Implication of length contraction

[pixel]

I had posted a similar question on another forum but didn't get much of a discussion. I'm interested to know what people here think.

So consider a spaceship midway between stars A and B and initially at rest in the reference frame of the stars. The ship then accelerates away from A to some fraction of the speed of light in a short time so it is still more or less midway between the stars (I'm thinking the details of this are not significant to the discussion). In the ship's reference frame, the distance between A and B is now less than it was before due to length contraction. Is it possible that in its reference frame the ship is now closer to A than before it started moving? That is, its motion moves it away from A but due to length contraction it is actually closer? That seems like a strange sequence of events.

 

[Nugatory]

That is, its motion moves it away from A but due to length contraction it is actually closer?

This is a really bad/confusing way of using the word "actually". If you're on the spaceship what you'll actually see is an increasing blueshift, indicating that you're moving away from A just as you'd expect. The length contraction only shows up when you calculate where A was at the same time that you started accelerating using the post-acceleration frame instead of the pre-acceleration frame.

The key here is that phrase "at the same time". This is a relativity of simultaneity problem, not a length contraction problem.

 

[Dale]

In the ship's reference frame, the distance between A and B is now less than it was before due to length contraction.

I am not sure this is true. Can you explicitly write down the transformation between the star’s inertial frame and A’s non-inertial reference frame?

 

[Nugatory]

I am not sure this is true. Can you explicitly write down the transformation between the star’s inertial frame and A’s non-inertial reference frame?

It may be that OP is comparing the length in the pre-acceleration inertial frame and the length in the pst-acceleration inertial frame.

 

[Dale]

It may be that OP is comparing the length in the pre-acceleration inertial frame and the length in the pst-acceleration inertial frame.

It could be, but then he is making a mistake. In neither inertial frame did the length change pre and post acceleration. Instead it would just be basic length contraction which I think is not what he is interested in.

 

[sweet springs]

Is it possible that in its reference frame the ship is now closer to A than before it started moving?

Why not? A is closer and moving away from the ship. Does it make any trouble?

Thanks for showing an interesting story that contradicts coommon sense of daily life "distance increase when we're leaving from it."

 

[Dale]

Thanks for showing an interesting story that contradicts coommon sense of daily life "distance increase when we're leaving from it."

But it doesn’t contradict it in any physically meaningful way. I mean, if I want to I can use a coordinate system where sitting here all the sudden the distance to Paris gets smaller even though I am just sitting on my couch at home, but that is just a matter of coordinates with no physical significance. I won’t be able to step outside and see the Eiffel Tower, regardless of how close my coordinate system says it is.

 

[Ibix]

Here are a couple of Minkowski diagrams illustrating this scenario. The red line is the ship, which accelerates instantaneously to 0.6c. The blue lines are the two planets. The dashed lines show the light cone of the acceleration event, and the two red crosses denote parties held by mission control on the two planets, simultaneous with when they believe the ship accelerated. The first one is in the rest frame of the planets:

The second is in the rest frame of the moving ship:

The important thing to do is look at the red crosses representing the parties. In the initial frame, they happen two light years either side of the ship at the same time as the ship accelerates. In the final frame, they happen further apart and not simultaneously. And this causes a problem.

If I want to stick these frames together and say "how does the ship interpret this" then I might naively approach it as follows. I would cut the top off the first diagram (because that's "after the acceleration") and the bottom off the second diagram (because that's "before the acceleration") and stick them together. But what about the parties? Depending on which side of the cut they fall, one of them never appears or one of them appears twice. Does the ship's crew really think this? How do they explain being able to hear the party in mission control over the radio if it never happened?

The problem is that the cuts through the two diagrams aren't parallel, because the two frames' notions of simultaneity aren't the same. The cut on the first diagram goes through the two party events - but just look at the slant on that line in the second diagram! My naive synthesised diagram covers some of spacetime twice and some of it not at all. And this is why you get funny effects like "the planet suddenly jumps closer". Before the acceleration, you said "now on the planet" was its red cross. Afterwards, you say it's where its blue line crosses the x' axis. You aren't comparing apples to apples.

You need a non-inertial frame to do the description properly. I'm rather partial to Dolby and Gull's radar coordinates for this - see https://arxiv.org/abs/gr-qc/0104077. The end result is that they take the past light cone from the first diagram (which everyone agrees is the past), the future light cone from the second diagram (which everyone agrees is the future), and synthesises a smooth transition between the two in the spacelike-separated regions.

 

[PeroK]

I had posted a similar question on another forum but didn't get much of a discussion. I'm interested to know what people here think.

So consider a spaceship midway between stars A and B and initially at rest in the reference frame of the stars. The ship then accelerates away from A to some fraction of the speed of light in a short time so it is still more or less midway between the stars (I'm thinking the details of this are not significant to the discussion). In the ship's reference frame, the distance between A and B is now less than it was before due to length contraction. Is it possible that in its reference frame the ship is now closer to A than before it started moving? That is, its motion moves it away from A but due to length contraction it is actually closer? That seems like a strange sequence of events.

Let's do some calculations. In the original rest frame of the ship, at time $t=0$, we have $A$ at x-coordinate $-x_0$ and $B$ at x-coordinate $+x_0$.

Now, in the rest frame of the accelerated ship, these two events transform to (using $c=1$ units):

A's coordinates are $(\gamma vx_0, -\gamma x_0)$. And B's coordinates are $(-\gamma vx_0, \gamma x_0)$.

So, as others have said, this tells us where A and B are/were at different times in the new "moving" ship frame. So, we need to calculate where A and B are at time $t'=0$ in the new frame. Both A and B are moving with velocity $-v$ in this frame so we have:

At $t' = 0$, $x'_A = -\gamma x_0 + v (\gamma vx_0) = -\gamma(1 - v^2)x_0 = -x_0/\gamma$

At $t' = 0$, $x'_B = \gamma x_0 - v (\gamma vx_0) = \gamma(1 - v^2)x_0 = x_0/\gamma$

And that all makes sense. In the new frame, the fixed distance between A and B is length-contracted. And at $t'=0$ in the new frame A is closer than at $t=0$ in the old frame.

But, as far as the spaceship is concerned, you could interpret this as:

In my old frame, the last data I have about planet A was that it was at $-x_0$ at time $t=0$.

In my new frame, the first data I have about planet A is that it was at $-\gamma x_0$ (further away) at some time $t' > 0$.

If we take the spaceship's changing inertial frame as something physically related to the spaceship, then it's not valid to run the clock back in the new frame, to calculate where planet A was at $t'=0$ because that frame was not the spaceship's rest frame for those earlier events at that point in space.

In particular, the spaceship could not have made any direct observations about earlier data: about events earlier that $t' = \gamma vx_0$ for planet A.

In that sense, the spaceship can say where planet A was at $t'=0$ in the new reference frame. But, you must remember, that that reference frame was not the spaceship's rest frame for some of those events. And, therefore, the time and place of those events do not represent something that the spaceship could have measured for itself.

 

[Dale]

Depending on which side of the cut they fall, one of them never appears or one of them appears twice.

And this is precisely why the naive approach is mathematically incorrect. The naive approach maps one spacetime event (the rearwards party) to two coordinates and the mapping is required to be one-to-one.

(I know @Ibix is well aware of this. This is mostly so @pixel understands the issues)

I'm rather partial to Dolby and Gull's radar coordinates for this - see https://arxiv.org/abs/gr-qc/0104077.

Me too. As far as I know it is the only method for constructing a non inertial observer’s reference frame that respects the second postulate.

 

[pixel]

It may be that OP is comparing the length in the pre-acceleration inertial frame and the length in the pst-acceleration inertial frame.

Yes, that's what I was doing, hence my comment that I was hoping the details of how it got to the final velocity were not important.

 

[Dale]

Yes, that's what I was doing

It is important to note that that is not and cannot be the same as “the ships frame”. That is two separate inertial frames, neither of which is the ships non inertial frame.

 

[Ibix]

Yes, that's what I was doing, hence my comment that I was hoping the details of how it got to the final velocity were not important.

The details are important. But it's perfectly acceptable to pretend that the acceleration was instantaneous - or at least occurred in negligible time. That's what I did in my Minkowski diagrams above.

The problem is that you cannot just stitch together two frames like that. You know the paper street atlases of a town that come in books? They usually have a bit of overlap between one page and the next. Stitching together two frames as you are trying to do is closely analogous to photocopying two pages and taping them together without thinking about the overlap, then wondering why the High Street is half a mile longer than it should be. The answer is that the High Street isn't longer. You just constructed a bad map.

If you want to construct a good map, as @Dale says, you need to use a non-inertial frame. I recommend radar coordinates.

 

[PeroK]

Yes, that's what I was doing, hence my comment that I was hoping the details of how it got to the final velocity were not important.

In my opinion it's quite a good question. I like what @Ibix posted in post #8. That resolves the issue as far as a "paradox" is concerned. It's quite important, however, to study the issues here and understand why planet A doesn't physically get any closer.

 

[pixel]

It is important to note that that is not and cannot be the same as “the ships frame”. That is two separate inertial frames, neither of which is the ships non inertial frame.

I appreciate the responses to my question and it will take me a bit of time to go through them before I can understand what was said. In the meantime, can you expand on your statement? What is the "that" that cannot be the same as the ship's frame?

 

[Dale]

What is the "that" that cannot be the same as the ship's frame?

That = “comparing the length in the pre-acceleration inertial frame and the length in the pst-acceleration inertial frame”.

 

[Ibix]

I appreciate the responses to my question and it will take me a bit of time to go through them before I can understand what was said. In the meantime, can you expand on your statement? What is the "that" that cannot be the same as the ship's frame?

If you are regarding lengths as having changed, you have to be doing it in a single coordinate system - the system used by the ship. But if you are comparing the length in the stationary and moving frames at the moment of acceleration, some of at least one of the lengths is in the past in one of the frames.

It's perfectly fine to compare the lengths between frames. But you can't construct a "frame of the ship" that contains both lengths, because the frame ends up having to describe some part of spacetime twice. Like my map analogy in #13, the resulting oddities are because you constructed your frame in an invalid way, not because anything odd is actually going on.

 

[jbriggs444]

But you can't construct a "frame of the ship" that contains both lengths, because the frame ends up having to describe some part of spacetime twice

I want to expand a bit on this.

For any finite rate of acceleration, you can construct a "frame of the ship" that encompasses some region around the ship without having any problems with "double-mapping", "aliasing", "folding over" or whatever you want to call the badness in question. You can do this in a conceptually simple way. For each instant on the ship's journey, you pick out the inertial frame of reference that is instantaneously co-moving with the ship (a "tangent inertial frame") at that instant. You use this to establish spatial coordinates (x, y, z) for events that are simultaneous (in this frame) with the instant in question. For the t coordinate of these events you use the ship clock's time. You repeat this process for all instants on the ship's world line.

As long as you restrict your attention to a sufficiently narrow tube around the ship's path through space-time, this gives you a perfectly good but non-inertial coordinate system with no aliasing problems. In this non-inertial coordinate system you may nonetheless see coordinate effects such as objects that change coordinates faster than c (but not faster than light) or that come closer as you "move away".

If you have large accelerations, then you will have to narrow the tube to avoid aliasing problems. With an impulsive acceleration, the tube narrows completely on one side and you have unavoidable aliasing problems on one side and un-mapped events on the other. So you either need to avoid using non-inertial coordinates altogether or use a different approach to construct a well behaved set of non-inertial coordinates.