# Physical meaning of the Rindler hyperbola vertex and the Rindler lines

Two questions regarding the Rindler diagram:

1) Does the vertex of a given hyperbola in the diagram have physical meaning? I know it is the inverse of the constant proper acceleration ($\alpha$) associated with that particular hyperbola, and I have heard mentioned several times (although never seen formally demonstrated) that acceleration is proportional to the gravitational potential. Can some sort of physical meaning be built out of these mathematical facts? (Even though the Rindler diagram describes flat space-time with no gravitational potential)

2) What sort of juicy information are we supposed to get out of the Rindler rays extending from the origin and intersecting all the hyperbolae? As far as I can tell these rays preserve the ratio of starting distances between the space ships taking off at the same time and traveling at different constant proper accelerations along different hyperbolae. But that is all. Do these rays tell us something about simultaneity?

Bonus points for whoever can throw in a formal proof that vector potential is equal to acceleration.

(Edit:) Oh, one more thing occurred to me--the thing people call the "Rindler time" ($\omega$) should actually be called the proper velocity ($\omega=\alpha \tau$). Why do people call it the Rindler time?

and I have heard mentioned several times (although never seen formally demonstrated) that acceleration is proportional to the gravitational potential.

This is the heart of general relativity. You cannot distinguish locally acceleration and gravitational field, or, if you prefer, inertial mass and gravitational mass are equals.

(Even though the Rindler diagram describes flat space-time with no gravitational potential)

In fact, in a Schwarzschild black hole, near the horizon, and in a very small angular region $\theta=0$, you may approximate the Schwarzschild metrics by the Rindler metrics $ds^2=\rho^2d\omega^2-d\rho^2-dx^2-dy^2$, with $\rho \approx 2\sqrt{2MG(r-2MG)}, \omega= \frac{t}{4MG}, x \approx2MG\theta \cos\phi, x \approx2MG\theta \sin\phi$

Do these rays tell us something about simultaneity?

With the tranformation $T=\rho \sinh \omega, Z = \rho \cosh \omega, X=x, Y=y$, the metrics is Minkowski : $ds^2= dT^2-dZ^2-dX^2-dY^2$.

The Rindler hyperbolae are in fact hyperbolae in ($Z,T$) axis. Now, a displacement in $\omega$, $\omega \to \omega + a$, corresponds to a Lorentz transformation for the $Z, T$ coordinates. So, if, for instance, $2$ events (with different $\rho$) correspond to the same $\omega = T=0$, for an other observer $Z', T'$ which has constant speed relatively to $T, Z$, the two events, seen by this new observer appear at different times $T'$, but will share the same Rindler time $\omega = a$

Ref : Susskind/Lindesay, An Introduction to Black Holes, Information, and the String Theory Revolution, The Holographic Universe, World Scientific, pages $8-10$