Who is moving faster?

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  • #26
Orodruin
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GR uses SR to create a 4d spacetime.
No it doesn't. GR is a generalisation of the space-time concept to curved space-times including a prescription for how the curvature relates to the energy momentum tensor.
 
  • #27
Ibix
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As usual at this point I disagree. A reference frame is something real. You can take the corner of your lab and the clock on the wall as a reference frame as the most simple example.
In that case, OP is using "imaginary frame" to mean some frame in which the corner of your lab is moving with some four-velocity. I don't think that "real" or "imaginary" are helpful labels, to be honest (my last post notwithstanding). You can define them directly in terms of actual physical things (Einstein's rods and clocks, for example) or in terms of derived quantities (e.g. the imaginary rods and clocks of a hypothetical observer in inertial motion). They're both either real or not depending on your definition of "real".
 
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  • #28
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A reference frame is something real.
Depends on what you mean by "something real".

You can take the corner of your lab and the clock on the wall as a reference frame as the most simple example.
You can use a material object to define infinitely many reference frames, but that doesn't make all those reference frames material objects.
 
  • #29
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I though we need to use SR to calculate the time difference and then wondering whether to use GR to add an extra value by adding gravity effect of its own mass. But, GR uses the spacetime concept which is already based on SR. Only SR is limited to certain problems. I was mixing gravity with SR hoping for a more accurate result, when if fact gravity doesn't even exist in this context, but the spacetime geometry makes it feel as it were a force. GR uses SR to create a 4d spacetime.
It would be best (not that people conform strictly tho what it's best) to study SR first, and GR second. So the answer to the GR part of your question can and should be put off.

Let's do a variant on your problem to illustrate where there is a potential for confusion. Let's have two identical spaceships in your universe, and no planet at all. We can either have them both at rest at some distance away, and have one of the spaceships accelerate towards the other, or we can have the two spaceships initially in relative motion, whichever seems simpler to you.

In either case, we start considering the time issue when both spaceships are at the same location in space. They compare their watches, and synchronize them. Then time passes, and the two spaceships separate.

The two spaceships cannot compare their watches directly. They have to do it through an indirect process, for instance they might watch the other spaceship through a large telescope and read a large clock, or they might exchange radio signals with encoded timestamps. The visual or radio readings obtained in this manner mean that the observed readings are always out-of-date. Mathematical computations based on some model have to be done to figure out what the clock is reading "now".

The tricky part of the problem is that each spaceship has a different concept of "now". This is frequently misunderstood, and it's hard to even get the initial idea across, as words are very slippery things, and the idea is so strange that people tend to reject the favored interpretation of what is meant by this in termis of some other unintended interpretation. Unfortunately, it's not clear how to fix this problem - you'd think there would be some set of words so precise that they'd prevent misunderstandings, but words that precise tend not to be read and fully understood.

The best way that I know of for conveying the details is to go through a rather elaborate description of exactly what signals are exchanged, and how, in order to compare the clock readings. I'll try that approach, though it's a bit long.

The following is a typical example. Let T be the time at which the two spaceships are co-located. At T+1hr, according to spaceship #1, spaceship #1 sends out a signal which encodes the time, that it is now 1 hour after T. This signal is received at spaceship #2 at T + 2hr, according to spaceship 2's clock. Spaceship #2 immediately sends a signal back to spaceship #1, encoding the fact that it was sent out at T+2hr. This signal is received by spaceship #1 at T+4hr.

You may notice a pattern here, if not I'll point it out. If either spaceship sends a signal out at time X, it is received by the other at time 2x. The factor of 2 here is for ease of exposition - the general rule is that if a spacehip sends out a signal at time X, it is received by the other spaceship at time k*X, where k is some constant that depends on the relative speed. If you'd like more detail on this general approach or justification of it, I'd suggest Bondi's very old book, "Relativity and Common Sense". The general approach is usually called k-calculus, though it only involves algebra, not calculus.

Here is how spaceship #1 interprets these facts. It sent out the signal at T+1hr, and received it at T+4hr, so the signal took 3 hours to get to spaceship #2 and come back. The speed of light is constant, spaceship #1 concludes that at T+2.5 hr, spaceship #2 was 1.5 light-hours away.

Spaceship #1 also concludes that at T+2.5 hr, spaceship #2's clock read 2.0 hours, rather than 2.5 hours, so it must have been slow.

Spaceship #2 can make exactly the same observations, and come to the same conclusion. So at this point , there is no answer to the question of which clock is running faster.

If spaceship #2 turns around and accelerates so that it rejoins spaceship #1, spaceship 2's clock will read the lower time when the reunite. If spaceship #1 turns around and accelerates so as to catch spaceship #2, spaceship #1's clock will read the lower when they re-unite.

The fundamental idea here is that each spaceship has a different concept of "now". This is called the relativity of simultaneity. There is a lot written about this - understanding this is one of the tricky points about understanding SR.

For one paper on the topic (aimed, however, at teachers rather than students), see http://cds.cern.ch/record/571967/files/0207081.pdf, "The Challenge of overcoming deeply held student beliefs about the relativity of simultaneity".
 
  • #30
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I was confused because I thought that if we use the reference frame in which the planet is at rest and the probe at a velocity the clock on board will tick slower. But if we use the frames the other way round, the clock on the planet will slow down. I thought it didn't make sense. To answer the topic question, we don't need the third frame of reference. We define a frame in which both the planet and the probe are initially at rest. Then we know that the probe will be moving in this reference frame.
If we go to the probe reference frame, the planet will appear as leaving the probe, and clock on board we can suppose is runs slower. But then the probe will need to do something to come back and it will need to catch the planet. This way in the probe reference frame the probe will not be at rest anymore but moving faster than the planet previously moved in the probe reference frame. So we get the same result. We can ignore the acceleration.
 
  • #31
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Neither time really slows down. Time as experienced by you is a measure of the length of your path through spacetime, and the planet and probe took different routes with different lengths. That's all there is to it. I think you are just confusing yourself thinking about frames, especially as you are implicitly using non-inertial frames, and talking about "imaginary frames" (which makes no sense - all frames are imaginary, when all's said and done).
Yes, you were right. I was using multiple frames attributed to objects that are always at rest in those frames of reference, except for the clocks and frames moving relative to each other. That generated confusion and contradictory ideas.
 
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If we go to the probe reference frame, the planet will appear as leaving the probe, and clock on board we can suppose is runs slower. But then the probe will need to do something to come back and it will need to catch the planet. This way in the probe reference frame the probe will not be at rest anymore but moving faster than the planet previously moved in the probe reference frame. So we get the same result. We can ignore the acceleration.
It's a good idea when starting with SR to think primarily of inertial reference frames. You can start with a frame in which the probe is initially at rest and the planet is moving, the planet-based clock therefore ticks at a lower rate in that frame. To catch up though, the probe must then accelerate and move faster than the planet which means that although both clocks are now "ticking slowly" in the chosen frame, the probe's clock is affected to a greater degree. When the probe catches up with the planet, it therefore shows a lesser total elapsed time, by the same amount as would be calculated in the planet's rest frame of course.
 
  • #33
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There is a different version of a thought experiment without using acceleration by introducing a third object, sincronizing clocks, like in a relay race. Basically, one probe passes by, synchronizes its clock with the planet, then meets another probe, in the opposite direction,same speed, synchronizes clocks again. The third probe, passes by the planet again and they read their clocks.
 
  • #34
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That's right, and if you consider the picture from the first probe, the second probe is moving much faster than the planet in order to catch up. It's the same as what I said but avoids the need for an engine burn.
 
  • #35
Mister T
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Suppose we have a single planet in the entire universe and we send a probe through space. There is a clock on the probe and one on the planet. The probe will fly at a constant speed relative to the planet then it comes back.
In order for the two clocks to end up at the same place at least one of them will have had to change direction. If only one of the two clocks changed its direction of motion (in your example it's the one on the probe) then that's the one for which a smaller amount of time will have elapsed.
 
  • #36
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As usual at this point I disagree. A reference frame is something real. You can take the corner of your lab and the clock on the wall as a reference frame as the most simple example. Physics is after all an empirical science and deals with the quantified description of real things you can measure with the adequate measurement devices. That can be a simple yard stick to measure distances up to a ultraprecise interferometer like LIGO to measure tiny distortions of its arms by hitting gravitational waves, but it's a material things that defines reference frames.
No, it isn't. It is convention which defines reference frames. You can take the same material objects making the same measurements and adopt a convention assigning any coordinates you like. As usual, one good example is the GPS earth centered inertial frame in which none of the material measurement devices are at rest.
 
  • #37
vanhees71
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Sure you can use one frame and relabel it with other coordinates, but to fix a frame to begin with you need material objects to define one frame.
 
  • #38
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to fix a frame to begin with you need material objects to define one frame.
The material objects alone certainly don't define the frame, it clearly also requires a convention in addition to the material objects.
 
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  • #39
Mister T
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Sure you can use one frame and relabel it witah other coordinates, but to fix a frame to begin with you need material objects to define one frame.
You can use one set of objects to define a coordinate system and relabel it with other coordinates to define a different coordinate system. But the coordinate system is the frame, not the objects. If the objects were the frame you couldn't use the same objects to define two different frames.
 
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