06.03.2008

see also the

Quotations from Crothers' papers are displayed in
**black**. Equation labels of type (n) refer to Crothers' papers.

**Abstract**

In the last years since 2005 S. Crothers has published a series of papers in the Journal
PROGRESS IN PHYSICS***)** (see [3]) which deal with the alleged fact that black holes are
not compatible with General Relativity. Crothers views stem
from certain dubious ideas on spacetime manifolds, especially in the case of
Hilbert/Schwarzschild metrics: His idea is that instead of the 2-sphere of the event horizon
there is merely *one single central* point.
It will be shown below that this assumption would lead to a curious world where Crothers'
''central point'' can be approximated in sense of distance by 2-spheres S_{r} of
radius r > α. Hence the event horizon cannot be a single point. −
Concerning the two validity regions of the Schwarzschild metric in contrast to Crothers'
claims the fact is recalled that both validity regions of the Schwarzschild metric
can be covered by introduction of the Eddington-Finkelstein coordinates.

***) NB.: The journal ''PROGRESS IN PHYSICS'' should not be mixed up with the prestigious and much older IOP-journal
''Reports on Progress in Physics''.**

Crothers bases his objection of Schwarzschild black holes on two statements: Concerning the Schwarzschild metric (2.1) below he asserts in the Introduction of [1]:

When the required mathematical rigour is applied it
is revealed that

1) **r _{o} =α denotes a point, not a 2-sphere,**
and that

2)

This assertion cannot be true: We consider the Schwarzschild/Hilbert metric

(2.1)
ds˛ = − (1 − ^{α}/_{r}) dt˛ +
(1 − ^{α}/_{r})^{−1} dr˛ + r˛(dθ˛ + sin˛θ dφ˛)

in the spacetime that is accessible for a physical observer, i.e. for r > α:
Here the metric (2.1) defines submanifolds S_{r} for each pair of fixed values
of t and r, the metric of which follows from (2.1) to be

(2.2) ds˛ = r˛ (dθ˛ + sin˛θ dφ˛) .

Hence S_{r} is a 2-sphere with radius r.
The set S_{α} of singularities of the Schwarzschild/Hilbert metric has the metric

(2.3) ds˛ = α˛ (dθ˛ + sin˛θ dφ˛).

and hence is a 2-sphere as well.

The distance between S_{r} and S_{α} is given by Crothers'
''proper radius'' (cf [1, eq. (14)] with C(r)= r˛)

(2.4)
R_{p}(r) =
[r(r−α)]^{½} +
α ln |(r^{½}+(r−α)^{½}) α^{−½}|

measurable in radial direction between arbitrary associated points of the concentric spheres.
Since R_{p}(r) is continuous at r=α the distance between S_{r} and
S_{α} tends to 0 for r → α:

(2.5)
lim_{r → α} R_{p}(r) = R_{p}(α) = 0 .

Therefore, the set S_{α} of the metric singularities can be approximated
with respect to the distance R_{p}(r) by concentric 2-spheres of radius r > α: Thus,

See also **Section 4**.

This claim is not true: As will be recalled here the region r > α,
accessible for human observers, can be extended to the region r > 0 by the introduction of
simple coordinates well-known as
**
Eddington-Finkelstein coordinates** (cf.[4, p.184]).
The additional part of the world - usually called ''black hole'' is not directly explorable by
human observers. We can only try to
extrapolate the rules that have been found in the accessible part of the world.

The special structure of the Schwarzschild metric (2.1) allows a simple extension from the obvervable region r > α to the region r > 0 crossing the former boundary r = α.

Before doing so it is advantageous to simplify the notation by an obvious transformation:
By applying the substitution ^{r}/_{α} → r we can simplify
the Schwarzschild metric (2.1) to

(3.1)
ds˛ = − (1 − ^{1}/_{r}) dt˛ +
(1 − ^{1}/_{r})^{−1} dr˛ + r˛(dθ˛ + sin˛θ dφ˛)

i.e. in case α>0 we are allowed to assume α=1 without loss of generality.

Now we rewrite eq. (3.1) to

(3.2)
ds˛ = (1 − ^{1}/_{r}) [ −dt˛ +
( ^{r dr}/_{r−1})^{˛}] + r˛(dθ˛ + sin˛θ dφ˛) .

Instead of t we introduce a new variable v by

(3.3) v = t + r + ln |r−1| ,

hence ^{r dr}/_{r−1} = dv − dt and

(3.4)
ds˛ = − (1 − ^{1}/_{r}) dv˛ + dv dr + dr dv
+ r˛(dθ˛ + sin˛θ dφ˛) ,

which metric form is free from singularities in the region {(v,r) | 0 < r < ∞,
v Î**R**}.

**Remark** Equ. (3.3) is valid for r < 1 *as well*, which *generally* yields
dv = dt + dr + ^{dr}/_{r−1}. Inserting this in eq. (3.4) leads back to
eq. (3.1) as the reader will check immediately. Therefore we have the result:

This result can be applied to again calculate the induced metric on the sphere S_{α}
to obtain eq. (2.3) again (with α=1).

We shall determine here a subset of the event horizon to show again that it cannot be only one central point:

The metric of an equatorial section θ = π/2 through an Euclidean space parametrized by spherical polar coordinates (r, θ, φ)

(4.1) ds˛ = dr˛ + r˛ (dθ˛ + sin˛θ dφ˛) Þ ds˛ = dr˛ + r˛ dφ˛ .

yields a *plane* with polar coordinates (r, φ), while θ = π/2.

A similar equatorial section for the Schwarzschild metric at constant time variable t yields the metric

(4.2)
ds˛ =
(1 − ^{α}/_{r})^{−1} dr˛
+ r˛ dφ˛

which is no longer plane, i.e. no longer representable in a plane, say z=0. However, instead of the plane z=0 we can define a surface z = z(r,φ) over a plane with polar coordinates (r,φ). Due to the spherical symmetry z cannot depend on φ, hence we have to consider a rotational surface z = z(r): The metric of this surface is given by

(4.3)
ds˛ =
(1 + z_{r}˛) dr˛ + r˛ dφ˛ .

Comparison with the metric (4.2) yields
z_{r} = (^{α}/_{r−α})^{½} , hence

(4.4)
z(r) = 2 [α(r−α)]^{½} .

This is a rotational surface generated by rotating the parabola
z = 2 [α(r−α)]^{½}
around the z-axis, see the figure of that surface
(cf. [6] Flamm's paraboloid) .

We see that z = 0 for r = α is the (**red marked**) boundary of
the *accessible* world, where z > 0.

The boundary (subset of the event horizon) is not a single point.

Let us compare the metric usually attributed to Schwarzschild

ds*˛ = (1 − ^{α}/_{r*}) dt˛ −
(1 − ^{α}/_{r*})^{−1} dr*˛
− r*˛ (dθ˛ + sin˛θ dφ˛)
(6)

with Crothers' "new" metric:

ds˛ = (^{C½−α}/_{C½}) dt˛
−
(^{C½}/_{C½−α})
^{C'˛}/_{4C}dr˛ − C (dθ˛ + sin˛θ dφ˛)
(7)

This metric has a certain blemish: the differential dr can be removed, such that the variable r is completely substituted by the new variable C using C'dr = dC, hence

(5.1)
ds˛ = (^{C½−α}/_{C½}) dt˛ −
(^{C½}/_{C½−α})
^{1}/_{4C}dC˛ − C (dθ˛ + sin˛θ dφ˛)

What Crothers did not mention in his papers [1] and [2]:

**Both metrics, defined by the eqs.(6) and (7)/(5.1)
are equivalent,
i.e. the associated manifolds are identical, merely represented by different
coordinates (t,r*,θ,φ) and (t,C,θ,φ) repectively,
associated by the coordinate transform
**

(5.2) C = C(r*) = r*˛ and r* = r*(C) = C^{½}.

So normally there is no reason for considering other than the STANDARD form (6) of the Schwarzschild metric. Other equivalent forms may be of historical interest merely. Crothers' question of correct naming of the different versions of equivalent metrics has become obsolete nowadays. For more see Section 6.

From the coefficients
g_{oo}
of the metrics (7) and (6) respectively
it can be seen directly that the metric (7) becomes singular at
C^{½} = α, while the metric (6) becomes singular at r* = α.

Crothers defines a value r_{o} by the equation C(r_{o}) = α˛.
From C(r*) = r*˛ we obtain r_{o} = α: While the metric (7) is singular at
C = C(r_{o}) = α˛ the equivalent metric (6) has its corresponding singularity at
r = r_{o} = α.

Crothers is interested in a *radial*
coordinate with an evident *geometrical* meaning. Therefore he introduces a new variable,
a "proper radius" R_{p} by *radial* integration of the line element ds of (7)
(dt=0, dθ=0, dφ=0)
starting from the singularity, which after some calculations yields

R_{p}(C) =
[C^{½} (C^{½}−α)]^{½} +
α ln |(C^{¼}+(C^{½}−α)^{½}) α^{−½}|
(14)

The same result would have been attained by radial integration of the line element ds* of (6) starting at its singularity r* = α:

(5.3)
R_{p}*(r*) =
[r*(r*−α)]^{½} +
α ln |(r*^{½}+(r*−α)^{½}) α^{−½}|

where r* = C^{½}. We then have R_{p}*(r*) =
R_{p}(r*˛).

**Conclusion** The use of the metric (7)/(5.1) instead of the technically simpler
Schwarzschild metric (6) is an *unnecessary* complication which cannot yield new results
exceeding those attained by use of the Schwarzschild metric.

Crothers' problems with the analysis of GRT are mainly caused by his misconceptions concerning the role of coordinates. In his paper [2] we read:

The black hole, which arises solely from an incorrect analysis of the Hilbert solution, is based upon a misunderstanding of the significance of the coordinate radius r. This quantity is neither a coordinate nor a radius in the gravitational field and cannot of itself be used directly to determine features of the field from its metric. The appropriate quantities on the metric for the gravitational field are the proper radius and the curvature radius, both of which are functions of r. The variable r is actually a Euclidean parameter which is mapped to non-Euclidean quantities describing the gravitational field, namely, the proper radius and the curvature radius.

Crothers expects a geometrical meaning always being attached to a coordinate. He insinuates
that the coordinate r, known from spherical polar coordinates as *radial distance* from the center,
should maintain its meaning when appearing in another context, e.g. as the parameter r of the
Schwarzschild metric. In [2, Sect.2] we read about an isotropic generalization of
the Minkowski line element:

ds˛ = A(r)dt˛ − B(r)dr˛ − C(r) (dθ˛ + sin˛θ dφ˛) ,
(2a)

A,B,C >0 ,

where A,B,C are analytic functions. I emphatically remark
that *the geometric relations between the components of the
metric tensor of (2a) are precisely the same as those of (1).*
The standard analysis writes (2a) as,

ds˛ = A(r)dt˛ − B(r)dr˛ − r˛ (dθ˛ + sin˛θ dφ˛) , (2b)

and claims it the most general, which is incorrect. The form of C(r) cannot be pre-empted ...

This renaming method is *somewhat lax* but often used in mathematics, though it
could be misunderstood if taken literally:
The setting C := r˛ means that a *new* meaning is assigned to the
variable r. Since r already occurs in eq.(2a), it would be *better* to use a *new*
symbol, say r*, not r, for the new variable: r*˛ := C(r). As a consequence the terms
A(r)dt˛ and B(r)dr˛ must be rewritten as functions of the new variable r* by introducing
new cofficients
A*(r*):=A(r) and B*(r*):=B(r)(dr/dr*)˛.
This yields

ds˛ = A*(r*)dt˛ − B*(r*)dr*˛ − r*˛ (dθ˛ + sin˛θ dφ˛) , (2b*)

Then, all *s are removed to obtain

ds˛ = A(r)dt˛ − B(r)dr˛ − r˛ (dθ˛ + sin˛θ dφ˛) , (2b)

To repeat it: **The terms A, B, r in (2a) and (2b) respectively have different meanings,
here precisely specified.** However, the rewriting (2a) as (2b) is perfectly
justified herewith.

[1] S. Crothers, *On the General Solution to Einstein's Vacuum Field and its Implications for Relativistic Degeneracy.
*, PROGRESS IN PHYSICS Vol. 1 , April 2005

http://www.ptep-online.com/index_files/2005/PP-01-09.PDF

[2] S. Crothers, *On the Geometry of the General Solution for the Vacuum Field of the Point-Mass*,
, PROGRESS IN PHYSICS Vol. 2 , July 2005

http://www.ptep-online.com/index_files/2005/PP-02-01.PDF

[3] S. Crothers, *The Published Papers of Stephen J. Crothers*,

http://www.geocities.com/theometria/papers.html

[4] S.M. Carroll, *Lecture Notes on General Relativity*,
http://xxx.lanl.gov/pdf/gr-qc/9712019

[5] N.N. , Eddington-Finkelstein coordinates, http://en.wikipedia.org/wiki/Eddington-Finkelstein_coordinates

[6] N.N. , Schwarzschild metric, http://en.wikipedia.org/wiki/Schwarzschild_solution#Flamm.27s_paraboloid

[7] A.J.S. Hamilton, More about the Schwarzschild Geometry, http://casa.colorado.edu/~ajsh/schwp.html

[8] N.N. , Gullstrand-Painlevé Coordinates, http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates