You begin by describing time translation invariance, even relating it to space translation invariance. This is all well and good, except that you then you ask:
“Does it make sense to say that the global rate of motion could slow down, or speed up, over the whole universe at once—so that all the particles arrive at the same final configuration, in twice as much time, or half as much time? You couldn’t measure it with any clock, because the ticking of the clock would slow down too.”
This one doesn’t make as much sense to me. This is not just a translation but is actually a re-scaling. If you rescale time separately from space then you will have problems because you will qualitatively change the metric (special relativity under t → 2t no longer uses a minkowski metric). This in turn changes the geometric structure of spacetime. If you rescale both time and space then you have a conformal transformation, but this transformation is not a lorentz transformation. I’m not so sure physics is invariant under such transformations.
The electroweak force has been observed to violate both charge conjugation symmetry and parity symmetry. However, any lorentz invariant physics must be symmetric under CPT (charge conjugation + parity + time reversal). Thus if our universe is lorentz invariant, it is not time-reversal invariant. So you will at least need to keep the direction of time, even if you are able to otherwise eliminate t.
“@Stirling: If you took one world and extrapolated backward, you’d get many pasts. If you take the many worlds and extrapolate backward, all but one of the resulting pasts will cancel out! Quantum mechanics is time-symmetric.”
Um… no. As I explained above, lorentz invariance plus CP violation in electroweak experiments indicate that the universe is not invariant under time-reversal. http://en.wikipedia.org/wiki/CP_violation
Eh… correction. Quantum Mechanics may be time-symmetric, but quantum field theories including weak interactions are not.
Does it make sense to say that the global rate of motion could slow down, or speed up, over the whole universe at once—so that all the particles arrive at the same final configuration, in twice as much time, or half as much time? You couldn’t measure it with any clock, because the ticking of the clock would slow down too.
This one doesn’t make as much sense to me. This is not just a translation but is actually a re-scaling. If you rescale time separately from space then you will have problems because you will qualitatively change the metric (special relativity under t → 2t no longer uses a minkowski metric). This in turn changes the geometric structure of spacetime. If you rescale both time and space then you have a conformal transformation, but this transformation is not a lorentz transformation. I’m not so sure physics is invariant under such transformations.
If you change the value of c as you scale time then physics will stay the same.
c is the speed of light. It’s an observable. If I change c, I’ve made an observable change in the universe --> universe no longer looks the same?
Or are you saying that we’ll change t and c both, but the measured speed of light will become some function of c and t that works out to remain the same? As in, c is no longer the measured speed of light (in a vacuum)? Then can’t I just identify the difference between this universe and the t → 2t universe by seeing whether or not c is the speed of light?
I also think you’re stuck on restricting yourself only to E&M using Special Relativity. If you take t → 2t you change the metric from minkowski space to some other space, and that means that you’ll have gravitational effects where there previously weren’t gravitational effects. You might be able to salvage that in some way, but it’s going to be a lot more complicated than just changing the value for c. The only thing I can think of is to re-define the 4-vector dot-product and the transformation laws for objects with Lorentz indeces, and even that might not end up being consistent.
A coupleof things:
You begin by describing time translation invariance, even relating it to space translation invariance. This is all well and good, except that you then you ask:
“Does it make sense to say that the global rate of motion could slow down, or speed up, over the whole universe at once—so that all the particles arrive at the same final configuration, in twice as much time, or half as much time? You couldn’t measure it with any clock, because the ticking of the clock would slow down too.”
This one doesn’t make as much sense to me. This is not just a translation but is actually a re-scaling. If you rescale time separately from space then you will have problems because you will qualitatively change the metric (special relativity under t → 2t no longer uses a minkowski metric). This in turn changes the geometric structure of spacetime. If you rescale both time and space then you have a conformal transformation, but this transformation is not a lorentz transformation. I’m not so sure physics is invariant under such transformations.
The electroweak force has been observed to violate both charge conjugation symmetry and parity symmetry. However, any lorentz invariant physics must be symmetric under CPT (charge conjugation + parity + time reversal). Thus if our universe is lorentz invariant, it is not time-reversal invariant. So you will at least need to keep the direction of time, even if you are able to otherwise eliminate t.
“@Stirling: If you took one world and extrapolated backward, you’d get many pasts. If you take the many worlds and extrapolate backward, all but one of the resulting pasts will cancel out! Quantum mechanics is time-symmetric.”
Um… no. As I explained above, lorentz invariance plus CP violation in electroweak experiments indicate that the universe is not invariant under time-reversal. http://en.wikipedia.org/wiki/CP_violation
Eh… correction. Quantum Mechanics may be time-symmetric, but quantum field theories including weak interactions are not.
If you change the value of c as you scale time then physics will stay the same.
Uh… what?
c is the speed of light. It’s an observable. If I change c, I’ve made an observable change in the universe --> universe no longer looks the same?
Or are you saying that we’ll change t and c both, but the measured speed of light will become some function of c and t that works out to remain the same? As in, c is no longer the measured speed of light (in a vacuum)? Then can’t I just identify the difference between this universe and the t → 2t universe by seeing whether or not c is the speed of light?
I also think you’re stuck on restricting yourself only to E&M using Special Relativity. If you take t → 2t you change the metric from minkowski space to some other space, and that means that you’ll have gravitational effects where there previously weren’t gravitational effects. You might be able to salvage that in some way, but it’s going to be a lot more complicated than just changing the value for c. The only thing I can think of is to re-define the 4-vector dot-product and the transformation laws for objects with Lorentz indeces, and even that might not end up being consistent.