Rindler coordinates in Minkowski spacetime

[spaghetti3451]

In an inertial coordinate system in two-dimensional Minkowski spacetime, the metric takes the form
$$(ds)^{2} = - (dt)^{2} + (dx)^{2},$$
and in an accelerating coordinate system in two-dimensional Minkowski spacetime, the metric takes the form
$$(ds)^{2} = - R^{2}(d\eta)^{2} + (dR)^{2}.$$
The coordinates $t$ and $x$ are called inertial coordinates and the coordinates $\eta$ and $R$ are called Rindler coordinates.

These coordinates describe the line element of Minkowski spacetime from the perspective of an inertial observer and of an accelerating observer respectively.
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Rindler coordinates in Minkowski spacetime are related to inertial coordinates in Minkowski spacetime by the change of coordinates
$$x = R\cosh\eta, \qquad t = R\sinh\eta.$$
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An inertial observer is an observer sitting at fixed $x$. This is not a geodesic - it is a stationary trajectory.

A Rindler observer is an observer sitting at fixed $R$. This is not a geodesic - it is a uniformly accelerating trajectory.
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1. How can you prove that an observer sitting at fixed $R$ has a fixed acceleration?

2. How can you prove that the Rindler coordinates only cover the right wedge of Minkowski space? Why don't Rindler coordinates cover the other three patches of Minkowski space?

3. Can a Minkowski observer ever communicate with a Rindler observer?

4. Why do Rindler observers see a horizon at $R = 0$? How is this horizon similar to a black hole horizon?

 

[robphy]

Did you try to map curves of constant-$R$ or constant-$\eta$ into an ordinary spacetime diagram?
Given the trajectory of a particle, can you determine the velocity and acceleration?

 

[Nugatory]

An inertial observer is an observer sitting at fixed $x$. This is not a geodesic - it is a stationary trajectory.

Are you sure of this?

 

[Dale]

. How can you prove that an observer sitting at fixed R has a fixed acceleration?

Just calculate the proper acceleration and show that it is constant wrt time.

3. Can a Minkowski observer ever communicate with a Rindler observer?

If you and a friend are in an elevator and one of you jumps do you expect an interruption in communication?

 

[pervect]

An inertial observer is an observer sitting at fixed $x$. This is not a geodesic - it is a stationary trajectory.

Huh? An observer in an inertial frame with a constant velocity (including a constant velocity of zero) is following a geodesic.