Comparison of Special Relativity with a Galilean "preferred frame" theory

Discussion in 'Physics & Math' started by James R, Jan 14, 2013.

  1. James R Just this guy, you know? Staff Member

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    In this thread, I aim to outline and discuss some of the differences between Einstein's Special Relativistic theory of space and time and a particular kind of "preferred frame" theory of "absolute" space and time which has been advocated from time to time by posters on sciforums.

    In this post, I will start by listing the basic postulates of each theory under discussion, then deriving some basic results from those postulates.

    Postulates

    Special Relativity (hereinafter referred to as SR)
    SR1. All of the laws of physics take the same form in all inertial frames of reference.
    SR2. The speed of light, c, is the same in all inertial frames of reference.

    "Preferred frame" theory (hereinafter referred to as PF)
    PF1. The laws of mechanics take the same form in all inertial frames of reference.
    PF2. There exists a single "preferred" frame of reference (hereinafter referred to as the "absolute frame"), in which the speed of light has a particular value, c.
    PF3. The measured time interval between any two specific events is the same in all frames of reference.

    Obvious implications of the postulates

    Comparing SR1 and PF1, we see that there is a "preferred" frame in the PF theory for electromagnetic phenomena. It can be shown that Maxwell's equations of electromagnetism predict the propagation of light as an electromagnetic wave only in the absolute frame. Maxwellian light waves are not possible in other frames in the PF theory. In contrast, under SR, the Maxwell equations of electromagnetism take the same form in any inertial frame, posing no problem for the propagation of light. Since the PF theory can't really handle the modern description of light, for the purposes of the discussion that follows I will assume that light is composed of "photons" which act essentially like any other particle (e.g. an electron or proton).

    Comparing SR2 and PF2, we see that in the PF theory the speed of light is frame-dependent. More on this below.

    Looking at PF3, we see that the PF theory has a notion of a universal time. This notion is not present in SR. If fact, it can be shown that both space and time intervals between events in SR are frame-dependent. In the PF theory, time is absolute, but spacial intervals are frame dependent, as we shall see.

    Spacetime transformations

    Here, I list the relevant transformations for one-dimensional motion. Here, I deal with three objects or observers, denoted A, B and C. Let A denote the (one-dimensional) spacetime coordinates of event E as (x,t); B denotes the coordinates of the same event as (x',t'). Let C travel in the positive x direction at speed v, as measured in A's frame. Let C travel in the positive x' direction at speed \(v_{CB}\), as measured in B's frame. We assume that the x and x' axes are parallel.

    SR

    The relevant transformations between A's and B's coordinates are the well-known Lorentz transformations:

    \(x' = \gamma (x - vt)\)
    \(t' = \gamma (t - vx/c^2)\)
    \(\gamma \equiv 1/\sqrt{1-(v/c)^2}\)

    The relevant derivations of these transformations from the postulates can be found in many places on the web, including in a number of threads on sciforums.

    It can also be shown that the speed of C in A's frame is

    \(v_{CA} = \frac{v_{CB} + v}{1+v_{CB} v/c^2}\)

    The inverse transformations are:

    \(x = \gamma (x' + vt')\)
    \(t = \gamma (t' + vx'/c^2)\)

    PF

    The relevant transformations here are the Galilean transformations:

    \(x' = x - vt\)
    \(t' = t\)

    and the speed of C in A's frame is:

    \(v_{CA} = v_{CB} + v\)

    The inverse transformations are, obviously:

    \(x = x' + vt'\)
    \(t = t'\)

    Application - a light in a moving box

    Here, I compare the values of various quantities for a particular example situation.

    Let object A be a stationary observer. In the SR situation, A's can be any inertial frame. In the PF situation, we assume that A is stationary in the absolute frame. Let object B be a cubical box of side length 2L, as measured when the box is at rest (i.e. in the box's rest frame). The box has a light at the centre that can emit a flash. Remember that the box, B, is taken to be travelling at speed v in A's frame. This is a relative speed in the SR case, and we take it to be an absolute speed in the PF case.

    At a particular time, which we designate as time zero in all inertial frames, the light emits photons in all directions. We consider the particular cases of two photons emitted in the direction of travel of the box (positive x) in frame A, and in the opposite direction (negative x).

    We consider the spacetime coordinates of several events. E1 is the emission event. E2 is the event that the photon emitted in the positive x direction hits the "front" wall of the box. E3 is the event that the photon emitted in the negative x direction hits the "rear" wall of the box.

    Consider first the rest frame of the box. We calculate the spacetime coordinates of the three events as follows. First, we examine the situation in frame B.

    SR
    In frame B, the photon travels at speed c (according to postulate SR2). The front wall of the box is at x'=L, and the rear wall is at x'=-L. In this frame (the rest frame of the box), the walls never move. So, the time taken by the photon to reach either wall (after emission) is t' = L/c. The spacetime coordinates are therefore:

    \(E1: (x',t') = (0,0)\)
    \(E2: (x',t') = (L, L/c)\)
    \(E3: (x',t') = (-L, L/c)\)

    Notice that in frame B, events E2 and E3 have the same time coordinate, which means they occur simultaneously.

    PF
    In frame B, the box as a whole is moving with an "absolute" speed v. In frame A (the absolute frame), the speed of light is c, so in frame B the speed of the "forward-moving" photon is c - v, and the speed of the "backwards-moving" photon is c + v. These adjustments are necessary to account for the fact that the box is moving in the absolute frame. They take into account the rule for adding velocities that is given above, and postulate PF2.

    In frame B, the walls do not move, so the time taken to reach the front wall is different from the time taken to reach the rear wall. The spacetime coordinates of the events are:

    \(E1: (x',t') = (0,0)\)
    \(E2: (x',t') = (L, \frac{L}{c-v})\)
    \(E3: (x',t') = (-L, \frac{L}{c+v})\)

    Notice that event E3 has a smaller time coordinate than event E2, which means that the photon emitted in the forwards direction reaches the front wall after the photon emitted in the backwards direction has reached the rear wall.
    ---

    Now, let us examine the same three events in frame A.

    SR

    To find the relevant (x,t) coordinates we use the inverse Lorentz transformations, which give:

    \(E1: (x,t) = (0,0)\)
    \(E2: (x,t) = (\gamma L(1 + \frac{v}{c}), \gamma (\frac{L}{c})(1 + \frac{v}{c}))\)
    \(E3: (x,t) = (\gamma(-L(1 - \frac{v}{c})), \gamma (\frac{L}{c})(1 - \frac{v}{c}))\)

    Looking at the spatial coordinates of E2 and E3, we see that in frame A the box moved along in the positive x direction after emission of the light, as expected. The \(\gamma\) factor has done something a little strange with the distance between the front and rear walls of the box, but we'd need to investigate further to see exactly what. In fact, it turns out that in frame A the distance between the walls of the box is no longer 2L, but \(2L/\gamma\), which means that the box appears length-contracted in frame A. I have not explicitly shown this here, but we can do that later in the thread if people want to see the maths.

    The time coordinates of E2 and E3 are also interesting. In frame A, E3 occurs before E2, so the photon is observed to hit the rear wall of the box before the other photon hits the front wall. This makes sense because in frame A the speed of light is the same in both directions (see SR2), but the box is moving along in the x direction, so that the rear wall "catches" the photon before the front wall.

    This is a similar result to the PF result, but something strange has happened. In frame B, the two photons hit the walls simultaneously (E2 and E3 have the same time coordinate), yet in frame A the hits were not simultaneous. We conclude that the notion of what is simultaneous is frame-dependent under SR.

    Let us compare PF. Again, we need the inverse transformations, from which we obtain:

    \(E1: (x,t)=(0,0)\)
    \(E2: (x,t) = (\frac{cL}{c-v}, \frac{L}{c-v})\)
    \(E3: (x,t) = (-\frac{cL}{c+v}, \frac{L}{c+v})\)

    In this case, the time interval between the two photons hitting their respective walls is the same in frame A as it was in frame B - not surprising since this is postulate PF3 of the theory. Notice also that in frame A, the absolute frame, each photon has travelled a distance equal to c multiplied by its time of flight since emission, as we expect from postulate PF2.

    It can also be shown (although I haven't done this here) that the distance between the walls of the box remains 2L in frame A. Hence we conclude that there is no length contraction in the PF theory.

    Conclusions (so far)

    Both theories are based on a small number of postulates. The PF theory suffers in comparison to SR in that it restricts the range of physical laws that are invariant under a change of reference frames. SR suffers in comparison to PF in that it sacrifices absolute time and space; both time and space become frame-dependent in SR. On the other hand, the speed of light is frame-dependent in PF, but constant in SR.

    Both theories are mathematically self-consistent, but they predict different physical outcomes. The only way to decide which is superior, of course, is to conduct actual experiments to test the predictions of the theories against reality. Whenever such tests are done, SR is found to provide an accurate description of reality as it is observed, and PF fails. On the other hand, PF works reasonably well provided that the speeds involved (such as the speed of frame B relative to frame A) are small compared to the speed of light.

    The most glaringly obvious problem with PF is postulate PF2. This is a problem because experiments have explicitly shown that the speed of light is the same in different inertial frames. In particular, the Michelson-Morley experiment shows that the speed of light is apparently unaffected by the orbital motion of the Earth, which is constantly changing its reference frame.
     
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  3. LaurieAG Registered Senior Member

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    Hi James R,

    Your comments are also relevant to the differences that you would expect to see between a space+time metric and a 3D + time metric.

    Which metric would be preferrable for something practical like galactic navigation?
     
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  5. RJBeery Natural Philosopher Valued Senior Member

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    Hi James, this is a great thread topic. One thing that I haven't explored but have been curious about...if all available methods for measuring the speed of light are themselves dependent on the speed of light would we even be able to discern between a PF and an SR world?
     
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  7. wellwisher Banned Banned

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    Say we assume that the speed of light is the same in all references. The question I ask myself is how is this possible and how does it occur? There needs to be a logical explanation for this.

    What everything suggests to me, is there is not only a speed of light reference, that is the same in all references, but there is also an absolute zero reference, that is the same in all references, so all reference experiments are comparing the speed of light to the same absolute zero reference, and not the velocity of their own reference.

    Here is some logic:

    At a speed of light reference, infinite time in finite reference, appears to lapse in an instant of time at the speed of light. If the speed of light interacts for less than infinite time, such as fractions of a second with an atom, this means this interactio will occur in less than an instant in the speed of light reference.

    Less than an instant in C, means finite reference does not appear to move unless it could move beyond infinity, which is a contradiction in terms. This results in the absolute zero reference which exists for less than an instant in C.
     
  8. Motor Daddy ☼☼☼☼☼☼☼☼☼☼☼ Valued Senior Member

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    Thanks for starting this thread, James. I am gonna stay clear and let the experts sort it out. I will be observing from the absolute frame!
     
  9. Tach Banned Banned

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    It is not very difficult to show that PF fails all the Ives-Stilwell types of experiments. Theories competing with SR must be able to explain the results of three classes of experiments[1]:
    1. Michelson-Morley
    2. Kennedy-Thorndike
    3. Ives-Stilwell
    PF fails all three.

    [1]http://rmp.aps.org/abstract/RMP/v21/i3/p378_1
     
    Last edited: Jan 14, 2013
  10. RJBeery Natural Philosopher Valued Senior Member

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    Tach, I posed this question to James but perhaps you have an opinion as well: if all available methods for measuring the speed of light are themselves dependent on the speed of light would we even be able to discern between a PF and an SR world? It's a sincere question, I'm not picking a fight here.
     
  11. Tach Banned Banned

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    The post just above yours states clearly that PF fails a set of tests that are passed by SR.
     
  12. Beer w/Straw Transcendental Ignorance! Valued Senior Member

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    I personally don't like SR.

    It's too special, not general enough.
     
  13. billvon Valued Senior Member

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    Many of them are not. Many of the M-M experiments, for example, do not even measure the speed of light; they just measure the difference in speed between two light beams, with no reliance upon the actual speeds involved.
     
  14. RJBeery Natural Philosopher Valued Senior Member

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    That's the specific issue I want to understand. Is it permissible to have a PF world in which that preferred frame, whatever it is, remains unidentifiable because all local inertial c measurements are invariant? I believe the answer is yes if all measurements taken are themselves reliant upon c as well.
     
    Last edited: Jan 14, 2013
  15. brucep Valued Senior Member

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    And what is the value of that difference in the local coordinate frame of the comparative measurement? 0 ms^-1. So it tells us that measurements of the local coordinate speed of light are invariant. It doesn't say what the speed is just that it's the same in the local coordinate frame where the comparison is made. Invariant in all local coordinate frames. Since physics is frame invariant the PF transformation didn't work so Lorentz figured out a way to make it work. Nobody seemed to want to admit what it meant. Except for Einstein.
     
  16. brucep Valued Senior Member

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    Why would the measurement of the speed of light depend on what the measurement result is?

    This is the local coordinate speed of light

    dr/dt = 1

    dr_distance/dt_tick rate.
     
  17. przyk squishy Valued Senior Member

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    Well the question sort of answers itself doesn't it? The whole point of a preferred frame is that the laws of physics differ from frame to frame and presumably take their simplest form in the "preferred" frame. So if the laws of physics differed from one frame to another you could use that to test whether you were in this preferred frame or not. And if there's no difference that can be tested in any way, on what basis could you say there was a preferred frame?
     
  18. Tach Banned Banned

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    It doesn't matter what you believe. You are mixing the PF with LET. What you are saying is true for LET but it is false for PF.
     
  19. brucep Valued Senior Member

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    James is discussing the difference between Newton and Einstein. Newton = space and time intervals are absolute and the speed of light is whatever it needs to be. Einstein = space and time intervals are relative and the local coordinate speed of light is invariant. Basically your question is irrelevant to the discussion.

    Edit: That's not what James is discussing. Bad assumption on my part.
     
    Last edited: Jan 15, 2013
  20. RJBeery Natural Philosopher Valued Senior Member

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    It appears PF essentially turns into LET if we consider that our measuring devices rely on c.
    Technically an LET world could reside atop of a PF backdrop, couldn't it? You might say the PF backdrop becomes irrelevant then, I suppose. The absolute speed of c could vary but that change could be locally undetectable. I'm working under the presumption that James R is trying to address Motor Daddy's view of the world (perhaps I'm wrong on this point). If Motor Daddy's view of the world were as elementary as detecting a change in c then convincing him of its falsehood would be much easier.
     
  21. Tach Banned Banned

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    No, it doesn't. LET uses Lorentz transforms, PF doesn't.
    Maxwell eqs. are invariant wrt. LET, they aren't invariant wrt PF.
    PF fails the three classes of experiments I mentioned, LET passes all of them.
     
  22. brucep Valued Senior Member

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    As you say LET makes the same predictions as SR and is theoretically equivalent to SR.
     
  23. brucep Valued Senior Member

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    Explain how measuring devices rely on c? Where did James make a comparison between LET and SR that would support your conclusion the PF he's discussing is equivalent to SR and LET? He showed the PF predictions are not equivalent to SR and LET.
     

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