affine lengths are finite must end in singularities is nothing but dogma14. This is the basis for all the singularity theorems of Hawking, Penrose and others and so these are at best unproven, at worst false. Even if they were true then all they would prove is that at least one light ray from the outside is asymptotic to an event horizon and is a FALL but one might have to wait for an infinite time to confirm it for accreting black holes. Proving this would make a good initial problem for a mathematically inclined doctoral student. The author’s opinion is that gravitational clumping leads inevitably to black holes in our universe, confirming what is observed, but this does not lead to sin- gularities. It is true that there are ”proofs” that the curvature of a non-rotating one is infinite at its central point.These all assume that matter is classical and that it satisfies whatever nineteenth century equation of state the proponents require to prove whatever it is that they wish to prove. Equations of state assume that all variables, such as pressure and volume, occur in the simplest al- gebraic fashion. This may be true for the low density laboratory or engineering experiments but perhaps not at black hole densities. The author has no doubt, and never did, that when Relativity and Quantum Mechanics are melded it will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong! There are no event horizons when a > m for a Kerr metric since there are no real roots of r2 − 2mr + a2 = 0. It still has a singular ring, radius a. The metric would need either a mass m rotating at the velocity of light at this radius (impossible!) or an actual star with greater radius and lesser velocity at its equator. Hardly anybody believes that real stars contain singularities (Penrose states this as a principal, counterbalancing his edict that all black holes have singularities!) and so it must be that centrifugal force combined with internal pressures can overcome the ”Newtonian attraction” inside such very fast rotating stars. Also, the inner region of Kerr allows movement both inwards and outwards towards the inner horizon, just like the neighbourhood of 14My experience from listening to graduate students discussing research papers is that they almost always give the mathematics a very cursory glance. This is also true for many professionals. Life is short and they are understandably more interested in physics. 15 a regular star. As a star shrinks its centrifugal forces rise rapidly. There is no known reason why there cannot be a fast rotating nonsingular star inside the horizons generating the Kerr metric outside. There is no published paper that even claims to prove that this is impossible and yet so many believe ”All black holes contain a singularity.”. The secondary PNV’s in Kerr, the k+, do not start at a Cauchy surface outside since they are tangential to the outer horizon in the past. They are not counterexamples to what Roger claims. We need to look at the other set of light rays, the k−. These can start on a Cauchy surface since they do cross the event shell coming in. They would be asymptotic to the inner horizon on the other side except that there is no matter inside, Kerr is singular in the middle and these rays do not connect. There are many reasons why I never believed that Penrose proved that black holes must be singular, e.g. 1. There are no trapped surfaces inside the inner horizon. One can al- ways move outwards from any point inside this, e.g. k+. Even time-like geodesics can do so. If a ray travels down the rotation axis from outside it can end up asymptotic to the inner horizon on the opposite side. This is how I proved in 1963 that Kerr has two horizons. A graduate student, Alex Goodenbour, recalculated this in 2021 with the same results. 2. It is not good enough to show that there is a FALL that is normal to the original trapped surface. It could finish tangential to the inner horizon rather than the interior of the central body. This has to be shown to be impossible, not just assumed or ignored. 3. It is very much a ”do it yourself” paper where the reader is supposed to prove the more difficult parts. Some of these may not even be true. Suppose we have an real star that is spinning fast but is also shrinking and is on the verge of forming a black hole. There will be a shell surrounding it that is difficult, but not impossible, to escape from. As the star contracts further this will become harder and harder until an event shell forms. Radial light rays that come down the axis through the North pole will pass through the star and die asymptotically on the inner horizon on the opposite side. I cannot prove this because there is no agreed solution for the metric inside the star. What has been shown by example is that FALL’s can exist asymptotic to event horizons. All would be black hole singularity theorems must prove that this is impossible. This has not been done and so all the proofs of the various singularity theorems are incomplete. they always were since nobody could prove that FALL’s imply singularities. In desperation, many will say that Kerr is just a special case. This is specious. Specific counterexamples are the standard way to disprove general claims in both Mathematics and Physics. The author does not have to prove that any ”apparent” inner horizon must have an asymptotic FALL but this is almost certainly true and is probably what the Penrose paper is trying to prove. 16

Finally, what do I believe happens to a real collapsing neutron star? Suppose its mass m is 5M⊙, larger than the Chandrasekhar limit for nonrotating ones, and that its radius is close to the Schwarzschild limit, 2m. It could be rotating so fast that the a-parameter is greater than m and no event horizon can form, so we will assume that a < m but close to it. Inside the star there will be an equilibrium between the spin, pressure, and Newtonian type forces15. As one moves away from the star, the spin forces will drop faster than the ”Newtonian” ones. Very high initial velocities will therefore be needed to escape from S, even though the spin helps greatly near its surface. If extra matter joins the star, e.g. from the initial supernova, then further collapse will probably take place. Unless the angular momentum to mass ratio rises very rapidly the outer shell will become impenetrable and a black hole will form. This is the moment when the singularity is believed to appear, according to all true believers. ”No singularity before event shell forms, inevitable singu- larity afterwards!” What the author has tried to show is that there is no reason why this should happen. Centrifugal forces will always dominate in the end as the radius of the body decreases. That is just Physics. The Kerr metric shows that there will be a region between the event shell and the central body where an eagle can fly if it flaps its wings hard enough. It will, of course, notice the outside universe spinning very quickly around it. It may also have a problem with the radiation building up between the star and inner horizon. This all ignores the ”maximal extensions” of the exact solutions. They are oddities with no physical significance and would require generating masses inside each inner section. They prove nothing more than the original empty Kerr or Schwarzschild. They cannot be generated in the infinite past or future. Furthermore, they do not counter the examples I have given here. Remember, one counterexample kills a universal claim. In conclusion, I have tried to show that whatever the Penrose and Hawking theorems prove has nothing to do with Physics breaking down and singularities appearing. Of course, it is impossible to prove that these cannot exist, but it is extremely unlikely and goes against known physics. 7 Appendix The purpose of this section is to calculate the contravariant PNV’s in Boyer- Lindquist and then Kerr-Schild coordinates. The cyclic coordinates, (t.φ) are the only ones that change in any application of Papapetrou’s theorem, neither r nor θ. The suffix s is used to show that the coordinates and the metric, KS, correspond to those used for Schwarzschild. The transformation from Kerr to Papapetrou, i.e. Boyer-Lindquist, coordinates is dts =dt− 2mrdr, dφs =dφ+ adr, rs =r, θs =θ. (16a) ∆∆ 15Astrophysicists may say, ”What is the equation of state?”. This is still a work in progress for a neutron star. 17 ∆=r2 −2mr+a2, Σ=r2 +a2cos2θ. (16b) The only partial derivative operator that changes is ∂r, (∂rs, ∂θs, ∂φs, ∂ts) = (∂r + 2mr∂t − a ∂φ, ∂θ, ∂φ, ∂t) (17) ∆∆ The covariant form of the metric, ds2, is a sum of squares of a suitably weighted orthonormal basis, ds2 = Σ dr2 − ∆ 􏰀dts + a sin2 θ dφs􏰁2, ∆Σ + Σdθ2 + sin2θ 􏰀(r2 + a2)dφs + adts􏰁2 (18) Σ The contravariant metric is a similar sum of squares of the orthogonal tetrad to that in (18), gμν∂∂ =∆∂ 2− 1 􏰀(r2+a2)∂ −a∂ 􏰁2 μνΣrs ∆Σ ts φs +1∂2+ 1 􏰀∂ −asin2θ∂􏰁2] Σθs Σsin2θ φs ts (19) Light rays are only defined up to a multiplicative constant which can depend on the ray, and so we will remove any overall factors. From (10a) the PNV k = k− is k− = (dts + asin2θdφs) + (Σ∆−1)dr (20) which a factor of the first two terms in 18. Because the Boyer-Lindquist metric, KS, is invariant under the inversion (ts → −ts, φs → −φs), the second PNV is the other root of the first two terms in (18) and (19). In the original Kerr-Schild coordinates, k± = ∓Σdr + [∆(dt + asin2θdφ) + (−2mr + a2 sin2 θ)dr] (21) k± = ∓(∆∂r + 2mr∂t − a∂φ) + ((r2 + a2)∂t − a∂φ) (22) The contravariant version of the original PNV, k = k−, is simpler than in its covariant form, k = ∂t − ∂r =⇒ dr = −1. (23) dt The other two variables, φ and θ, are constant along this PNV. The second PNV, k∗, is more complicated, Using t as the best physical parameter along these rays dt = r2 +2mr+a2, dt =−r2 +2mr+a2. (24) which shows that k+ points inwards between the horizons but outwards else- where. lies on each event horizon when ∆(r) = 0 and is also parallel to its 18 dr r2 − 2mr + a2 dφ 2a Outgoing 0 r r+ r Figure 2: r ̇ plotted against r for the ‘outgoing’ null geodesics on the axis. r− and r+ are the inner and outer event horizons, respectively. normal there. From section 2, whenever ∂t is a Killing vector on a light ray then any affine parameter on the ray is an exponential function of t. For Kerr, it approaches a constant as t −→ +∞ on the inner horizon or as t −→ −∞ on the outer horizon. These affine parameters can be chosen to be constant on each horizon so that a is a smooth function throughout. References [1] R. Penrose, ”Gravitational collapse and space-time singularities”, Phys. Rev. Lett. 14, p. 57 (1965). [2] R. Penrose, [3] S.W. Hawking ”Black Holes in General Relativity”, Commun. Math. Phys. 25, p. 152-166 (1972). [4] K. Schwarzschild, “U ̈ber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”, Sitzung. Preuss. Acad. Wiss. 7, p. 189 (1916) [5] A.S. Eddington, “A comparison of Whitehead’s and Einstein’s formula”, Nature, 113, p. 2832 (1924) [6] D. Finkelstein, “Past-future asymmetry of the gravitational field of a point particle”, Phys.Rev., 110 (4), p. 965 (1958) [7] J.R. Oppenheimer, H. Snyder, Phys. Rev. 56, p. 455 (1939). 19 r

View publication stats [8] R. P. Kerr, ”Gravitational field of a spinning mass as an example of a algebraically special metric”, Phys. Rev. Lett. 11, p. 237 (1963). [9] R. P. Kerr and A. Schild, ”A new class of vacuum solutions of the Ein- stein field equations”, Atti del convegno sulla relativit`a generale; problemi dell’energia e onde gravitationali, G. Barbera, Ed., p. 173 (Firenze, 1965). [10] R. P. Kerr and A. Schild, ”Some algebraically degenerate solutions of Ein- stein’s gravitational field equations”, Proc. Symp. Appl. Math, R. Finn, Ed., Am.Math.Soc. p. 173 (1965). [11] D.L. Wiltshire, M. Visser and S.M. Scott, ”The Kerr Spacetime,” Camb. Univ. Press, p. 38 (2009) [12] A. Papapetrou, Champs gravitationals stationaires ˆ‘a symm ́etric axial, Ann. Inst. H. Poincar ́e 4 83 (1966) [13] A.K. Raychaudhuri, ”Relativistic Cosmology”, Phys. Rev. 98 (4), p. 1123- 1126 (1955). [14] M. Kruskal, “Maximal extension of Schwarzschild metric”, Phys. Rev. 119, p. 1743 (1959). [15] G. Szekeres, “On the singularities of a Riemannian manifold”, Pub. Math. Deb. 7, p. 285 (1959). [16] J.N. Goldberg and J.N. Sachs, ”A theorem on Petrov Types”, Acta. Phys. Polon., suppl. 22, p. 13 (1962). [17] I. Robinson and A. Trautman, ”Some spherical gravitational waves in gen- eral relativity”, Proc. Roy. Soc.Lond., A 265, p. 463-473 (1962) [18] E. T. Newman, L. Tamburino and T. Unti, ”Empty space generalisation of the Schwarzschild metric”, J.Math.Phys., 4, p. 915-923 (1963) [19] G.E. Ellis and S.W. Hawking, The Large Scale Structure of Space-Time, Cambridge University Press, (2009) 20

“Singularities don’t exist,” claims black hole pioneer Roy Kerr The brilliant mind who discovered the spacetime solution for rotating black holes claims singularities don't physically exist. Is he right?

This visualization shows what the interior of a rotating (Kerr) black hole looks like, from the perspective of an observer who has crossed over the inner event horizon in that spacetime. The pink region illustrates the view down inside the alleged ring singularity that is present in the mathematical formulation of the Kerr spacetime. Whether this represents a physical (curvature) singularity or not has recently been reopened for debate. Credit: David Madore KEY TAKEAWAYS
 * Way back in 1963, Roy Kerr became the first person to write down the exact solution, in general relativity, for a realistic, rotating black hole. 60 years later, it's still used everywhere.  Although Roger Penrose won the Nobel Prize in physics just a few years ago for demonstrating how black holes come to exist in our Universe, singularities and all, the subject isn't closed.  We've never peered beneath the event horizon, and have no way of detecting what's inside. Using a powerful mathematical argument, Kerr argues that singularities shouldn't physically exist. He may be right.

Ethan Siegel

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Here in our Universe, whenever you gather enough mass together in a small enough volume of space, you’re bound to eventually cross a threshold: where the speed at which you’d need to travel to escape the gravitational pull within that region exceeds the speed of light. Whenever that occurs, it’s inevitable that you’ll form an event horizon around that region, which looks, acts, and behaves exactly like a black hole as seen from the outside. Meanwhile, inside, all that matter gets inexorably drawn toward the central region inside that black hole. With finite amounts of mass compressed to an infinitesimal volume, the existence of a singularity is all but assured.

The predictions for what we should observe outside the event horizon match extraordinarily well with observations, as we’ve not only seen many luminous objects in orbit around black holes, but have even now imaged the event horizons of multiple black holes directly. The theorist who laid the foundation for how realistic black holes form in the Universe, Roger Penrose, subsequently won the Nobel Prize in Physics in 2020 for his contributions to physics, including for the notion that a singularity must exist at the center of every black hole. But in a surprising twist, the legendary physicist who discovered the spacetime solution for rotating black holes — Roy Kerr, way back in 1963 — has just written a new paper challenging that idea with some very compelling arguments. Here’s why, perhaps, singularities may not exist within every black hole, and what the key issues are that we should all be thinking about.

Once you cross the threshold to form a black hole, everything inside the event horizon crunches down to a singularity that is, at most, one-dimensional. No 3D structures can survive intact. That’s the conventional wisdom, and has been treated as proven for over 50 years. But with rotation added into the mix, one of the assumptions of the “proof” seems to fall apart. Credit: vchalup / Adobe Stock Making an ideal black hole If you want to make a black hole, in Einstein’s general relativity, all you have to do is take any distribution of pressureless mass — what relativists call “dust” — that starts in the same vicinity and is initially at rest, and let it gravitate. Over time, it will contract down and down and down to smaller volumes, until an event horizon forms at a specific distance from the center: dependent solely on the total amount of mass that you began with. This produces the simplest type of black hole known: a Schwarzschild black hole, which has mass, but no electric charge or angular momentum.

Einstein first put forth general relativity, in its final form, in late 1915. Just two months later, in early 1916, Karl Schwarzschild had worked out the mathematical solution for a spacetime that corresponds to this situation: a spacetime that’s completely empty except for one point-like mass. In reality, the matter in our Universe isn’t pressureless dust, but rather is made of atoms and subatomic particles. Nevertheless, through realistic processes like: black holes certainly form in our Universe. We’ve observed them, and we’re certain they exist. However, a big mystery remains: what happens inside of them, in their interiors, where we cannot observe?
 * the core-collapse of massive stars,
 * the mergers of two massive-enough neutron stars,
 * or the direct collapse of a large amount of matter, either stellar or gaseous,

Size comparison of the two black holes imaged by the Event Horizon Telescope (EHT) Collaboration: M87*, at the heart of the galaxy Messier 87, and Sagittarius A* (Sgr A*), at the center of the Milky Way. Although Messier 87’s black hole is easier to image because of the slow time variation, the one around the center of the Milky Way is the largest as viewed from Earth. These black holes are certain to have event horizons, as we’ve imaged them. Credit: EHT collaboration (Acknowledgment: Lia Medeiros, xkcd) The argument for a singularity There’s a simple argument you can make to understand why we think that all black holes, at least under the Schwarzschild set of assumptions, ought to have a singularity at their centers. Imagine you’ve crossed over the event horizon, and are now on the “inside” of the black hole. Where can you go from here? The reason why? Because space itself is flowing: like a waterfall or a moving walkway beneath your feet. Even if you speed yourself up so that you’re moving arbitrarily close to the speed of light, the rate at which space is flowing is so great that no matter which direction you move in, the singularity appears to be “down” in all directions. You can draw out the shape of where you’re allowed to go, and even though it forms a mathematically interesting structure known as a cardioid, all paths lead to you winding up at the center of this object. Given enough time, these black holes should all have a singularity at their centers.
 * If you fire your thrusters directly at the singularity, you’ll just get there faster, so that’s no good.
 * If you fire your thrusters perpendicular to the direction of the singularity, you’ll still get drawn inward, and there’s no way to get farther from the singularity.
 * And if you fire your thrusters directly away from the singularity, you’ll find that you’re still approaching the singularity faster and faster as time goes on.

When matter collapses, it can inevitably form a black hole. Roger Penrose was the first to work out the physics of spacetime, applicable to all observers at all points in space and at all instants in time, that governs a system such as this. His conception has been the gold standard in general relativity ever since. However, while it robustly applies to non-rotating black holes, there may be a flaw with the reasoning that predicts it for realistic, rotating black holes. Credit: J. Jarnstead/Royal Swedish Academy of Sciences The Kerr advance: adding rotation But here in the real Universe, the ideal case of having a mass with no rotation to it isn’t exactly a good physical model of reality. Consider that: it makes sense that all physically realistic black holes would be rotating. It turns out that while asking the question of what does a spacetime look like if you have only a single point mass in your Universe is a relatively straightforward problem to solve in Einstein’s general relativity — after all, Karl Schwarzschild solved it in just a couple of months — the question of what spacetime looks like if you have a mass that rotates is much more complicated. Indeed, many brilliant physicists worked on this problem and were unable to solve it: for months, years, and even decades.
 * there are many masses in the Universe,
 * these masses, over time, gravitationally attract one another,
 * causing them to move relative to one another,
 * which leads to the clumping and clustering of matter in a non-uniform way,
 * and that as clumps of matter move relative to one another and gravitationally interact, they’ll exert not just forces but torques on one another,
 * that torques cause rotation,
 * and that as rotating objects collapse, their rotation rate increases due to the conservation of angular momentum,

But then, in 1963, New Zealand physicist Roy Kerr finally cracked it. His solution for the spacetime describing realistic, rotating black holes — the Kerr metric — has been the gold standard for what relativists have used to describe it ever since.

The exact solution for a black hole with both mass and angular momentum was found by Roy Kerr in 1963, and revealed, instead of a single event horizon with a point-like singularity, an inner and outer event horizon, as well as an inner and outer ergosphere, plus a ring-like singularity of substantial radius. An external observer cannot see anything beyond the outer event horizon, and if you replace the ring singularity with a non-singular object, the spacetime outside the horizon is unaffected. Credit: M. Visser, The Kerr Spacetime, 2007 Rotation and reality When you add rotation in, the situation for how spacetime behaves suddenly becomes a lot more complicated than it was in the non-rotating case. Instead of a spherical event horizon marking the delineation between where it’s possible to escape the black hole (outside) versus where escape is impossible (inside), and instead of all “inside” paths leading to a singularity at the center, the mathematical structure of a rotating (Kerr) black hole looks extremely different.

Instead of a single, spherical surface describing the event horizon and a point-like singularity at the center, the addition of rotation causes there to be several important phenomena that aren’t apparent in the non-rotating case.
 * Instead of a single solution for the location of the event horizon, as in the Schwarzschild case, the equation you wind up with in the Kerr case is quadratic, giving two separate solutions: an “outer” and “inner” event horizon.
 * Instead of the event horizon marking the location where the timelike component of the metric flips sign, there are now two surfaces that are different from the inner and outer event horizons — the inner and outer ergospheres — that delineate those locations throughout space.
 * And instead of a zero-dimensional, point-like singularity at the center, the angular momentum present smooths that singularity into a one-dimensional surface: a ring, with the rotational axis of the black hole passing perpendicular through the center of the ring.

In the vicinity of a black hole, space flows like either a moving walkway or a waterfall, depending on how you want to visualize it. Unlike in the non-rotating case, the event horizon splits into two, while the central singularity gets stretched out into a one-dimensional ring. Nobody knows what occurs at the central singularity, but it only must exist if all possible paths inevitably lead to it. This is true in the non-rotating case, but is it true in the rotating case? Credit: Andrew Hamilton/JILA/University of Colorado This leads to a variety of, shall we say, less-than-intuitive effects that occur within a Kerr spacetime that don’t occur within a Schwarzschild (non-rotating) spacetime. Because the metric itself has an intrinsic rotation to it and couples to all of space outside the event horizons and ergospheres, all outside inertial reference frames will experience an induced rotation: a frame-dragging effect. This is similar to electromagnetic induction, but for gravitation. Because of the non-spherically symmetric nature of the system, where we now have one of our three spatial dimensions representing a rotational axis and where there’s a direction (clockwise or counterclockwise, for example) to that rotation, a particle that orbits this black hole won’t make a closed ellipse that remains in the same plane (or a slowly-decaying and precessing ellipse, if you account for all of general relativity’s effects), but rather will move throughout all three dimensions, eventually filling in a volume enclosed by a torus.

And, perhaps most importantly, if you track the evolution of any particle that falls into this object from outside, it won’t simply cross over to the inside of the horizon and head inexorably toward the central singularity. Instead, other important effects occur that may work to “freeze” these particles in place, or to otherwise prevent them from traveling all the way to the theoretical “ring” singularity at the center. That’s where we owe it to ourselves to take a good look at what Roy Kerr, who’s been thinking about this puzzle for longer than anyone else alive, has to say about it.

An animation of the orbit of a single test particle just outside of the innermost stable orbit for a Kerr (rotating) black hole. Note that the particle has a differing radial extent from the black hole’s center dependent on the orientation: whether you’re aligned or perpendicular to the spin axis of the black hole. Note also that the particle doesn’t remain in a single plane, but rather fills the volume of a torus as it orbits the black hole. Credit: Yukterez (Simon Tyran, Vienna)/Wikimedia Commons Revisiting the argument for a singularity The biggest argument for why a singularity must exist inside of black holes comes from two titanic figures in 20th century physics: Roger Penrose and Stephen Hawking. 1. The first part of the argument, from Penrose alone, is that wherever you have what’s called a trapped surface — a boundary from which nothing physical can escape, e.g., an event horizon — any light rays interior to that trapped surface will possess a mathematical property known as having finite affine length. 2. This “finite affine length light,” or FALL, for each light ray then implies that the light must terminate in an actual singularity, which is the second part of the argument from Penrose and Hawking. 3. You can then show that any object that enters the region between the outer and inner event horizons must “fall through” to the interior. 4. And, because you need a source to generate the spacetime, the existence of a ring singularity is required. At least, that’s how the traditional argument goes. The third and fourth parts of the argument are airtight in general relativity: if parts one and two are true, then you need a singularity at the core. But are parts one and two both true? That’s where Kerr’s new paper comes into play, asserting that no, this is a mistake that we’ve been making for over half-a-century.

A mathematical simulation of the warped space-time near two merging neutron stars that result in the creation of a black hole. The colored bands are gravitational-wave peaks and troughs, with the colors getting brighter as the wave amplitude increases. The strongest waves, carrying the greatest amount of energy, come just before and during the merger event itself. What occurs outside the event horizon is not practically affected by whether there is a ring singularity at the center, or some other, extended object that is non-singular. (Credit: SXS Collaboration) What Kerr showed is that if you go all the way back to his original, generalized coordinate formulation for Kerr black holes, the Kerr-Schild coordinates, through every single point in the interior of the Kerr black hole, you can draw light rays that are: Moreover, if you ask the key question, “How common are these light rays?” the answer is that there are an infinite number of them, and that half of these rays are in the region between the two event horizons, with at least two through every point in that region.
 * tangential (i.e., approach but do not intersect) to one of the two event horizons,
 * do not have endpoints (i.e., they continue to travel forever),
 * and yet still have finite affine lengths (i.e., they are FALLs).

The problem, as Kerr has been able to show, is with point #2 in the aforementioned argument. Sure, you have a trapped surface in Kerr spacetime, and all the light rays within that trapped surface have finite affine length. But is that light required to terminate in a singularity? Not at all. In fact, by demonstrating the presence of these light rays that are tangential to an event horizon and that do not have endpoints, he has provided a counterexample to that notion. In Kerr’s own words: “It has not been proved that a singularity, not just a FALL, is inevitable when an event horizon forms around a collapsing star.”

Shadow (black), horizons and ergospheres (white) of a rotating black hole. The quantity of a, shown varying in the image, has to do with the relationship of angular momentum of the black hole to its mass. Because actual matter must collapse to form this black hole, and because the conditions that necessarily lead to a singularity are not met under this scenario, the existence of a singularity is not guaranteed. Credit: Yukterez (Simon Tyran, Vienna)/Wikimedia Commons The problem with Hawking & Penrose It’s kind of remarkable, if you go back in history, to realize how much of our acceptance of the existence of a singularity depends on an unproven assertion. Back in 1970, Hawking and Penrose wrote a paper called The singularities of gravitational collapse and cosmology, and within it note that there are other possibilities to consider than the traditional (curvature) singularities when it comes to realistic black holes. With the disproof that Kerr has demonstrated, some people have instead asserted that you need to consider the maximal extensions of the Kerr space, and you’ll find the need for a singularity there. For example, in the Boyer-Lindquist extension of the Kerr spacetime, you have a collection of copies of the separate parts of the original Kerr metric, and because there are no interior collapsed stars inside, it’s certain to be singular. But again, as Kerr points out, you must assume that each interior section of the spacetime, even in the Boyer-Lindquist extension, contains a (collapsed) star within it, and therefore encounters the same problem. Other extensions (such as Kruskal) have been proposed, but Kerr shot those attempts to evade this problem down as well, by demonstrating that Kerr is its own maximal extension. As Kerr puts it:

“These extensions may be analytic, but at best they are constructed using copies of the original spaces together with some fixed points. These will be nonsingular inside each copy of the original interior if the same is true inside the original Kerr and therefore the extensions are irrelevant to the singularity theorems. Anyone who does not believe this needs to supply a proof. They are all physically irrelevant since real black holes start at a finite time in the past with the collapse of a star or similar over-dense concentration of matter, not as the white hole of the Kruskal or Boyer-Lindquist extensions.” Put simply: a FALL does not necessarily mean a singularity, and Kerr chalks the confusion up to physicists conflating geodesic distance/length with affine distance/length: two concepts that are not, in fact, identical. Kerr also points out that if there were a nonsingular object, like a stretched-out neutron star corpse, inside the Kerr black hole, it too would generate the Kerr spacetime we observe. In other words, there are good reasons to revisit the notion that a singularity must exist inside each realistic, rotating black hole.

When an observer enters a non-rotating black hole, there is no escape: you get crushed by the central singularity. However, in a rotating (Kerr) black hole, passing through the center of the disk bounded by the alleged ring singularity is possible, and while it might take you to an extended part of the space known as an antiverse, it could also be that the “ring singularity” is just a phantasm. Credit: Andrew Hamilton, JILA, University of Colorado-Boulder Final thoughts We have to remember an important aspect of general relativity that almost everyone — laypersons and physicists alike — often overlook: “general relativity is about forces, not geometry.” The person who said that wasn’t some crackpot; it was Einstein himself. general relativity is not simply pure mathematics; it’s a description of the physical Universe, placed on a firm mathematical footing. You can’t simply “write down a spacetime” and expect that to describe reality, you have to start from a physically motivated set of conditions and show how that spacetime solution (e.g., a rotating black hole) comes to be. If the only way you can “prove” the existence of a singularity is by ignoring the physical creation of the object in the first place, your proof is not valid. However, demonstrating a counterexample to your attempted proof, both physically and mathematically, is an excellent way to falsify any assertion that gets made. With Kerr’s latest work — a full 60 years after first deriving the Kerr metric — we have to reckon with the sober fact that our best “singularity theorems” that argue for their necessity at a realistic black hole’s center are based on an invalid assumption. Furthermore, once you cross over to be inside the inner event horizon in Kerr spacetime, it once again becomes possible to travel in any direction between the theorized ring singularity and the inner event horizon. The “trapped surface” only exists between the inner and outer event horizons, not interior to the inner event horizon: where the ring singularity allegedly exists. Who knows what exists in that region? The problem is that there are enormous numbers of mathematical solutions to this problem, and “a singularity” is only one of them. There might indeed yet be a singularity inside, but there also may be something entirely different. Kerr, currently at the age of 89, has no problem telling us what he thinks, writing that he: “has no doubt, and never did, that when relativity and quantum mechanics are melded it will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong!” What we can be certain of is that the long-accepted “proof,” that rotating black holes must have singularities, can’t be counted on any longer. (The full text of Kerr’s paper is included above this article.)

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“Singularities don’t exist,” claims black hole pioneer Roy Kerr The brilliant mind who discovered the spacetime solution for rotating black holes claims singularities don't physically exist. Is he right?

This visualization shows what the interior of a rotating (Kerr) black hole looks like, from the perspective of an observer who has crossed over the inner event horizon in that spacetime. The pink region illustrates the view down inside the alleged ring singularity that is present in the mathematical formulation of the Kerr spacetime. Whether this represents a physical (curvature) singularity or not has recently been reopened for debate. Credit: David Madore KEY TAKEAWAYS
 * Way back in 1963, Roy Kerr became the first person to write down the exact solution, in general relativity, for a realistic, rotating black hole. 60 years later, it's still used everywhere.  Although Roger Penrose won the Nobel Prize in physics just a few years ago for demonstrating how black holes come to exist in our Universe, singularities and all, the subject isn't closed.  We've never peered beneath the event horizon, and have no way of detecting what's inside. Using a powerful mathematical argument, Kerr argues that singularities shouldn't physically exist. He may be right.

Ethan Siegel

Copy a link to the article entitled http://“Singularities%20don’t%20exist,”%20claims%20black%20hole%20pioneer%20Roy%20Kerr

Here in our Universe, whenever you gather enough mass together in a small enough volume of space, you’re bound to eventually cross a threshold: where the speed at which you’d need to travel to escape the gravitational pull within that region exceeds the speed of light. Whenever that occurs, it’s inevitable that you’ll form an event horizon around that region, which looks, acts, and behaves exactly like a black hole as seen from the outside. Meanwhile, inside, all that matter gets inexorably drawn toward the central region inside that black hole. With finite amounts of mass compressed to an infinitesimal volume, the existence of a singularity is all but assured.

The predictions for what we should observe outside the event horizon match extraordinarily well with observations, as we’ve not only seen many luminous objects in orbit around black holes, but have even now imaged the event horizons of multiple black holes directly. The theorist who laid the foundation for how realistic black holes form in the Universe, Roger Penrose, subsequently won the Nobel Prize in Physics in 2020 for his contributions to physics, including for the notion that a singularity must exist at the center of every black hole. But in a surprising twist, the legendary physicist who discovered the spacetime solution for rotating black holes — Roy Kerr, way back in 1963 — has just written a new paper challenging that idea with some very compelling arguments. Here’s why, perhaps, singularities may not exist within every black hole, and what the key issues are that we should all be thinking about.

Once you cross the threshold to form a black hole, everything inside the event horizon crunches down to a singularity that is, at most, one-dimensional. No 3D structures can survive intact. That’s the conventional wisdom, and has been treated as proven for over 50 years. But with rotation added into the mix, one of the assumptions of the “proof” seems to fall apart. Credit: vchalup / Adobe Stock Making an ideal black hole If you want to make a black hole, in Einstein’s general relativity, all you have to do is take any distribution of pressureless mass — what relativists call “dust” — that starts in the same vicinity and is initially at rest, and let it gravitate. Over time, it will contract down and down and down to smaller volumes, until an event horizon forms at a specific distance from the center: dependent solely on the total amount of mass that you began with. This produces the simplest type of black hole known: a Schwarzschild black hole, which has mass, but no electric charge or angular momentum.

Einstein first put forth general relativity, in its final form, in late 1915. Just two months later, in early 1916, Karl Schwarzschild had worked out the mathematical solution for a spacetime that corresponds to this situation: a spacetime that’s completely empty except for one point-like mass. In reality, the matter in our Universe isn’t pressureless dust, but rather is made of atoms and subatomic particles. Nevertheless, through realistic processes like: black holes certainly form in our Universe. We’ve observed them, and we’re certain they exist. However, a big mystery remains: what happens inside of them, in their interiors, where we cannot observe?
 * the core-collapse of massive stars,
 * the mergers of two massive-enough neutron stars,
 * or the direct collapse of a large amount of matter, either stellar or gaseous,

Size comparison of the two black holes imaged by the Event Horizon Telescope (EHT) Collaboration: M87*, at the heart of the galaxy Messier 87, and Sagittarius A* (Sgr A*), at the center of the Milky Way. Although Messier 87’s black hole is easier to image because of the slow time variation, the one around the center of the Milky Way is the largest as viewed from Earth. These black holes are certain to have event horizons, as we’ve imaged them. Credit: EHT collaboration (Acknowledgment: Lia Medeiros, xkcd) The argument for a singularity There’s a simple argument you can make to understand why we think that all black holes, at least under the Schwarzschild set of assumptions, ought to have a singularity at their centers. Imagine you’ve crossed over the event horizon, and are now on the “inside” of the black hole. Where can you go from here? The reason why? Because space itself is flowing: like a waterfall or a moving walkway beneath your feet. Even if you speed yourself up so that you’re moving arbitrarily close to the speed of light, the rate at which space is flowing is so great that no matter which direction you move in, the singularity appears to be “down” in all directions. You can draw out the shape of where you’re allowed to go, and even though it forms a mathematically interesting structure known as a cardioid, all paths lead to you winding up at the center of this object. Given enough time, these black holes should all have a singularity at their centers.
 * If you fire your thrusters directly at the singularity, you’ll just get there faster, so that’s no good.
 * If you fire your thrusters perpendicular to the direction of the singularity, you’ll still get drawn inward, and there’s no way to get farther from the singularity.
 * And if you fire your thrusters directly away from the singularity, you’ll find that you’re still approaching the singularity faster and faster as time goes on.

When matter collapses, it can inevitably form a black hole. Roger Penrose was the first to work out the physics of spacetime, applicable to all observers at all points in space and at all instants in time, that governs a system such as this. His conception has been the gold standard in general relativity ever since. However, while it robustly applies to non-rotating black holes, there may be a flaw with the reasoning that predicts it for realistic, rotating black holes. Credit: J. Jarnstead/Royal Swedish Academy of Sciences The Kerr advance: adding rotation But here in the real Universe, the ideal case of having a mass with no rotation to it isn’t exactly a good physical model of reality. Consider that: it makes sense that all physically realistic black holes would be rotating. It turns out that while asking the question of what does a spacetime look like if you have only a single point mass in your Universe is a relatively straightforward problem to solve in Einstein’s general relativity — after all, Karl Schwarzschild solved it in just a couple of months — the question of what spacetime looks like if you have a mass that rotates is much more complicated. Indeed, many brilliant physicists worked on this problem and were unable to solve it: for months, years, and even decades.
 * there are many masses in the Universe,
 * these masses, over time, gravitationally attract one another,
 * causing them to move relative to one another,
 * which leads to the clumping and clustering of matter in a non-uniform way,
 * and that as clumps of matter move relative to one another and gravitationally interact, they’ll exert not just forces but torques on one another,
 * that torques cause rotation,
 * and that as rotating objects collapse, their rotation rate increases due to the conservation of angular momentum,

But then, in 1963, New Zealand physicist Roy Kerr finally cracked it. His solution for the spacetime describing realistic, rotating black holes — the Kerr metric — has been the gold standard for what relativists have used to describe it ever since.

The exact solution for a black hole with both mass and angular momentum was found by Roy Kerr in 1963, and revealed, instead of a single event horizon with a point-like singularity, an inner and outer event horizon, as well as an inner and outer ergosphere, plus a ring-like singularity of substantial radius. An external observer cannot see anything beyond the outer event horizon, and if you replace the ring singularity with a non-singular object, the spacetime outside the horizon is unaffected. Credit: M. Visser, The Kerr Spacetime, 2007 Rotation and reality When you add rotation in, the situation for how spacetime behaves suddenly becomes a lot more complicated than it was in the non-rotating case. Instead of a spherical event horizon marking the delineation between where it’s possible to escape the black hole (outside) versus where escape is impossible (inside), and instead of all “inside” paths leading to a singularity at the center, the mathematical structure of a rotating (Kerr) black hole looks extremely different.

Instead of a single, spherical surface describing the event horizon and a point-like singularity at the center, the addition of rotation causes there to be several important phenomena that aren’t apparent in the non-rotating case.
 * Instead of a single solution for the location of the event horizon, as in the Schwarzschild case, the equation you wind up with in the Kerr case is quadratic, giving two separate solutions: an “outer” and “inner” event horizon.
 * Instead of the event horizon marking the location where the timelike component of the metric flips sign, there are now two surfaces that are different from the inner and outer event horizons — the inner and outer ergospheres — that delineate those locations throughout space.
 * And instead of a zero-dimensional, point-like singularity at the center, the angular momentum present smooths that singularity into a one-dimensional surface: a ring, with the rotational axis of the black hole passing perpendicular through the center of the ring.

In the vicinity of a black hole, space flows like either a moving walkway or a waterfall, depending on how you want to visualize it. Unlike in the non-rotating case, the event horizon splits into two, while the central singularity gets stretched out into a one-dimensional ring. Nobody knows what occurs at the central singularity, but it only must exist if all possible paths inevitably lead to it. This is true in the non-rotating case, but is it true in the rotating case? Credit: Andrew Hamilton/JILA/University of Colorado This leads to a variety of, shall we say, less-than-intuitive effects that occur within a Kerr spacetime that don’t occur within a Schwarzschild (non-rotating) spacetime. Because the metric itself has an intrinsic rotation to it and couples to all of space outside the event horizons and ergospheres, all outside inertial reference frames will experience an induced rotation: a frame-dragging effect. This is similar to electromagnetic induction, but for gravitation. Because of the non-spherically symmetric nature of the system, where we now have one of our three spatial dimensions representing a rotational axis and where there’s a direction (clockwise or counterclockwise, for example) to that rotation, a particle that orbits this black hole won’t make a closed ellipse that remains in the same plane (or a slowly-decaying and precessing ellipse, if you account for all of general relativity’s effects), but rather will move throughout all three dimensions, eventually filling in a volume enclosed by a torus.

And, perhaps most importantly, if you track the evolution of any particle that falls into this object from outside, it won’t simply cross over to the inside of the horizon and head inexorably toward the central singularity. Instead, other important effects occur that may work to “freeze” these particles in place, or to otherwise prevent them from traveling all the way to the theoretical “ring” singularity at the center. That’s where we owe it to ourselves to take a good look at what Roy Kerr, who’s been thinking about this puzzle for longer than anyone else alive, has to say about it.

An animation of the orbit of a single test particle just outside of the innermost stable orbit for a Kerr (rotating) black hole. Note that the particle has a differing radial extent from the black hole’s center dependent on the orientation: whether you’re aligned or perpendicular to the spin axis of the black hole. Note also that the particle doesn’t remain in a single plane, but rather fills the volume of a torus as it orbits the black hole. Credit: Yukterez (Simon Tyran, Vienna)/Wikimedia Commons Revisiting the argument for a singularity The biggest argument for why a singularity must exist inside of black holes comes from two titanic figures in 20th century physics: Roger Penrose and Stephen Hawking. 1. The first part of the argument, from Penrose alone, is that wherever you have what’s called a trapped surface — a boundary from which nothing physical can escape, e.g., an event horizon — any light rays interior to that trapped surface will possess a mathematical property known as having finite affine length. 2. This “finite affine length light,” or FALL, for each light ray then implies that the light must terminate in an actual singularity, which is the second part of the argument from Penrose and Hawking. 3. You can then show that any object that enters the region between the outer and inner event horizons must “fall through” to the interior. 4. And, because you need a source to generate the spacetime, the existence of a ring singularity is required. At least, that’s how the traditional argument goes. The third and fourth parts of the argument are airtight in general relativity: if parts one and two are true, then you need a singularity at the core. But are parts one and two both true? That’s where Kerr’s new paper comes into play, asserting that no, this is a mistake that we’ve been making for over half-a-century.

A mathematical simulation of the warped space-time near two merging neutron stars that result in the creation of a black hole. The colored bands are gravitational-wave peaks and troughs, with the colors getting brighter as the wave amplitude increases. The strongest waves, carrying the greatest amount of energy, come just before and during the merger event itself. What occurs outside the event horizon is not practically affected by whether there is a ring singularity at the center, or some other, extended object that is non-singular. (Credit: SXS Collaboration) What Kerr showed is that if you go all the way back to his original, generalized coordinate formulation for Kerr black holes, the Kerr-Schild coordinates, through every single point in the interior of the Kerr black hole, you can draw light rays that are: Moreover, if you ask the key question, “How common are these light rays?” the answer is that there are an infinite number of them, and that half of these rays are in the region between the two event horizons, with at least two through every point in that region.
 * tangential (i.e., approach but do not intersect) to one of the two event horizons,
 * do not have endpoints (i.e., they continue to travel forever),
 * and yet still have finite affine lengths (i.e., they are FALLs).

The problem, as Kerr has been able to show, is with point #2 in the aforementioned argument. Sure, you have a trapped surface in Kerr spacetime, and all the light rays within that trapped surface have finite affine length. But is that light required to terminate in a singularity? Not at all. In fact, by demonstrating the presence of these light rays that are tangential to an event horizon and that do not have endpoints, he has provided a counterexample to that notion. In Kerr’s own words: “It has not been proved that a singularity, not just a FALL, is inevitable when an event horizon forms around a collapsing star.”

Shadow (black), horizons and ergospheres (white) of a rotating black hole. The quantity of a, shown varying in the image, has to do with the relationship of angular momentum of the black hole to its mass. Because actual matter must collapse to form this black hole, and because the conditions that necessarily lead to a singularity are not met under this scenario, the existence of a singularity is not guaranteed. Credit: Yukterez (Simon Tyran, Vienna)/Wikimedia Commons The problem with Hawking & Penrose It’s kind of remarkable, if you go back in history, to realize how much of our acceptance of the existence of a singularity depends on an unproven assertion. Back in 1970, Hawking and Penrose wrote a paper called The singularities of gravitational collapse and cosmology, and within it note that there are other possibilities to consider than the traditional (curvature) singularities when it comes to realistic black holes. With the disproof that Kerr has demonstrated, some people have instead asserted that you need to consider the maximal extensions of the Kerr space, and you’ll find the need for a singularity there. For example, in the Boyer-Lindquist extension of the Kerr spacetime, you have a collection of copies of the separate parts of the original Kerr metric, and because there are no interior collapsed stars inside, it’s certain to be singular. But again, as Kerr points out, you must assume that each interior section of the spacetime, even in the Boyer-Lindquist extension, contains a (collapsed) star within it, and therefore encounters the same problem. Other extensions (such as Kruskal) have been proposed, but Kerr shot those attempts to evade this problem down as well, by demonstrating that Kerr is its own maximal extension. As Kerr puts it:

“These extensions may be analytic, but at best they are constructed using copies of the original spaces together with some fixed points. These will be nonsingular inside each copy of the original interior if the same is true inside the original Kerr and therefore the extensions are irrelevant to the singularity theorems. Anyone who does not believe this needs to supply a proof. They are all physically irrelevant since real black holes start at a finite time in the past with the collapse of a star or similar over-dense concentration of matter, not as the white hole of the Kruskal or Boyer-Lindquist extensions.” Put simply: a FALL does not necessarily mean a singularity, and Kerr chalks the confusion up to physicists conflating geodesic distance/length with affine distance/length: two concepts that are not, in fact, identical. Kerr also points out that if there were a nonsingular object, like a stretched-out neutron star corpse, inside the Kerr black hole, it too would generate the Kerr spacetime we observe. In other words, there are good reasons to revisit the notion that a singularity must exist inside each realistic, rotating black hole.

When an observer enters a non-rotating black hole, there is no escape: you get crushed by the central singularity. However, in a rotating (Kerr) black hole, passing through the center of the disk bounded by the alleged ring singularity is possible, and while it might take you to an extended part of the space known as an antiverse, it could also be that the “ring singularity” is just a phantasm. Credit: Andrew Hamilton, JILA, University of Colorado-Boulder Final thoughts We have to remember an important aspect of general relativity that almost everyone — laypersons and physicists alike — often overlook: “general relativity is about forces, not geometry.” The person who said that wasn’t some crackpot; it was Einstein himself. general relativity is not simply pure mathematics; it’s a description of the physical Universe, placed on a firm mathematical footing. You can’t simply “write down a spacetime” and expect that to describe reality, you have to start from a physically motivated set of conditions and show how that spacetime solution (e.g., a rotating black hole) comes to be. If the only way you can “prove” the existence of a singularity is by ignoring the physical creation of the object in the first place, your proof is not valid. However, demonstrating a counterexample to your attempted proof, both physically and mathematically, is an excellent way to falsify any assertion that gets made. With Kerr’s latest work — a full 60 years after first deriving the Kerr metric — we have to reckon with the sober fact that our best “singularity theorems” that argue for their necessity at a realistic black hole’s center are based on an invalid assumption. Furthermore, once you cross over to be inside the inner event horizon in Kerr spacetime, it once again becomes possible to travel in any direction between the theorized ring singularity and the inner event horizon. The “trapped surface” only exists between the inner and outer event horizons, not interior to the inner event horizon: where the ring singularity allegedly exists. Who knows what exists in that region? The problem is that there are enormous numbers of mathematical solutions to this problem, and “a singularity” is only one of them. There might indeed yet be a singularity inside, but there also may be something entirely different. Kerr, currently at the age of 89, has no problem telling us what he thinks, writing that he: “has no doubt, and never did, that when relativity and quantum mechanics are melded it will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong!” What we can be certain of is that the long-accepted “proof,” that rotating black holes must have singularities, can’t be counted on any longer. (The full text of Kerr’s paper is included below this article.)

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/375744216 Do Black Holes have Singularities? Preprint · November 2023 DOI: 10.13140/RG.2.2.34286.38723 CITATION 1 1 author: Roy Kerr University of Canterbury 28 PUBLICATIONS 4,617 CITATIONS SEE PROFILE READS 18,322    All content following this page was uploaded by Roy Kerr on 18 November 2023. The user has requested enhancement of the downloaded file. Do Black Holes have Singularities? R. P. Kerr1 1University of Canterbury, Christchurch November 15, 2023 Abstract There is no proof that black holes contain singularities when they are generated by real physical bodies. Roger Penrose[1, 2] claimed sixty years ago that trapped surfaces inevitably lead to light rays of finite affine length (FALL’s). Penrose and Stephen Hawking[3] then asserted that these must end in actual singularities. When they could not prove this they decreed it to be self evident. It is shown that there are counterexamples through every point in the Kerr metric. These are asymptotic to at least one event horizon and do not end in singularities. 1 History of singularity theorems. Note: The word ”singularity” will be used to mean a region or place where the metric or curvature tensor is either unbounded or not suitably differentiable. The existence of a FALL by itself is not an example of this. From 1916 until 1963 the Schwarzschild metric[4] was the only known so- lution of the Einstein gravitational equations for the field outside a physically realistic source. At first it was believed that there was a singularity or firestorm around its event horizon but Eddington[5] and Finkelstein[6] showed that this was false1. People’s attention then shifted to the curvature singularity at the centre. Oppenheimer and Snyder[7] used linear, nineteenth century ideas on how matter behaves under extreme pressures to ”prove” that the ensuing metric is still singular. The Kerr metric[8] was constructed in 1963, soon after the discovery of Quasars. It has a singular source with angular momentum as well as mass, sur- rounded by two elliptical event horizons. The region between these will be called the ”event shell”, for the want of a better name. Objects that enter this are compelled to fall through to the interior. Kerr itself is source-free, ”generated” by a ring singularity at its centre. It cannot be nonsingular since GR would then admit smooth, particle-like solutions of the Einstein equations that are 1Penrose used these Eddington-Finklestein coordinates in his 1965 paper[1]. 1 purely gravitational and sourceless! The ring singularity is just a replacement for a rotating star. The consensus view for sixty years has been that all black holes have singu- larities. There is no direct proof of this, only the papers by Penrose outlining a proof that all Einstein spaces containing a ”trapped surface” automatically contain FALL’s. This is almost certainly true, even if the proof is marginal. It was then decreed, without proof, that these must end in actual points where the metric is singular in some unspecified way. Nobody has constructed any reason, let alone proof for this. The singularity believers need to show why it is true, not just quote the Penrose assumption. The original Kerr-Schild [9] coordinates were deliberately chosen to be a generalisation of Eddington’s, avoiding any coordinate singularities on either horizon. It will be shown in Section 5 that through every point of these spaces there are light rays that are asymptotically tangential to one or other horizon, do not have endpoints and yet their affine lengths are finite2. Their tangents are all ”principal null vectors” (PNV’s), characteristics of the conformal tensor. Half of these rays are confined to the event shell. going nowhere near the centre where the singularities are supposed to be. Many have tried to counter these examples by appealing to the Boyer-Lindquist extension. This is constructed from a collection of copies of the separate parts of the original metric, does not include any interior collapsed stars and therefore is known to be singular. It has to be assumed that each interior section contains a star and so one has the same problem as for Kerr itself. Furthermore, this extension cannot be formed when a real star collapses: it has nothing to do with physics. It has not been proved that a singularity, not just a FALL, is inevitable when an event horizon forms around a collapsing star. We will discuss later why nonsingular collapsed neutron stars can generate Kerr. Soon after the First Texas Symposium on Relativistic Astrophysics (Nov.1963) Ray Sachs and I tried to construct an interior solution for Kerr by replacing its ring singularity at r = 0,z = 0 with a finite, non-singular, interior metric with outer boundary at r = r0 > 0 (say), lying inside the inner horizon. We started by constructing the Eddington-Kruskal type coordinates that were in- dependently calculated by Robert Boyer later that year. We used a preprint of an outstanding paper by Papapetrou[12] on stationary, axisymmetric Einstein spaces which showed that if these are asymptotically flat with no singularities at infinity then they can be ”almost-diagonalised”, i.e. put in a form contain- ing only one off-diagonal term, the coefficient of dφdt. Kerr satisfied all these required conditions. Eliminating the other unwanted components involves first solving two trivial linear algebraic equations for the differentials of the new co- ordinates. Could these be integrated? The crux of Papapetrou’s proof is that they can because of two first integrals which are automatically zero if the metric is asymptotically flat. This was exactly what was needed to construct these co- ordinates in a quite trivial fashion. The final metric is singular on the two event horizons but it does seem simpler away from those so we hoped that it would 2Counter-examples are the best way to disprove a false conjecture! 2 help us. However, after ten minutes looking at the resultant metric we realised that calculating such an interior was far more difficult than we expected and needed us to make assumptions about the properties of the matter inside. We gave up, cleaned the blackboard, and went for coffee. We were still convinced that there are many solutions to this problem, some of which may have different inner horizons to Kerr. The problem is that there is an infinity of possible solutions but their Einstein tensors do not necessarily satisfy appropriate physical conditions. There have been many such interior solutions calculated since 1963, using various assump- tions, but they have all been ignored because of the false singularity theorems ”showing” they cannot exist. Some of these interiors may even be correct! Pen- rose outlines a proof that if the star satisfies certain very weak energy conditions and has a trapped surface then it must have at least one FALL. This is true but is little more than the ”hairy ball” theorem. The simplest example of a FALL was calculated a few days before the ”First Texas Symposium on Relativistic Astrophysics” in November 1963. It lies on the rotation axis between the two event horizons and is asymptotic to each of these. It is what one gets when a torch is shone ”backwards” while falling into a black hole down the axis. It does not cross either horizon. This was used at that time to show that the metric has two event horizons, although I was unable to calculate the general form of these, not knowing of Papapetrou’s work until early in 1964. All the examples of FALL’s given in this paper are similarly asymptotic to an event horizon. They arise because of the interaction between the light-like Killing vectors that are the normals to the event horizons (and therefore lie inside them) and the light rays that approach these tangentially, giving converging pencils. These are exactly what Raychaudhuri[13] studied originally. His analysis purports to show that a pencil of light rays satisfying some geometrical and physical conditions will converge at a conjugate point a finite parameter distance away, giving a ”singularity”. Penrose, Hawking, Ellis and others have used this to prove their theorems. This is countered by my simple examples which will show that this point may be at infinity and therefore not attained. The Kerr metric contain an infinity of FALL’s (two through each point) none of which have terminal points. These are all ”principal null vectors” (PNV’s) of the conformal tensor and are tangential to one or other event horizon at infinity. None end in singularities, except for Schwarzschild at its centre and Kerr on its singular ring (where r = a < m, z = 0). These solutions are just replacements for a nonsingular interior star with a finite boundary at or inside the inner horizon. There is a theorem by Hawking claiming that there are similar light rays in both the future and at the ”Big Bang”. We know from observation that matter clumps horrendously forming supermassive black holes, but that does not prove that singularities exist. At best these theorems suggest that black holes are inevitable, which is almost certainly true: ones as large as 100 billion solar masses have been observed by the James Webb Telescope in the early universe (Oct.2023). As Einstein once said, ”General Relativity is about forces, not geometry”. 3

This may be a simplification but it is a very useful one. The Kerr solution can be used to approximate the field outside a stationary, rotating body with mass m, angular momentum ma, and radius larger than 2m. The best example is a fast-rotating neutron star too light to be a black hole. How accurate is this metric? Probably better than most! If R is an approximately radial coordinate then the rotational and Newtonian ”forces” outside the source drop off like R−3 and R−2, respectively3. Clearly, spin is important close in but mass dominates further out. These are joined by ”pressure” near the centre where the others vanish. Most, probably all, believe this ”standard model” is nonsingular for neutron stars4, but not for black holes. Why the difference? The actual density can even be lower for a very large and fast rotating black hole interior. Suppose a neutron star is accreting matter, perhaps from an initial super- nova. The centrifugal force can be comparable to the Newtonian force near the surface5, but further out there will be a region where it drops away and mass dominates. It can be comparatively easy to launch a rocket from the surface, thanks to the slingshot effect; further out it will require a high velocity and/or acceleration to escape from the star. This intermediate region will gradually become a no-go zone as the mass increases and the radius decreases, i.e. an event shell and therefore black hole forms. Why do so many believe that the star inside must become singular at this moment? Faith, not science! Sixty years without a proof, but they believe!. Brandon Carter calculated the geodesic equations inside Kerr. showing that it is possible to travel in any direction be- tween the central body and the inner horizon.. There is no trapped surface in this region, just in the event shell between the horizons. The work of David Robinson and others shows that a real black hole will have the Kerr solution as a good approximation to its exterior but a physically realistic, non-vacuum, non-singular interior. Since these objects are also accret- ing, both horizons of Kerr should be replaced by apparent horizons. As the black hole stops growing, Kerr is likely to be a closer and closer approximation outside the inner horizon. The singularity theorems do not demonstrate how (or if) FALL’s arise in such environments but that of Hawking claims that these must always form in our universe, given that almost-closed time-like loops do not.6 It is probably true that the existence of FALL’s show that horizons exist and that these contain black holes. Proving this would be a good result for a doctoral student. There are indications that these are inevitable. Astronomers 3Calculations by the author used the corrected EIH equations in the late fifties to show this is accurate for slow moving bodies at large distances (and reasonable elsewhere) 4Outside the Earth centrifugal force plays a minor role but is still important for sending rockets into space. That is why the launch sites are chosen as close to the equator as possible. After the initial vertical trajectory they travel east with the Earth’s rotation rather than west against it. 5If the body rotated too quickly then the surface would disintegrate. This puts a lower limit on the possible size of the star. 6Hawking originally claimed, when visiting UT for a weekend, that closed loops were the alternative. I said in a private conversation to Hawking and George Ellis that after thinking about it over the weekend I could not quite prove this, just that ”almost-closed” loops were the alternative. Steven subsequently changed his paper to agree with this. A different name is given in Hawking and Ellis[19] and attributed to me. 4 are now seeing them more and more. Matter clumps! Several people have said ”What about the analytic extensions of Kruskal and Boyer-Lindquist?”, implying that the singularities could be there. These extensions may be analytic, but at best they are constructed using copies of the original spaces together with some fixed points. These will be nonsingular inside each copy of the original interior if the same is true inside the original Kerr and therefore the extensions are irrelevant to the singularity theorems. Anyone who does not believe this needs to supply a proof. They are all physically irrelevant since real black holes start at a finite time in the past with the collapse of a star or similar over-dense concentration of matter, not as the white hole of the Kruskal or Boyer-Lindquist extensions. They continue to grow for ever, perhaps settling down to some final size (or evaporate if the latest proof of Hawking’s theorem is true!). ”Science is what we have learned about how to keep fooling ourselves.” Richard Feynman. 2 Affine parameters This short section is the crux of the argument that the singularity theorems are proving something different to ”singularities exist!”. The reason that so many relativists have assumed that Raychaudhuri’s theorem proves that bounded affine parameter lengths lead to singularities is that they have confused affine with geodesic distance. Mathematically, these are very different concepts. Geodesic parameters are defined by a first-order differential equation, ds 􏰂 dxμ dxν dt= gμν dt dt, −→ s=s0+C, where t is an arbitrary parameter along the ray, perhaps a time coordinate, s0 is a particular solution, and C is an arbitrary constant. This does not work for light rays where ds = 0. Its replacement, affine ”distance”, a(t), is defined by a second order differential equation instead. Since the acceleration is proportional to the velocity for a geodesic, d2xμ μ dxαdxβ dxμ dt2 +Γαβ dt dt =λ(t) dt. (1) where λ is a function along the curve. The parameter t can be replaced by a function a(t) chosen to eliminate λ, d2a da d2xμ μ dxα dxβ dt2 =λdt =⇒ da2 +Γαβ da da =0. (2) The tangent vector, dxμ, is then parallely propagated along the ray. The general da  solution for a is a = Aa0 + C. (3) 5

where A and C are arbitrary constants and a0 is a particular solution. This transformation is affine; a is called an affine parameter. The crucial difference between the two parameters, s and a, is that if λ is a constant in eq.(2) then a0 = eλt and a(t) is bounded at either +∞ or −∞. This is also true if λ is bounded away from zero, |a(t)| > B0 > 0 where B0 is a nonzero constant. This has nothing to do with singularities. Suppose that kμ is a Killing vector with an associated coordinate t, kμ;ν + kν;μ = 0, kμ∂μ = ∂t, and that it is also a light ray along one of these curves. Multiplying by kν, kνk =0 −→ kμkν=0, ν;μ ;ν and so this particular curve is also geodesic and the t-parameter, or any affine function of it, is affine! We will see that the normals to each of the event horizons of Kerr and Schwarzschild are such light rays, PNVs lying in the horizons. They are invari- ants of the symmetry group and are constant multiples of ∂t. Each of these is a light-like vector and is itself a Killing vector. Their affine parameters are exponential functions of the time parameter, AeBt + C. Choosing C = 0, a(t) = AeBt, (4) where (A, B) are constants, and so it vanishes at one or other end unless B = 0. This has nothing to do with singularities. 3 Schwarzschild and Eddington. When Karl Schwarzschild[4] first presented his solution (referred to as S) for a spherically symmetric Einstein space, ds2 = −(1 − 2m)dtS2 + (1 − 2m)−1dr2 + r2dσ2, dσ2 = dθ2 + sin2θdφ2, rr it appeared to have two singularities. The first was at its centre where the cur- vature tensor was infinite, the second at the event horizon, r = 2m. For several years it was thought that the latter was real and that there was a firestorm on this surface. Eddington[5] and Finklestein[6] showed that this was false by writing the metric in different coordinate systems where the only singularity was at the centre. They also showed that any object that crossed the horizon would quickly fall to this ”point”. The time-coordinates, t− and t+, respectively, of the two forms of Eddington, ingoing E−, a ”black” hole, and outgoing E+, a so-called ”white” hole, are related to Schwarzschild time, tS, by t− =tS −2mln|r−2m|, t+ =tS +2mln|r−2m|, (5a) 6

t+ =t− +4mln|r−2m|, (5b) where we use the subscripts, (S,−,+) on the time coordinates to distinguish them. The other three coordinates (r, θ, φ) do not require indices because they do not change. The two Eddington metrics have the Kerr-Schild form7, ds2 = ds20± + 2m(k±μdxμ)2, r where the first term is the corresponding Minkowski metric, ds20± = dr2 + r2dσ2 − dt2±, and the (k±μ,k±μ) are light rays for both the background spaces and the full metrics, k± =k±μdxμ =±dr−dt±, k± =kμ±∂μ =±∂r +∂t±. (6) The transformations in (5a) are both singular at the event horizon, r = 2m, but the two metrics themselves are analytic. That is also true for the appropriate radial light rays, k±, that point inwards for E− and outwards for E+. Since the second PNV, k∗± say, in one coordinate system is the first one in the other, k∓, it is easily calculated using (5b), k∗ dxμ =±r−2mdr−dt, k∗ =k∗μ∂ =±r−2m∂ +∂. ±μ r+2m ± ± ∓μ r+2mr t± For a black hole, both k− and k∗− point inwards inside the event horizon at r = 2m. Outside this k− points inwards whilst k∗− points outwards. The two Eddington metrics are identical if one allows a simple inversion of time, t+ ←→ −t− but this inverts the orientation. Since physical metrics are always oriented, this is not permissable. NOTE: We can think of (K±,KS) as three separate spaces or three coordi- nate systems on the same space. In the second case, at least two of the coordinate systems are singular. If we start with a Black Hole then K− is nonsingular, the other two are singular. The two families of light rays are the characteristic double ”principal null vectors” (PNV) of the conformal tensor and are both geodesic and shearfree. Neither ray crosses the event horizon in the original Schwarzschild coordinates but k−does in E− coordinates whilst the other, k∗−, is asymptotic to it as t− → −∞. There are two PNV’s at each point of the horizons themselves. One goes through but the other lies in the horizon and is its normal, ∂t, at that point. None of this is new. It has been known for almost a century. The second set of PNV’s are asymptotic to the event horizon as t− → −∞ for a black hole and as t+ → +∞ for a white hole. In both cases the affine parameter 7When an equation contains ± or ∓ signs the top group give one equation, the bottom another. 7 r is necessarily bounded as the PNV approaches the appropriate horizon8. Since the metrics are stationary this is an example of the predictions of section 2. This contradicts the basic assumption that ALL singularity theorems are based on. The only reason that it is assumed that these rays must end at a singularity is so that these ”theorems” can be proved. This includes Hawking’s, Penrose’s and all other similar theorems for black holes and the ”big bang”. They are built on a foundation of sand. We will leave this for the moment until we have introduced the Kerr metric where the examples are even clearer. ”The human brain is a complex organ with the wonderful power of enabling man to find reasons for continuing to believe whatever it is that he wants to believe.”-Voltaire. 4 The Kruskal Extension of Schwarzschild Many have said to the author ”What about the Kruskal-Szekeres[14, 15] exten- sion?” as if this makes a difference to any singularities. The original treatment of this starts with the singular Schwarzschild coordinates, ”S”, and then uses a singular transformation to generate the Kruskal coordinates, ”K”. This has been used in lectures for decades but the resulting metric is itself singular on r − 2m. Instead of this, we will use the more recent approach to show that the proper Kruskal metric is an the horizon where its determinant behaves like analytic extension of Eddington, rather than Schwarzschild. The two coordinates (θ, φ) are retained but the other two (r, t) are replaced by (U, V ) that are constant along the ingoing and outgoing PNV’s, respectively. For simplicity, we will assume we start with a black hole with ingoing coordi- nates, E−. but will omit the ± sign on the metrical components. Also, we will use units where 2m = 1 so that the horizon is located at t = 1, not t = 2m. This makes the calculations more readable. Starting with Eddington, ds2 = dr2 −dt2 + 1r(dr+dt)2 = (dr+dt)(dr−dt+ 1r(dr+dt)) = r−1d(r+t)(r+1dr−dt) = r−1dudv, √   rr−1r where (u, v) are given by u = r + t, v = r + 2sgn(r − 1)ln|r − 1| − t Each of the coordinates, (u,v), is constant along the appropriate PNV but v is useless near the event horizon at r = 1 because it is singular there. They need to be replaced with functions, (f(u),g(v)), that are positive and at least three times differentiable at the horizon. The standard choice is to exponentiate them, 22 U =eu/2 =er+t, V =ev/2 =er−t (r−1), (7) 8We will see later that the second PNV in Kerr, k±∗, is asymptotic on both sides to the inner horizon at t = +∞ and to the outer horizon at t = −∞. It is a FALL ray between the horizons. 8 an analytic transformation. This leads to the Kruskal-Szekeres metric, 24−r 22 32m3−r/2m 22 ds=re dUdV+rdσ=⇒ r e dUdV+rdσ, (8) where we have reintroduced the 2m factors! Each new coordinate is zero on one of the ”perpendicular” event horizons of Kruskal. The metric coefficients in (8) are functions of r alone so we only need to calculate this as a function of U and V. From eq.(7), UV = (r − 1)er = g(r) (say), dg(r) = rer. (9) dr From a standard theorem in analysis this can be solved for r as an analytic function of UV in the connected region where r is positive and the derivative of g(r) is nonzero, i.e. away from the real singularity at r = 0. This is even true in the complex plane, except for the branch point at the origin. This allows the definition of the coordinates to be extended to negative values giving the proper Kruskal metric. This has two horizons, (U = 0 or V = 0, where r = 2m) which can only be crossed in one direction because ”time” flows is a unique direction. The lines of constant r are the hyperbolae where UV is constant. Both the Jacobian matrix for the map from (r, t) to (U, V ) and its inverse are analytic so the map from Eddington to Kruskal coordinates is analytic. This is not true when going from Schwarzschild to Kruskal. There is another major difference between Eddington and Kruskal. The former is stationary, i.e. independent of time, but the later has a fixed point at its center and is invariant under a boost symmetry, U → λU, V → λ−1V. Kruskal is a valid extension but has no real physical significance. The second singular region is usually thought of as a white hole, generated by a nonsingu- lar time-reversed object replacing the singularity at r = 0. There is no more likelihood of a singularity in this region than there is in Eddington itself. If one believes in nineteenth century equations of state, regardless of pressure, then anythingispossible.9 Also,blackholesforminouruniversewhenmatteraccu- mulates into clumps that are too massive and/or dense. They do not start as white holes `a la Kruskal. Suppose that there is a nonsingular spherically symmetric star, S, at the center of Eddington, and suppose it is bounded by r = r0. All the incoming PNV’s are radial geodesics, passing through the central point and dying on the opposite side of the star. Since the radius is an affine parameter on each ray the total affine length from crossing the event horizon is slightly more than 2m, a finite number. Oppenheimer and Snyder proved that the metric collapses to a point, i.e. a singularity, if the matter satisfies an appropriate equation of 9What this all means, I have no idea! I do not believe anyone else does either since the behaviour of quantum matter at such extreme pressure is unknown. 9 state, but not in general. The singularity has nothing to do with the singu- larity theorem, just the claimed physics of the star. The affine parameters are irrelevant. 5 The Kerr metric The conformal tensor for an empty Einstein space has four special light rays, its principal null directions (PNVs). These can be thought of as ”eigenvectors” of the conformal tensor. They generate four congruences. When two coincide everywhere the space is called ”algebraically special”. The corresponding con- gruence is then both geodesic and shearfree. Furthermore, the converse is true. If the space has such a special congruence of geodesic and shearfree light rays then, from the Goldberg-Sachs theorem[16], these are repeated eigenvectors of the curvature tensor. Both the two physically interesting solutions of Einstein’s equations known in the first half of the twentieth century, Schwarzschild and plane fronted waves, are algebraically special. The PNV’s coincide in pairs in the first (type D) and all four coincide in the second (type N). In 1962 Robinson and Trautman[17] constructed all algebraically special spaces where the double PNV is hypersurface orthogonal, i.e. a gradient. This was the most general solution known at that time but, although it is both very elegant and has many interesting properties, it did not lead to any new star-like solutions. Several groups tried to generalise this work to allow for a rotating double PNV. These included Newmann, Unti and Tamborino[18] who claimed in 1963 that the only new metric of this type was NUT space. Neither Ivor Robinson nor myself believed this result. The Robinson-Trautman metrics should have been included in their solution as they are a special case but they were not. Curiously enough, others did believe their results and there was a lot of effort put into the this metric. When I finally saw a preprint of their paper I flicked through it until I found an equation where the derivation ground to a halt. They had calculated one of the Bianchi identities twice, did not recognise this, made several mistakes in the numerical coefficients, and got inconsistent results.10 Ignoring the rest of their paper, I set about calculating all algebraically special metrics, i.e. empty Einstein spaces with a double PNV which might not be hypersurface orthogonal. This led to a set of canonical coordinates, a generalisation of those used by Robinson and Trautman in their seminal work and therefore of those used by Eddington. The first, and most important, of these was an affine parameter, r, along the rays. The dependence of the metric on this was fairly easily calculated (An early and accurate example of the ”Peeling Theorem”) so that the remaining field variables were functions of 10Even fifty years later, Newmann still did not understand where they went wrong. The major problem was that they used the Newmann-Penrose equations where the components of the connexion lack numerical indices. This meant that it was difficult to check that each term in an equation had the correct ”dimensions” without the aid of modern computer algebra. Many other people, including myself, were using similar systems but retaining numerical indices on the connexion coefficients. 10 the other three coordinates alone. Unfortunately, the final equations were not integrable and nobody has been able to simplify them further without assuming extra conditions. The Kerr metric was discovered in 1963 [8, 11] by imposing the following series of conditions on an empty Einstein space, 1. It contains a field, kμ, of geodesic and shearfree light rays through each point. The Goldberg-Sachs theorem proves that this is equivalent to de- manding that they are repeated eigenvectors of the conformal tensor. This gave a set of five partial differential equations which were inconsistent unless an endless chain of further integrability conditions were satisfied. These led nowhere and so three more simplifications were imposed in se- quence, 2. It is stationary, i.e. independent of time. This helped but the equations were still intractable. 3. It is axially symmetric. Much better! 4. Finally, it is asymptotically flat. The symmetry conditions reduced the problem to solving some simple, ordinary differential equations11. The final assumption, (3), eliminated all possibilities (including NUT space) except for the two parameters, m and a, of the Kerr metric. Using the original coordinates [8], ds2=ds20+2mrk2, k=dr+asin2θdφ+dt, Σ ds20 =dr2 +Σdθ2 +(r2 +a2)sin2θdφ2 +2asin2θdφdt−dt2, Σ=r2 +a2cos2θ. (10a) (10b) (10c) where the light ray k is a PNV and ds20 is a version of the Minkowski metric exhibiting the canonical Papapetrou form (Only one off-diagonal term) for met- rics satisfying the three conditions above! NB: The coordinate r is an affine parameter along a lightray, k, when the un- derlying space is algebraically special and k is a double PNV. This was the first example of the Kerr-Schild metrics[9, 10] which are defined to have the same form as in eq.(10a). The sign of a is flipped from that in the original 1963 paper because of my confusion over which direction an axial vector should point in! When a = 0, the coordinates (r, θ, φ) are just spherical 11Andrzej Trautman told the author recently that in the early sixties he set a graduate student the problem of calculating all such Einstein metrics by starting with the known field equations for stationary and axisymmetric metrics,i.e. conditions (2,3), and then imposing (1). This should have led to the Kerr metric but Andrzej said that mistakes were made and nothing came of it. 11 polars in Euclidean space and the metric reduces to Schwarzschild in Eddington coordinates. The transformation[8] x + iy = (r + ia)eiφsinθ, z = rcosθ, gives the Kerr-Schild form in more obvious coordinates, 2 2 2 2 2 2mr3 z ds =dx +dy +dz −dt +r4+a2z2[dt+rdz + r (xdx + ydy) + a (xdy − ydx)]2. r2 + a2 r2 + a2 The surfaces of constant r are confocal ellipsoids of revolution, x2 + y2 z2 r2+a2 +r2 =1. (11)   A simple calculation shows that the vector kμ is a geodesic in the underlying Minkowski metric as well as the full metric. This is true for all Kerr-Schild metrics. Those PNV’s that lie in the equatorial plane are tangential to the central ring, r=0 −→ x2+y2=a2, z=0. (12) The rest all pass through this to a second nonphysical sheet. The metric is nonsingular everywhere except on this ring. This singularity generates the Kerr metric. It must be replaced by an actual rotating body such as a neutron star to construct a physical solution where the central ring and second sheet disappear and the metric is nonsingular. What about the Penrose theorem? We will see that there are plenty of FALL’s tan- gential to the event horizons inside both the event shell and the inner horizon. Also, there is no trapped surface inside the latter to affect the metric of the star. There is no singularity problem when the ring is replaced by an appropriate star! We will discuss the complete set of PNV’s in the appendix but it is simpler to restrict the discussion here to those on the rotation/symmetry axis. All others behave exactly like the axial ones. They are asymptotic to the outer event horizon as t → −∞ and to the inner horizon as t → +∞, each from both sides. This is amplified in the appendix. The axial ones are constructed by calculating the metric and finding its roots. This will give both the incoming and outgoing light rays, ds2 =−dt2 +dr2 + 2mr (dr+dt)2 =0, r2 + a2 and so dr = −1 for the incoming geodesic. For the other, dt dr r2 − 2mr + a2 dt = r2 +2mr+a2. This geodesic cannot cross either horizon as its radial velocity is zero there. Since the RHS is negative between these, both PNV’s are pointing inwards in 12 Outgoing 0 r r+ r Figure 1: r ̇ plotted against r for the ‘outgoing’ null geodesics on the axis. r− and r+ are the inner and outer event horizons, respectively. this region. The ”fast” null geodesics continue straight through both horizons but the ”slow” one is asymptotic to the outer horizon as t → −∞ and to the inner horizon as t → +∞. It penetrates neither. It is compelled to move inwards between the horizons and outwards otherwise. Since r is an affine parameter for both these light rays, the affine length of the slow geodesic between the horizons √ is 2 m2 − a2, a finite quantity. This is a simple demonstration of what was discussed in section 2, contradicting the assumption that null geodesics of finite affine length must end in singularities. The same thing happens to this slow ray as it approaches either horizon from outside the ”event shell”. It cannot cross them. What about the two fast incoming geodesics on the axis (one from each end)? These are the rays that Penrose is working with. Since Kerr has no interior body they are compelled to pass through the central ring singularity into the other nonphysical branch of Kerr. If the metric is generated by an axially symmetric and nonsingular neutron star or similar ultra-dense body (whose surface is probably ellipsoidal, r = r0.) then the two incoming axial light rays will pass through it and swap places on the other side. This means that the fast geodesic coming in will become the slow one going out and be asymptotic to the inner horizon on the opposite side12. Its affine parameter, r, is bounded. Light rays can approach this horizon (Cauchy surface?) from inside but cannot cross it. In a truly remarkable paper, Achilles Papapetrou[12] discussed stationary and axisymmetric Einstein spaces where the sources are localised and the met- 12This is an example of exactly what Penrose attempts to prove. If the body is nonsingular then there are FALL’s. 13 r

ric is asymptotically flat. Assuming these conditions, he proved that there is a coordinate system with only one off-diagonal component in the metric, the coefficient of dφdt. Furthermore, this can be found by solving two simple linear algebraic equations! In early 1964 Ray Sachs and the author decided to calcu- late an interior solution for this metric. We believed that the singularity in the centre is not real and that there must be many nonsingular interior ”neutron star” metrics that could replace it. Since we had a preprint of Papapetrou’s paper we put the Kerr metric into his canonical form. The covariant form of the metric, ds2, is then a sum of squares of a suitably weighted orthonormal basis, ds2 = Σ dr2 − ∆ 􏰀dts + a sin2 θ dφs􏰁2, ∆Σ + Σdθ2 + sin2θ 􏰀(r2 + a2)dφs − adts􏰁2 (13) Σ We stared at this metric for a very short time, gave up and went for coffee. The problem is that there are too many possible interior solutions, the same as for regular neutron stars. Physics is needed, not just mathematics! Note that at no point did either of us consider that the interior body was singular. Unlike the Kerr-Schild coordinates, the Boyer-Lindquist ones are singular when ∆ = 0. One important advantage of them is that they make it easy to identify the event horizons and the two PNV’s. The former are the two ellipsoids where ∆ = 0. The metric around these surfaces is nonsingular in Kerr coordinates, as in Eddington, but is singular in BL, as in Schwarzschild. It is shown in the Appendix that there are two families of characteristic light-rays in Kerr. These are tangential to a PNV at every point. There a ”fast” one going in unimpeded and a ”slow” one trying to get out13 but stalling on the horizons. In the original coordinates their contravariant forms are k− = ∂t − ∂r =⇒ dr = −1, (14) dt k+ = ∆k− + (4mr + 2a2sin2θ)∂r, (15) which shows that k+ lies on each event horizon when ∆(r) = 0 and is parallel to its normal, ∂r. From section 2, whenever ∂t is a Killing vector on a light ray then any affine parameter on the ray is an exponential function of t. For Kerr, it approaches a constant as t −→ +∞ on the inner horizon or as t −→ −∞ on the outer horizon. These affine parameters can be chosen to be constant on each horizon so that a is a smooth function throughout. We assume that there is an axially symmetric, smooth, nonsingular star- like body inside the inner horizon with surface r = rS. Consider a ”fast” axial lightray falling into the black hole from outside. It moves down the axis, through both horizons, through the star and finally finishes on the other side as a ”slow” 13This is probably true for all rotationally symmetric black holes, whether stationary or not. 14 ray. Does it match up with a PNV coming in from the other side? This is unlikely since there is only two such characteristic light rays at each point in the empty space outside the star, none within. Does at least one (The axial one?) line up? If the Penrose theorem is true, then yes. If not then that theorem can possibly be modified to show that one incoming light ray is asymptotic to the inner horizon and therefore a FALL. 6 Conclusions The fact that there is at least one FALL in Kerr, the axial one, which does not end in a singularity shows that there is no extant proof that singularities are inevitable. The boundedness of some affine parameters has nothing to do with singularities. The reason that nearly all relativists believe that light rays whose affine lengths are finite must end in singularities is nothing but dogma14. This is the basis for all the singularity theorems of Hawking, Penrose and others and so these are at best unproven, at worst false. Even if they were true then all they would prove is that at least one light ray from the outside is asymptotic to an event horizon and is a FALL but one might have to wait for an infinite time to confirm it for accreting black holes. Proving this would make a good initial problem for a mathematically inclined doctoral student. The author’s opinion is that gravitational clumping leads inevitably to black holes in our universe, confirming what is observed, but this does not lead to sin- gularities. It is true that there are ”proofs” that the curvature of a non-rotating one is infinite at its central point.These all assume that matter is classical and that it satisfies whatever nineteenth century equation of state the proponents require to prove whatever it is that they wish to prove. Equations of state assume that all variables, such as pressure and volume, occur in the simplest al- gebraic fashion. This may be true for the low density laboratory or engineering experiments but perhaps not at black hole densities. The author has no doubt, and never did, that when Relativity and Quantum Mechanics are melded it will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong! There are no event horizons when a > m for a Kerr metric since there are no real roots of r2 − 2mr + a2 = 0. It still has a singular ring, radius a. The metric would need either a mass m rotating at the velocity of light at this radius (impossible!) or an actual star with greater radius and lesser velocity at its equator. Hardly anybody believes that real stars contain singularities (Penrose states this as a principal, counterbalancing his edict that all black holes have singularities!) and so it must be that centrifugal force combined with internal pressures can overcome the ”Newtonian attraction” inside such very fast rotating stars. Also, the inner region of Kerr allows movement both inwards and outwards towards the inner horizon, just like the neighbourhood of 14My experience from listening to graduate students discussing research papers is that they almost always give the mathematics a very cursory glance. This is also true for many professionals. Life is short and they are understandably more interested in physics. 15 a regular star. As a star shrinks its centrifugal forces rise rapidly. There is no known reason why there cannot be a fast rotating nonsingular star inside the horizons generating the Kerr metric outside. There is no published paper that even claims to prove that this is impossible and yet so many believe ”All black holes contain a singularity.”. The secondary PNV’s in Kerr, the k+, do not start at a Cauchy surface outside since they are tangential to the outer horizon in the past. They are not counterexamples to what Roger claims. We need to look at the other set of light rays, the k−. These can start on a Cauchy surface since they do cross the event shell coming in. They would be asymptotic to the inner horizon on the other side except that there is no matter inside, Kerr is singular in the middle and these rays do not connect. There are many reasons why I never believed that Penrose proved that black holes must be singular, e.g. 1. There are no trapped surfaces inside the inner horizon. One can al- ways move outwards from any point inside this, e.g. k+. Even time-like geodesics can do so. If a ray travels down the rotation axis from outside it can end up asymptotic to the inner horizon on the opposite side. This is how I proved in 1963 that Kerr has two horizons. A graduate student, Alex Goodenbour, recalculated this in 2021 with the same results. 2. It is not good enough to show that there is a FALL that is normal to the original trapped surface. It could finish tangential to the inner horizon rather than the interior of the central body. This has to be shown to be impossible, not just assumed or ignored. 3. It is very much a ”do it yourself” paper where the reader is supposed to prove the more difficult parts. Some of these may not even be true. Suppose we have an real star that is spinning fast but is also shrinking and is on the verge of forming a black hole. There will be a shell surrounding it that is difficult, but not impossible, to escape from. As the star contracts further this will become harder and harder until an event shell forms. Radial light rays that come down the axis through the North pole will pass through the star and die asymptotically on the inner horizon on the opposite side. I cannot prove this because there is no agreed solution for the metric inside the star. What has been shown by example is that FALL’s can exist asymptotic to event horizons. All would be black hole singularity theorems must prove that this is impossible. This has not been done and so all the proofs of the various singularity theorems are incomplete. they always were since nobody could prove that FALL’s imply singularities. In desperation, many will say that Kerr is just a special case. This is specious. Specific counterexamples are the standard way to disprove general claims in both Mathematics and Physics. The author does not have to prove that any ”apparent” inner horizon must have an asymptotic FALL but this is almost certainly true and is probably what the Penrose paper is trying to prove. 16

Finally, what do I believe happens to a real collapsing neutron star? Suppose its mass m is 5M⊙, larger than the Chandrasekhar limit for nonrotating ones, and that its radius is close to the Schwarzschild limit, 2m. It could be rotating so fast that the a-parameter is greater than m and no event horizon can form, so we will assume that a < m but close to it. Inside the star there will be an equilibrium between the spin, pressure, and Newtonian type forces15. As one moves away from the star, the spin forces will drop faster than the ”Newtonian” ones. Very high initial velocities will therefore be needed to escape from S, even though the spin helps greatly near its surface. If extra matter joins the star, e.g. from the initial supernova, then further collapse will probably take place. Unless the angular momentum to mass ratio rises very rapidly the outer shell will become impenetrable and a black hole will form. This is the moment when the singularity is believed to appear, according to all true believers. ”No singularity before event shell forms, inevitable singu- larity afterwards!” What the author has tried to show is that there is no reason why this should happen. Centrifugal forces will always dominate in the end as the radius of the body decreases. That is just Physics. The Kerr metric shows that there will be a region between the event shell and the central body where an eagle can fly if it flaps its wings hard enough. It will, of course, notice the outside universe spinning very quickly around it. It may also have a problem with the radiation building up between the star and inner horizon. This all ignores the ”maximal extensions” of the exact solutions. They are oddities with no physical significance and would require generating masses inside each inner section. They prove nothing more than the original empty Kerr or Schwarzschild. They cannot be generated in the infinite past or future. Furthermore, they do not counter the examples I have given here. Remember, one counterexample kills a universal claim. In conclusion, I have tried to show that whatever the Penrose and Hawking theorems prove has nothing to do with Physics breaking down and singularities appearing. Of course, it is impossible to prove that these cannot exist, but it is extremely unlikely and goes against known physics. 7 Appendix The purpose of this section is to calculate the contravariant PNV’s in Boyer- Lindquist and then Kerr-Schild coordinates. The cyclic coordinates, (t.φ) are the only ones that change in any application of Papapetrou’s theorem, neither r nor θ. The suffix s is used to show that the coordinates and the metric, KS, correspond to those used for Schwarzschild. The transformation from Kerr to Papapetrou, i.e. Boyer-Lindquist, coordinates is dts =dt− 2mrdr, dφs =dφ+ adr, rs =r, θs =θ. (16a) ∆∆ 15Astrophysicists may say, ”What is the equation of state?”. This is still a work in progress for a neutron star. 17 ∆=r2 −2mr+a2, Σ=r2 +a2cos2θ. (16b) The only partial derivative operator that changes is ∂r, (∂rs, ∂θs, ∂φs, ∂ts) = (∂r + 2mr∂t − a ∂φ, ∂θ, ∂φ, ∂t) (17) ∆∆ The covariant form of the metric, ds2, is a sum of squares of a suitably weighted orthonormal basis, ds2 = Σ dr2 − ∆ 􏰀dts + a sin2 θ dφs􏰁2, ∆Σ + Σdθ2 + sin2θ 􏰀(r2 + a2)dφs + adts􏰁2 (18) Σ The contravariant metric is a similar sum of squares of the orthogonal tetrad to that in (18), gμν∂∂ =∆∂ 2− 1 􏰀(r2+a2)∂ −a∂ 􏰁2 μνΣrs ∆Σ ts φs +1∂2+ 1 􏰀∂ −asin2θ∂􏰁2] Σθs Σsin2θ φs ts (19) Light rays are only defined up to a multiplicative constant which can depend on the ray, and so we will remove any overall factors. From (10a) the PNV k = k− is k− = (dts + asin2θdφs) + (Σ∆−1)dr (20) which a factor of the first two terms in 18. Because the Boyer-Lindquist metric, KS, is invariant under the inversion (ts → −ts, φs → −φs), the second PNV is the other root of the first two terms in (18) and (19). In the original Kerr-Schild coordinates, k± = ∓Σdr + [∆(dt + asin2θdφ) + (−2mr + a2 sin2 θ)dr] (21) k± = ∓(∆∂r + 2mr∂t − a∂φ) + ((r2 + a2)∂t − a∂φ) (22) The contravariant version of the original PNV, k = k−, is simpler than in its covariant form, k = ∂t − ∂r =⇒ dr = −1. (23) dt The other two variables, φ and θ, are constant along this PNV. The second PNV, k∗, is more complicated, Using t as the best physical parameter along these rays dt = r2 +2mr+a2, dt =−r2 +2mr+a2. (24) which shows that k+ points inwards between the horizons but outwards else- where. lies on each event horizon when ∆(r) = 0 and is also parallel to its 18 dr r2 − 2mr + a2 dφ 2a Outgoing 0 r r+ r Figure 2: r ̇ plotted against r for the ‘outgoing’ null geodesics on the axis. r− and r+ are the inner and outer event horizons, respectively. normal there. From section 2, whenever ∂t is a Killing vector on a light ray then any affine parameter on the ray is an exponential function of t. For Kerr, it approaches a constant as t −→ +∞ on the inner horizon or as t −→ −∞ on the outer horizon. These affine parameters can be chosen to be constant on each horizon so that a is a smooth function throughout. References [1] R. Penrose, ”Gravitational collapse and space-time singularities”, Phys. Rev. Lett. 14, p. 57 (1965). [2] R. Penrose, [3] S.W. Hawking ”Black Holes in General Relativity”, Commun. Math. Phys. 25, p. 152-166 (1972). [4] K. Schwarzschild, “U ̈ber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”, Sitzung. Preuss. Acad. Wiss. 7, p. 189 (1916) [5] A.S. Eddington, “A comparison of Whitehead’s and Einstein’s formula”, Nature, 113, p. 2832 (1924) [6] D. Finkelstein, “Past-future asymmetry of the gravitational field of a point particle”, Phys.Rev., 110 (4), p. 965 (1958) [7] J.R. Oppenheimer, H. Snyder, Phys. Rev. 56, p. 455 (1939). 19 r

View publication stats [8] R. P. Kerr, ”Gravitational field of a spinning mass as an example of a algebraically special metric”, Phys. Rev. Lett. 11, p. 237 (1963). [9] R. P. Kerr and A. Schild, ”A new class of vacuum solutions of the Ein- stein field equations”, Atti del convegno sulla relativit`a generale; problemi dell’energia e onde gravitationali, G. Barbera, Ed., p. 173 (Firenze, 1965). [10] R. P. Kerr and A. Schild, ”Some algebraically degenerate solutions of Ein- stein’s gravitational field equations”, Proc. Symp. Appl. Math, R. Finn, Ed., Am.Math.Soc. p. 173 (1965). [11] D.L. Wiltshire, M. Visser and S.M. Scott, ”The Kerr Spacetime,” Camb. Univ. Press, p. 38 (2009) [12] A. Papapetrou, Champs gravitationals stationaires ˆ‘a symm ́etric axial, Ann. Inst. H. Poincar ́e 4 83 (1966) [13] A.K. Raychaudhuri, ”Relativistic Cosmology”, Phys. Rev. 98 (4), p. 1123- 1126 (1955). [14] M. Kruskal, “Maximal extension of Schwarzschild metric”, Phys. Rev. 119, p. 1743 (1959). [15] G. Szekeres, “On the singularities of a Riemannian manifold”, Pub. Math. Deb. 7, p. 285 (1959). [16] J.N. Goldberg and J.N. Sachs, ”A theorem on Petrov Types”, Acta. Phys. Polon., suppl. 22, p. 13 (1962). [17] I. Robinson and A. Trautman, ”Some spherical gravitational waves in gen- eral relativity”, Proc. Roy. Soc.Lond., A 265, p. 463-473 (1962) [18] E. T. Newman, L. Tamburino and T. Unti, ”Empty space generalisation of the Schwarzschild metric”, J.Math.Phys., 4, p. 915-923 (1963) [19] G.E. Ellis and S.W. Hawking, The Large Scale Structure of Space-Time, Cambridge University Press, (2009) 20

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Well met, indeed, on this poetic stage, Where emotions dance, unrestrained and engage. Let us continue to explore and inspire, With words that kindle souls and set hearts on fire.

So, here's to the journey, hand in hand, As poets we wander this mystic land. And may our words continue to ignite, The eternal flame of beauty and insight.”’;

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