Rindler coordinates in Minkowski spacetime

In summary, Minkowski spacetime can be described using inertial coordinates and accelerating coordinates, also known as Rindler coordinates. These coordinates represent the perspectives of an inertial observer and an accelerating observer, respectively. The change of coordinates from inertial to Rindler is given by x = Rcosh(eta) and t = Rsinh(eta). An observer in Rindler coordinates experiences a fixed acceleration and can only see the right wedge of Minkowski space. Communication between Minkowski and Rindler observers is not possible due to the different perspectives. Rindler observers also see a horizon at R = 0, similar to a black hole horizon, and their trajectories are not geodesic.
  • #1
spaghetti3451
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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?
 
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  • #2
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?
 
  • #3
spaghetti3451 said:
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?
 
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  • #4
spaghetti3451 said:
. 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.

spaghetti3451 said:
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?
 
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  • #5
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.
 
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1. What are Rindler coordinates in Minkowski spacetime?

Rindler coordinates are a set of non-inertial coordinates used to describe the spacetime near a uniformly accelerating observer in Minkowski spacetime. They are useful for studying special relativity in accelerated frames of reference.

2. How are Rindler coordinates related to Minkowski spacetime?

Rindler coordinates are a coordinate system within Minkowski spacetime, which is the flat, four-dimensional spacetime described by special relativity. They allow us to describe the effects of acceleration on an observer within this spacetime.

3. What is the significance of Rindler coordinates?

Rindler coordinates are significant because they allow us to study the effects of acceleration on an observer in Minkowski spacetime. They also provide a way to connect special relativity with general relativity, as general relativity describes the effects of acceleration on spacetime.

4. How are Rindler coordinates used in physics?

Rindler coordinates are used in physics to study the effects of acceleration on an observer within Minkowski spacetime. They are also used in theoretical physics to connect special relativity with general relativity.

5. Are Rindler coordinates only applicable to uniformly accelerating observers?

While Rindler coordinates were originally developed for uniformly accelerating observers, they can also be used to describe the spacetime for non-uniformly accelerating observers, as long as the acceleration is small enough that the observer can be approximated as being uniformly accelerating.

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