August, 7, 2008

_{Quotations from Roessler's papers in} black

. . . Specifically, the Einstein equations imply the famous Schwarzschild solution which describes how light and everything else behaves in greater and greater proximity to a heavy mass if the latter is condensed down to its Schwarzschild radius. Into the radial Schwarzschild metric one can, for example, enter the simultaneous positions of two stationary outside points r_o and r_i (“outer“ and “inner“) while r is the radial parameter (not the radial distance) and r_s = 2GM/c² is the Schwarzschild radius (with G being Newton‘s gravitational constant, M the gravity-generating mass in question and c the standard speed of light). The metric then allows one to calculate the so-called “coordinate time difference“ Δt for light sent between points r_i and r_o and vice versa,

                Δt = ¹/_c ∫_ri^r_o (1 − r_s/r)⁻¹ dr = ¹/_c (r_o − r_i + ln ^{r_o − r_s}/_{ri − rs})                 (1)

[8] (p. 130). Multiplication of this time interval by the standard velocity of light, c, then formally generates a distance:

                c Δt = r_o − r_i + ln ^{r_o − r_s}/_{ri − rs} .                 (2)

This vertical distance near a black hole has no name so far. One sees that it diverges (becomes infinite) when r_i approaches the Schwarzschild radius r_s. This reflects the well-known fact implicit in Eq.(1) that light emerging from the Schwarzschild radius (at r_i = r_s) takes an infinite time to reach an outside point r_s and vice versa [8]. The reason is also well known: the speed of light, c as a function of r, approaches zero as r approaches r_s in the Schwarzschild metric [8].

This is somewhat misleading: c usually denotes the ''standard'' speed of light. There is no c as function of r. What is meant here by the author is the speed of light at r=r_i as seen by an observer located at r=r_o when considering the difference Δt meant relative to his local time: Δt = t_o − t (t=t_i) From eq.(1) it follows by differentiating with respect to t by assuming r_i = r_i(t)

                −1 = ¹/_c (1 − r_s/r_i)^{−1 dr_i}/_dt ,

or

                ^dr_i/_dt = − c (1 − r_s/r_i) → 0         if         r_i → r_s         (r_i > r_s) .

However, ^dr_i/_dt is not the speed of light seen by an observer fixed at r=r_o as we shall see below.

What could be shown in reference [7] is that the distance described by Eq.(2) is real. That is, the infinite time it takes by Eq. (1) to cover the distance between r_s and r_o is in accord with the interpretation that c is constant throughout. This “constant-c interpretation“ of Eq.(2) is compatible with a proposal made by Max Abraham in 1912 [9] in response to the first nonconstant- c theory proposed by Einstein in 1911 [10]. The variable-c feature got then incorporated four years later into general relativity – which indeed might never have been found without it. Now, the variable-c property unexpectedly turns out to be redundant in a special case: Eq.(2) formally implies that closer and closer to the horizon, space gets more and more strongly dilated in compensation for the lacking decrease in c [7]. The same locally isotropic size change had been demonstrated before in the much more special context of the equivalence principle [11].

Evidently Roessler has not understood the meaning of the ''constant-c property'' and the physical meaning of the parameters t, r, θ and φ which are given by the Schwarzschild metric:

                dσ² = −(1 −^r_s/_r) c² dt² + (1 −^r_s/_r)⁻¹ dr² + r² (dθ² + sin²θ dφ²)

In a frame at rest at r=r_o measurements of spatial distances are performed at constant time, dt=0, therefore, the local length metric is given by

                dλ² = dσ²_{(at dt=0)} = (1 −^r_s/_r)⁻¹ dr² + r² (dθ² + sin²θ dφ²)

whilst the time measurement requires fixed space coordinates. The local time element dτ is given by

                dτ² = − dσ²/c²_{(at dr=0, dθ=dφ=0)} = (1 −^r_s/_r) dt² .

We may conclude from this that the parameter t is NOT the local time τ, r is NOT the local radial length. The propagation of light is given by the light cone defined by dσ²=0. Hence we have

                0 = dσ² = − c² dτ² + dλ²

to obtain the local speed of light to be |dλ/dτ| = c.

The new taking-literally of Eq.(2) is tantamount to an infinite downward-extension of the Einstein-Rosen funnel (the upper half of the famous Einstein-Rosen bridge). Three previously unknown facts follow from the re-interpretation of the unchanged mathematics:
1) infinite proper in-falling time;
2) infinitely delayed Hawking radiation;
3) infinitely weak chargedness of black holes.
All 3 contradict accepted wisdom, so the standard calculations must have involved an undiscovered false step at some point since the mathematics is unchanged. Indeed for one of the three (the first), a straightforward proof could be found that the non-constant-c traditional interpretation makes the same prediction [7].

The author ignores here that there is no absolute time in the Schwarzschild spacetime. Time is frame dependent. Therefore, an infalling object (or an object expelled with speed of light as well) needs only a finite time measured in its proper time to pass the space between the Schwarzschild radius r_s and some outer radius r_o > r_s [3], [4]. The finiteness of proper time Δτ for outgoing/infalling light follows from the fact that light travels along the light cone dσ²=0 , hence

                Δτ = ∫ dσ/c = 0.
(Thanks to an attentive reader who pointed to a flaw in my former argumentation.)

Therefore, his points above do not apply. Especially Point 1) is substantially wrong. And ad 2): Hawking radiation will not be hindered by time delay to leave the black hole since for the proper time τ there is no ''infinite'' time delay.

Conclusion:

O.E. Roessler's misinterpretation of the Schwarzschild metric lets become his further considerations in [1] and [2] null and void. These are no papers that could be taken into account when problems of black holes are discussed.