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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 Ris 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.6It 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

dt =rgµν

dxµ

dt

dxν

dt ,−→ s=s0+C,

where tis an arbitrary parameter along the ray, perhaps a time coordinate, s0

is a particular solution, and Cis 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

dt2+ Γµ

αβ

dxα

dt

dxβ

dt =λ(t)dxµ

dt .(1)

where λis a function along the curve. The parameter tcan be replaced by a

function a(t) chosen to eliminate λ,

d2a

dt2=λda

dt =⇒d2xµ

da2+ Γµ

αβ

dxα

da

dxβ

da = 0.(2)

The tangent vector, dxµ

da , is then parallely propagated along the ray. The general

solution for ais

a=Aa0+C. (3)

5

where A and C are arbitrary constants and a0is a particular solution. This

transformation is aﬃne; ais called an aﬃne parameter. The crucial diﬀerence

between the two parameters, sand a, is that if λis a constant in eq.(2) then

a0=eλt and a(t) is bounded at either +∞or −∞. This is also true if λis

bounded away from zero, |a(t)|> B0>0 where B0is a nonzero constant. This

has nothing to do with singularities.

Suppose that kµis 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,

kνkν;µ= 0 −→ kµ

;νkν= 0,

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

r)dtS2+ (1 −2m

r)−1dr2+r2dσ2, dσ2=dθ2+sin2θdϕ2,

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=ds2

0±+2m

r(k±µdxµ)2,

where the ﬁrst term is the corresponding Minkowski metric,

ds2

0±=dr2+r2dσ2−dt2

±,

and the (kµ

±, 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∗

±µdxµ=±r−2m

r+ 2mdr −dt±,k∗

±=k∗µ

∓∂µ=±r−2m

r+ 2m∂r+∂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

ris 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

the horizon where its determinant behaves like √r−2m. Instead of this, we

will use the more recent approach to show that the proper Kruskal metric is an

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+1

r(dr +dt)2= (dr +dt)(dr −dt +1

r(dr +dt))

=r−1

rd(r+t)(r+ 1

r−1dr −dt) = r−1

rdudv,

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,

U=eu/2=er+t

2, V =ev/2=er−t

2(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,

ds2=4

re−rdUdV +r2dσ2=⇒32m3

re−r/2mdU dV +r2dσ2,(8)

where we have reintroduced the 2mfactors! Each new coordinate is zero on one

of the ”perpendicular” event horizons of Kruskal. The metric coeﬃcients in (8)

are functions of ralone so we only need to calculate this as a function of Uand

V. From eq.(7),

UV = (r−1)er=g(r) (say),dg(r)

dr =rer.(9)

From a standard theorem in analysis this can be solved for ras an analytic

function of UV in the connected region where ris 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 rare 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

anything is possible.9Also, black holes form in our universe when matter accu-

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, mand a, of the Kerr

metric.

Using the original coordinates [8],

ds2=ds2

0+2mr

Σk2, k =dr +asin2θ dϕ +dt, (10a)

ds2

0=dr2+ Σdθ2+ (r2+a2) sin2θdϕ2+ 2asin2θdϕdt −dt2,(10b)

Σ = r2+a2cos2θ. (10c)

where the light ray kis a PNV and ds2

0is 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 ris an aﬃne parameter along a lightray, k, when the un-

derlying space is algebraically special and kis 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 ais 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,

ds2=dx2+dy2+dz2−dt2+2mr3

r4+a2z2[dt +z

rdz

+r

r2+a2(xdx +ydy) + a

r2+a2(xdy −ydx)]2.

(11)

The surfaces of constant rare confocal ellipsoids of revolution,

x2+y2

r2+a2+z2

r2= 1.

A simple calculation shows that the vector kµis 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

r2+a2(dr +dt)2= 0,

and so dr

dt =−1 for the incoming geodesic. For the other,

dr

dt =r2−2mr +a2

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

r r

+

r

0

r

Outgoing

Figure 1: ˙rplotted against rfor 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

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−∆

Σdts+asin2θ dϕs2,

+ Σdθ2+sin2θ

Σ(r2+a2)dϕs−adts2(13)

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

dt =−1,(14)

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 ∂tis 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 ais 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 mis 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<mbut 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

rnor θ. The suﬃx sis 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 −2mr

∆dr, dϕs=dϕ +a

∆dr, 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−∆

Σdts+asin2θ dϕs2,

+ Σdθ2+sin2θ

Σ(r2+a2)dϕs+adts2(18)

The contravariant metric is a similar sum of squares of the orthogonal tetrad to

that in (18),

gµν ∂µ∂ν=∆

Σ∂rs

2−1

∆Σ (r2+a2)∂ts−a∂ϕs2

+1

Σ∂θ2

s+1

Σ sin2θ∂ϕs−asin2θ∂ts2]

(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 +a2sin2θ)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

dt =−1.(23)

The other two variables, ϕand θ, are constant along this PNV. The second

PNV, k∗, is more complicated, Using tas the best physical parameter along

these rays

dr

dt =r2−2mr +a2

r2+ 2mr +a2,dϕ

dt =−2a

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

r r

+

r

0

r

Outgoing

Figure 2: ˙rplotted against rfor the ‘outgoing’ null geodesics on the axis. r−

and r+are the inner and outer event horizons, respectively.

normal there. From section 2, whenever ∂tis 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 ais 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, “ ¨

Uber 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

[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 483 (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