md65536

But that's incorrect. Times on B's clock don't add up to times on X's clock. Do you at least understand that you're talking about 2 different clocks? I know it's only page 12 but do you understand that much so far? Agreed, there are so many ways to describe the concepts, and different people "get it" different ways, plus I often make mistakes. It's too bad this isn't in the relativity forum and might be read by others who'll get it. There's always something that makes more sense with someone else's explanation. I don't agree, as worded. Different observers do measure the ends of the stick simultaneously in their respective frames, and they get the correct relative length of the stick. But it's true those measurement times are different for different observers. The proper length is measured when the stick's at rest, it's not enough to measure at a single time, because any observer can do the latter. (Not mentioned here, but also it's no good to measure the length at times that are simultaneous in the stick's rest frame, because in other frames the stick moves between those two times.)

The thread's question needs interpretation, and I might be interpreting it differently than others. I think that what you're asking is how much mass you would need to make everything in the universe gravitationally bound to it, despite the current rate of expansion. If I'm thinking about it right, any constant rate of expansion will result in a constant-size cosmic horizon, beyond which it is impossible for matter to be gravitationally bound across that distance. The reason is that the matter would have to be falling in faster than the speed of light, to overcome the expansion of space between it and the mass. The horizon is determined by expansion alone, so making a more massive BH won't help... except... If you had matter right on the cosmic horizon, you'd need to basically have it falling in at a speed of c to overcome expansion. That would imply a BH with an event horizon at the same radius as the cosmic horizon??? (assuming Schwarzschild BH) But then, if you had a BH even close to that size, matter on the cosmic horizon would be a lot closer to it than if it were a point mass, so wouldn't it fall in anyway? Or does it work out that the gravitational influence of a BH is still the same as if it were a point mass? This seems really weird, because even if we completely ignore expansion for a moment, wouldn't this mean that the gravitational influence of a BH is roughly proportional to 1/r^2, while the mass is proportional to 1/r, no matter how big it is? That seems to imply that if you could have a BH of unlimited mass, you could make it so that the Schwarzschild radius is so large that an object outside it is so extremely far away from the center of the BH that the gravitational acceleration is small, even if it is near the horizon. Am I thinking about this correctly? How would an infalling observer describe the BH? It seems that the event horizon (a lightlike surface) would still pass by it at the speed of light, despite minimal acceleration. Meanwhile it seems like another observer, hovering farther away, outside the horizon, could easily avoid falling in, and see that same event horizon as stationary. Where's the error in my thinking? Back to expansion, would it even make sense to talk about a constant rate of expansion of spacetime, in a volume that is entirely occupied by a massive BH? The BH curves the spacetime so extremely that the volume inside the horizon is not a part of the same spacetime outside??? Does curved spacetime expand the same as flat spacetime? Would a volume containing a large BH expand the same as a volume of empty space?

Another answer based only on the sketch: 27+48=75 basically says "The amount of time that X's clock ticks while B travels, plus the time on X's clock when B begins, equals the arrival time on X's clock of 1:15" 27+48=75 expresses the sentence from B's frame. 75+0=75 expresses the exact same sentence, from the E+X frame of reference. Those are sensible statements because they're adding times measured by the same clock. 45+?=75 says "The amount of time that B's clock ticks while B travels, plus ??? equals the time on X's clock then B arrives." That doesn't make sense, because those times are measured by different clocks.

The 3 "stages" sound good to me... Please don't go backward from this point, where the details are no longer fine! Answers: 1. RoS concerns the simultaneity of separated clocks or events, in this case the time at Earth relative to the time at X. It's not showing up yet because you're not comparing the times of things at E and X. However, your stages 2 and 3 are describing the same calculations. They're just "what is measured in B's inertial frame." RoS can be used in stage 2 to explain why, when B is arriving at X, B can find that X's clock is at 1:15 while E's clock is only at 0:27. Both have recorded 27 minutes passing (according to inertial B) during B's trip, but X started with a clock 48 minutes ahead (according to inertial B). Meanwhile, in the E+X frame, these same clocks are synchronized. That's relativity of simultaneity. 2. There's no reason, the "stages" are just calculations from different frames. You can calculate from any frame in any order that you want, and you don't have to do them all. Any one frame of reference can make all the measurements necessary, assuming all the clocks are accessible to them, to calculate the proper times mentioned here (B arrives at X at 1:15 on X's clock and 0:45 on B's clock, all frames of reference can show that on their own, using the initial conditions described above. 0:27 on E's clock is a coordinate time (measured by B from a distance) and is a frame dependent measure, differing in different frames again due to RoS). It doesn't make sense to line up all the stages as you did in the diagram. They're times measured by different clocks. By analogy, imagine if you made a map lining up all the countries in the world and attaching them vertically. It would show relative lengths of countries, but it wouldn't show how they're actually connected.

Actually, that does make intuitive sense in retrospect (cheating, OR if your common sense considers enough information). The intuition is that an electron radiating EM energy, is associated with "change" rather than proper acceleration. An electron at rest relative to Earth isn't changing relative to an EM field. Instead it should be expected to radiate if you moved an EM field around it. In freefall, the electron isn't changing in terms of inertia due to spacetime curvature, but it is (or can be) changing with respect to the EM field. So, no proper acceleration, but radiation is possible. But I agree about the maths; intuition is useless if it doesn't reflect what the maths say.

This is an argument similar to Mach's principle. I think scientific theories neither support nor refute it. How would you even conceptually accelerate everything? You could for example use a uniform gravitational field throughout the universe. But then, you could detect gravitational time dilation between different points, where there currently is none. On the other hand, could you use frame dragging to cancel out those effects? I don't know enough about frame dragging to say anything, except that it's conceivable to me that if you rotated the entire universe around your body, frame dragging might cause your arms to pull away from your body, ie. a possible way to make Mach's principle true. I can think of two opposing ideas here. One is that physical quantities are typically defined according to things that can be measured. Eg. time is defined as what a clock measures. It might be okay to say proper acceleration is what an accelerometer measures? Another is that measuring devices generally are not perfect. For example if you apply force directly to the cantilever in an accelerometer, it says that the body itself is accelerating, which it isn't. A test for whether an effect is "real" might be, "Does this effect affect *every* measuring device, or just individuals?" For example, a pendulum clock on a ship might tick slowly, but that doesn't mean time is slowing down, because a cesium clock would not tick slowly. On a rocket traveling at relativistic speed, time really is slower than a stationary clock, because all clocks on the rocket tick slowly. If you manipulated everything in the universe so that everything accelerated, but all possible things that can measure acceleration are adjusted so that they don't detect it, then practically I'd say that there is no difference between that and things not accelerating at all. Philosophically I think this is called empiricism. If there really is no possible way to detect a difference between two things, I'd say there's no difference. Would an electron in freefall in a gravitational field radiate? I'm guessing it wouldn't, so it emits when it is properly accelerated? But that would still give you a way (eg. using specific gravitational fields) to accelerate things and have them not detect proper acceleration? Could you say that for practical purposes, proper acceleration is a locally measurable difference in acceleration (or force) applied to different nearby points?

Coordinate speed is relative, but proper speed is absolute. Coordinate acceleration is relative, but proper acceleration is absolute. Even relative speeds can have absolute consequences. For example, if two objects have a relative speed and they collide, that collision is absolute. When things have a speed relative to other things, that can be measured. Acceleration can involve different parts of things having different relative speeds at different times. For example, if a cantilever in an accelerometer momentarily has a non-zero speed relative to the rest of the device, the device measures a proper acceleration. If you think of a "cosmic stage" there may be a mystery, but the practical measurements aren't mysterious.

That's metaphysics. If you look at just the physics, you can see in the equations how the change in velocity adds up (given enough time and distance). The "something more", if it isn't in the equations, is probably not going to be something measurable (unless you discover something new experimentally), so you can interpret it however you want and invent whatever "something more" you want, without it having any bearing on observed reality, which in my opinion is why metaphysics is philosophy, not science. There are 2 possible results. One is, you manipulate the universe so that the two twins experience the exact same thing as they do in the simpler case, for example using gravitational fields and gravitational time dilation instead of just Lorentz time dilation. You might also use frame dragging and calculate the effect of all the mass in the entire universe on the twins. In this case, by making the twins experience exactly the same thing as before, by definition they're going to experience the same difference in aging, and you end up with the same result. Even though you moved the entire universe around instead of the twin to get an exact equivalent of simple acceleration, you still get that the twin that felt the effect of acceleration is the one that ages less. Second possibility is that you modify things so that they don't have the same experience as in the simpler case. In this case, depending on gravitational time dilation etc., you can get different results, eg. make it so the other twin ages less. But that should be expected; if you change the experiment you can change the results. If "something moving relative to the rest of the universe" is measurably exactly the same as "the universe moving relative to something", then you can't experimentally say that one is really happening and the other isn't. To say they're different would be metaphysics. I think this would be the Machian viewpoint that MigL mentioned. I think an anti-Machian viewpoint would be that speculatively, there must always be a measurable difference between accelerating something around the universe, vs. accelerating the universe around the something? That said, I think it would be very difficult to figure out how to manipulate the entire real universe so that there's no measurable difference from just accelerating a twin. In a toy universe containing only the twins, it should be easier.

What's the paradox? The title implies that "yourself" played both black and white. White won the game. You were both black and white. You won as white and lost as black. There's nothing paradoxical about that, and not even a feeling of doubt about those answers, let alone a sense that it's unsolvable. What am I missing? Is there some assumption that is somehow a tautology, something like "You can't be your own adversary"? "One entity cannot be two different things at once?" I can't think of any assumption that I agree with that would require a contradiction in answering the questions. What is the gist of the argument that it's unsolvable? I would bet money that you're right! I'm curious about your reasoning though. White won, "you" were playing white. No other person was mentioned. The title implies your opponent was yourself. If you are calling "yourself" two persons, that's fine, I see nothing paradoxical about that. Ignoring the title, the story describes an opponent who is not seen when the game is won. It could describe a game against someone else, who stood up and walked around after playing their final move. Not only is there no paradox, there are multiple consistent interpretations. Is it "unsolvable" because you've left out information? But I suspect that the two people are both you, and that somehow that seems impossible to you. I suspect that one person is described as "two persons" for the meaning of "playing a game for two persons", and then the meaning of "two persons" is switched, without it being obvious to you, to mean "two persons refers to separate individuals".

It's a strawman argument, and his entire argument. "These numbers don't make sense; there is something wrong with relativity." > Those aren't the numbers that special relativity predicts. "But they have to be! It's the only way that relativity can be right!"

So that's 3 clocks, each measuring different spacetime intervals between two events? Each interval corresponds to the world line of the respective clocks? In GR the interval isn't just a separation of time and of space, but along a path? If you use one clock to measure the spacetime interval of another clock's world line, the first clock is measuring coordinate time, and the second is measuring proper time. I think what Halc was asking is, if you express the second clock's world line in terms of the first's coordinate measures of time and distance, do you still get the same invariant spacetime interval? I suspect that the answer is generally 'no', because the coordinate time measured by the first clock, which is generally distant from the second clock, is measuring something different than what the second clock is measuring locally. The clocks, traveling on different world lines, do not each measure the same local effects of curvature as the other. But isn't it also possible to express that same world line, in different coordinates where they are also separated in space? The difference here is that instead of talking about 2 clocks on 2 different world lines, measuring different things, I'm talking about one clock, but described using different coordinates. I suppose you would say, (as you explained before) the components of the spacetime interval for the 4 dimensions could be different, but the resulting spacetime interval is invariant, which simply means that all observers agree on the proper time that the clock measures between the two events, even if they describe the clock differently in their own coordinates. I think both Halc and I are struggling with the meaning of that, maybe in different ways? I think that the meaning of different observers having different time coordinates for the single clock's world line, yet calculating the same proper time is this: Even if different observers locally measure the space and time coordinates of the clock, over an infinitesimal line element, the different observers can still describe that invariant line element using different mixes of space and time. Same local measurements, different coordinates = same invariant spacetime interval. Remote measurements = different world lines, with different spacetime intervals. Sorry if I'm repeating things, it's not yet clear to me.

What's the point of working through the details of relativity, like length contraction, but not accepting the predictions of SR, and instead replacing them with your own opinions? Does the length contraction as you're describing it, actually make sense to you? Or are you hoping to show that if you replace a few details with nonsense, then all of SR can be clearly seen as nonsense? Have you convinced yourself of that? Do you expect to convince others? Do you actually believe what you're saying, do you really believe that if length contraction as predicted by SR is applied, then Earth sees the traveler having gone only 0,6 LY in the example above? Or is the above example meant to show that length contraction must be nonsense? If it's the latter... Can I convince you that addition is nonsense, by adding 3 plus 2 together and getting 9? Are you convinced? Do you think I should spend 20 years telling people that addition is nonsense because I think addition, in my opinion, gives a ridiculous answer of 9?

All from the Earth's viewpoint: Is the traveling clock moving? Is a ruler attached to the traveling clock moving? Is a ruler attached to the Earth moving? Do you know which are length contracted?

There are two things that could mean, without context. Two different objects passing through the same pair of events can have different world lines (eg. twin paradox, with 2 paths of different proper time (geometric length)). Or, a single object passing through two events. In the latter case, there's only one world line. The object passes through a specific set of events, and everyone agrees that it passes through those events. The shape of that world line is different in different coordinates. For example, in a coordinate system that moves along with the object, the object is stationary all along the single world line. In other coordinates, the same world line might span a great distance. However, the 4d length of each infinitesimal part of that world line (representing local measurements) is the same no matter whose coordinates you use, and the geometric length of the whole world line is invariant, even though it can be made up of different spacial lengths and coordinate time spans. I didn't mean that, but I think what I said didn't make sense. In Minkowski spacetime, the interval s^2 = (ct)^2 - r^2 is invariant. The value r is the spacial length between say two distant events in flat spacetime, say in the coordinates of some distant observer. I have this notion that there's no such measure r in curved spacetime, because the local measurements of length along a world line between the two events will be different than the local measure of lengths near the observer. In SR there's not that problem, because a ruler has the same length throughout all of a given inertial frame's spacetime. In GR, the observer could make up a meaning of a distance between two remote points in its coordinates, but that wouldn't necessarily correspond with anything meaningful in local measurements of distance along the world line. So (ct)^2 - r^2 doesn't have enough information, and the notion of distance between two points in curved spacetime is not meaningful on its own, because it depends on the path between those points. For events that are "nearby" each other, with a single geodesic between the points, there is enough info, because the length would be along that geodesic, so an interval like (ct)^2 - r^2 only applies locally (to events that are near each other) in GR. (Even if the observer is distant from the events and has a different local measure of length, she can still meaningfully describe the distance between the two events that are near each other.) Does that make sense?

Credit where due, you seem to be making progress. Using conventions like time on the y-axis and light signals shown at 45 degree angles really helps in communicating ideas. In cases where you start with bad assumptions and then show what happens, the conclusions aren't going to be useful. If you put "garbage in", you get "garbage out." Eg. the Doppler factor of 5 from assuming the clocks tick at the same rate, is not useful. You *can* assume the clocks tick at the same rate, but then you'll find that the speed of light isn't constant etc., which doesn't match reality. Where you lined up "lines of simultaneity" by connecting moving clocks reading the same time, is not correct. That's not what simultaneity means. If you want to make further progress, try this: In diag 2, where it shows a light signal from Planet X to Earth when the traveler turns around, draw a light signal from Earth to Planet X at that same time, and see where (and when) it reaches the traveler. You'll see that Earth and the traveler can't see the same thing, and there must be asymmetry. That diagram would show what each observer actually sees. Any other diagram would have to agree with it. Diag 5 does not show the same thing as diag 2. If diag 5 showed the same scenario but from a different perspective, it would still show that the traveler sees the same things that diag 2 shows the traveler seeing. In my opinion, that interpretation is not incompatible with SR. But, however you interpret what is "really" happening, it still has to match the measurements, and the twins "really" do measure a difference in age. Earth doesn't see that distance length-contracted, the clock does. To wrap your head around it, you can ask "is the ruler that I'm using to measure this distance, moving?" If yes, length contraction will apply, otherwise it won't. For example if you measure the distance between Earth and Planet X is 1 LY according to Earth, that ruler is not moving. If something travels from Earth to X, the ruler doesn't move. Earth measures distances along that ruler without length contraction. Meanwhile if a traveler going from Earth to X would see that same ruler connecting Earth and X moving along with them; the distance between Earth and X is length-contracted.

Sure, they can measure different coordinate times. Those times can be a component of the invariant spacetime interval, without being invariant themselves. Different observers have different components that combine to the same interval. It's the proper time that is the invariant length of a spacetime interval. Everyone agrees on the time that a clock measures on its world line between two events. But to different observers, the clock's path with have different coordinate times, and different coordinate lengths. Consider an infinitesimal line element of such a world line. For some observers, the clock can be stationary over that line element, with spacial components of 0 and an infinitesimal time component. For other observers the clock is moving, and the line element contains infinitesimal spacial component. They disagree on the components, but agree on its invariant length that results each set of those components.

You're right, though it could but not really but actually it could, but not really. "All parts accelerating equally and simultaneously" in a single frame basically describes Bell's paradox, but kind of the inverse because you're slowing it instead of speeding it. Definitely you can't do it as a rigid object. If you stop a ruler all at once, you change its rest length. Its former length-contracted length becomes its new proper length, it must be compressed. To do this would require an opposite rule, that strain or stress doesn't affect the ruler's behavior, so it could be arbitrarily compressed. I guess a normal ruler would disintegrate if it was forced to stop quickly from relativistic speeds all at once, or if slowed gradually enough, it would have internal forces pushing it to resist being compressed, which would cause different parts of it to have different speeds. A realistic material ruler could survive not being perfectly Born rigid.

True... if Earth alone stopped the ruler and the ruler approached the limit of perfect rigidity, it would take at least 2 years for Earth to see part of the ruler come to rest 1 LY away, twice as long as if the ruler stopped all at once. But that's why I specified that the entire ruler was "made to stop" simultaneously in a given frame, rather than that it just stopped. I avoid worrying about practical details because SR isn't limited to what's practical. But it would mean this is a poor example to use as a thought experiment to explain what can be expected in real experiments. Not that anyone's argued that exactly! Practically, you could stop a very long ruler all at once (in a given frame), eg. with a very long synchronized brake. But, yikes... what happens if the traveler's 1 LY-long ruler is stopped by the traveler, and the rest of it stopped like a wave propagating across it at a speed of c? Before the traveler moving at .6c stops, it sees the ruler behind it at rest, 1 LY long, and Earth is .8 LY away (1/gamma) just before stopping, but appears .5 LY away (Doppler, receding). Then just after the traveler stops, but before the moving ruler stops, the ruler appears 2 LY long (Doppler, approaching). But it's length-contracted to .8 LY which means the far end has already passed the Earth (not yet seen) and really is still traveling at .6c... The closing speed of the far end of the ruler and the "stop" wave is 1.6c which means half a year for it to stop, it stops .2+.3 LY = halfway between Earth and the stopped traveler, as measured in the Earth's frame. Which means that stopping an object at its front would physically compress it by the Doppler factor (or more)! This agrees with what Earth would see: If the moving ruler stopped as if by a light signal traveling from its far end, it would appear to stop all at once, and it would appear .5 LY long (Doppler, receding) before that happened. This seems like an even worse "impractical example", sorry! Is it pointless to even consider such a thing? (an object in which the speed of sound equals c) Back to this thread........... the above example would be what a traveler might see if a ruler accelerated along with the traveler. However, using long rulers as representing lengths in different inertial frames, the rulers would remain inertial; they'd remain in their respective inertial frames, and no part of any ruler would be seen moving at a different speed than the rest of the ruler. A moving ruler would have different parts appear distorted differently (compressed moving away from you, stretched moving toward you). Using these inertial rulers, it is obviously (with hand waving) paradoxical to measure an inertial Earth appearing to move closer to you after you've come to a rest relative to it.

So instead of saying the spacetime interval isn't invariant in curved spacetime, I should have said the interval defined for Minkowski spacetime, ie. the quantity (ct)^2 - r^2 isn't invariant in GR. I was going to ask if the spacetime interval in GR is a local thing only, that applies only to intervals between nearby events, but if it implies world line lengths are invariant, it might apply to any arbitrarily separated events? Oh but then, there can be multiple world lines between two events in GR, so the spacetime interval is local only??? and a world line's length depends on local variances in spacetime curvature. Conversely, in Minkowski spacetime there is only one straight line between any 2 points, so the spacetime interval equation for SR is invariant no matter how far apart the events are. Am I on the right track? The interval in GR is based on a set of values of g for each pair of the 4 space and time dimensions??? Different observers would have different components for the 4 dimensions, but the interval itself would be invariant. Can that be paraphrased as, "The spacetime interval in GR is invariant because curved spacetime is locally Minkowskian", so that any variations in the interval's components are like those seen in SR's interval, or is that wrong or incomplete? I don't understand. If you have two events in spacetime in a binary star system, where you might carry a mass from one event to the other, isn't there a fixed gravitational potential energy difference between the masses at the two events? Doesn't that imply a meaningful notion of gravitational potential?

The funny thing is, he's already accepted that a Doppler effect is acceptable in his definition of reality, and it's an easy modification of a twin paradox setup to make neither twin inertial and make it truly symmetric. Then just say "that's what they both see." It's also funny because for me, seeing the asymmetry in the Doppler analysis of the twin paradox is probably what fully sold me on the predictions of SR, and I never doubted the resolution of the paradox after that, even though I still would have struggled with "the Earth's clock jumps forward with the traveler's change in inertial frame." Then if you cherry pick some predictions of SR, you can get something that fits Michel's reality and doesn't add up (which is not a problem for Michel). For example, if you let the traveling twin have a lightyear-long ruler attached behind it, and you make it so the entire ruler stops simultaneously in Earth's frame, then you can see something like this: Say v=.6c, from Earth the receding ruler appears compressed by the Doppler factor of 1/2. Then when the 1 LY mark on the ruler reaches Earth, that part of the ruler stops, but the traveler appears to keep moving until it reaches 1 LY rest distance. All along the ruler, a "wave" of successive lengths of the ruler being seen coming to stop and returning to normal length spreads down the ruler, the wave moving at an "apparent rate" of c, so that it takes 1 year to see the traveler and the end of the ruler coming to a stop 1 LY away. That's something SR predicts and sounds similar to what Michel has described the traveler seeing (instead of SR's prediction of the traveler seeing Earth's entire ruler appearing normal instantly, when the traveler---not Earth's ruler---stops and comes to rest in Earth's frame). I'm not positive I got the details right. Course, you'd have to sell your soul to argue that the predictions of SR describe reality and show that SR is wrong.

The Pythagorean theorem seems to crop up a lot, and if you move one of the terms to the other side it looks like an equation for a hyperbola. Often you can see that in diagrams, where you have eg. a length in the x dimension, and one in ct, and the hypotenuse is meaningful in some way. Anyway I'm still figuring out things about that. The numbers I used were just an interval s^2=1, with (ct)^2=1,r^2=0 in one frame, and (ct)^2=4, r^2=3 in another. A common speed used in examples is approx 0.866 c ie. sqrt(3)/2, because the Lorentz factor in that case is a simple factor of 2. You might try setting gamma to 2 and solve gamma=1/sqrt(1-(v/c)^2) for v, if you can, to see why those numbers come about. Or get Wolfram Alpha to solve it for you if not. Otherwise, you can get away with following a lot of examples just using v=.866c, gamma=2. (v=.6c, gamma=1.25, Doppler factor=2 is also common.)

It's more about the coordinates (an inertial frame is 3 Euclidean space dimensions and a time that is the same everywhere within the frame), the clocks just represent a measure of the frame's time at different points. If you never had to consider different frames, you could use a single clock to represent time everywhere. I'll just keep talking because I hope more people talk about the meaning of the spacetime interval being invariant! But with respect to that, do you know the 3 types of interval: space-like, light-like, and time-like? The type is invariant, and depends on if s^2 is respectively negative, zero, or positive (in the sign convention you used, s^2 = (ct)^2 - r^2). For any spacelike interval, there's a set of inertial frames where the two events are simultaneous. For a timelike interval, there's an inertial frame where the two events occur at the same place. I think the interval being invariant means that if S1 and S2 are more distant in M than E, then they must also be farther apart in time in M than E. An example is that a clock at rest ticks faster (smallest time between ticks) than if measured from any frame in which it is moving. If you have an interval where r=0, ct=1 (ie. a clock at rest ticking once, with appropriate choice of units) then s^2=1. In some other frame where it takes ct=2 for the moving clock to tick once, s^2 also must equal 1 = 4 - 3, so r must equal sqrt(3) units of distance. In that frame, there's more time between the two events, and more distance. And indeed, in such a frame, gamma=2, v=(sqrt(3)/2)c, and the moving clock moving for 2/c units of time moves a distance of sqrt(3). Does that make sense? I'm not sure I got it right because I have no experience dealing with intervals.

While waiting for a better answer... Having the events on the sun unnecessarily complicates things because of spacetime curvature. You're measuring distances from afar, in a different gravitational potential, so there's not one "correct" way to measure those distances. I don't think curved-spacetime intervals are invariant. However, if you're treating the sun as just a location in flat spacetime, that's fine. You wouldn't directly compare the arrival time of light signals from the events, you'd want the time those signals were sent. Basically you'd subtract the travel time of light to find that. Edit: I think I'm misunderstanding "time the difference in signal capture between S(1) and S(2)". Signals aren't really a necessary part. You could use any clock(s) synchronized with the observer's, to measure the events' times. Typically a clock at S1 and one at S2 would be used, but in this example a single clock at E and then compensate for light travel time would be more practical. Then M in a different reference frame would measure its different set of times, either using a different set of clocks or by transforming the times from another reference frame. To measure distance, you can imagine all of space being covered in a lattice of rulers, at rest in an observer's inertial frame. Another observer would use a different lattice of rulers. Then the events are located at some place on those rulers, and you can measure the distance between them. Simply knowing the position of the sun in the observer's coordinates, or timing distances using light signals, or other ways, tells you distances and/or the locations of the events. An event has a time and a location in any reference frame's coordinates, ie. a place and a time on a lattice of rulers and synchronized clocks. Then r is the spatial distance between the two events, and t is the time between the two events.

The speed would change in different reference frames, exactly as the speed of an object would. To see this, imagine light through some medium, and an object traveling along with it at the same speed, so that they both pass through the same set of events. In another frame, if their speeds were no longer equal, they wouldn't pass through the same events, which would be a paradox. As swansont mentioned, because of the velocity addition (composition) formula, changing the observer's speed by non-relativistic v will change the speed of light in the medium by a LOT less than v.

This is literally your thread now, I'm curious how this adds up in your examples, without relativity. You said that when traveling twin A stops, it doesn't immediately see Earth as stopped because of the delay of light. Relativity says Earth doesn't see A stop immediately, and you're saying A sees Earth appearing the same. For example, suppose Earth is at rest at the 0 mark on a ruler, and A travels to a 1 LY mark, and then stops. The whole time while traveling, A sees that Earth appearing to stay at the 0 mark, agreed? When A stops at the 1 LY mark, how far away does it see the Earth? I think it should be 1 LY. It sees Earth at 0 and itself at 1 LY and the distance is "normal", not contracted, and it is 1 LY. If A sees Earth continuing to move away, how is it appearing to move? Does it appear to move farther than 1 LY away? How do those numbers add up, in your view?