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

Gerhard W. Bruhn, Darmstadt University of Technology

06.03.2008

see also the

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

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''.

1. Crothers' basic views

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 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,

S_α 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 S_α 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 + 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.

5. Further comments on Crothers' paper [1]

6. Some comments on Crothers' paper [2]