Accelerated obserevers in special (not general!) theory of relativity

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  • #26
Hurkyl
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There are two reasonable notions of "locally Minkowski".

(1) The most commonly used is that at each individual point, the metric has signature +---. This means the tangent space at a point has the same geometry as (the tangent space to) Minkowski spacetime. However, any region of space-time may exhibit (second-order) deviations from the predictions of special relativity.

(2) The other reasonable notion is that each point in space-time is contained in a region that has the same geometry as Minkowski spacetime. Within such regions, there is absolutely no deviation from the predictions of special relativity.

(3) The inverse notion is that of being "globally Minkowski" -- the entirety of spacetime has the geometry (and topology) of Minkowski spacetime.

I am pretty sure that most people mean (3) when they talk about special relativity. I would like to emphasize that (2) is more general than three, even in 'completed' spacetimes: for example, you can have cylinder-like and torus-like spatial slices. In such spacetimes, you get additional global phenomena that arise from the topology.

Forgetting about these global phenomena that distinguish case (2) from case (3) can lead to paradoxes, such as the cosmological twin paradox.
 
  • #27
robphy
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Regarding the relationship between SR and GR...

If you set the stress-energy tensor (and the cosmological constant) to 0 in Einstein's equation, the solution you get is the Minkowski metric. So SR can be described as "what GR says about a universe that's completely empty". I think the limit G→0 accomplishes the same thing. It removes the coupling between the geometry and matter, and the effect is the same as removing the matter.
Setting the stress-tensor to 0 admits Minkowski as one solution... but not uniquely.
You get a bunch of other solutions called the "vacuum solutions", which includes [exterior-]Schwarzschild, for example. (To get Minkowski, you'll need the whole Riemann curvature tensor to be zero... not just Ricci [or Einstein].)
 
  • #28
Al68
Unlike for inertial observers, there is not really a "standard" convention for defining the rest frame of an accelerating observer, and thus no standard simultaneity convention either (and remember that simultaneity is just that, a convention, the question of what events are simultaneous from a given observer's 'point of view' is not really a physical question at all).
It seems to me that we could just define an event co-local to the accelerated observer and a co-moving inertial frame, then transform that event into any other inertial frame. Without spacial separation, there would be no lack of simultaneity between the frames, even if one is accelerated. And we could define as many such events as we choose, as often as we choose, and just use the standard simultaneity convention for inertial frames to transform from one inertial frame to another.

Einstein didn't specify a simultaneity convention when he analyzed accelerated frames in his writings, but he must have assumed that others would infer that he was using the same definition as in SR, and a local co-moving inertial frame for each event. Is there any other practical way to do it without inventing a new convention?

Al
 
  • #29
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It isn't a great idea to attach the calculation in a pdf file. It's too hard to include bits of the math in quotes.
Thank you for your answer to my question. Because the forum software eat one of my previous messages (non saved to local disk) I made the new post in [tex]\LaTeX[/tex], posted it as PDF with [tex]\LaTeX[/tex] source in the attached zip file. You can use the source for citing without typesetting formulas. I know for better forum software (very much).

I'll just comment on what you did in #1.
I do not understand what is #1, but my feeling is that it is not important for understanding of the rest of your post.

So far so good. Note that t0 is just the time coordinate in frame S of some event on A's world line.
Yes, I know that [tex]t_0[/tex] is the moment in the system S. But, what is wrong. I have think that it is correct formula for the proper time of bird i.e. time of the bird's clock from its pocket (the clock flying together with the bird). If this formula is incorrect what is correct. Interpretation is as follows

The bird flies with motion law [tex]x=A(t)[/tex] in inertial system S of the ground. In the moment [tex]t_1[/tex] of S, the bird meets a bridge. In the moment [tex]t_2[/tex] of S (after [tex]t_2[/tex]) the bird meets a stone. How much the bird is biologic older from meeting bridge, to meeting the stone? If the formula

[tex]\int_{t_1}^{t_2}\sqrt{1-(A'(t)/c)^2}\,dt[/tex]

is incorrect, what formula is correct?

But before you say that this is what A measures you should think about how he measures it. If he e.g. measures the time it takes to bounce a radar signal off the other bird, the result isn't going to be b'-a'. The coordinate distance in the co-moving inertial frame only agrees with the radar distance in the limit where the size of the region of spacetime where the measurement is performed goes to zero.
Yes, I know it, and I am expected this question. It is related to this remark:

This is a little misleading, because you're making it sound like there is a single "correct" way for an accelerating observer to define simultaneity at each point on his worldline. It's certainly true that you can construct a "rest frame" for the accelerated observer in such a way that his definition of simultaneity at each moment will coincide with that of the inertial frame where he's instantaneously at rest (I think Rindler coordinates, which work as a rest frame for an observer experiencing constant proper acceleration, would have this property), and in that case everything you say is correct, but you don't have to construct the accelerating observer's "frame" in this way. Unlike for inertial observers, there is not really a "standard" convention for defining the rest frame of an accelerating observer, and thus no standard simultaneity convention either (and remember that simultaneity is just that, a convention, the question of what events are simultaneous from a given observer's 'point of view' is not really a physical question at all).
I am interesting for following: If I use the same ideas for computing frame-invariant quantities in another examples, can I obtain incorrect results?

Thank you very much!
 
  • #30
George Jones
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Thank you for your answer to my question. Because the forum software eat one of my previous messages (non saved to local disk) I made the new post in [tex]\LaTeX[/tex], posted it as PDF with [tex]\LaTeX[/tex] source in the attached zip file. You can use the source for citing without typesetting formulas.
You got caught by a safety feature; users are logged out automatically after a certain period of time. This is done because some users might use public computer, and might inadvertently walk away from the computer without logging out, leaving their Physics Forum account active. If a user is automatically logged out while composing a post, and the user then clicks Submit Reply, the post, unfortunately, is lost.

If you use your own private computer, you can avoid this by checking the "Remember Me?" box that is to the right of "User Name" at the time of log-in. Then, as long as you don't clear your cookies, you will stay logged in permanently. If you don't want to check the "Remember me?" box, there are a couple of work-arounds.

Keep a second Physics Forums browser window open. After finishing your post, but before submitting the post, switch back to the other window, refresh the browser, and look at the top right corner to see if you're still logged in. If you are not, then log in. Switch back to the post window and submit the post. You can ensure that you stay logged in by periodically switching to the non-post Physics Forums window and refreshing.

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Nedeljko said:
I do not understand what is #1
Post #1. At the top right of each post (not window), there is a post #. This post is post #30. The numbers give a convenient way to reference particular posts within a thread.

Another tip: when using latex within blocks of text, use the itex and /itex tags instead of the the tex and /tex tags. Use the tex and /tex tags for equations that are between, not within, bolcks of text.

For example in post #1, you wrote

"We will fix a moment [tex]t_0'[/tex] of the bird's A clock and compute [tex]t_0[/tex] such that [tex]t_0'=\int_0^{t_0}\sqrt{1-(A'(t)/c)^2}\,dt[/tex]. Then we compute the vector [tex]v=A'(t_0)[/tex] and fix the Innertial system S' such that S' has constant velocity [tex]v[/tex] relative to system S."

Using the itex and /itex tags, this looks like

"We will fix a moment [itex]t_0'[/itex] of the bird's A clock and compute [itex]t_0[/itex] such that [itex]t_0'=\int_0^{t_0}\sqrt{1-(A'(t)/c)^2}\,dt[/itex]. Then we compute the vector [itex]v=A'(t_0)[/itex] and fix the Innertial system S' such that S' has constant velocity [itex]v[/itex] relative to system S."

So, please give latex another try.

[edit]Any further discussion of these issues should take place either in the Forum Feedback & Announcements forum (the last forum on the main Physics Forums page), or by Private Message.[/edit]
 
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  • #31
Fredrik
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Yes, I know that [tex]t_0[/tex] is the moment in the system S. But, what is wrong. I have think that it is correct formula for the proper time of bird i.e. time of the bird's clock from its pocket (the clock flying together with the bird). If this formula is incorrect what is correct.
I didn't say that it's wrong. It's not. I just explained what t0 is because you didn't.

Yes, I know it, and I am expected this question. It is related to this remark:
Yes, JesseM explained it very well.

I am interesting for following: If I use the same ideas for computing frame-invariant quantities in another examples, can I obtain incorrect results?
I'm not sure exactly what you mean by "the same ideas", but it clearly doesn't matter what frame you use to compute something coordinate independent like "proper time".
 
  • #32
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I'm not sure exactly what you mean by "the same ideas", but it clearly doesn't matter what frame you use to compute something coordinate independent like "proper time".
Let we assume that the observer's motion law is given in inertial system S. The ideas are:

1. Computing the time of inertial system as a function depending on the observer's proper time.
2. Computing values in inertial system S' co-moving with the observer in any fixed moment.
3. Considering these values as values measured by the observer in this fixed moment of the observer's time.
4. Expressing these values as functions depending on the observer's time.

That are brief description of the ideas. Consult the attached PDF for the example of computation. Can I use these ideas safely?
 

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