- #1

cianfa72

- 1,939

- 213

- TL;DR Summary
- Clarification about the definition of Rindler coordinates with respect to the observers's proper acceleration profile.

Hi,

starting from this post Basic introduction to gravitation as curved spacetime I would ask for a clarification about Rindler coordinates.

From this wiki entry Rindler coordinates I understand that the following transformation (to take it simple drop ##y,z##)

$$T = x\sinh{(\alpha t)} , X=x\cosh{(\alpha t)}$$

with ##\alpha## parameter fixed leads to the worldlines of a family of accelerating observers (bodies) each with the given proper acceleration ##\alpha##. For example the following picture (from the above link) represents the case ##\alpha=0.5##. Each observer in this family (congruence) is identified by a different value of the ##x## coordinate.

Now, if I understand correctly, the proper time of each observer (i.e. the proper time read by its wristwatch) is set to ##0## when their worldlines intersect the X axis of the Lorentzian coordinate chart. Thus if we consider for example the curve ##t=1## it intersects the observers hyperbolas in a point (an event) in which the proper time ##\tau## read by each observer's wristwatch is actually ##\tau=1##.

If the above a correct, I'm not sure if it actually makes sense to call that family of observers as "Rindler observers" since they have the

Thank you.

starting from this post Basic introduction to gravitation as curved spacetime I would ask for a clarification about Rindler coordinates.

From this wiki entry Rindler coordinates I understand that the following transformation (to take it simple drop ##y,z##)

$$T = x\sinh{(\alpha t)} , X=x\cosh{(\alpha t)}$$

with ##\alpha## parameter fixed leads to the worldlines of a family of accelerating observers (bodies) each with the given proper acceleration ##\alpha##. For example the following picture (from the above link) represents the case ##\alpha=0.5##. Each observer in this family (congruence) is identified by a different value of the ##x## coordinate.

Now, if I understand correctly, the proper time of each observer (i.e. the proper time read by its wristwatch) is set to ##0## when their worldlines intersect the X axis of the Lorentzian coordinate chart. Thus if we consider for example the curve ##t=1## it intersects the observers hyperbolas in a point (an event) in which the proper time ##\tau## read by each observer's wristwatch is actually ##\tau=1##.

If the above a correct, I'm not sure if it actually makes sense to call that family of observers as "Rindler observers" since they have the

*same*profile of proper acceleration (e.g. ##\alpha=0.5## as in the above picture).Thank you.

Last edited: