Question

Coordinates of Rindler.
One way to study accelerated observers uniformly in 1 + 1 dimensional plane-spacetime is using the Rindler coordinates given by:

with α the own acceleration of the observer.

(a) Show that the interval in these coordinates is:

(b) Consider purely radial trajectories in a Schwarzschild space with dθ2 = dφ2 = 0. Show that very close to the event horizon of a Schwarzschild black hole, the Rindler coordinates are a good approximation to the Schwarzschild coordinates. Why should not we be surprised that this happens?

(a) Show that the interval in these coordinates is: ds² = α²ρ²dχ²−dρ².

sinh (αχ) , x.cosh(ax), t- sinh (αχ) , x.cosh(ax), t-

Answer 1

(a)

$cdt=d\rho\sinh\alpha\chi+\alpha\rho\cosh\alpha\chi d\chi$

$dx=d\rho\cosh\alpha\chi+\alpha\rho\sinh\alpha\chi d\chi$

$-c^2dt^2+dx^2=d\rho^2-\alpha^2\rho^2d\chi^2$

(b)

For radial trajectories,

$ds^2=-\left ( 1-\frac{r_s}{r} \right )dt^2+\left ( 1-\frac{r_s}{r} \right )^{-1}dr^2$

in units of c=1 and $r_s=2GM$

For $r=\epsilon+r_s,\epsilon\ll 1$

$1-\frac{r_s}{r}=\frac{\epsilon}{r_s+\epsilon}\approx \frac{\epsilon}{r_s}$

and

ar de

so that

2 2