# Special Relativity vs Alternatives

1. Jan 8, 2005

### yogi

In deference to ElectroPhysics request to end posting to the Twin Paradox thread, those who wish to make further comments on this passionately discussed subject may make their views known here. I will not answer anything posted on the Electrophysics thread, but will be happy to respond to all courteous comments made here. One more comment. The fact that I have reservations about frame reciprocity does not mean I don't understand what you are saying, nor does it mean I don't understand SR.

2. Jan 8, 2005

### JesseM

I'll repost what I said in my last post on that thread:
By the way yogi, I just posted this thread with some illustrations of a scenario like this, with two rulers moving in parallel next to each other, and clocks placed at regular intervals along each ruler, sychronized within the ruler's reference frame. You may find it helpful to look at the diagrams and see exactly how things work out consistently in this scenario.

It would be pretty easy to extend this sort of thing to the "triplet paradox" you were describing earlier on this thread. Imagine each triplet is riding along a ruler, and all the rulers are moving at constant velocity relative to one another. We can say triplet A corresponds to the earthbound triplet in your scenario, triplet B corresponds to the one travelling away from earth, and triplet C corresponds to the one travelling towards the earth. At the initial moment when A and B are right next to each other, their clocks read the same time; then we can also specify that at the moment B and C pass next to each other, their clocks must also read the same time. In this case, when C passes A, C's clock will be behind A's clock. In A's frame, the clocks of both B and C are ticking slower than his own, by the same amount; but both B and C see A's clocks ticking slower then their own clocks, also by the same amount. However, B sees all the clocks along A's ruler to be out-of-sync, just like in my illustrations--he sees clocks closer to where A is sitting being behind clocks farther from where A is sitting. On the other hand, C also sees the clocks along A's ruler being out-of-sync, except he sees clocks closer to where A is sitting being ahead of clocks farther from where A is sitting. This is the explanation for why there is no inconsistency in the fact that C's clock will be behind A's clock when they finally pass, despite the fact that both B and C saw A's clocks running slower throughout the journey. If you're still having trouble seeing this, even after looking at the illustrations in my other post, I can make up a more detailed scenario and calculate some numbers explicitly to show how this will work.

3. Jan 8, 2005

### yogi

JesseM Thanks - I had briefly read your posts in the other thread and will now take the time to go over them in depth.

4. Jan 9, 2005

### pervect

Staff Emeritus
The main comment I have is that special relativity is actually a lot easier to deal with than non-isotropic coordinate systems. Sometimes, unfortunately, one does have to deal with non-isotropic coodinate systems, but they aren't particularly fun to deal with.

Really the only thing that one needs to know to understand SR is that clocks can only be unambiguously synchornized when they are at the same point in space-time. When they are at different points, different observers will synchronize clocks in different ways.

Exactly why some people seem to have trouble with this simple viewpoint isn't clear to me.

For those diehards who are not content with my statement, and would like to consider or explore the complexities of a non-isotropic coordinate system, I'll repeat my earlier observation and try and make it a bit more clear and pointed this time around.

To make the point very very clear, we are dealing here with arbitrary clock synchronizations. I'm saying that anyone who is prpeared to take non-isotropic coordinate systems seriously (or in general any alternate clock synchronization scheme) should be prepared to defend in detail the notion that the speed of an airplane measured by measuring it's takeoff and landing in different timezones (PST and CST) and subtracting the clock readings is a perfectly valid measurement of the airplane's speed, and should also be willing to work out the consequences to the calcuation of the airplane's momentum, the conservation of momentum, and all of the other effects that happen due to the dynamics of this unusual choice of coordinate systems.

5. Jan 9, 2005

### yogi

JesseM - That is a well worked out depiction of how things behave according to the time slippage formula vx/c^2. The observed time difference in relatively moving frames is 1) a function of their different rates and 2) the time slippage effect illustrated by your clocks and rulers. Every clock in the other frame is seen to tick slower whereas time observed on a succession of passing clocks is faster.

While these formulations are correct, they are observational. The addition of a third traveler to the twins was of course only to avoid the commonly relied upon explanation of acceleration as a means of distinguishing the frames. My own view of how the aging arises is that the start and finish at the halfway (turn around point) of the journey are simply two spacetime events in each frame - in other words no time is added to any clocks anywhere at any time in any abrupt manner - in reaching the turn around point, the travelers clock has for some physical reason, logged less time - and upon turning around and returning at the same velocity, the total age difference will be double what the traveler lost on the outward leg for the same physical reason. So even though SR gives you the correct end game age difference based upon observational reasons - in denying the existence of a preferred frame it consequently requires some means for distinguishing the frames to avoid the "so called" paradox.

Last edited: Jan 9, 2005
6. Jan 9, 2005

### yogi

Pervect - no question re the ease of dealing with isotropic frames. In the theories that I am familiar with, the analysis invariably starts with an observer in an isotropic frame. The second frame is then considered (or calculated from some experiment or premise) to be an over and back isotropic frame. Only SR requires both frames to be one way isotropic.

7. Jan 9, 2005

### wisp

There is no issue with SR mathematically showing that moving clocks run slow. However, the proof for this is one-sided. There are lots of examples of moving clocks running slow (including muons, etc) relative to clocks on the surface of the earth, but there is no real evidence to prove that an observer moving fast passed the earth will see earth based clocks running slow.

If you consider an ether theory that supports relativistic effects, it shows that an observer moving fast passed the earth will see earth clocks run faster.
Experimental proof is needed to resolve this issue.
A simple test would be for the international space station to measure a frequency from a point on the earth's surface. As the space station passed overhead, according to SR it should see a lower frequency from the source, by a factor of gamma (transverse Doppler effect for moving observer). However, if it detects a higher frequency, by a factor of gamma, then this means also that earth clocks relative to the faster moving observer will run faster not slower.

Last edited: Jan 9, 2005
8. Jan 9, 2005

### Hurkyl

Staff Emeritus
One important issue here is this: if Special Relativity is right in one coordinate chart, then Special Relativity is correct in all of the reference frames it defines.

Since those advocating alternative theories usually start off by saying something like "Sure, SR has it right in one reference frame, but it's got all the other ones wrong", I have good reason to be immediately skeptical of their grasp of the problem.

Furthermore, those proposing alternative theories tend to be extremely vague about the process of measurement, which is unfortunate because it's a central issue.

As far as I have been able to tell, at least from a mathematical point of view, "alternative theories" are nothing more than restricting one's self to some class of coordinate charts that is not the inertial reference frames defined by Special Relativity -- there is no difference in physical content.

Let me restate this so it can sink in -- if two theories entirely agree in one coordinate chart, then there is no difference in the physical content of the two theories.

And, a very underappreciated fact about SR is that it is fully capable of dealing with other coordinate charts. (In the same way that one can study rotating, or otherwise accelerating reference frames in Newtonian kinematics) So, it's unclear that these "alternative theories" are anything that can't be done entirely within SR.

9. Jan 9, 2005

### wisp

This is true. But I don't believe that a true relativistic ether based theory would entirely agree with SR in earth-based inertial frames. There would be very close agreement, because the speed of the earth relative to the ether is very small compared to c, but at higher relative ether speeds the theories would part company.

Only by carrying out experiments in faster frames can the theories be properly tested.

10. Jan 9, 2005

### Staff: Mentor

You're saying the Earth's motion through the ether isn't fast enough to test ether theories? Seriously? You're entire objection to Relativity is based on the hope that oneday we'll find that FTL travel really is possible?

Last edited: Jan 9, 2005
11. Jan 9, 2005

### Chronos

Aether? Has aether been discovered and I missed it?

12. Jan 9, 2005

### Hurkyl

Staff Emeritus
The more important question, though, is if SR and such a theory agree near earth, when very high speeds are involved. (We already deal with speeds very near the speed of light)

We already know, from GR, that SR is only valid locally. The universe is modelled as a particular type of differentiable manifold whose very definition is: "something which looks locally like Minowski geometry". (Minowski geometry is the geometry of SR). Similarly, the physics of GR are assumed to be locally that of SR.

In other words, if you have a theory which says that SR is locally correct, then your theory must agree with GR.

*sigh* Actually, if you replace "SR" with "Minowski geometry" and "GR" with "differential manifold", then I'm very confident in the above statement, but my differential geometry isn't strong enough to be entirely sure about what I actually wrote.

13. Jan 9, 2005

### yogi

Hurkyl - I am obviously missing something that you are very certain about - but why is it impossible to start with one frame that everyone could agree upon as a true isotroptic inertial frame (say we do this by running around the universe to find a 0 gravity field with isotroptic CBR.. whatever. We will call this the S frame. We do not introduce any postulates to get this frame isotropic - we do it experimentally making sure the one way velocity of light is the same in every direction. From S, we look at another frame S' which is attached to a high speed particle (P') and we measure its lifetime to be extended. We conclude from this experiment that clocks are running slower in S' relative to clocks in S vis a vis Gamma. The S frame also contains the same type of particle (P) which decays in accordance with its characteristic half life as per clocks in S. Why, w/o further postulation, are we not logically entitled to make the following inquiry

Whether the observer on S' will see the lifetime of the particle P in S extended and conclude as per SR that S clocks run slow relative to his own clock, or as wisp suggests, the S' observer will measure the lifetime of P as shorter and conclude that his own clock is running slower.

In other words, as I see it, there can be a useful transform from S to S' via SR but what dictates why the transform needs to be reciprocal as per your statement:

"One important issue here is this: if Special Relativity is right in one coordinate chart, then Special Relativity is correct in all of the reference frames it defines."

14. Jan 9, 2005

### Hurkyl

Staff Emeritus
Something that just struck me as a possible source of confusion is this: reference frames are a non-physical -- they're man-made constructions.

The reason I say that SR is right in every frame it defines is this: measurements in those frames can be defined as the result of a well-defined physical experiment. (For example, attaching a laser to a clock, firing the laser at an object, and measuring how long before the laser light returns)

If SR is correct in just one reference frame (call it S), that means it correctly predicts the outcome of any physical experiments.

However, all measurements in the frame S' are the result of a physical experiment.

Therefore, the behavior of S' must be exactly what SR predicts.

From here, you can mathematically prove things (by doing the analysis in S!) like if a beam of light travels from point A to point B, that the experiments performed to do measurements in S' will determine that the change in distance over the change in time will be c.

The key point here is that, in SR, is that its reference frames are very well defined -- you can precisely outline an experiment that can be used to compute the coordinates of an event in a reference frame.

I usually don't see such precision in the presentation of aether theories. Someone could present a thought experiment, give me the exact space-time coordinates, in S, of every event that occurs in the thought experiment, but I would have no idea how to determine anything relative to S'.

Now, this is just geometry, and doesn't touch on the facts that make SR an interesting theory, which is that every law of physics is described by the exact same equation in both S and S'. (Where S' is an inertial reference frame as defined by SR)

15. Jan 9, 2005

### yogi

Hurkyl: "Something that just struck me as a possible source of ...
The reason I say that SR is right in every frame it defines is this: measurements in those frames can be defined as the result of a well-defined physical experiment. (For example, attaching a laser to a clock, firing the laser at an object, and measuring how long before the laser light returns"

ok - that gives you the average over and back velocity - but isn't it possible that the one way velocity in one frame may be isotropic but only the over and back velocity in a relatively moving frame may will be isotropic - in other words in the example above with pions, we defined an S frame that would satisfy SR as well as all of the alternatives - and we have many experiments that all verify that the over and back velocity in the earth frame is isotroptic - but hazy data on one way isotrophy. Therefore can we really conclude that because we have a good over and back experiment in one frame that light in a relatively moving frame will have one way isotrophy?

To me this is an extrapolation that is of course consistent with SR, but isn't that a bootstrap argument in light of the theory under review.

Last edited: Jan 9, 2005
16. Jan 9, 2005

### JesseM

That should be gamma*v*x/c^2. I wasn't actually using a specific time slippage formula, though, I was just using the Lorentz transformation equations:

x' = gamma*(x - v*t)
t' = gamma*(t - v*x/c^2)

x = gamma*(x' + v*t')
t = gamma*(t' + v*x'/c^2)

where gamma = 1/(1 - v^2/c^2)

Here, x and t represent space and time coordinates in ruler A's frame, x' and t' represent space and time coordinates in ruler B's frame, and v represents ruler B's velocity as seen in ruler A's frame.
The Lorentz transformation equations are not exactly "observational"--they can be derived mathematically from the assumptions that the laws of physics should work the same way in every observer's frame, and that light should be defined to move at c in each frame (note that this is a starting assumption rather than an empirical result).

The only place where observations come in is in checking that all the laws of physics remain unchanged under this type of coordinate transformation. Formally, this is equivalent to saying that all the equations of physics have the mathematical property of "Lorentz-invariance". To explain exactly what this means, it may be a bit easier to first explain the concept of "Galilie-invariance" since this is a little simpler mathematically. Here is the Galilei transformation for transforming between coordinates of different inertial reference frames in Newtonian physics:

x'=x - vt
y'=y
z'=z
t'=t

x=x' + vt'
y=y'
z=z'
t=t'

To say a certain physical equation is "Galilei-invariant" just means the form of the equation is unchanged if you make these substitutions. For example, suppose at time t you have a mass m1 at position (x1, y1, z1) and another mass m2 at position (x2, y2, z2) in your reference frame. Then the Newtonian equation for the gravitational force between them would be:

F = Gm1m2/[(x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2]

Now, suppose we want to transform into a new coordinate system moving at
velocity v with respect to the first one. In this coordinate system, at
time t' the mass m1 has coordinates (x1', y1', z1') and the mass m2 has
coordinates (x2', y2', z2'). Using the Galilei transformations, we can
figure how the force would look in this new coordinate system, by
substituting in x1 = x1' + vt', x2 = x2' + vt', y1 = y1', y2 = y2', and
so forth. With these substitutions, the above equation becomes:

F = Gm1m2/[(x1' + vt' - (x2' + vt'))^2 + (y1' - y2')^2 + (z1' - z2')^2]

and you can see that this simplifies to:

F = Gm1m2/[(x1' - x2')^2 + (y1' - y2')^2 + (z1' - z2')^2]

In other words, the equation has exactly the same form in both coordinate systems. This is what it means to be "Galilei invariant". More generally, if you have any physical equation which computes some quantity (say, force) as a function of various space and time coordinates, like f(x,y,z,t) [of course it may have more than one of each coordinate, like the x1 and x2 above, and it may be a function of additional variables as well, like m1 and m2 above] then for this equation to be "Galilei invariant", it must satisfy:

f(x'+vt,y',z',t') = f(x',y',z',t')

...for all possible values of v.

From this, it's pretty simple to see what it must mean for a given physical equation to be "Lorentz invariant" as well. Here are the Lorentz transformation equations in three dimensions:

x'=gamma*(x - vt)
y'=y
z'=z
t'=gamma*(t - v*x/c^2)

x=gamma*(x' + v*t')
y=y'
z=z'
t=gamma*(t' + v*x'/c^2)

So, if you have some physical equation f(x,y,z,t), then for it to be "Lorentz-invariant" it just must have the following property:

f(gamma*(x'+v*t'),y',z',gamma*(t'+v*x'/c^2)) = f(x',y',z',t')

...for every v<c.

This is just a mathematical property of a given equation or set of equations, it is simply a matter of calculation to check if the equation satisfies it (the equation for Newtonian gravity would not have this property, so it would not be Lorentz-invariant). Maxwell's laws have this property of Lorentz-invariance, as do all the most fundamental laws currently known (such as the laws of quantum field theory).

As long as all the laws of physics are Lorentz-invariant, this also means that any physical ruler must appear to shrink as it moves relative to a given observer, and every physical clock must appear to slow down, since the laws which determine the length of the ruler (such as the laws determining the spacing of atoms in a solid) and the laws which determine the length of a clock-tick (such as the laws governing the rate that quartz crystals oscillate) are themselves Lorentz-invariant, and thus must have the same form when you transform to different Lorentzian reference frames.

Furthermore, as long as all the laws of physics are Lorentz-invariant, then even if you believe there is a single absolute truth about which reference frame has the "correct" definition of length, time, and simultaneity (ie even if you believe in Absolute Space and Absolute Time), there can be no possible experiment that will tell you which frame this is. So although relativity does not say you can't believe in some sort of "metaphysically preferred reference frame", it does say there is no physically preferred reference frame.
Well, as long as the equations governing the triplets have this mathematical property of Lorentz-invariance, then it is inevitable that if each triplet assigns coordinates to events using a system of rulers and clocks at rest relative to himself, with the clocks synchronized based on the assumption that light will travel at c in all directions relative to himself, then the Lorentz transformation equations will correctly translate between different triplets' coordinate systems.

Last edited: Jan 10, 2005
17. Jan 9, 2005

### yogi

hurkyl: "I usually don't see such precision in the presentation of aether theories. Someone could present a thought experiment, give me the exact space-time coordinates, in S, of every event that occurs in the thought experiment, but I would have no idea how to determine anything relative to S'."

I would agree that the coordinates of the events are murky in alternative theories. Because they lack the symmetry of SR each thought experiment must be worked through to see where it will end. What I was trying to do in the previous thread was to set up the starting event to be consistent with SR method of synchronization, letting the chips fall where they may upon arrival at the turnaround point. In a true sense, any alternative must borrow heavily upon SR and its many experimental successes.

18. Jan 9, 2005

### yogi

JesseM - one of the issues raised by alternatives is Lorentz Invarience, and to what it is to be applied. Almost all alternatives embrace galileon invariance - and some such as LET rely upon LI but then proceed from the notion that the ether is a preferred frame. Other theories reject LI. Einstein was firmly committed to the proposition that it was impossible to measure motion wrt space as was Poincare and others. Accordingly, it was a requirement of his theory of SR that all inertial frames be indistinguishable. The CBR anisotrophy raises questions that have not been fully answered. Moreover, the notion of a preferred location defined by a gravitational field is a rather new idea - which at this point cannot be dismissed as non-meritorious. The inertial transforms derived by Selleri have been applied against all presently available SR experiments and they are consistent therewith. Of course - one free space experiment to establish one-way isotropy of light in an inertial frame should end all dissenting speculation in favor of SR.

19. Jan 9, 2005

### Hurkyl

Staff Emeritus
(I'm going to assume that, in the first frame, the rate of a moving clock are divided by gamma)

Yes. For instance, we could use the inertial frames of Gallilean Relativity instead of Special Relativity.

In fact, there is a great variety in the ways one could go about selecting reference frames that vary from the mundane to the extremely bizarre.

But, an inertial traveller will always able to select a coordinate chart in which he is at rest and the one-way speed of light is isotropic -- one of the ones defined by SR.

Incidentally, the same argument applies for sound waves too. If you have a reference frame in which the speed of sound is one-way isotropic throughout your medium, then any observer travelling at a constant speed can select a rest frame in which the speed of sound has one-way isotropy. (Though, the physics of this one would be a little weird -- the relativity postulate would fail, because the laws of physics are not invariant under speed-of-sound Lorentz transforms... though sound-related formulae might take a nicer form)

20. Jan 11, 2005

### yogi

Hurkyl - going back to one of the examples above and applying it to the twin's aging - let us again start with a traveling twin which we will represent by a particle P' that decays 10 usec after being created. P' is awaiting its creation on earth in a lab at rest wrt to earth and alpha which are both at rest in the S frame. There is a clock on alpha and a clock attached to p' and one on earth ....we sync the earth alpha clocks using the method of sending a light flash in all directions from a point midway between earth and alpha as measured in the S frame. When the light flash arrives, both clocks are started and our twin (the particle P') is created and launched toward alpha.

At this point I would presume you would agree that we can in principle define a precise space-time coordinate for the beginning of the interval in both S' (which is the frame at rest wrt to P' as it travels toward alpha) and the S frame. P' travels at v relative to earth (near c) velocity until it reaches alpha at which point P disintegrates. We will assume alpha is 5 LY from earth, so when the decay (arrival at alpha) occurs, 10 years (+ a little more since P' is not a photon) will have passed according to alpha's clock. 10 usec have been logged on the clock which escorts P' ...So we now have defined the coordinates required to compute the interval in both frames. Since the interval is invariant during transformation, then in the S frame the [square of the time (5+ years) - the square of the ct distance (5 light years)] must equal the interval in the S' frame, but since P' is at rest wrt to S' the interval in the S' frame is simply the proper time [10 usec] that corresponds to the lifetime of P'

So we have a well defined event to mark the spacetime coordinate of the beginning of the interval in both frames, and because we have in advance specified the proper time in S' as 10 usec, we have a well defined terminating event for the interval in both in frames. The fact that we have clocks in S and S' that have logged different local times at the end of the experiment is consequent to the invariance of the interval.

Before going further - do you have any problems with how the spacetime coordinates of the events are defined?

Last edited: Jan 11, 2005