Implikationen von G&M

[Barney]

[Solkar]

Hi!

About G&M eq.(3.15) to eq.(3.18):

I think it'd be useful to analyze those equations in reverse order

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Starting with (3.18):

The $\Gamma$-Function in the Denominator is basically a real (even complex) continuation of the natural factorial; e.g

$\Gamma(n) = (n-1)!$ (III.1)

for $n \epsilon \mathbb{N}\setminus \{0\}$

This yields the Denominator of (3.18) for even $(D-1)$, and the famous

$\Gamma(\frac{1}{2}) = \sqrt{\pi}$ (III.2)

together with

$\Gamma(z+1) = z \cdot \Gamma(z)$ (III.3) (*),


will yield $\Gamma$ for odd $(D-1)$ in (3.18)

Thus we get

$\Gamma(\frac{1}{2}) = \sqrt{\pi}$ (III.4)
$\Gamma(1) = 0! \equiv 1$ (III.5)
$\Gamma(\frac{3}{2}) = \Gamma(\frac{1}{2} + 1) = \frac{1}{2}\Gamma(\frac{1}{2}) = \frac{1}{2}\sqrt{\pi}$ (III.6)
$\Gamma(2) = 1! = 1$ (III.7)
$\Gamma(\frac{5}{2}) = \Gamma(\frac{3}{2} + 1) = \frac{3}{2}\Gamma(\frac{3}{2}) = \frac{3}{4}\sqrt{\pi}$ (III.8 )
...

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When we use this for (3.18) we get

For$D=4$
$ \Omega_{4-2} = \frac{2 \pi^{(4-1)/2}}{\Gamma((4-1)/2)}=\frac{2 \pi^{3/2}}{\frac{1}{2} \sqrt{\pi}} = 4 \sqrt{\frac{\pi^3}{\pi}} = 4 \pi $ (III.9)

For$D=5$
$ \Omega_{5-2} = \frac{2 \pi^{(5-1)/2}}{\Gamma((5-1)/2)}=\frac{2 \pi^{4/2}}{1} = 2 \pi^2 $ (III.10)

Both are apparently the D-1 unit sphere surface areas, so G&M's term "volume of the unit D-2 sphere" (above eq.(3.18), atop their p.13; PDF p.14 respectively) is imo at least a little misleading.

What do you think?


Best regards,

Solkar

(*) e.g. here:

[Barney]

[Solkar]

Hi Barney!

A wild guess of mine:

General Stokes' Theorem

$\int \limits_{V}^{} d\omega = \oint \limits_{\partial V} \omega $ (IV.1)

resp one of its consequences

$\int \limits_{V}^{} \langle \vec{\nabla} , \vec{\Psi} \rangle dV = \oint \limits_{\partial V} \vec{\Psi} d\vec{\Sigma} $ (IV.2) (Gauss' Theorem, e.g Maxwell I)

or - even less likely

$\int \limits_{V}^{} \vec{\nabla} \times \vec{\Psi} d\vec{\Sigma} = \oint \limits_{\partial V} \vec{\Psi} d\vec{r} $ (IV.3) (Classical Stokes' Theorem, e.g. Maxwell-Faraday)

applied in any tricky way on Einstein's "space-field" (here labelled $\Psi$ by me)



It would still not be a Lebesgue-Volume, but maybe it has some hidden ART-(or ST-)meaning, which does not disclose itself here.



Greetings,

Solkar

[Solkar]

At least with

$D=4$
$ \Omega_{4-2} = 4 \pi $ (from III.9)

we might get for (3.17)

$k_4 = \frac{2{(2 \pi)}^{(4-4)}}{(4-2) 4 \pi} = \frac{1}{4 \pi} $ (V.1)

thus for (3.16) expressed in NU:

$ R_4(M) = {\frac{1}{\frac{1}{sqrt{8 \pi}}} \lbrace { \frac{ \frac{1}{4 \pi}M}{\frac{1}{sqrt{8 \pi}} } \rbrace^{1/(4-3)} = \frac{{(sqrt{8 \pi})}^2}{4 \pi} M = 2M [NU] \underbrace{\equiv}_{? well, maybe} \frac{2 G M }{c^2} [SI]$ (V.2)

There are samples when NU don't provide much clarity after all....

[Solkar]

About

$ds^2 = -\left[ 1- \left({R(M)\over r}\right)^{D-3}\right] dt^2 +{1\over 1- \left(\frac{R(M)}{r}\right)^{D-3}}dr^2 + r^2 d\Omega^2$ (G&M eq.(3.15))

This is said to be a D-dimensional Schwarzschild solution ("SE") ; and obviously it formally yields the well-known 4-dimenssional Schwarzschild metric for D=4.

Question is, what validity this equation will have for higher dimensions D>4.

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G&M do not directly cite a source for this generalization, but they do mention

Myers, R.C., and Perry, M.J., “Black Holes In Higher Dimensional Space-Times”, Ann. Phys. (N.Y.), 172, 304, (1986). (their ref [29], my ref [1] resp "M&P" subsequently)

with reference to Kerr-solutions (and the end of their Chapter 3.2.1).

Unfortunately, neither [1] nor

Tangherlini, F.R., “Schwarzschild field in n dimensions and the dimensionality of space problem”, Nuovo Cimento, 27, 636, (1963). ([2] or "S/T" subsequently)
("S/T" because "Schwarzschild-Tangherlini-" model/solution/equation seems to be a usual term in higher-dim ART)

are freely available.

I'm working on getting access to these papers.

---

There are, however, some interesting free sources covering the topic

"Black Holes in Higher Dimensions"
Roberto Emparan and Harvey S. Reall
http://www.livingreviews.org/lrr-2008-6
([3] resp "E&R" subsequently)

and


Teo, E., “Black diholes in five dimensions”, Phys. Rev. D, 68, 084003, (2003). Related online version (cited on 14 February 2008):
http://arxiv.org/abs/hep-th/0307188
([4] resp "arXiv:hep-th/0307188v2" resp "T-BDH" subsequently)

===

It is, however, somewhat tempting trying to verify G&M (3.15) by own efforts.

A)Complexification
==================

Given the standard form

$ds^2 = c^2 B(r) dt^2 - A(r) dr^2 - r^2 (d\theta + sin^2\theta d\phi^2) $ (VI.2)

used for deduction of 4-dim SE

Given A and B were (real-valued) components of a complex function

$f(z) = A(|z|) + i B(|z|) $ (VI.3)

one might get some useful features for A's and B's existence and uniqueness from that; and

$f(r+it) = A(r,t) + i B(r,t) $ (VI.4)

could be a good approch to get to non-stationary solutions (e.g. Kerr)

But that is just a guess of mine.

B)Induction over $D\varepsilon\mathbb{N}$ and Gaussian Elimination
========================================

Let the basis be SE for 4 dim.

For the step:

With

$R_{\mu\nu} = R^{\alpha}_{\mu\nu\alpha}=\frac{\partial{\Gamma^{\alpha}_{\mu\alpha}}}{\partial x^{\nu}} - \frac{\partial{\Gamma^{\alpha}_{\mu\nu}}}{\partial x^{\alpha}} + \Gamma^{\alpha}_{\beta\nu}\Gamma^{\beta}_{\alpha\mu}-\Gamma^{\alpha}_{\beta\alpha}\Gamma^{\beta}_{\mu\nu}$ (VI.5)

and

$\Gamma^{\beta}_{\mu\nu}=\frac{1}{2}g^{\alpha\beta} \left(\frac{\partial g_{\alpha\nu}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) $ (VI.6)

On could do away with plenty of the terms by either the steps's premise ("valid for D-1")

and the classical SE premises SE-metric being

- vacuum
- spherical
- stationary

That would presumably not be not very difficult, but very "lengthy"; maybe MAPLE or another high-end CAS would be helpful; "Maxima" (which I use) does not seem overly appropriate for supporting this task

===============

In any case, it would be efficient to wait for papers [1] and [2] to become available; otherwise we could easily "reinvent the wheel".

===============

There is, however, something we can discuss right now:

SE and Kerr are both "vacuum solutions", meaning that the stress-energy tensor $T_{\mu\nu}$vanishes.

For G&M (3.1)-(3.5) $T_{\mu\nu}$ is not "allowed" to vanish, because

With $T_{\mu\nu}=0$ we'd get

$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \pm \kappa T_{\mu\nu} = 0$ (VI.7)
$\Rightarrow g^{\mu\nu} R_{\mu\nu} - \frac{1}{2} g^{\mu\nu} R g_{\mu\nu} = g^{\mu\nu} \quad \cdot 0 $
$\Rightarrow R - \frac{1}{2} R \cdot 4 = 0 $

thus

$R = 0 $ (VI.8)

which would make the Integrals of G&M(3.1)-(3.4) useless.

What's the justification for G&M deploying SE/Kerr in 3.2.1 nevertheless?



Best regards,

Solkar

[Erik]

[Solkar]

[Erik]

[Solkar]

[Erik]

[Solkar]

[Erik]

[Erik]

[Solkar]

@Erik: