Discussion of S. Crothers' Views on Black Hole Analysis in GRT

Gerhard W. Bruhn, Darmstadt University of Technology


see also the

Open letter to Dr. D. Rabounski, Editor-in-Chief of the Journal PROGRESS IN PHYSICS *)

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

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 Sr 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''.

1. Crothers' basic views

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) ro =α denotes a point, not a 2-sphere, and that
2) 0 < r < α is undefined on the Hilbert metric.

2. Objections to claim 1)

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 Sr 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 Sr is a 2-sphere with radius r. The set Sα 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 Sr and Sα is given by Crothers' ''proper radius'' (cf [1, eq. (14)] with C(r)= r˛)

(2.4)                 Rp(r) = [r(r−α)]½ + α ln |(r½+(r−α)½) α−½|

measurable in radial direction between arbitrary associated points of the concentric spheres. Since Rp(r) is continuous at r=α the distance between Sr and Sα tends to 0 for r → α:

(2.5)                 limr → α Rp(r) = Rp(α) = 0 .

Therefore, the set Sα of the metric singularities can be approximated with respect to the distance Rp(r) by concentric 2-spheres of radius r > α: Thus,

Sα cannot be a single point.

See also Section 4.

3. Objections to claim 2)

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 ÎR}.

The singularities of the Schwarzschild metric (1.1) at r=α=1 are spurious merely, i.e. no singularities of spacetime.

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:

The metric (3.4) is an extension of each of the two validity regions of the Schwarzschild metric (3.1) to the other one.

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

4. Somewhat elementary differential geometry

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 + zr˛) dr˛ + r˛ dφ˛ .

Comparison with the metric (4.2) yields zr = (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.

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

5. Further comments on Crothers' paper [1]

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'˛/4Cdr˛ − 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) 1/4CdC˛ − 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) = C½.

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 goo of the metrics (7) and (6) respectively it can be seen directly that the metric (7) becomes singular at C½ = α, while the metric (6) becomes singular at r* = α.

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

Crothers is interested in a radial coordinate with an evident geometrical meaning. Therefore he introduces a new variable, a "proper radius" Rp 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

                Rp(C) = [C½ (C½−α)]½ + α ln |(C¼+(C½−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)                 Rp*(r*) = [r*(r*−α)]½ + α ln |(r*½+(r*−α)½) α−½|

where r* = C½. We then have Rp*(r*) = Rp(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.

6. Some comments on Crothers' paper [2]

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.

Without loss of generality the coefficient C(r) in eq. (2a) can be assumed as C(r)=r˛.


[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

[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

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

[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