pervect

Thank you for a nice list. Looking at this list, I realized that thermodynamic time may be the only one that cannot be defined quantitatively. For example, is it possible to define one second of thermodynamic time?

Not sure if it fits in your list or would be considered one of already mentioned classes, but in celestial mechanics different forms of dynamical time has historically been used a lot.

pervect said:

• Proper time. This is the sort of time measured by a clock. Proper time is an interval between two events, but in special and general relativity, this interval, as mesured by a clock, depends on the path the clock takes. This dependency is sometimes referred to as the "twin paradox", though it's not inherently paradoxical.
• Coordinate time. The basic purpose of coordinate time is to assign a unique label (generally, a real number) to the time at which an event occurs. While we usually assign time coordinates based on the reading one or more clocks, proper time alone is not sufficient to define coordinate time as proper time alone has no notion of how to synchronize clocks.
• [...]
• Cosmological time. This is a specific type of coordinate time in general realtivity, but I believe it is important enough to get a brief mention and a place on the list.

Are coordinate time and cosmological time the proper time of a given clock?

Jaime Rudas said:

Are coordinate time and cosmological time the proper time of a given clock?

No. Proper time does not have a notion of simultaneity like coordinate time does.

anuttarasammyak said:

Thank you for a nice list. Looking at this list, I realized that thermodynamic time may be the only one that cannot be defined quantitatively. For example, is it possible to define one second of thermodynamic time?

It is just the arrow of time, never heard of thermodynamics time.

Dale said:

No. Proper time does not have a notion of simultaneity like coordinate time does.

Understood, but let me rephrase the question slightly: Is coordinate time equivalent to the proper time of a given clock?

Jaime Rudas said:

Understood, but let me rephrase the question slightly: Is coordinate time equivalent to the proper time of a given clock?

In general, no. We can choose any coordinate system we please according to whatever amuses us, and there's no reason that the coordinates we choose have to match the readings of any clock anywhere (for a trivial example, consider the coordinates found by applying the transformation ##t'=\ln t## in flat spacetime where ##t## is an ordinary Minkowski time coordinate). It's hard to imagine why we might want to do that, but we can.

In practice, we often choose coordinates that correspond to proper time along the worldline of some clock, like say a clock that is at rest relative to us and our lab equipment, because it it's convenient. There are exceptions though: for example the Schwarzschild time coordinate is what a physically unrealizable clock an infinite distance from the gravitating body would read.

Ibix

Ibix said:

I'd say that the proper time of a clock is only defined along its wordline, whereas coordinate and cosmological times are defined over a 4d region.

This is also what I would say is the key difference. They have different domains. It can be that on the worldline of a given clock they have the same value, but even in that case they are defined over different domains.

Nugatory said:

In general, no. We can choose any coordinate system we please according to whatever amuses us, and there's no reason that the coordinates we choose have to match the readings of any clock anywhere (for a trivial example, consider the coordinates found by applying the transformation ##t'=\ln t## in flat spacetime where ##t## is an ordinary Minkowski time coordinate).

Would conformal time, as defined by Davis & Lineweaver in equation (29), be an example of this?

Ibix

pervect