- #1
wLw
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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.
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.