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}^{ro}
(1 − r_{s}/r)^{−1} dr
=
^{1}/_{c}
(r_{o} − r_{i} +
ln ^{ro − rs}/_{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 ^{ro − rs}/_{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}
^{dri}/_{dt} ,

or

^{dri}/_{dt} = − c (1 − r_{s}/r_{i})
→ 0
if
r_{i} → r_{s}
(r_{i} > r_{s}) .

However, ^{dri}/_{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 −^{rs}/_{r}) c² dt²
+ (1 −^{rs}/_{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 −^{rs}/_{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 −^{rs}/_{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.

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.

[1] O.E. Rössler, *Abraham-Solution to Schwarzschild Metric Implies That CERN Miniblack Holes Pose a
Planetary Risk*,

http://www.wissensnavigator.ch/documents/OTTOROESSLERMINIBLACKHOLE.pdf

[2] O.E. Rössler, *Abraham-like return to constant c in general relativity:
“*R*-theorem“ demonstrated in Schwarzschild metric*,

http://www.wissensnavigator.ch/documents/OTTOROESSLERMINIBLACKHOLE.pdf

[3] Wikipedia, *Black Hole*,

http://en.wikipedia.org/wiki/Black_hole

[4] Wikipedia, *Schwarzschild metric*,

http://en.wikipedia.org/wiki/Schwarzschild_metric

[5] Wikipedia, *Eddington-Finkelstein coordinates*,

http://en.wikipedia.org/wiki/Eddington-Finkelstein_coordinates