Anyhow, here's some of the pretty stuff:
It's traditional to explain Einstein's special theory of relativity as a theory that says that the speed of light is fixed (globally) in our own frame of reference, and that objects moving with respect to our frame are time-dilated and length-contracted, by the famous Lorentz factor.
And that characterisation certainly generated the appropriate predictions for special relativity, just as it did for Lorentzian Ether Theory ("LET"). But we can't verify that this time-dilation effect is physically real in cases where SR applies the principle of relativity (i.e. cases that only involve simple uniform linear motion). Thanks to its application of Lorentz-factor relationships, Special Relativity doesn't allow us to physically identify the frame that lightspeed is supposed to be constant in. When we make proper, context-appropriate calculations within SR, we have the choice of assuming that lightspeed is globally constant in our frame, or in the frame of the object we're watching, or in the frame of anybody else who has a legal inertial frame – it's usually a sensible choice to use our own frame as the reference, but really, it it doesn't matter which one we pick, and sometimes the math simplifies if we use someone else's frame as our reference (as Einstein did in section 7 of his 1905 paper).
Some people who've learnt special relativity through the usual educational sources have expressed a certain amount of disbelief (putting it mildly) when I mention that SR allows observers a free choice of inertial reference frame, so let's try a few examples, to get a feel of how special relativity really works when we step away from the older "LET" descriptions that spawned it.
Some Mathy Bits:
1: Physical prediction
Let's suppose that an object is receding from us at at a velocity of four-fifths of the speed of light, v = 0.8c
Special relativity predicts that the frequency shift that we'll see is given by
frequency(seen)/frequency(original) = SQRT[ (c-v) / (c+v) ], so according to SR, we should see the object's signals to have one third of their original frequency. This is special relativity's physical prediction. The object looks to us, superficially, as if it's ageing at one third of its normal rate, but we have a certain amount of freedom over how we choose to interpret this result.
= SQRT[ (1-0.8) / (1+0.8) ]
= SQRT[ 0.2/1.8 ] = SQRT[ 1/9 ]
2: "Motion plus time dilation"
It's usual to break this physical SR prediction into two notional components, a component due to more traditional "propagation-based" Doppler effects, calculated by assuming that lightspeed's globally constant in somebody's frame, and an additional "Lorentz factor" time dilation component based on how fast the object is moving with respect to that frame.
The "simple" recession Doppler shift that we'd calculate for v = 0.8c by assuming that lightspeed was fixed in our own frame would be
frequency(seen) / frequency(original) = c/(c+v), and the associated SR Lorentz-factor time-dilation redshift is given by
= 1/1+0.8 = 1/1.8
freq'/freq = SQRT[ 1 - vv/cc ]Multiplying 0.6 by 1/1.8 gives
= SQRT[ 1 - (0.8)² ] = SQRT[ 1 - 0.64 ] = SQRT[ 0.36 ]
0.6/1.8 = 6/18Same answer.
3: Different frame
Or, we can do it by assuming that the selected emitter's frame is the universal reference.
This gives a different propagation Doppler shift result, of
freq'/freq = (c-v)/cWe then assume that because we're time dilated (because we're moving w.r.t. the reference frame), and that because our clocks are slow, we're seeing everything to be Lorentz-blueshifted, and appearing to age faster than we'd otherwise expect, by the Lorentz factor.
= 1 - 0.8 = 0.2
The formula for this is
freq'/freq = 1/SQRT[ 1 - vv/cc ]Multiplying these two components together gives a final prediction for the apparent frequency shift of
= 1/0.6 = 5/3
0.2× (1/0.6) = 0.2/0.6 = 2/6Same answer.
So although you sometimes see physicists saying that thanks to special relativity, we know that the speed of light is globally fixed in our own frame, and we know that particles moving at constant speed down an accelerator tube are time-dilated, actually we don't. In the best-case scenario, in which we assume that SR's physical predictions are actually correct, the theory says that we're entitled to assume these things as interpretations of the data, but according to the math of special relativity, if we stick to cases in which SR is able to obey the principle of relativity, it's physically impossible to demonstrate which frame light "really" propagates in, or to prove whether an inertially-moving body is "really" time-dilated or not. It's interpretative. Regardless of whether we decide that we're moving and time-dilated or they are, the final physical predictions are precisely the same, either way. And that's the clever feature that we get by incorporating a Lorentz factor, that George Francis Fitzgerald originally spotted back in the Nineteenth Century, that Hendrik Antoon Lorentz also noticed, and that Albert Einstein then picked up on.
4: Other frames, compound shifts, no time dilation
But we're not just limited to a choice between these two reference frames: we can use any SR-legal inertial reference frame for the theory's calculations and still get the same answer.
Let's try a more ambitious example, and select a reference-frame exactly intermediate to our frame and that of the object that we're viewing. In this description, both of us are said to be moving by precisely the same amount, and could be said to be time-dilated by the same amount ... so there's no relative time dilation at all between us and the watched object. We can then go ahead and calculate the expected frequency-shift in two stages just by using the simpler pre-SR Doppler relationships, and get exactly the same answer without invoking time dilation at all!
The "wrinkle" in these calculations is that velocities under special relativity don't add and subtract like "normal" numbers (thanks to the SR "velocity addition" formula), so if we divide our recession velocity of 0.8c into two equal parts, we don't get (0.4c+ 0.4c), but (0.5c+0.5c)
(under SR, 0.5c+0.5c=0.8c – if you don't believe me, look up the formula and try it)
So, back to our final example. The receding object throws light into the intermediate reference frame while moving at 0.5c. The Doppler formula for this assumes "fixed-c" for the receiver, giving
freq'/freq = c/(c+v)Having been received in the intermediate frame with a redshift of f'/f = 66.66'%, the signal is then forwarded on to us. We're moving away from the signal so it's another recession redshift.
=1/1.5 = 2/3
The second propagation shift is calculated assuming fixed lightspeed for the emitting frame, giving
freq'/freq = (c-v)/cThe end result of multiplying both of these propagation shift stages together is then
=1 - 0.5/1 = 0.5/1 = 1/2
2/3 × 1/2Again, exactly the same result.
No matter which SR-legal inertial frame we use to peg lightspeed to, special relativity insists on generating precisely the same physical results, and this is the same for frequency, aberration, apparent changes in length, everything.
So when particle physicists say that thanks to special relativity we know for a physical fact that lightspeed is really fixed in our own frame, and that objects moving w.r.t. us are really time-dilated ... I'm sorry, but we don't. We really, really don't. We can't. If you don't trust the math and need to see it spelt out in black and white in print, try Box 3-4 of Taylor and Wheeler's "Spacetime Physics", ISBN 0716723271. IF special relativity has the correct relationships, and is the correct description of physics, then the structure of the theory prevents us from being able to make unambiguous measurements of these sorts of things on principle. We can try to test the overall final physical predictions (section 1), and we can choose to describe that prediction by dividing it up into different nominal components, but we can't physically isolate and measure those components individually, because the division is totally arbitrary and unphysical. If the special theory is correct, then there's no possible experiment that could show that an object moving with simple rectilinear motion is really time-dilated.
If you're a particle physicist and you can't accept this, go ask a mathematician.