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In physics, we have two categories of quantities:

  1. Fundamental: These quantities are defined by a rule how to measure them.
  2. Non-fundamental: These quantities are (directly or indirectly) defined in terms of type #1 quantities.

In his Special Relativity (Zur Elektrodynamik bewegter Körper) Einstein defines time as "what is displayed by a clock". This definition applies

  • to an inertial observer (one for which Newton's postulates hold true) with
  • a clock that is at rest relative to her and that is close to her

Starting from that definition, he defines the time for places that are (far) away from the observer:

He imagines that each point of space is equipped with a similar clock, all at rest relative to that inertial observer, and synchronized to the clock close to this observer by means of Poincaré-Einstein-Synchronization, i.e. by exchanging signals of light under the assumption that light travels at a speed of $c$ for every inertial observer. This defines time (now called "proper time") for each point in space for that specific observer.

Starting from this basis, Einstein starts deriving results like how different observers not at rest respective to each other perceive space, time, angles, speeds, etc.

Question: How is this done in General Relativity? What is the definition of time there?

Using Poincaré-Einstein-Synchronization might be tricky or not work at all in a curved spacetime, maybe even changing with time (spacetime being non-static) or there might be more than one path for light (like around a heavy object) or be non-symmetric (like objects on different sides of a horizon). And only after you derived that theory you might notice that it's basic definition of time from Special Relativity does not work.

Other approach would be to develop General Relativity without worrying about the definition of time and then define time by some other means, for example via the metric tensor. This would imply that time is non-fundamental and the metric tensor is the fundamental quantity, which leaves us with defining a rule how to measure the metric tensor. I doubt this even possible without having already means to measure time intervals and distances in the first place.

Or one could define time only in close proximity to the observer, assuming that the definition of Special Relativity works fine, where "close" means "spacetime curvature can be neglected". All this only comes clear after you established the theory and curvature and such. Then one has to establish a definition of time for locations that are "far" away (in terms of curvature) but it is not even clear to me whether that is well-defined or unique.

Related questions like Time, what is it?, Concept of time in General Relativity, understanding time: Is time simply the rate change? address different issues like metaphysics. The best match appears to be Time in general relativity, but the answers are more about how to draw information from the metric tensor.

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    $\begingroup$ Your definition of what makes a quantity "fundamental" seems antiquated. In modern physics, the fundamental quantities often are not (directly) observable. $\endgroup$ – mmeent Jan 29 '20 at 23:32
  • $\begingroup$ Yes, modern physics is mostly axiomatic. So was already Newton's machanics $\endgroup$ – emacs drives me nuts Jan 30 '20 at 8:57
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For fundamental questions about time you must refer to the fundamental notion of proper time instead of coordinate time.

One essential difference between general relativity and Newton's system of space and time is the fact that instead of one absolute time concept there are two time concepts - coordinate time and proper time. Both time concepts are linked by the equation of velocity-dependent time dilation $$dt = \gamma (v) d\tau$$ and by the equation of gravitational time dilation. Coordinate time is time after time dilation, and proper time is time before time dilation.

The fundamental concept of proper time is defined as "the time measured by a clock following a given object". You can see that this definition does not refer to spacetime, it refers only to the object, to the particle. In a spacetime diagram, the proper time of worldlines cannot be observed, it can only be calculated. And this is the definition of (proper) time: Proper time is the aging of a particle.

This definition of time is part of a twofold time concept: In a first step, time is generated in the form of proper time by the particles of the universe. In a second step, observers are observing the worldlines of the particles and by this their respective coordinate time.

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