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11th May 2022, 12:37 PM  #1 
Illuminator
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The Relativity of Simultaneity
The relativity of simultaneity is a consequence of Einstein's special theory of relativity.
In the Newtonian conception of space and time, time is both absolute and independent of space. In Newtonian physics, for example, whether two events happen at the same time is an objective question, completely independent of your point of view: They do or they don't, and that's the end of the matter. Special relativity tells us the Newtonian conception of time isn't quite right. According to special relativity, time is neither absolute nor fully independent of space. That linkage between time and space is why, when we change our point of view via the Lorentz transformation known as a Lorentz boost, we must transform our time coordinates as well as our spatial coordinates. One consequence of relativity's merger of time and space into a unified spacetime is that simultaneity is not absolute. If two distinct events are simultaneous in one coordinate system (of the special type known as an inertial reference frame), then there are equally valid coordinate systems of that type in which those events do not occur at the same time. Whether those events are simultaneous is therefore not an objective question. Its answer depends upon the coordinate system you prefer to use. Outline of this essay
I will use the phrase "Recall that...", together with hyperlinks, to state some basic principles and definitions of special relativity. Readers who are unfamiliar with those principles should follow the links. Recall that special relativity applies only to flat (Minkowski) spacetime, which I will assume throughout this essay. (The relativity of simultaneity becomes more general in general relativity, but that generalization involves substantially more advanced mathematics. The basic ideas of special relativity are accessible to anyone with a solid grasp of high school algebra and an interest in physics.) For the reasons given in Part 3, I will assume special relativity is a correct theory. My goal here is to explain how the relativity of simultaneity follows from that theory. Part 1: Precise statements of the relativity of simultaneity Recall that a Cartesian coordinate system for 3dimensional Euclidean space assigns numbers (coordinates) x, y, and z to every point of the space in such a way that the distance ds between two points of the space satisfies the equation ds^{2} = dx^{2} + dy^{2} + dz^{2}where dx is the difference between the x coordinates of the points, dy the difference between the y coordinates, and dz the difference between the z coordinates. That equation is of course the Pythagorean theorem generalized to three dimensions. Taking the positive square root of both sides of that equation defines the Euclidean metric (also known as the Euclidean distance function). Recall that, in special relativity, an inertial reference frame is a 4dimensional coordinate system for Minkowski spacetime that assigns numbers t, x, y, and z to every point of the space in such a way that the spacetime interval between two points is given by the equation ds^{2} = − c^{2} dt^{2} + dx^{2} + dy^{2} + dz^{2}where dt is the difference between the t coordinates of the points, dx the difference between the x coordinates, dy the difference between the y coordinates, and dz the difference between the z coordinates. That equation is also known as the Minkowski pseudometric or (by a minor abuse of terminology) the Minkowski metric. An inertial reference frame is said to be inertial because the straight line obtained by interpolating between the coordinates of two distinct points represents a path through spacetime of some particle moving at constant velocity, under its own inertia, hence free from acceleration. Recall that the signs of the terms on the right hand side of that equation form a metric signature. The spoiler explains my preference for the (− + + +) signature I'm using throughout this essay. Recall that, with the metric signature I'm using (which is the opposite of the metric signature used by this Wikipedia page), a negative value of the spacetime interval is said to be timelike, a positive spacetime interval is said to be spacelike, and a zero value is lightlike.
With all that background out of the way, we can now give two precise statements of the relativity of simultaneity. Relativity of Simultaneity (Version A) If E_{1} and E_{2} are distinct events that occur at the same time in some inertial reference frame R, then there is another inertial reference frame R' in which E_{1} and E_{2} occur at different times.Relativity of Simultaneity (Version B) If events E_{1} and E_{2} are separated by a spacelike interval, then there is an inertial reference frame R_{1} in which E_{1} occurs before E_{2}, an inertial reference frame R_{2} in which E_{2} occurs before E_{1}, and an inertial reference frame R_{0} in which E_{1} and E_{2} occur at the same time. Part 2: Proof that the relativity of simultaneity is a consequence of basic principles of special relativity I'll put the proofs in spoilers, because some readers might want the fun of proving these things themselves, while others might be intimidated by the sight of a mathematical proof. As a warmup, I'll start with a couple of help theorems (lemmas). Lemma. If E_{1} and E_{2} are distinct events that occur at the same time in some inertial reference frame R, then E_{1} and E_{2} are separated by a spacelike interval. Lemma. If Version B is true, then Version A is true. The following lemma is a fact about Cartesian coordinate systems for 3dimensional space, with the time coordinate just going along for the ride, but everyone should read the last two sentences of the spoiler. (ETA: My original statement of this lemma was incorrect. I think it's okay now.) Lemma. If E_{1} and E_{2} are events with coordinates (t_{1}, x_{1}, y_{1}, z_{1}) and (t_{2}, x_{2}, y_{2}, z_{2}) in some inertial reference frame R, then there is an inertial reference frame R' in which E_{1} has coordinates (t_{1}, 0, 0, 0) and E_{2} has coordinates (t_{2}, x_{2}', 0, 0), where x_{2}' > 0. Recall that the spacetime interval between two events is an invariant, which is to say that spacetime interval has the same value in all inertial reference frames. Applying that fact to the previous lemma, the spacetime interval between E_{1} and E_{2} has the same value in R as in R'. Now, finally, I can prove my more general statement of the relativity of simultaneity. Relativity of Simultaneity (Version B) If events E_{1} and E_{2} are separated by a spacelike interval, then there is an inertial reference frame R_{1} in which E_{1} occurs before E_{2}, an inertial reference frame R_{2} in which E_{2} occurs before E_{1}, and an inertial reference frame R_{0} in which E_{1} and E_{2} occur at the same time. Part 3: Empirical evidence for the relativity of simultaneity Part 2 proved the relativity of simultaneity without making any assumptions beyond basic concepts of special relativity and high school algebra. If the parts of special relativity that are used in those proofs are real, then the relativity of simultaneity must be real. The only way the relativity of simultaneity could fail is for some of special relativity's basic ideas to be wrong. That is extremely unlikely. Relativity is one of the most thoroughly tested theories in all of modern science. The parts of special relativity used in the above proofs have been subjected to many experimental and observational tests. Because special relativity is a special case of general relativity, all of the experimental and observational tests of general relativity also count as evidence for the correctness of special relativity, and therefore count as evidence for the relativity of simultaneity. Although it is easy to give examples of experimental tests that the entire theory of relativity has passed, it is much harder to give specific examples of experiments that offer a direct demonstration of the relativity of simultaneity. The reason for that should be obvious. The relativity of simultaneity basically says you can't find a demonstration of simultaneity that is independent of the arbitrary reference frame(s) you choose to use in a demonstration. Furthermore, the only way to demonstrate nonsimultaneity as an absolute concept (independent of reference frame) is to use events that are separated by a timelike interval, which implies all reference frames will agree on which came first. Part 4: What do "observers" have to do with the relativity of simultaneity? Parts 1, 2, and 3 of this essay don't mention observers at all. It was possible to state the relativity of simultaneity in mathematically precise form, and to give mathematically rigorous proofs of the relativity of simultaneity while assuming only basic principles of relativity. That tells you the concept of "observers" is extraneous. There is no need to mention observers when discussing the relativity of simultaneity. Yet many expositions of the relativity of simultaneity do talk about observers, especially when the exposition is aimed at a general audience. Here is part of what a very good freshmanlevel textbook on special relativity has to say about observers:
Originally Posted by A P French
One of the dangers not mentioned in that quotation is the danger of thinking an observer is forced to reason in terms of an inertial reference frame in which the observer is at rest. Not so. Observers are free to think in terms of any inertial frame they find convenient, and may transform back and forth between frames whenever they like. Someone traveling on a train, for example, may prefer to think of himself as being at rest while reading a book or talking to another passenger, but prefer to think of himself as moving when he looks out the window at passing scenery. The fact that you can think in terms of whatever inertial frame you like is literally the first postulate of special relativity. Part 5: Is the relativity of simultaneity a perceptual phenomenon? No. Emphatically no. Parts 1, 2, and 3 of this essay were able to state and to prove the relativity of simultaneity without mentioning anything remotely connected to perception. That tells you perception has nothing at all to do with the relativity of simultaneity. Why, then do some people make the mistake of thinking the relativity of simultaneity has something to do with perception? Partly because some people overlook the difference between "subjective" and "perceptual". The relativity of simultaneity tells us simultaneity is either absolute and objective (in the case of events that are separated by a timelike interval) or completely relative and subjective (in the case of events separated by a spacelike interval). (In the latter case, the subjectivity comes from the fact that everyone gets to select his own preferred frame.) Perception isn't involved with either situation. Another reason people make this mistake is the sloppy talk about observers that so many authors throw into discussions of relativity to make the exposition feel less formal and intimidating to a general audience. Even Einstein's thought experiments talked about observers, although Einstein's descriptions of those experiments were nowhere near as sloppy as some of the expositions you'll find on today's WorldWide Web. Here is an example, written by a computer programmer who cofounded Autodesk, showing how excessive talk about observers within a somewhat confused exposition of the relativity of simultaneity can lead to a gratuitous (but fairly harmless) mention of perception:
Originally Posted by John Walker
Although I'm a bit critical of Walker's exposition, my search for popular expositions of the relativity of simultaneity that described the effect as "perceptual" or a mere matter of perception turned up many that were far worse than Walker's, including several written by antirelativity cranks. 
11th May 2022, 12:47 PM  #2 
Penultimate Amazing
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No it's not.
Seriously, well done, man. Clears up a ton from the other discussion. 
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11th May 2022, 01:26 PM  #3 
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Quote:

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11th May 2022, 06:27 PM  #4 
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Thanks for this. Most of the OP is way beyond my understanding, though.

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11th May 2022, 06:50 PM  #5 
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11th May 2022, 08:14 PM  #6 
Penultimate Amazing
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I think that the baseline for this discussion should be:
1. Acceptance of the postulates of Special Relativity and; 2. Acceptance that the postulates of Special Relativity imply that a Lorentz Transform is the correct way to translate between inertial frame. Every theory in science is challengeable and if anyone wants to challenge 1 and/or 2 above that is fine, but for clarity it might be better to challenge these in a different thread. And this is just a suggestion and not telling anyone what they should say or not, especially as this is not my thread. Postulates of Special Relativity 1. First postulate (principle of relativity) The laws of physics take the same form in all inertial frames of reference. 2. Second postulate (invariance of c) As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference. 
11th May 2022, 08:18 PM  #7 
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11th May 2022, 08:30 PM  #8 
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Actually, I guess the starting point for all of this is the fact that c is constant. Necessarily, then, time isn't how we think it is.
I realize that I don't really know how exactly people (Einstein?) first even came up with such a counterintuitive idea in the first place. (Don't want to derail the thread, and I haven"t looked, maybe Google might have a ready answer: but still, if anyone actually knows already, perhaps a single quick post in response won't hurt: How did this weirdsounding idea, that c is constant, first even come to be conceived of?) 
11th May 2022, 08:39 PM  #9 
Observer of Phenomena
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There's an explanation for that in the book I'm reading which I have mentioned a couple of times in our recent discussions. It's to do with Maxwell's equations of electromagnetism, but it's a bit complex for me to go into right now.

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11th May 2022, 08:43 PM  #10 
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Suppose that Bob and Alice, instead of clapping, each emit a photon.
Bob and Carol are standing near each over, each motionless with respect to the other. Alice flies close by at considerable speed. Each Alice and Bob emit a photon. Carol is standing midway between Bob and Alice at the time Bob emits the photon. Carol registers photons from Bob and Alice at the same time. This implies that Bob and Alice emitted the photons simultaneously. Now transform the frame where Alice is motionless and Bob and Carol zip by her at considerable speed. In this frame Carol still registers the photons at the same time. But in the transformed coordinates Carol was not midway between Bob and Alice at the time Bob emitted the photon. So in this frame they did not emit the photons simultaneously. It wouldn't take any more than high school maths to establish that this really does follow from the basic postulates and the acceptance of a Lorentz Transform as the correct way to transform coordinates. 
11th May 2022, 08:51 PM  #11 
Penultimate Amazing
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Maxwell's equations of electromagnetism are considerably harder to grasp than special relativity, they certainly stopped me in my tracks. Which is why I suggested taking the postulates as given for the purposes of this discussion.
I believe that there was already experimental confirmation of the constancy of the speed of light by the end of the 18th century, even before the Aether had been dealt a death blow, or at least results which made more sense under the assumption that the speed of light was constant for all inertial observers. 
11th May 2022, 10:38 PM  #12 
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11th May 2022, 11:01 PM  #13 
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The negative result of the MichelsonMorley experiment was a big clue.
https://en.wikipedia.org/wiki/Michel...ley_experiment 
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12th May 2022, 12:20 AM  #14 
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Thanks for the hint. The suggests a specific direction in which I might nose around a bit, online, to see if something readily comes up. Right. I've read about that experiment I remember (perhaps even actually studied it in physics class, I don't rightly remember that), but all of that had gone clean out of my mind, until I read up on your link again. Yep, like you say, that must have been a big clue. Actually, although it turned out to be false, the ether (aether?) theory does sound perfectly plausible, in terms of being intuitive. How utterly crazy the nature of reality turned out to be, following on that experiment that junked the aether thing and was instrumental in eventually bringing in relativity. (And of course, that was an age innocent of the utter madness that is QM.) 
12th May 2022, 05:49 AM  #15 
Illuminator
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I'll have to let the physicists answer that. For the purposes of my exposition, it's mostly a matter of whether a consensus of physicists regards the speed limit as part of special relativity.
Feel free to replace the highlighted part of
Maxwell's equations played an important role in the history of special relativity. Einstein knew he was onto something important when he understood how the magnetic field is explained by the Lorentz transformation.
Originally Posted by Wikipedia

12th May 2022, 01:42 PM  #16 
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Einstein himself wrote a book, appropriately titled “Relativity: The Special and the General Theory”, that discusses a lot about how he came up with the idea and walks through a lot of his thought experiments. It’s available on Amazon )or your bookstore of choice).
I found it useful and informative, and would recommend it for anyone with an interest. Sent from my volcanic island lair using carrier pigeon. 
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12th May 2022, 03:31 PM  #17 
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I found that book really useful too. He pitched it at a general audience without dumbing it down too much. No bowling balls on trampoline style analogies.
But others have said the read it and found it difficult to follow but could more easily read other books that I found challenging. It seems that finding the right popular exposition of scientific ideas is a matter of finding the one that works for you. I would recommend the Einstein book, I found it engaging and simplifies the concepts without too much dumbing down and, well he is Einstein and he ought to know. The AP French one that WD Clinger recommends above is also good for a text book style treatment of it. 
12th May 2022, 06:24 PM  #18 
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I find it helps me to make these things concrete with numbers, at least initially. It isn't any kind of rigorous proof but it helps to get across how the transforms work.
I created a Desmos graph for the Alice, Bob and Carol. I have made it Alice and Carol who emit the photons and Bob as the observer, comoving with Alice, just because it made it neater to have A,B,C across the page representing the events of A=Alice emits a photon, B=Bob checks his watch and sees 0 seconds and C=Carol emits a photon. The first graph is Alice's inertial frame, the second is Carol's inertial frame The coordinates are (t,x) and note these are the other way round to the normal convention for Cartesian coordinates of (horizontal,vertical) The world lines are Alice, Bob and Carol respectively left to right. The vertical axis is time and the the horizontal axis is the distance in the direction Carol is moving relative to Alice and Bob. It is scaled so that the speed of light forms a 45 degree angle. The distances on the graph can be metres, in which case the time axis is in 1/c seconds, or the distances can be light seconds, in which case the vertical axis is in seconds. The graph, for anyone interested is here: https://www.desmos.com/calculator/41jsgfyhen You can calculate the spacetime interval and confirm it does not change between the frames (note that the speed of light is set to 1 for this, hence the c^{2} term is left out of the spacetime interval calculation in Desmos) 
12th May 2022, 07:10 PM  #19 
Observer of Phenomena
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I'd like to see someone who knows what they're talking about provide a response to this, if possible:
I know that there were recent clickbaity headlines saying that time might not exist, but they were all about a new book explaining Loop Quantum Gravity (which I might have to get hold of a copy of), one aspect of which is that time is an emergent property rather than an inherent characteristic of reality. 
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12th May 2022, 07:28 PM  #20 
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Quote:
About the only thing I can think of that we can really demonstrate about it's nature is that we can exchange distance for time. That's what the Twin Paradox demonstrates. One twin chooses to experience more time, the other more distance. 
12th May 2022, 07:29 PM  #21 
Penultimate Amazing
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I would be interested in the answer. I suspect it might be, no one knows at this point in time.
I would just point out that we don't need to assume that time is an actual thing to be independent of or unified with anything for the purposes of this thread. So that stone wall isn't a stone wall. I am guessing that none of us assume that time is an actual thing to be independent of or unified with anything. We are merely open to the possibility that science will show us what time is. 
12th May 2022, 07:45 PM  #22 
Observer of Phenomena
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I've heard that entropy defines the arrow of time.
Let's see if my explanation of entropy passes muster. There is only one way that you can stand a pencil on its point. You have to arrange the pencil so its centre of balance is precisely over its point. You can rotate the pencil around its axis, but the number of states that the pencil can be balanced is limited to those in which its centre of balance is directly above its point. But there are a lot of ways in which a pencil can be not on its point. It can be on its side, pointing in any direction, it can be angled on a slope, it can even be on its flat end. The pencil on its point is in a state of low entropy, and its natural inclination is for it to fall over, into a state of higher entropy. This tendency for a system to change to higher entropy states is what defines the arrow of time. This explanation is almost certainly inadequate, and I look forward to finding out why. 
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12th May 2022, 08:16 PM  #23 
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12th May 2022, 08:57 PM  #24 
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12th May 2022, 08:58 PM  #25 
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12th May 2022, 10:38 PM  #26 
Observer of Phenomena
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I meant both so it's nice to hear that I succeeded at one of them, at least. I find that one good way to test my understanding of a subject is to try to explain it to someone else.
So why is entropy not a good explanation for the arrow of time? 
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12th May 2022, 11:30 PM  #27 
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Basically there is the "why does it go that way?" Question. If the laws of physics are reversible then why can't entropy increase in the other direction, timewise?
If you take the basic laws of Newtonian Physics and change the delta t to a minus delta t you can also see entropy increase. If you model a heat engine step by step and about half way to equilibrium you multiply the.delta t by 1 it just continues to run down to equilibrium. So that is the part which goes against the idea of time as a dimension. 
13th May 2022, 12:11 AM  #28 
Observer of Phenomena
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I was under the impression that entropy wasn't timereversible. If you take a jar full of ball bearings (low entropy) and pour it out on the floor so that they go everywhere (high entropy), it won't look at all the same if you run that film in the projector backwards, unlike, say, an electromagnetic interaction which is timereversible.

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13th May 2022, 12:15 AM  #29 
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The way that I heard of this thought experiment is that Tom is standing next to a railway track and he is exactly half way between two light sources A and B which are also next to the railway track. Dick is on a train that is moving from light source A to light source B.
At the instant when Dick is directly adjacent to Tom, both light sources flash simultaneously (in Tom's reference frame). Since Tom is exactly the same distance from both light sources, he will see the light from both sources at the same instant and will conclude that both light sources flashed simultaneously. However, because Dick is moving towards light source B, he will see B's flash before he see A flash. Since the speed of light is the same in both reference frames, Dick's (equally valid) conclusion is that light source B flashed before light source A. I don't know what Harry was doing all this time. 
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13th May 2022, 12:24 AM  #30 
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Brian Greene describes a peace treaty between the presidents of Forwardland and Backwardland. They only agree to a treaty if they sign the document simultaneously  neither wants to be the first to sign. They sit at either end of a table with a lamp placed directly in the middle. When the lamp goes on, they sign the document.
Now put the table, the presidents, and the lamp on a train, moving smoothly and steadily and undergoing no acceleration. By Einstein's principle of equivalence, no experiment done inside the train will produce any result that indicates that the train is moving (inertial reference frames are equivalent), so both presidents still see the light go on at the same time, and from their point of view the documents are signed simultaneously. But from the point of view of a spectator on the platform as the train whizzes past, the president of Forwardland signs the document first, since she is moving towards the photons that are from her point of view coming towards her. She travels less distance before seeing the light than does the president of Backwardland, who is rushing away from the photons that the lamp emits. The citizens of Forwardland are outraged at being tricked and the war between the two countries continues. 
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