- #1
Altabeh
- 657
- 0
Hello
In the Kruskal extension of the Schwarzschild metric, where the metric is now transformed into the new coordinates (T,X,\theta,\phi), the line element is
ds^2=-\frac{32M^3e^{-r/2m} }{r}(-dT^2+dX^2)+r^2(d\theta^2+\sin^2(\theta)d\phi^2),
with
\left (r/2m-1 \right ) e^{r/2m}=X^2-T^2 (1)
and t/2m=2\tanh^{-1}(T/X). (2)
From the equation (1) we see that \nabla_{\alpha}r=0 at X=T=0 and this is obvious. Okay, but we know that the static Killing field \xi^{\alpha} becomes collinear with \nabla_{\alpha}r=0 thus requiring \xi^{\alpha} to also vanish there. My question is that how is this possible? Looking at the static Killing vectors of the Schwarzschild metric in a general Cartesian-like coordinate system x^{\mu},
\xi^0=0 and {\xi}^{i}={\epsilon}^{{ik}}{x}^{k}
where
\epsilon^{ik}=-\epsilon^{ki} =\left[ \begin {array}{ccc} 0&a&-b\\ \noalign{\medskip}-a&0&c<br /> \\ \noalign{\medskip}b&-c&0\end {array} \right],
with a,b,c being all arbitrary constants, how come the above requirement (X=T=0) gets all Killing vectors to vanish? My own view on the problem is that we must find the explicit expressions for r and t from (1) and (2), respectively, and then calculate the Killing vectors and put X=T=0. Am I on the right track or what?
Thanks in advance
In the Kruskal extension of the Schwarzschild metric, where the metric is now transformed into the new coordinates (T,X,\theta,\phi), the line element is
ds^2=-\frac{32M^3e^{-r/2m} }{r}(-dT^2+dX^2)+r^2(d\theta^2+\sin^2(\theta)d\phi^2),
with
\left (r/2m-1 \right ) e^{r/2m}=X^2-T^2 (1)
and t/2m=2\tanh^{-1}(T/X). (2)
From the equation (1) we see that \nabla_{\alpha}r=0 at X=T=0 and this is obvious. Okay, but we know that the static Killing field \xi^{\alpha} becomes collinear with \nabla_{\alpha}r=0 thus requiring \xi^{\alpha} to also vanish there. My question is that how is this possible? Looking at the static Killing vectors of the Schwarzschild metric in a general Cartesian-like coordinate system x^{\mu},
\xi^0=0 and {\xi}^{i}={\epsilon}^{{ik}}{x}^{k}
where
\epsilon^{ik}=-\epsilon^{ki} =\left[ \begin {array}{ccc} 0&a&-b\\ \noalign{\medskip}-a&0&c<br /> \\ \noalign{\medskip}b&-c&0\end {array} \right],
with a,b,c being all arbitrary constants, how come the above requirement (X=T=0) gets all Killing vectors to vanish? My own view on the problem is that we must find the explicit expressions for r and t from (1) and (2), respectively, and then calculate the Killing vectors and put X=T=0. Am I on the right track or what?
Thanks in advance
