## Acceleration effect on light speed

The idea that the experience of an accelerated observer might be approximated by considering its worldline as comprising many small inertial pieces is a good one. And during each inertial step the speed of light seems to be constant everywhere. But at the velocity boosts or “frame jumps” between the steps, the apparent coordinates of all events (including those on the world line of a light signal) get shifted, so the light seems to jump ahead or back. Taking the limit of these approximations leads to the conclusion that the light signal does not seem to have constant velocity from the point of view of the accelerated observer. (Since the “frame jumps” lead to coordinate changes that are proportional to the distance of the event from the observer, this does not change the fact that every light signal seems to have the same speed when it reaches the observer, so there is no local change and it is just when the signal is far away from the observer that its velocity appears to vary.)

## What is relative in Relativity?

What is relative in any physical theory of “relativity” are the space-time coordinates of events from the perspectives of different observers.

One problem, I think, with the names using ‘Theory of Relativity’ is that they seem to suggest theories about what is relative, rather than (more correctly) about how the coordinates used by different observers need to be related in order to ensure that the laws of physics are invariant (ie NOT relative).

In fact the coordinates that seem most natural to any observers for the purpose of expressing their experience in quantitative terms are always to some extent relative to the observers, so just saying that they are relative without specifying how is not telling us much (though in the new theories there is “more” relativity in the sense that time as well as the spatial coordinates becomes relative).

Our intuitively expected relationship between the coordinates of relatively moving observers allows all observers to use the same time coordinate, and so to agree on which events are simultaneous (ie constitute the same moment in time). It also preserves the form of Newton’s equations of motion for observers moving at constant relative velocity – which, as Galileo noted, has the consequence that observers moving with constant relative velocities cannot, by mechanical experiments, identify any particular one as being stationary. So the question of who is moving can only be answered relative to a particular observer – but this is just one particular instance of the relativity of coordinates.

[Sometimes observers moving relative to some larger object such as the Earth might choose to agree on a fixed Origin based on that object rather than on their own positions. But Galileo noted that if they are all moving together inside a moving vessel without any view of the outside, then it makes sense for them to use the vessel itself as their frame of reference – and relative to that, anything outside would appear to be moving in the opposite direction. In the world of Galilean/Newtonian physics there is nothing aside from its greater size which makes us prefer the Earth’s frame to that of the vessel, nor anything besides Earth’s proximity which makes us prefer its frame to that of the Sun. The answer to whether or not anything is or is not actually moving was thus, even in classical mechanics, entirely relative to the observer’s arbitrary choice of a frame of reference; and so that certainly was NOT anything new in Einstein’s theory.]

The above noted preservation of form of the equations of motion is perhaps confusingly called both “Galilean invariance” and “Galilean relativity”. The confusion could be avoided by making it clear that the word “relativity” applies to coordinates and “invariance” to the laws of physics. But I think that the practice of using “relativity” for the invariance itself rather than for the coordinate transformations under which it holds was indeed a misnomer which I believe precedes Einstein (though as an aside I must add that it seems surprisingly difficult to find out who was actually the first to do this).

Einstein’s special theory describes how the spacetime coordinates must be related in order for the laws of electromagnetism to have the same form for all inertial (ie unaccelerated) observers in the absence of any gravitational field. It turns out that for this to work, observers in relative motion will not be able to use the same time coordinates, and indeed will have different notions of simultaneity; so in this theory there is indeed something more that is “relative” than in the Galilean theory (but I don’t think that is why the theory got its name).

Einstein’s theory derives the relativity of simultaneity, and the formulas relating spacetime coordinates of different observers, from the principle of invariance of Maxwell’s equations (and so in particular, invariance of the speed of light) from the points of view of all inertial observers. But in my opinion Einstein’s reference to that principle as the “principle of relativity” (as opposed to the “principle of invariance” as suggested for example by Felix Klein) was indeed a misnomer, and apparently even Einstein eventually expressed some agreement with this  (but too late to actually change it).

[The special theory of relativity also includes modifications of the laws of mechanics (excluding gravity) which are necessary for them to remain invariant under the same transformations as those which preserve Maxwell’s equations – but this has nothing to do with the name except for the fact that perhaps the thinking was that the “principle” in question was that all physical laws need to be invariant under the same relativity of coordinates.]

The general theory goes on beyond the special theory to describe how the coordinates should be related in order to preserve an invariant form for both electromagnetic and gravitational forces under more general conditions (including accelerated observers and gravitational fields). So it’s not that more things are relative in the general theory, but rather that the relativity of the same things is explored under a more general range of conditions.

P.S. It should perhaps be noted that, just as the special theory has no distinguished inertial frame, the general theory does not provide any purely local way to distinguish inertial from accelerated frames as no accelerated observer can distinguish the experience of being accelerated from that of being prevented from falling freely in some “fictitious” gravitational field – which can only be identified as truly fictitious by observing the absence of possible sources (mass-energy distributions) out to an arbitrarily great distance. So there is some sense in which acceleration vs gravitation distinction is not quite absolute in the general theory but I don’t think that this (or the absence of any distinguished inertial frame in the special theory) was ever the reason for our use of the word “relativity”.

## Experimental Confirmation of SR

Fitzgerald and Lorentz showed how if we assume that the structure and dynamics of all matter arises from electromagnetic forces which obey Maxwell’s equations in some particular (“aether”) frame of reference (not necessarily that of the lab itself), then the result would be that moving bodies experience length contraction, slowed vibration, and increased inertial mass – all in such a way that a moving observer would be unable to detect any of these effects on itself and would instead think that objects stationary with respect to the aether were exhibiting them instead.

All experimental results so far (and also, I am sure, the modified Hafele-Keating that I suggested) are consistent with the Fitzgerald-Lorentz prediction of undetectability of the aether and symmetric apparent effects of length contraction, time dilation, and increased inertial mass.

I thought your question was about the symmetry of the situation rather than the existence of a special “aether” frame.

But if you are asking whether any experiment can prove the absence of an aether frame the answer is no. The reason we reject the assumption of an aether frame is just because we don’t need it (and so by Ockham’s Razor we don’t make it).

## Days of Future Past?

Has the future already happened according to special relativity? – NO.

In fact, in special relativity, the question of whether or not an event has “already happened” depends on the observer and has no meaning if the observer is not specified.

I find it so hard to believe!! – THEN DON’T.

Believe this instead (but only after making sure that you understand it):

What is true according to special relativity is that for any distant observer relative to whom you are moving sufficiently rapidly, some events in your future may be seen as in their past relative to the time on their clock at which you think they are now (or rather at which you will think they were now when you eventually see that “now” event in their lives).

[And for every event in your future there are some possible observers in your “now” (though you will not have actually seen them yet) who, when they finally see that event, will judge it to have happened in their past relative to the time on their clock at which you (will) think they are now.]

So in the world of special relativity, there is no time-ordering of events that all observers will agree on.

## Why should the twin on the spaceship be younger than the other on earth if each of them is supposed to observe the time dilation of the other in his own frame?

The question of which is younger when they are apart and in relative motion has no answer unless we specify the observer who is making the comparison (which could be either of them – or perhaps some other arbiter such as one who is stationary with respect to the Cosmic Microwave Background radiation).

Once they reunite they, and everyone else, will agree that the one who ends up younger is the one who experienced more acceleration towards the other when they were far apart (or more precisely for whom the integral of distance times the negative of its second derivative is greatest). But even though they will agree on the end result, they won’t agree on a moment-by-moment accounting of how their ageing rates compared.

## In the twin paradox it is often stated that the clocks can only be compared at the same location. Why can’t the clocks be compared at space stations synchronized with the earth clock on the travelling twin’s journey?

The traveller’s clock can indeed be unambiguously compared with each space station clock at the event where they pass by one another, but that is still only comparing clocks when they are at the same location. And the problem with saying that comparing one’s time with that on a space station is equivalent to comparing it with the one on Earth is that it depends on agreeing that the space station clocks are properly synchronized. But if the space station clocks appear synchronized with the Earth clock in its own frame, then they will not appear synchronized to the traveller who is passing by them. So the time on the space station clock does not match the traveller’s idea of what is the current time back on Earth.
One can indeed go through the process of keeping track of the space-station clock times compared to the traveller’s clock, and will find that those recorded times are all greater on the space-station clocks by the same Lorentz gamma factor. But so long as the velocity remains constant, the traveller could be part of a lined up fleet of ships all moving at the same velocity past the Earth (and so stationary with respect to one another with the Earth and space stations moving past them), and if they all synchronize their clocks with the traveller then the Earth and space station clocks will record the intervals between successive ships of the fleet as greater than the time differences between the clocks on those ships. In other words the Earth (and space station) observers see the ship times as more closely spaced than their own and the traveller (and fleet ship) observers see the times on space station clocks as more closely spaced than the times (on their own ship-based clocks) at which they pass by them. At first sight perhaps this looks like a paradox, but we need to note that each observer of either kind is comparing times on different clocks of the other kind with successive times on the same clock of their own and each can attribute the effect to an assumption that the other set of clocks is not properly synchronized. So this isn’t really a paradox, but there is still no way of deciding which team is actually synchronized and which is not – and without being sure of that the traveller can’t rely on the space stations as true representatives of the time back on Earth.
Making the traveller turn around and return to Earth is just one way of getting some particular pair of clocks back together for an unambiguous comparison of time intervals. (Another would be to have the Earth chase after the traveller and compare notes when she catches up, and yet another would be to do things symmetrically.) But they all involve having someone change their inertial frame (ie accelerate) and the result depends on the acceleration pattern but is always basically that the one who experienced the most acceleration towards the other when they were far apart is the one who will end up younger.

[In the symmetrical twins story both end up the same age, and are not surprised because each has seen the other age first more slowly and then more rapidly but ending up with exactly the same total amount of ageing as they themselves have experienced. If they use the light travel time to infer when each tick of the other’s clock actually occurred (as opposed to when they see it), then each will infer that the other’s clock was running more slowly during both constant speed parts of the trip, but more rapidly during the period when they felt the force of acceleration during the turn-around process – with the same final result.]

## Where in the universe can we find such an inertial frame? Certainly not on the surface of earth!

SR only applies exactly in the absence of gravity. So in the real world it is just an approximation that works well enough for predicting things where the effect of gravity is small (such as interactions between small high velocity particles in accelerators near the Earth’s surface, or between spacecraft and small bodies like asteroids far from planets, but not for things like apples falling out of trees on Earth).
In regions where it does provide a good approximation, it works just as well for accelerated as unaccelerated frames, but for accelerated frames the formulas needed to express physical laws in terms of the observer’s coordinates are more complicated.

## So the excuse used NOT to apply relativity theory in the twin paradox is a brief period of zero seconds at the turnaround point?

No one who knows what they are talking about has suggested “NOT to apply relativity theory”. On the contrary, the correct application of relativity theory leads to the conclusion that when the twins re-unite they agree on the fact that they have both seen the traveller age less. They just disagree on when during the trip the Earth-based twin aged faster. The one on Earth thinks it happened at a steady rate throughout the trip and the traveller (after actually seeing it during the return trip) thinks (after making the light travel time correction) that it happened quickly during the turn-around.

Prior to the turn around, each sees the other ageing more slowly (due to the Doppler effect) and, even after making the light travel time correction, thinks that part of that slowdown remains unexplained (and so in some sense is “really” happening).

But any claim that during the outbound journey “we know for a fact that the travelling twin is younger than the earth twin” (or vice versa) is completely false. There is nothing that is absolutely true about the relative ages of the twins until they are at rest with respect to one another.

## Lorentz Expansion!

It depends on who thinks the guns fired simultaneously.

If, as seems most likely, the question means that those Earth-based guns were synchronized by someone on Earth with them, then the distance between the two holes, as measured by any observer stationary with respect to the Earth, is (and remains) exactly equal to the distance between the two guns. Since the plate appears contracted to these Earth-based observers, if it was marked with units of length in its own rest frame those markings would appear closer together to the Earth-based observers and so there would be more of them between the holes than the number of length units measured on Earth.
In other words the distance between the holes would appear to be
greater from the point of view of someone travelling with the plate.

It may seem puzzling that this happens despite the fact that from the traveller’s point of view the distance between the guns is “length contracted” and so appears to be less than that measured between them on Earth.

The puzzle is resolved by the fact that from the point of view of the traveller the guns did NOT fire simultaneously. The one making the front hole appears to the traveller to have gone off earlier. (And it is a worthwhile exercise for anyone seeking to learn about relativity to work through the calculation needed to show that the delay is by exactly the right amount for the forward movement of the plate to create the observed bigger distance between the holes).

Alternatively, if the traveller thinks the guns went off simultaneously, then the Earth-based observers think there is a delay. (And again, working out the details is a worthwhile exercise for any beginning student of the subject.)

P.S. The question of whether or not a length or object is “length contracted” does not really make sense without any mention of which observer is doing the measurement.