The static Killing vectors in Kruskal coordinates

[Altabeh]

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
$$\epsilon^{ik}=-\epsilon^{ki} =\left[ \begin {array}{ccc} 0&a&-b\\ \noalign{\medskip}-a&0&c \\ \noalign{\medskip}b&-c&0\end {array} \right],$$

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

 

[George Jones]

Sorry, I haven't tried to read your post in detail.

I would use (1) and (2), and the chain rule for ordinary partial derivatives to calculate the static Killing vector field in Kruskal coordinates.

 

[Altabeh]

Sorry, I haven't tried to read your post in detail.

I would use (1) and (2), and the chain rule for ordinary partial derivatives to calculate the static Killing vector field in Kruskal coordinates.

Would you mind going a little bit deep into details of the calculation!?

Thanks
AB

 

[George Jones]

In Schwarzschild coordinates, the static Killing vector is $\partial/\partial t$, and

$$\frac{\partial}{\partial t} = \frac{\partial T}{\partial t} \frac{\partial}{\partial T} + \frac{\partial X}{\partial t} \frac{\partial}{\partial X},$$

so the components of Killing vector $\partial/\partial t$ with respect to the $\left\{ T, X, \theta, \phi \right\}$ coordinates are

$$\left\{ \frac{\partial T}{\partial t}, \frac{\partial X}{\partial t}, 0, 0 \right\}.$$

Differentiating (1) and (2) with respect to $t$ gives two linear equations in the two non-zero components.