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Relativistic time and mass-energy phasors

  1. Jan 26, 2016 #1
    I’m trying to get a better handle on reconciling the concept of time as it relates to the quantum wave function and the relativistic 4-momentum. To put it simply, do we look at the coefficient of the time variable in each as something separate from the time variable itself, or do we treat these as a unit?

    For example, in relation to QM where E=energy and M=mass, we have the energy function (psi)=e^-i(Et) which is equivalent to (psi)=e^-i(Mt) setting c and h to zero. The energy-mass coefficient of time in this equation determines the rotation rate of a “phasor” in complex space which, presumably, represents not only some indication of the energy-mass of, say, the object, it also represents, I assume, some characterization as to how that object is traveling through time. More specifically, It would seem that a higher energy-mass object would translate into the representation of an object moving through time faster than a lower energy-mass object would. Why? Because whatever time variable we are presented with is multiplied by the energy-mass coefficient. Therefore, a larger-mass object would possess phasors that are spinning more quickly through complex space than a smaller-mass object which translates into that object moving through time more quickly than the smaller-mass object.

    This idea seems to be the same when we look at the 4-momentum. Again, as in the above example, if we consider a stationary object with zero momentum, everything that is the 4-momentum is contained in the “time slot.” If we’re looking at the 4-velocity, this figure amounts to c, the speed of light. So implicitly this is telling us that object A in its own reference frame with unit mass “moves through time” at the speed of c. However, object B with 10x unit mass moves through time at 10 times the speed of the unit mass object, A.

    So, as a first pause, am I correct in my above interpretation? I’m coming at this from a “B-prefix” perspective but this is what the math is telling me. Please correct me if I’m wrong.

    Now on to the follow-up question. In the above discussion, we were talking about two objects, A and B, each in their own reference frame. Let’s say we now set these two objects moving at a velocity relative to one another. What do we get then? Let’s say we set object B moving relative to object A and make A the inertial reference “lab” frame. What do we get here? Well, it would seem to me that we could do a Lorentz transform on the time variable of B’s wave function and get a time dilation effect. However, we could also argue that, even so, B is still traveling through time faster than A because the coefficient (energy-mass) of its time variable is much larger than A’s. In fact, it’s mc^2 times faster.

    The upshot of this argument is that, even though B is traveling relative to A, B is aging faster than A due to its larger mass. Again, this is what the math seems to be telling me. Maybe I’m missing something here.

    Of course, I could state the problem much more straightforwardly and ask, does an object with a larger mass travel through time more quickly than an object with a smaller mass? When looking at objects A and B we can look at it two ways assuming each are in the same inertial reference frame: 1) we can look at it as they are both traveling through time at the same rate as defined by the variable t in their energy wave function and that the only difference between the two is the mass coefficient of that time variable. In this case, the difference would manifest itself only in the frequency difference of the rate of rotation of their phasors, which only relates to their respective energies. On the other hand, we can eschew the distinction between the energy coefficient and the time variable and simply say that B is traveling through time faster than A, period. Which one is it?
     
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  3. Jan 26, 2016 #2

    PeterDonis

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    Staff: Mentor

    I would say no, because it's leading you to an incorrect inference. See below.

    Yes, you are. "Aging" in relativity is not described by mass or energy. It's described by the proper time along an object's worldline. The proper time does not depend on the object's mass or energy; it only depends on the arc length along the worldline, which is a purely geometric quantity.

    Yes, this is better because it doesn't bring in all the extraneous stuff about QM (see further comments below). It just poses a straightforward question in the framework of relativity, which has a straightforward answer: no. See above.

    As far as relativity is concerned, energy/mass is just energy/mass; the concept of "phasors" with "frequencies" is a quantum concept and doesn't appear in classical relativity at all. If you want to ask about how that works (your picture of it is not necessarily wrong, but it seems to me to be too simplistic), you should start a separate thread in the Quantum Physics forum. The only real answer I can give from the standpoint of relativity is the one I gave above.
     
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