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OK, this is a spin off from another thread, but it's an interesting topic in its own right.
What I'm going to try to do is to give a rather physical interpretation of accelerated frames, specifically the Rindler metric, and a physical (rather than mathematical) interpretation of the Rindler coordinates and the associated Rindler distance.
We start out with an observer with a constant proper acceleration through a flat, Minkowski space-time. It can be shown that the equations of motion for such an observer are:
x^2 - c^2 t^2 = c^4 / a^2
see http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)
(assuming nobody vandalizes it - I'll have to learn the trick of linking to specific versions sometime) and
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html
which gives the same equation in alternate form, solving for x(t) as d(t).
We will call this observer our "reference observer". We can imagine this observer as being the "bottom" of the elevator.
Next, we need to construct a set of observers who are "stationary" with respect to this accelerating observer. There are several ways that we could accomplish this, but one reasonable way of doing this is to say that an observer is "stationary" relative to our accelerating observer if there is a constant two-way time delay for light - i.e. if the "radar distance" from our reference observer is constant. This is the definition we will use for the remainder of the post.
Note that the radar distance will turn out to be different from our Rindler distance, but they will share the feature that if the Rindler distance is constant, the radar distance is also constant.
One can work out the paths through space-time given this approach. One will get the same answer as the Wikipedia gets
http://en.wikipedia.org/wiki/Image:HyperbolicMotion.PNG
that the "stationary" observers all follow hyperbolic trajectories, with the same asymptotes as shown in the diagram. I.e in the above diagram, x=1 is the path followed by our reference observer, and x=2 is the path of an observer at some constant distance "above" our reference observer. We can call observer 2 the "top of the elevator".
I think this has been done in some old posts (with a certain amount of discussion going on before it was realized that two different seeming results were actually identical.
A couple of points need to be made:
1) It's a bit confusing to have the curve labeled as x=1 our "reference observer". This is related to the fact that our equations of motion do not have a solution for (x=0, t=0). One can re-write the equations to get around this confusion, but it's more trouble than its worth. I hope people can just deal with the fact that the curve for x=1 is our "reference observer" or "the bottom of the elevator", and that the curve for x=2 is "the top of the elevator".
2) The observers who maintain the same distance will not have the same accelerations! The observer at x=2 in our diagram, for instance, will be accelerating at half the rate of observer 1. This is necessary to maintain a constant distance. Why do we maintain a constant distance? Because we want to. The point of being in an elevator is that the roof is stationary (in some sense) with respect to the floor.
3) Clocks for the different observers will not run at the same rate. While the radar distance from our the bottom to the top of the elevator is a constant (and so is the Rindler distance), signals sent from the bottom to the top of the elevator will always be redshifted, and signals sent from the top of the elevator back to the bottom will be blue-shifted. These red and blueshifts are an indication that the clocks do not tick at the same rates (I hope this is clear as written and doesn't need more explanation. But I'm afraid that it might not be. But I'm not going to go into more detail unless asked.)
Now that we've set up the notion of a 'stationary observer', let's talk about how we measure distances.
The standard unit of distance is one wavelength of some particular cesium source. We can imagine a sequence of cesium sources, all stationary (per our "radar distance" notion of stationary) with respect to our reference observer. These observers will also be radar stationary with respect to each other. These observers form the basis of our Rindler coordinate system - we will give each of them a constant "distance coordinate" via a procedure outlined below.
The way we get the distance from the top to the bottom of the elevator is conceptually simple. Each of these stationary observers measures the distance to the next nearest observer in the chain, arranged as a straight line. (It's easiest if we make the problem only two dimensional, with one space dimension and one time dimension, then all our observers will automatically be in a straight line.) (expand: Basically, the defintion of distance in terms of wavelength means that our observers use radar methods to calculate distance. The distance will be equal to the total round trip time of the radar signal (as measured by a local cesium clock!), divided by 2, and multiplied by the speed of light.
Note that because of "gravitational time dilation", or the red/blue shifts that I mentioned, each of these observers thinks that the other observers clocks are off. That's why we have to go into such detail of specifying which clock we use. We do not use a single clock - we divide the distance up into a large number of segments so that our clocks are all very close together, and we use a clock that's co-located at the position where we measure distance.
Note also that nothing substantial would change if we replaced our radar and cesium clock setup with a series of physical platinum iridium bars. Atoms are atoms, and the atoms in the platinum-iridium bar would experience the same relativistic effects that the cesium atoms do.
The main difference would be that our radar-bar would be much more rigid than any physical bar could realistically be. This is because the speed of propagation in a physical bar is limited to the speed of sound in a material, and this is much much slower than the speed of light. The signals in our radar setup are the maximum physically possible - so our radar setup acts a like an idealized physical bar would be if the speed of sound in the bar were equal to the speed of light.
The fact that all our clocks tick at different rates leads to the differences between the "radar distance" and the "Rindler distance". For short distances, it's important to note that the two methods agree - it's only for long distances that there is a difference. If we restrict ourselves to elevators such that acceleration*height << c^2, we will not see any significant difference between Rindler distance and radar distance.
And that's there is all to it. The distance (measured in this elaborate way, with an infinite sequence of observers) from our reference observer to any stationary observer is the Rindler coordinate of that observer. Having a notion of distance, we can also note that the speed of light is still isotropic in the elevator (sometimes people get confused about that, if so I may have to write more, but I won't unless asked), and that we can synchronize nearby clocks by a light signal emitted by a source at the midpoint. To synchronize clocks that are far apart, we need to construct a chain of intermediaries, as we did to measure distance.Actually there could be more to say :-) but this post is already long enough, and we've hopefully met the main goal about setting up a physical interpretation of the Rindler coordinate system.
What I'm going to try to do is to give a rather physical interpretation of accelerated frames, specifically the Rindler metric, and a physical (rather than mathematical) interpretation of the Rindler coordinates and the associated Rindler distance.
We start out with an observer with a constant proper acceleration through a flat, Minkowski space-time. It can be shown that the equations of motion for such an observer are:
x^2 - c^2 t^2 = c^4 / a^2
see http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)
(assuming nobody vandalizes it - I'll have to learn the trick of linking to specific versions sometime) and
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html
which gives the same equation in alternate form, solving for x(t) as d(t).
We will call this observer our "reference observer". We can imagine this observer as being the "bottom" of the elevator.
Next, we need to construct a set of observers who are "stationary" with respect to this accelerating observer. There are several ways that we could accomplish this, but one reasonable way of doing this is to say that an observer is "stationary" relative to our accelerating observer if there is a constant two-way time delay for light - i.e. if the "radar distance" from our reference observer is constant. This is the definition we will use for the remainder of the post.
Note that the radar distance will turn out to be different from our Rindler distance, but they will share the feature that if the Rindler distance is constant, the radar distance is also constant.
One can work out the paths through space-time given this approach. One will get the same answer as the Wikipedia gets
http://en.wikipedia.org/wiki/Image:HyperbolicMotion.PNG
that the "stationary" observers all follow hyperbolic trajectories, with the same asymptotes as shown in the diagram. I.e in the above diagram, x=1 is the path followed by our reference observer, and x=2 is the path of an observer at some constant distance "above" our reference observer. We can call observer 2 the "top of the elevator".
I think this has been done in some old posts (with a certain amount of discussion going on before it was realized that two different seeming results were actually identical.
A couple of points need to be made:
1) It's a bit confusing to have the curve labeled as x=1 our "reference observer". This is related to the fact that our equations of motion do not have a solution for (x=0, t=0). One can re-write the equations to get around this confusion, but it's more trouble than its worth. I hope people can just deal with the fact that the curve for x=1 is our "reference observer" or "the bottom of the elevator", and that the curve for x=2 is "the top of the elevator".
2) The observers who maintain the same distance will not have the same accelerations! The observer at x=2 in our diagram, for instance, will be accelerating at half the rate of observer 1. This is necessary to maintain a constant distance. Why do we maintain a constant distance? Because we want to. The point of being in an elevator is that the roof is stationary (in some sense) with respect to the floor.
3) Clocks for the different observers will not run at the same rate. While the radar distance from our the bottom to the top of the elevator is a constant (and so is the Rindler distance), signals sent from the bottom to the top of the elevator will always be redshifted, and signals sent from the top of the elevator back to the bottom will be blue-shifted. These red and blueshifts are an indication that the clocks do not tick at the same rates (I hope this is clear as written and doesn't need more explanation. But I'm afraid that it might not be. But I'm not going to go into more detail unless asked.)
Now that we've set up the notion of a 'stationary observer', let's talk about how we measure distances.
The standard unit of distance is one wavelength of some particular cesium source. We can imagine a sequence of cesium sources, all stationary (per our "radar distance" notion of stationary) with respect to our reference observer. These observers will also be radar stationary with respect to each other. These observers form the basis of our Rindler coordinate system - we will give each of them a constant "distance coordinate" via a procedure outlined below.
The way we get the distance from the top to the bottom of the elevator is conceptually simple. Each of these stationary observers measures the distance to the next nearest observer in the chain, arranged as a straight line. (It's easiest if we make the problem only two dimensional, with one space dimension and one time dimension, then all our observers will automatically be in a straight line.) (expand: Basically, the defintion of distance in terms of wavelength means that our observers use radar methods to calculate distance. The distance will be equal to the total round trip time of the radar signal (as measured by a local cesium clock!), divided by 2, and multiplied by the speed of light.
Note that because of "gravitational time dilation", or the red/blue shifts that I mentioned, each of these observers thinks that the other observers clocks are off. That's why we have to go into such detail of specifying which clock we use. We do not use a single clock - we divide the distance up into a large number of segments so that our clocks are all very close together, and we use a clock that's co-located at the position where we measure distance.
Note also that nothing substantial would change if we replaced our radar and cesium clock setup with a series of physical platinum iridium bars. Atoms are atoms, and the atoms in the platinum-iridium bar would experience the same relativistic effects that the cesium atoms do.
The main difference would be that our radar-bar would be much more rigid than any physical bar could realistically be. This is because the speed of propagation in a physical bar is limited to the speed of sound in a material, and this is much much slower than the speed of light. The signals in our radar setup are the maximum physically possible - so our radar setup acts a like an idealized physical bar would be if the speed of sound in the bar were equal to the speed of light.
The fact that all our clocks tick at different rates leads to the differences between the "radar distance" and the "Rindler distance". For short distances, it's important to note that the two methods agree - it's only for long distances that there is a difference. If we restrict ourselves to elevators such that acceleration*height << c^2, we will not see any significant difference between Rindler distance and radar distance.
And that's there is all to it. The distance (measured in this elaborate way, with an infinite sequence of observers) from our reference observer to any stationary observer is the Rindler coordinate of that observer. Having a notion of distance, we can also note that the speed of light is still isotropic in the elevator (sometimes people get confused about that, if so I may have to write more, but I won't unless asked), and that we can synchronize nearby clocks by a light signal emitted by a source at the midpoint. To synchronize clocks that are far apart, we need to construct a chain of intermediaries, as we did to measure distance.Actually there could be more to say :-) but this post is already long enough, and we've hopefully met the main goal about setting up a physical interpretation of the Rindler coordinate system.
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