## Friday, 14 August 2009

### Fun with Special Relativity This is where I surprise everyone by saying something nice about Einstein's Special Theory of Relativity for a change. Considered as a piece of abstract geometry, special relativity (aka "SR" or "STR") is prettier than even some of its proponents give it credit for. The problems only kick in when you realise that the basic principles and geometry of SR considered as physics don't correspond well to the rules that real, physical observers and objects appear to follow in real life.

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) ]
= SQRT[ (1-0.8) / (1+0.8) ]
= SQRT[ 0.2/1.8 ] = SQRT[ 1/9 ]

=
1/3
, 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.

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)
= 1/1+0.8 = 1/1.8
, and the associated SR Lorentz-factor time-dilation redshift is given by
freq'/freq = SQRT[ 1 - vv/cc ]
= SQRT[ 1 - (0.8)² ] = SQRT[ 1 - 0.64 ] = SQRT[ 0.36 ]
= 0.6
Multiplying 0.6 by 1/1.8 gives
0.6/1.8 = 6/18
= 1/3

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)/c
= 1 - 0.8 = 0.2

We 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.
The formula for this is
freq'/freq = 1/SQRT[ 1 - vv/cc ]
= 1/0.6 = 5/3
Multiplying these two components together gives a final prediction for the apparent frequency shift of
0.2× (1/0.6) = 0.2/0.6 = 2/6
= 1/3

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)
=1/1.5 = 2/3
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.
The second propagation shift is calculated assuming fixed lightspeed for the emitting frame, giving
freq'/freq = (c-v)/c
=1 - 0.5/1 = 0.5/1 = 1/2
The end result of multiplying both of these propagation shift stages together is then
2/3 × 1/2
= 1/3
Again, 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. Anonymous said...

Your blog entry expresses several misconceptions and misunderstandings. It is an empirical fact that the speed of light has the same value in terms of any standard inertial coordinate system, from which it follows that relatively moving systems of inertial coordinates are related by Lorentz transformations. It is also an empirical fact that a material particle’s velocity (and hence its “time dilation”) is not an absolute quantity, but rather depends on the chosen system of coordinates. This is a fundamental tenet of relativity, so your criticism of what “some physicists say” is fallacious. We do indeed know that a particle moving with a certain velocity in terms of a certain system of inertial coordinates is moving with that velocity in terms of those coordinates. To suggest otherwise (as you do) is quite silly. It is also true that the particle is moving with completely different velocities (and therefore exhibits completely different time dilation) with respect to other inertial coordinate systems. These facts do not conflict with each other in any way. Time dilation, like velocity, is not an absolute quantity, it is a relative quantity. If you don’t understand this (which you obviously don’t), then you don’t understand the first thing about special relativity.

By the way, your claim that you can compute the Doppler shift “using the simpler pre-SR Doppler relationships without invoking time dilation at all” is obviously specious, because you make use of the SR velocity composition relation, which of course entails all the time dilation and length contraction features implicit in the Lorentz transformation.

ErkDemon (Eric Baird) said...

Hang on ... you're saying that the blog post "expresses several misconceptions and misunderstandings" ... even though its math is correct for SR, and you can't find any technical mistakes in it? :)

This is physics! (and math!). If I take an existing model (in this case SR), and can generate the appropriate answers for its physical predictions right every time, then I've already won. If you disapprove when I deliberately try to calculate the results of a theory in an unconventional way, and think that this means that I "obviously don't understand the first thing about" the theory in question, then I'd suggest that you don't consider theoretical physics (or mathematics) as a career.

This is part of what theoretical physicists and research mathematicians Are Supposed To Do.
We take existing problems and turn them upside down and inside out to find alternative ways of looking at them that might generate fresh insights and views of a problem that don't always correspond with the boring old cliche'd approaches that people rote-learn from the the textbooks.

If you feel that any unfamiliar approach is automatically wrong, and shows that someone "hasn't understood" a problem, because they're solving it in a slightly different way then ... please don't become a teacher! Anonymous said...

You evidently misread not only my comment but also your own original post. Your post did not consist of legitimately calculating well known results in various ways (which have been pointless but unobjectionable). Rather it consisted of false claims that were explicitly pointed out and falsified in my comment. For example, you said you were calculating the doppler shift using pre-SR physics, but then you used the relativistic velocity addition formula, which falsifies your claim. And so on. I pointed out several specific false statements, and explained WHY they were false. In response, you simply ruffel your feathers and act agreived. I can't help but think that if you had any substantive rebuttal to make, you would have made it.

ErkDemon (Eric Baird) said...

Anonymous, my "feathers were ruffled" because you'd confidently announced that
"Your blog entry expresses several misconceptions and misunderstandings"
, and yet you couldn't successfully identify a single one. You were the one here making claims that couldn't be supported, and since your claims were about me, I was entitled to be a bit narked.

In your second comment, you say that you "pointed out several specific false statements, and explained WHY they were false."

Again, not true. You made a number of personal interpretations of what you decided was my position, you got some of those conveniently wrong, and you critiqued those misinterpreted positions, rather than what I'd actually written. You didn't really find any faults in the actual post. The math was right. The facts were right. The problems that you identified were with your extra layer of dodgy extrapolation, not with the raw data.

For instance, I thought that I'd argued quite strongly against the bad idea that some people have that SR's velocity-based time dilation effect was absolute and isolatable for simple straightline motion. I argued the opposite view:

"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."

If that wasn't clear enough, I also directed people to a passage in Spacetime Physics that was broadly in agreement with what I'd just said, as evidence that this wasn't just my personal view, but was arguably a legitimate mainstream view of special relativity.

You then wrote:
"Time dilation, like velocity, is not an absolute quantity, it is a relative quantity. If you don’t understand this (which you obviously don’t), then you don’t understand the first thing about special relativity."

In other words, you just slagged me off for "obviously" holding a view that's the opposite of the one that I actually hold, that I'd just tried to explain, and had provided a supporting reference for ... and you then declared that this meant that I didn't understand the first thing about the subject!

Now, Anon, is it any wonder that I find it frustrating dealing with you? Because you seem to be continually making stuff up to win arguments. That's a Bad Thing.

Another from the same set of comments:
Anon: "For example, you said you were calculating the doppler shift using pre-SR physics, but then you used the relativistic velocity addition formula, which falsifies your claim.”

What I actually wrote was:

“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!”

, which is all quite correct.
We can.

I didn't "claim" in the post to be rederiving the SR equations using only "pre-SR physics", because, as I pointed out

“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 I'd explained that there was more to it than just the Doppler relationships. It was explicitly an SR exercise. The title of the article was "Fun with Special Relativity". The point that I was trying to make was that even if we believe that the SR equations are correct, we aren't required to believe in the physical existence of velocity-dependent time dilation to explain SR's physical predictions in these situations.

My (appropriate) use of the SR VAF didn't "falsify my claim", because I wasn't making the claim that you attributed to me in the first place. You invented that element yourself.

You keep projecting your own expectations onto a situation, and changing the facts to fit your conclusion. If the facts in question are about another person, then that person is entitled to get annoyed with you for doing it. Anonymous said...

I’m not projecting my expectation, I’m just commenting on what you’ve written. Let’s start with the first example you chose to illustrate my erroneous projection. You stated in a previous message “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!” I commented that this exclamatory claim is incorrect, and explained why: The relativistic velocity composition formula entails relativistic time dilation, so by invoking the former, you are invoking the latter. To confirm that I correctly interpreted your claims, you helpfully re-stated it in your reply, when you said “The point that I was trying to make was that even if we believe that the SR equations are correct, we aren't required to believe in the physical existence of velocity-dependent time dilation to explain SR's physical predictions in these situations.” You see, this is precisely the point that I “projected” you to be making, i.e., you are claiming that physics could dispense with time dilation, if only the world could free itself from the irrational dogma of special relativity. I am pointing out that you are mistaken. Of course, one can always make an arbitrary metaphysical stipulation that one particular time coordinate is “true” time, so there is no “true” time dilation, but this doesn’t change the empirical fact that the time coordinates of two relatively moving systems of inertial coordinates are mutually time-dilated. Hence I “projected” your position accurately, and correctly explained why your position is wrong.

Let’s take your other example. You wrote “For instance, I thought that I'd argued quite strongly against the bad idea that some people have that SR's velocity-based time dilation effect was absolute and isolatable for simple straightline motion. I argued the opposite view...” As usual, there are multiple fallacies in your position. First, you claim that “some people” say that time dilation is absolute... but no one claims any such thing, any more than they claim that velocity is absolute, since time dilation depends on velocity. Second, you say time dilation is not isolatable for simple straight line motion. Well, that is simply incorrect, just as it is incorrect to say that velocity is not isolatable for simple straight line motion. You conflate absolute with isolatable, but that is a fallacy. Velocity is not absolute, but it is isolatable. Same for time dilation. These are purely kinematic attributes of the relationships between two relatively moving systems of inertial space-time coordinates. I say again, if you don’t understand this, you don’t know the first thing about special relativity.