Solving Møller Energy Distribution Problem in Paper

[wLw]

https://arxiv.org/abs/gr-qc/0306101
I am now reading this attached paper. But i can not get energy result(2.8), and I calculated it and found it is zero. here is my process: firstly, i use Gauss law and rewrite the (2.6): $E=\frac{1}{8 \pi} \iint \chi_{0}^{0 \beta} \mu_{\beta} d S$
where µβ is the outward unit normal vector over an infinitesimal surface element dS,For a surface given by parametric equations x = rsinθcosφ, y = rsinθsinφ, z = rcosθ (where r is constant) one has µβ = {x/r, y/r, z/r} and dS = r^2sinθdθdφ

and the author get only one component:

$\chi_{0}^{01}=\frac{2 M}{r}(r-\alpha) \sin \theta$and I let β=0 and have:

$E=\frac{1}{8 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{2 M}{r}(r-\alpha) \sin \theta \frac{r \sin \theta \cos \varphi}{r} r^{2} \sin \theta d \theta d \varphi$

finally, I factor out all term that have nothing to do with θ,φθ,φ and get the integral:
$E=r^{2} \frac{2 M}{r}(r-\alpha) \frac{1}{8 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \sin ^{3} \theta \cos \varphi d \theta d \varphi=\mathrm{o} ! ! !$!

which shows the zero, i want to know what is wrong, could you help me.

 

[Orodruin]

The coordinates used are not Cartesian. You cannot impose Cartesian coordinates either.

 

[wLw]

so what coordinates should i use , could you provide the process of calculation , thanks a lot, this question perplex me for a long time

 

[Orodruin]

What is wrong with the coordinates used by the author? What is your background in GR?

 

[wLw]

he used the polar coordinates,because moller prescription is not limited to quasi- Cartesian coordinates, and what i used is also polar coordinates

 

[pervect]

he used the polar coordinates,because moller prescription is not limited to quasi- Cartesian coordinates, and what i used is also polar coordinates

Gad gives the line element in 2.1, and as Misner remarks in "Precis of General Relativiity", for the purpose of physics, the line element defines the coordinates used, at least as far as physicists are concerned.

<<link>>

Equation (1) [[the line element]] defines not only the gravitational field that is assumed, butalso the coordinate system in which it is presented. There is no other sourceof information about the coordinates apart from the expression for the met-ric. It is also not possible to define the coordinate system unambiguously in any way that does not require a unique expression for the metric. In mostcases where the coordinates are chosen for computational convenience, the expression for the metric is the most efficient way to communicate clearlythe choice of coordinates that is being made.

Unfortunately, I can't really follow Gad's paper, not without reading the references. It appears to me that X is some sort of pseudo-tensor due to Moeller. I'd have to track down his references to be sure, though.

From Gad's paper

Virbhadra[6] investigated the most general non-static spherically symmetric space-times, using the Einstein, Landau and Lifshitz, Papapetrou, and Wein-berg prescriptions, and he found that the definitions of energy distribution disagree in general. Recently, Xulu [7] used the Møller energy momentum expression to compute the energy distribution

This sounds to me like Gad is indeed using a pseudo-tensor approach originating with Moeller, but it's not clear why he is calling what he computes "the" energy distribution when it's not, in general, unique.

It's also not clear to me what he means when he says to "use Gauss' law". He's writing E as some sort of 3-volume integral, but $dx^1\,dx^2\,dx^3$ isn't a volume element, there's a missing factor of the square root of the determinant of the 3-metric. For instance, in spherical coordinates the 3-volume elelent is $r^2 \sin \theta \, dr \, d\theta \, d\phi.$, but as already noted these coordinates aren't spherical, and the factor of $\sqrt{-g}$ would be different. There is a factor of $\sqrt{-g}$ in the definition of X, which would then be included in $\Theta$, but I'd think that would be the 4-metric g, not the 3-metric g.

 

[wLw]

Thanks for your respond, as you said now all the energy complex is a pseudo-tensor due to nonlocalizable gravitational energy in GR。

, but it's not clear why he is calling what he computes "the" energy distribution when it's not, in general, unique.

and there are some papers show different results by using different prescriptions in a given space-time.

Gad use moller complex because moller complex is not limited to quasi- Cartesian coordinates , besides there are some papers that used other complexes and they should be transformed to quasi- Cartesian coordinates firstly. likehttps://arxiv.org/abs/gr-qc/0304081 . I do not care that because it is so troublesome,and I am interested in moller
and there are also some paper that used moller's prescription likehttps://arxiv.org/abs/gr-qc/0110058 and https://www.researchgate.net/publication/327133794_Energy-Momentum_Distribution_in_General_Relativity_for_a_Phantom_Black_Hole_Metric(eq. 18,19 in section 3)
but all above paper do not show the details of calculation,but just said like this"
sing Gauss’s theorem, the energy E can be written as...", i am confused to that how they get the result and i always get zero by following their steps as i showed in the first thread

 

[pervect]

but all above paper do not show the details of calculation,but just said like this"
sing Gauss’s theorem, the energy E can be written as...", i am confused to that how they get the result and i always get zero by following their steps as i showed in the first thread

I'm also a bit confused. I think my main issue is I don't know what volume the energy "E" is contained in. I'm assuming it's some shell of radius r, but is it unclear how thick the shell is. It could be normalized to the shell thickness (dr / $\sqrt{g_{rr}}$), or it could be normalized to the change in the r-coordinate, dr. Or is it not a shell, but the total energy contained in the region r < r_0?

I suppose I could try to work out the various alternatives and see if any of them match the paper, but it seems to me the answer should be straightforwards from the integral in 2.6 once one decides which quantity one wants to compute.