We are considering a stationary curved spacetime fabric. Temporal separation[Physical]is given by: [tex]{T}_{2}{-}{T}_{1}{=}{\int \sqrt {g}_{00}{dt}[/tex] [Limits of integration extending from t1 to t2which are of course the coordinate times] The above integral is path dependent,in the general case[depending on the nature of g(00)].So the physical separation of time in general is not unique for a pair of events. To reconcile the matter ,g(00) should not depend on more than one coordinate[leaving aside t]or else[rather in a generalized way] the above integral should be independent of path.
In the above thread the following ideas are tacitly present: 1) We have two space-time points[events] on the spacetime surface. A stationary gravitational field is being considered.t1 and t2 are the time components 2)We have considered several paths connecting the two events[spacetime points]. These paths lie on the spacetime surface. These points are inherently present in the above thread.But I have made them explicit now with a view towards garnering responses
What's the use of this notion? Not only is it path-dependent, but even given a particular path, it's dependent on your choice of coordinate system as well (unlike the proper time). We discussed a similar notion which you called "physical time" a while ago on another thread, my analysis of the notion's physical meaning in [post=2845736]this post[/post] applies here too:
We may consider a pair of events occurring in the distant galaxies. We ascribe to them the coordinates (t,x1,x2,x3) and (t',x1',x2',x3'). The coordinate separation of time is given by (t'-t). Corresponding to this value there may exist several values for physical time[Its separation].Which one should I consider if I am to carry out a theoretical investigation from the earth? Which one should correspond to my experimental observation and why?
What "experimental observations", specifically? Can you describe the details of the experiment you're thinking of? I can't think of any known experiment that would measure the quantity you call "physical time", which as I said it seems to be a fairly odd and useless notion. Physicists may be interested in the coordinate time between a pair of events in some coordinate system, and they may be interested in the proper time between a pair of events on the same worldline, but I don't know of any situation in which they would be interested in your "physical time" or where they would design an experiment to measure it.
So far as the problem is concerned there is absolutely no need to describe the experiment. The point is we are trying to measure the time difference between a pair of events in curved spacetime--and this is of course the physical time difference. There is another vital point. It is important to have a theoretical estimate of the physical time difference[which does not seem to have a unique value in the general case].
In the previous thread "we are trying to measure the time difference" [in the first paragraph]may be replaced by "we are trying to make an estimate of the time difference, by experiment" for the sake of clarity.
Why "of course"? Again, no physicist I'm aware of considers your "physical time difference" to be a useful quantity that's worth calculating, and the physical interpretation of this quantity is very odd as I described it in post #3. (Do you disagree with my physical interpretation there? Also, do you disagree that even given a specific choice of path, the value of the 'physical time difference' along that path will depend on the choice of coordinate system?) So I think you really need to provide some reasoned argument for why you think we should "of course" define the "time difference" in your unusual way, rather than seeing it as a rather arbitrary definition that wouldn't be useful in any practical calculation (either in a textbook or a real-world experiment). Personally I would define the "time difference" between two events either as the difference in coordinate time between them in some useful coordinate system, or else in terms of the proper time along some worldline between them. Why do you think your "physical time difference" is preferable to either of these definitions? Incidentally, I notice you have a habit of referring to your "previous thread" when really you are talking about your previous post, a "thread" is a series of posts on the same topic--each title on the main forum page, like "On the Physical Separation of Time", is the title of a thread, so when you click a title and see a series of posts discussing that subject, the whole series is a single "thread".
The physical time interval is,of course , more important.Clocks run at different rates at places with different values of the gravitational potential.The physical intervals are different at different altitudes though the coordinate intervals are the same.This plays a crucial role with the GPS.The rate of transmission form the satellites and the rate of reception at the ground are different.If this effect is not taken care of the GPS is supposed to turn into a failure in its accuracy. In the above example we have two sequences of events: 1)Transmission of information--one set of intervals 2)Reception of events--another set The above mentioned intervals are unequal. If one considers the interval between transmission and reception he gets a unique value since g(00) is dependent only on one coordinate--r[for the Schwarzschild metric].But can we assume such simplicity for complicated types of metrics that is for the general case?For such metrics can we assume that the integral in posting #1 is independent of path?
If a particle is travelling along a space-time world line: [tex] x^{\mu} = x^{\mu}(p) [/tex] where p is a parameter, then the proper time is ([itex]c \equiv 1[/itex]): [tex] d\tau^{2} = g_{\mu \nu} \, \dot{x}^{\mu} \, \dot{x}^{\nu} \, dp^{2} [/tex] [tex] \tau[x^{\mu}(p)] = \int_{p_{0}}^{p_{1}}{\sqrt{g_{\mu \nu}(x) \, \dot{x}^{\mu} \, \dot{x}^{\nu}} \, dp} [/tex] The requirement for path independence: [tex] \frac{\delta \tau}{\delta x^{\mu}(p)} = 0 [/tex] leads to the equation for a geodesic.
No, I can't. That is just an integral. In general, [itex]g_{0 0} = g_{0 0}(t, x, y, z)[/itex]. After you integrate with respect to [itex]t[/itex] from [itex]t_{1}[/itex] to [itex]t_{2}[/itex], you are left with a function that still depends on the spatial coordinates [itex]x, y, z[/itex]. What do you mean by path independence?
I simply wanted to say that the above integral is not path independent in the general case.So we have different values for physical separation[time] for a particular value of coordinate separation[that is,for a fixed pair of spacetime events]. Do you think it is necessary to have a unique value for the physical separation[temporal] for a fixed pair if events[spacetime points]?
In case you can do it please suggest the method or perform the calculation. The following points are to be noted: 1) We may connect a pair of spacetime points by several paths, not necessarily by geodesics. 2)In many cases we can have several geodesics connecting a pair of points[conjugate points]
By integrating w.r.t. [itex]t[/itex] (coordinate time), you had already specified a particular path, namely: [tex] x, y, z = \mathrm{const}, \; t_{1} \le t \le t_{2} [/tex] How can your integral be path dependent or independent when it is over a particlar path?