Does the relativity of simultaneity exist 'at a distance'?

[JTT10]

I have been studying special relativity and in working with Minkowski diagrams I have come to wonder if when simultaneity is shifted based on relative movement, does this 'shiftting' of simultaneity happen instantaneously across all of space? When a reference frame moves relative to something else there is an adjustment of spacetime coordinates in Minkowski space for that observer relative to one at rest. Is this to be interpreted as 'rippling out' at the speed of light or automatically being 'computed'?

 

[Matterwave]

The new coordinates are instantly computed everywhere when the boost is taken. This is why these Lorentz transformations are global. This leads to such things as the Andromeda paradox:

http://en.wikipedia.org/wiki/Andromeda_paradox

 

[JTT10]

Fascinating. This also then means that the slightest possible movement (we'll go with a quantum fluctuation) instantly 'registers' on the furthest possible location, the boundary of the universe. Just me or does this in some sense ring of holography?

 

[pervect]

I have been studying special relativity and in working with Minkowski diagrams I have come to wonder if when simultaneity is shifted based on relative movement, does this 'shiftting' of simultaneity happen instantaneously across all of space? When a reference frame moves relative to something else there is an adjustment of spacetime coordinates in Minkowski space for that observer relative to one at rest. Is this to be interpreted as 'rippling out' at the speed of light or automatically being 'computed'?

I would describe it as computed. It's rather similar to the way A7 on one map (I'm imagining a typical street map, the coordinates are basically similar except that they use numbers rather than a letter-number combo) might be equivalent to C9 on another, different map. If you change maps (frames), you don't physically "move" from A7 to C9, you just change the description. This change of description doesn't happen at any particular time because its more of a metaphysical event then a physical one.

If you are interested in accelerating observers, which might be said to be physically moving from one frame of reference to another, more needs to be said. I won't try to say it all here, it gets rather technical but I can give a few highlights. The first issue is that though space-time is flat for accelerated observers, it's no longer Minkowskian, so you need to use the mathematical concept of the metric to mathematically describe what's going on.

Introductory treatments of special relativity often use them metric -c^2 dt^2 + dx^2 + dy^2 + dz^2, this approach needs to be refined to handle coordinate systems for an accelerated observer.
An additional consequence of the introduction of the metric is that coordinates become different from distances, and it's cleaner to talk about "coordinates" rather than "frames" as frames usually carry the implication that a unit change in a coordinate represents a constant change in distance, which is no longer the case. This can be confusing to people who have conflated the separate notions of coordinates and distances.

The second issue is that the differing lines of simultaneity that are to be "stiched together" into a global coordinate system do eventually intersect. Since by the common understanding of what a coordinate system is one event must only have one set of coordinates, the coordinate system for an accelerated observer is limited as to what regions of space-time it can cover.

 

[bcrowell]

Setting up a Minkowski coordinate system in practice is going to be a lengthy and time-consuming process analogous to surveying a plot of land. You send signals back and forth and analyze the results. If you want to redo it using a different frame of reference, you can, but you have to start over again (or just recompute it using a Lorentz transformation). None of this is instantaneous.

Similarly, effects like relativistic length contraction and time dilation aren't what you *see* optically.

 

[Dale]

Fascinating. This also then means that the slightest possible movement (we'll go with a quantum fluctuation) instantly 'registers' on the furthest possible location, the boundary of the universe.

note that an inertial coordinate system covers all of minkowski spacetime with a non accelerating grid which, by definition, does not follow the acceleration of any object.

 

[JTT10]

Thanks for the responses. Bcrowell, I get what you are saying, but I am more curious about the metaphysical aspect that pervect had addressed. I know we must analyze the results to see what happened, but they happen nonetheless before we compare them. I just find it fascinating that it must be the case that movement in one place instantaneously 'changes' the ordering of events, or what we consider now, across all of space. The andromeda paradox expresses this well, but does no one else think it's even more interesting that an 'infinitesimal movement' is reflected on the boundary of reality? I know there is no causal effects or signaling of information, but the fact that relativity requires changes of description at-a-distance is quite interesting. I listened to Lenny Susskinds recent Messenger Lectures and he spoke of how 'everything becomes almost instantly entangled with everything else in the universe, and its nearly impossible for that to not be so'. Now, if we have nearly everything entangled, and all movements are reflected in some kind of change across all of reality, the universe seems 'very quantum' to be particularly imprecise. Isn't that fascinating? If we take this metaphor further and imagine viewing the universe from the outside, we'd have a thing governed by an internal structure (relativity) which can't access outside of its boundary (due to expansion) that in a way is fully expressed and reflected on the boundary. The mind jumps to black holes and Hawking radiation!

 

[PeterDonis]

when simultaneity is shifted based on relative movement, does this 'shifting' of simultaneity happen instantaneously across all of space?

Simultaneity is a convention, not a physical thing. "Shifting of simultaneity" just means re-labeling events with different time coordinates. You don't change any physical facts by re-labeling events with different coordinates.

When a reference frame moves relative to something else there is an adjustment of spacetime coordinates in Minkowski space for that observer relative to one at rest.

It's important when thinking of questions like this to clearly distinguish between coordinates, which are just mathematical labels put on events, and "reference frames", which should be thought of as actual physical rods and clocks, or the equivalent, that actually give measurements of times and lengths. Coordinates are just conventions, as I said above; but if you want the numbers assigned by coordinates to match up with the actual physical readings given by rods and clocks, then the properties that the coordinates have will depend on the state of motion of the rods and clocks.

For example, say you're flying through empty space with a lattice-work of rods and clocks attached to you, and you change your state of motion--you fire rockets and accelerate. Obviously the rods and clocks that are spatially separated from you in the lattice-work won't instantaneously change their state of motion with you; the change in motion will propagate through the lattice-work at the speed of light (if not slower--a real lattice-work will propagate the change at the speed of sound in the material of which it is made, which will be orders of magnitude slower than the speed of light). So if you are using the lattice-work of rods and clocks to assign coordinates to events, the rods and clocks will not define an inertial frame, because your state of motion is not inertial. So you can't use the coordinates assigned by the rods and clocks to define a "simultaneity" in the sense of special relativity, because only rods and clocks that define an inertial frame can be used to define that kind of simultaneity. Or, if you insist on using coordinates that do define a simultaneity in the sense of special relativity, then those coordinates won't match up with the readings of the rods and clocks (or with your own motion--you won't be at rest in these coordinates).

 

[PeterDonis]

I just find it fascinating that it must be the case that movement in one place instantaneously 'changes' the ordering of events, or what we consider now, across all of space.

But this "now", this "ordering of events", is not a physical thing; it affects no physical measurement and changes the results of no experiment. It's just a convention.

The andromeda paradox expresses this well, but does no one else think it's even more interesting that an 'infinitesimal movement' is reflected on the boundary of reality?

I think the Andromeda paradox is basically a red herring; it appears to make a claim about "reality", but all it's really making a claim about is a certain convention about assigning coordinates that, as I said above, corresponds to nothing physical. I think it's a shame that so many pop physics presentations, including ones written by well-known physicists, don't make this clear. If you want to read a longer rant about this , check out this post on my PF blog:

https://www.physicsforums.com/blog.php?b=4744

I know there is no causal effects or signaling of information, but the fact that relativity requires changes of description at-a-distance is quite interesting.

Relativity does *not* require any such change. You don't have to change the coordinates you use just because you've changed your state of motion. Do you change the coordinates you use when you drive to the grocery store, so that you think of the entire Earth as moving relative to you, instead of you moving relative to the Earth? Nobody is ever *required* to use coordinates in which they are always at rest. Again, I think it's a shame that so many pop physics presentations don't make this clear.

I listened to Lenny Susskinds recent Messenger Lectures and he spoke of how 'everything becomes almost instantly entangled with everything else in the universe, and its nearly impossible for that to not be so'.

I don't think he was talking about coordinates or reference frames or relativity of simultaneity here. Quantum entanglement is a different issue; it's still there even if you always use a single set of coordinates and never change them, even when you change your state of motion.

 

[pervect]

Fascinating. This also then means that the slightest possible movement (we'll go with a quantum fluctuation) instantly 'registers' on the furthest possible location, the boundary of the universe. Just me or does this in some sense ring of holography?

GR is a classical theory, so it doesn't handle quantum fluctuations. Speaking classically, the "slightest possible movement" doesn't require you to recompute anything, you are perfectly free to use coordinates in which you are moving. I'm not sure why you are feeling forced.

Slight changes in your coordinate system may cause large changes in coordinates, for an analogy consider rotating a map 1 degree. The usefulness of this anology is heightened by the fact that formally, a Lorentz boost (the mathematical term for a change in velocity) is very similar to a Euclidean rotation (*).

Objects close to you won't change coordinates much for a boost or rotation , while objects far away may appear to have large coordinate changes. The large changes will still be small relative to the total distance they are away from you however. For instance, if you rotate your map 1 degree, an object 1000 miles away might have coordinate changes that would make it appear to "instantly move" about 15 miles.

Except that nothing actually has physically moved. You've simply rotated your map 1 degree. Also nobody can "force" you to rotate your maps, you might do that because you think it's a good idea. I.e you are perfectly free to rotate 1 degree. And you are perfectly free after rotating yourself 1 degree to also change your maps from the old map to the new map with a different orientation, though the two decisions are separate. But neither action actually has any physical effect, it only has an effect on your description of the territotry, the map of the territory you are using. It does not have an actual effect on the territory.

I don't think the fact that rotating a map by 1 degree can causes large coordinate changes says anything about the holographic principle, therefore I also don't think that the fact that a small boost causes similar changes in coordinates says anything about the holographic principle

Basically, I suspect you are reading something into the math that isn't there. I would like to stress the usefulness of the analogy between the Lorentz boost and a Euclidean rotation in understanding the issue, the reference for that is below. I would also like you to reconsider carefully the significance of coordinates. I don't seem to get the sense that you agree with mt position, that coordinates are human inventions, like the labels on a map, with no direct physical significance, and that changing coordinates doesn't have any direct physical significance either.

I am curious if you also ponder the "mystery" that distant object "move so fast" when you turn your head, or if you are happy with that and just have an issue with the formally similar space-time equivalent.

(*) See Taylor & Wheeler "Space Time Physics", the "Parable of the surveyor" , to see the formal basis for comparing a Lorentz boost to a Euclidean rotation. This is at the very start of the book, you can download the relevant chapter (old edition) from EF Taylor's website. http://www.eftaylor.com/special.html

 

[Dale]

I just find it fascinating that it must be the case that movement in one place instantaneously 'changes' the ordering of events, or what we consider now, across all of space

This is simply not true that it must be the case. It is never true in the inertial coordinate systems which form the bulk of special relativity pedagogy. It is not even true for all non inertial frames.