Is there a good mathematical definition of time?
Time is what clocks measure. That may sound flippant but I do not mean it so. That is exactly the definition that is used in physics and has been since Einstein.
Unlike work or speed, time is not a mathematically derived quantity, so no. Similarly, you couldn't mathematically define length.
The simplest and most operationally correct definition of time is just "what clocks record". Or only slightly more detailed: a non-spatial interval between two events. (in that way, similar to length)
Could time be explained by entropy and/or cosmic inflation?
No, on the contrary, time is something that is used in MEASURING cosmic inflation and entropy. That's like asking if length could be explained by long objects..
Time is defined through the operation used to measure it.
Philosophers and physicists have been asking what time is for a very long time. Still no answer.
Well, then it's about time.
EDIT: this could go on for a long time. Someone should just shoot this thread
Long time? Wait, have you defined length yet?
No, entropy is one of the few formulas that is asymmetric with respect to time.
Time is the rate at which things change, it can be measured by different kinds of clock.
Similarly there are instruments which can measure space in terms of distance.
These measurements are what they are, a mathematical proof is not required.
Those measurements are relative though. A meter stick traveling at half the speed of light is not the same length as a meter stick sitting stationary relative to you. Same thing with a tick on a clock. Furthermore, two different observers can disagree over the timing of events. Its tough to talk about time without getting into spacetime.
Yes. though that is what appears to be so for somebody else not in the same refrrence frame as yourself.
Your own local clock and meter stick won't change as far as you can tell.
I consider Time to be a measure of the sequence of events, or of how the rate at which regular events occur compare to other regular events. 'Proper time' is what is measured by a clock and can vary between clocks depending on gravity and relative motion. Considering Time as only what is measured by clocks ignores many complexities which can then lead to misunderstanding and 'paradoxes'. To see the full picture you have to compare the rates of time of objects in a system and have a way of translating these to determine how they behave relative to each other. That is the purpose of Special and General Relativity.
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