Preferred direction in time?

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I had a question about michelson morley and about sr. Did michelson morley show that there is no preferred direction in space only? Is there a preferred direction in time? Seems like in sr, there is no preferred frame of reference, but does this mean that there is no preferred direction in time? Does time always flow forward? And if so, does this mean that there is a preferred direction in time? Thanks!
 

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Did michelson morley show that there is no preferred direction in space only?

The Michelson-Morley did not show that there is not a preferred direction in space. It showed that the speed of light is the same regardless of the state of motion of the interferometer.
 
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Is there a preferred direction in time?

There isn't in any of our fundamental theories (with the possible exception of some aspects of weak interactions, but those don't have anything to do with our everyday experience). However, there does appear to be in our experience. Where the apparent "arrow of time" in our experience comes from is an open question. One common hypothesis is that it comes from our universe having a very special set of initial conditions.
 
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Is there a preferred direction in time? Seems like in sr, there is no preferred frame of reference, but does this mean that there is no preferred direction in time?
There is no preferred direction in time in SR, but it is important to note that the future and the past are distinct, even if neither is preferred.

If you consider all of the events at a fixed spacetime interval from the origin, that set of events forms a hyperboloid. For spacelike intervals it is a hyperboloid of one sheet, meaning that you can smoothly rotate “east” into “west”.

In contrast, for timelike intervals it is a hyperboloid of two sheets, so you cannot smoothly rotate “future” into “past”. Future-directed timelike vectors are distinct from past-directed ones.
 
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Would having a preferred direction in time in sr mess up the theory in any way? Would sr still be consistent and viable even if there was a preferred direction in time? or in space?
 
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Would having a preferred direction in time in sr mess up the theory in any way?

This was already answered.

There is no preferred direction in time in SR
 
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If you reversed time, this wouldn't reverse entropy?
 
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Yes, but that probably doesn't mean what you think it means. Feynman, in his Caltech lecture on entropy, gives an example of a mixture of ink and water unmixing itself spontaneously. There's nothing physically wrong with that happening, it's just highly improbable. As Peter said in post #3, none of our physical laws have a preferred direction in time. It's just that the universe at one end of time seems to have been in a highly improbable state.
 

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