I need an operative definition of "measuring time in general relativity" that takes in consideration also the presence of strong gravitational fields between me and clock, able to deviate the light that I see coming from the hands in such a way to see an irregular, non periodic movement of them: in special relativity is extremely important to define how someone measure time, because is in the comparison of two measurements that appear the dilation of time, but in general relativity nobody cares about it. Now, if there is no operative definition of "measuring time", how is it possible to say that the proper time is the one "measured" if I don' t have a standard definition?
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2And how is this question different from the one you asked yesterday? Please state that clearly. – Jon Custer Jun 01 '16 at 13:41
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2Possible duplicate of "Measure of time in general relativity" – Jon Custer Jun 01 '16 at 13:42
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1This is a duplicate of the question you asked yesterday (which itself was a duplicate). Time is what good clocks measure, and you measure it by reading the clock. It does not matter whether, to you, the clock runs fast or slow or in some irregular way (all of these can happen in special relativity too, by the way): the time it read at a given event is the time on the clock's world-line at that event. – Jun 01 '16 at 14:17
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On the contrary, it is even more important to be careful "how we define and measure time" in general relativity than it is in special relativity.
In special relativity, the speed of the clocks affects the rate – as seen from another coordinate system (it's the time dilation). In general relativity, the gravitational potential influences the rate (gravitational red shift).
Both parts of the theory of relativity are based on the careful realization that time simply cannot be defined independently of the measurement apparatuses. So in special and general relativity and everywhere else, time is always what is measured by clocks (ideally the best clocks you can have).
Because it's measured by a particular gadget that is localized in space and that may move in space, we must be careful about its speed and its location in the gravitational field because those will affect the time shown by the clock at the end. By definition, the duration measured by a clock that moves from point A to point B along a trajectory C in spacetime is the "proper time" of the clock.
By using the equivalence principle and separating the trajectory to many pieces, it may be seen that this proper time – something that is measured experimentally using clock – may be identified with the integral $\int ds$ along the curve C in general relativity. Here, $$ ds = \sqrt{-\sum_{\mu\nu} g_{\mu\nu} dx^\mu dx^\nu} $$ If the curve were different, however, the clock would show something else and one must be careful about this fact. The theory's prediction for the time would be different, too. But there can't exist any "clock-independent" way of measuring or defining time. This is really the main point of relativity – or at least one of the main points. In special relativity, only inertial systems were allowed for the "simple formulation of the laws of physics", and it was enough to specify a speed to determine the frame. In general relativity, all nonlinear transformations of coordinates are allowed so the choice of the coordinate $t$ is basically a pure convention.
Concepts only exist to the extent to which they may be measured – i.e. "operationally defined". 100 years ago, this realization allowed Einstein to make the relativity breakthroughs. At that time, this philosophy was associated with "positivism", independently promoted by various philosophers for largely unphysical reasons. The same "positivism" also made it possible for other people to discover the laws of quantum mechanics where all facts only exist to the extent to which they are observed by an observer.
Luboš Motl
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