Analyzing Dynamics in Constant Acceleration w/Rindler & Equivalence

[e2m2a]

TL;DR

When to use RIndler coordinates or the principle of equivalence

Not sure when to use Rindler coordinates to analyze dynamics in a constant accelerating reference system. Rindler coordinates seem messy because they are always changing. Wouldn't it be easier to invoke the principle of equivalence and treat the environment of an accelerating system as a gravitational field? For example, suppose an observer in an accelerating frame with constant acceleration throws a ball at a 45 deg angle with respect to his system. According to this observer he will see the path of the ball as a parabola. With respect to an inertial observer wouldn't it be unnecessary complex to try and figure out the path the accelerating observer sees my using RIndler coordinates to determine this?

 

[Ibix]

With respect to an inertial observer wouldn't it be unnecessary complex to try and figure out the path the accelerating observer sees my using RIndler coordinates to determine this?

Why would an inertial observer want to use Rindler coordinates?

 

[e2m2a]

Why would an inertial observer want to use Rindler coordinates?

To determine what the observer in the accelerated frame observes for the motion of the ball or am I really confused about this?

 

[PeterDonis]

Rindler coordinates seem messy because they are always changing.

No, they aren't. Look at the metric in Rindler coordinates. None of the metric coefficients are a function of Rindler coordinate time.

 

[PeterDonis]

Wouldn't it be easier to invoke the principle of equivalence and treat the environment of an accelerating system as a gravitational field?

Um, that's what you are doing when you use Rindler coordinates. Rindler coordinates are the natural coordinates for an observer at rest in a rocket with constant proper acceleration, and an object inside the rocket that is dropped in free fall will appear to accelerate downward due to the (pseudo-)gravitational field inside the rocket.

 

[PeterDonis]

Moderator's note: Thread level changed to "I".

 

[Ibix]

To determine what the observer in the accelerated frame observes for the motion of the ball or am I really confused about this?

OK - but all the Rindler coordinates are is the "at rest" coordinates of the accelerated observer. You don't really need to use them except to express the final result if you want.

There's a close analogy with rotating coordinates in ordinary Euclidean geometry. It's often easier to work in Cartesian coordinates and solve a problem, write an object's path in terms of $x,y,z$ coordinates and then translate to polars and then rotating coordinates. There's nothing stopping you working entirely in the rotating coordinates, including all the inertial forces (Coriolis, etc) in your calculations, but you aren't obliged to do so.

Similarly, you can take your results in your inertial frame and transform them into the accelerated frame, or work in the accelerated frame where there's an inertial force (uniform gravitational field). Do whichever is easier.

Does that help?

 

[e2m2a]

OK - but all the Rindler coordinates are is the "at rest" coordinates of the accelerated observer. You don't really need to use them except to express the final result if you want.

There's a close analogy with rotating coordinates in ordinary Euclidean geometry. It's often easier to work in Cartesian coordinates and solve a problem, write an object's path in terms of $x,y,z$ coordinates and then translate to polars and then rotating coordinates. There's nothing stopping you working entirely in the rotating coordinates, including all the inertial forces (Coriolis, etc) in your calculations, but you aren't obliged to do so.

Similarly, you can take your results in your inertial frame and transform them into the accelerated frame, or work in the accelerated frame where there's an inertial force (uniform gravitational field). Do whichever is easier.

Does that help?

Yes. Making things as simple as possible is always the most efficient method. Thank you.

 

[Dale]

Summary:: When to use RIndler coordinates or the principle of equivalence

Rindler coordinates seem messy because they are always changing.

Maybe you are doing something wrong. Rindler coordinates are both stationary and static. And Rindler observers form a timelike Killing vector field (which I guess is implied by the previous sentence). Can you explain what you mean here?