On the Physical Separation of Time

[Anamitra]

We are considering a stationary curved spacetime fabric.
Temporal separation[Physical]is given by:
$${T}_{2}{-}{T}_{1}{=}{\int \sqrt {g}_{00}{dt}$$

[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.

 

[Anamitra]

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

 

[JesseM]

We are considering a stationary curved spacetime fabric.
Temporal separation[Physical]is given by:
$${T}_{2}{-}{T}_{1}{=}{\int \sqrt {g}_{00}{dt}$$

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:

I'm not sure what physical meaning could be assigned to these "physical" measures calculated using these altered metrics, which are different from the normal spacetime metric. Maybe if you divide an arbitrary worldline into a lot of short segments, and then for each segment you consider a short timelike worldline of constant position coordinate which goes through the midpoint of the segment and whose endpoints have the same time coordinates as the endpoint of the segment, then if you add up the proper time along all the little timelike worldlines (calculated using the normal metric), in the limit as the size of each segment approaches zero (so the number of segments approaches infinity) the sum of proper times will approach the "physical time" calculated with the altered metric? So it's sort of like approximating the smooth worldline by a "pixellated" line and then adding the vertical height of all the pixels, and considering the limit as the number of pixels goes to infinity.

 

[Anamitra]

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?

 

[JesseM]

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.

 

[Anamitra]

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].

 

[Anamitra]

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.

 

[JesseM]

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.

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".

 

[Anamitra]

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?

 

[Dickfore]

If a particle is traveling along a space-time world line:

$$x^{\mu} = x^{\mu}(p)$$

where p is a parameter, then the proper time is ($c \equiv 1$):

$$d\tau^{2} = g_{\mu \nu} \, \dot{x}^{\mu} \, \dot{x}^{\nu} \, dp^{2}$$

$$\tau[x^{\mu}(p)] = \int_{p_{0}}^{p_{1}}{\sqrt{g_{\mu \nu}(x) \, \dot{x}^{\mu} \, \dot{x}^{\nu}} \, dp}$$

The requirement for path independence:

$$\frac{\delta \tau}{\delta x^{\mu}(p)} = 0$$

leads to the equation for a geodesic.

 

[Anamitra]

Proper time and physical time are not identical concepts.

 

[Dickfore]

What do you mean by path indpendence?

 

[Anamitra]

You may refer to the integral in posting#1

 

[Dickfore]

You may refer to the integral in posting#1

No, I can't. That is just an integral. In general, $g_{0 0} = g_{0 0}(t, x, y, z)$. After you integrate with respect to $t$ from $t_{1}$ to $t_{2}$, you are left with a function that still depends on the spatial coordinates $x, y, z$. What do you mean by path independence?

 

[Anamitra]

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]?

 

[Dickfore]

I repeat my question: What does path independent mean? What do you consider a path?

 

[Anamitra]

Can you evaluate the integral in posting #1 without considering SOMETHING CALLED PATH?

 

[Dickfore]

Yes.

 

[Anamitra]

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]

 

[Dickfore]

By integrating w.r.t. $t$ (coordinate time), you had already specified a particular path, namely:

$$x, y, z = \mathrm{const}, \; t_{1} \le t \le t_{2}$$

How can your integral be path dependent or independent when it is over a particlar path?

 

[Anamitra]

I have not specified the path. I have simply given you the points, I mean the spacetime points or events[a pair of fixed points,spacetime points].

 

[Dickfore]

$t = t_{1}$ and $t = t_{2}$ do not define spacetime points, but hyperplanes.

 

[Anamitra]

Just think of the events (t,x1,x2,x3) and (t',x1',x2'x3'). I have specified them in #4

 

[Dickfore]

Oh, I think I see what you mean. I think the requirement is:

$$\frac{\partial g_{; i}}{\partial x^{k}} = \frac{\partial g_{; k}}{\partial x^{i}}$$

where

$$g_{; i} = -\frac{g_{0 i}}{g_{0 0}}$$

 

[Anamitra]

But for orthogonal systems $${g}_{\mu\nu}{=}{0}$$ if $${\mu}{\neq}{\nu}$$

 

[Dickfore]

But for orthogonal systems $${g}_{\mu\nu}{=}{0}$$ if $${\mu}{\neq}{\nu}$$

I don't know what orthogonal system means as it relates to GR.

 

[Passionflower]

Proper time and physical time are not identical concepts.

What, in your mind, is the difference between physical and proper time?

 

[Anamitra]

Physical time interval has been defined in #1. It is different from proper time.

 

[Anamitra]

Clocks run at different rates at different points in curved spacetime.This relates to the fact : Physical time intervals are different at different points and not to the concept of proper time. You may just think of the celebrated Pound and Rebca Experiment

 

[JesseM]

The physical time interval is,of course , more important.

Why?

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.

I'm pretty sure all GPS calculations use coordinate time and proper time. You haven't explained why you think your notion of "physical time" would be useful here.

In the above example we have two sequences of events:
1)Transmission of information--one set of intervals
2)Reception of events--another set

What do you mean "set of intervals"? If transmission is a single event, what "interval" would be associated with it? I can only see how there would be an interval of some quantity--say, coordinate time--between two events, like the event of transmission and the event of reception.

The above mentioned intervals are unequal.

No idea what "intervals" you're talking about, you haven't specified the events you want to take an interval between, nor have you specified what quantity you want to calculate an interval of (coordinate time, proper time, 'physical time', something else?)

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].

You never really specified what variable was being integrated in your integral, so I assumed that, as with an integral for proper time, the variable to be integrated would be the parameter of a parametrized worldline. For example, if we have some parameter p and functions r(p), t(p), theta(p), and phi(p), then each value of p corresponds to a particular r, t, theta and phi along a worldline (assuming we're using Schwarzschild coordinates), and every point on the worldline corresponds to some value of p (that's just what it means to 'parametrize' a worldline). In that case, if p₀ and p₁ are two values of p corresponding to events which lie on the worldline and which we want to calculate the "physical time" between, then the detailed form of the integral (assuming a Schwarzschild metric where $$g_{tt} = (1 - r_s / r )$$) would be $$\int_{p_0}^{p_1} \sqrt{(1 - r_s / r(p)) * (dt/dp)^2} \, dp$$, which will obviously depend on the choice of worldline which determines the exact equations for the functions r(p) and dt/dp (the first derivative of t(p)). This would be analogous to a calculation of the proper time, which for a path parametrized by p would be $$\int_{p_0}^{p_1} \sqrt{ g_{tt} * (dt/dp)^2 + g_{rr} * (dr/dp)^2 + g_{\theta \theta} * (d\theta/dp)^2 + g_{\phi \phi} * (d\phi /dp)^2 } \, dp$$ (see the wikipedia entry on proper time). If you didn't mean for the integral to be taken along a parametrized worldline in this way, then please write the integral in more detail, showing exactly what variable is being integrated as well as how $$g_00$$ depends on this variable.

 

[Anamitra]

I would request you to consider #29 with a view to understanding physical time and its importance in relation to the GPS. I have talked of the difference of physical time and proper time there. Coordinate separation of time interval is same at the satellites as well as on the ground. The physical separations are different.The idea has been clearly explained in #9.

 

[Anamitra]

The variable in the integral mentioned in #1 has been clearly stated. Coordinate time is the independent variable while the result of the integration is physical time.The values of the integrand depend on the choice of path.The limits of integration also have been stated.

 

[JesseM]

The variable in the integral mentioned in #1 has been clearly stated. Coordinate time is the independent variable while the result of the integration is physical time.

It's still unclear because one can easily use t as the parameter with which to parametrize any specific timelike worldline (at least ones outside the horizon, since the t coordinate becomes spacelike inside the horizon), you just need functions r(t), theta(t), and phi(t), then any point on the worldline will have coordinates of the form [t, r(t), theta(t), phi(t)]. In this case the integral I wrote before, $$\int_{p_0}^{p_1} \sqrt{(1 - r_s / r(p)) * (dt/dp)^2} \, dp$$ (which is analogous to the integral for proper time that I wrote down afterwards, it just drops all the parts of the metric aside from $$g_{tt}$$), would reduce to $$\int_{t_0}^{t_1} \sqrt{(1 - r_s / r(t))} \, dt$$. Is this what the more detailed form of your integral would be? or do you want the r in (1 - r_s/r) to be a constant rather than a function r(t)? (and if it's a constant, what constant value should it take? After all the two events we're interested in may have different r-coordinates)

 

[Anamitra]

$$\int_{t_0}^{t_1} \sqrt{(1 - r_s / r(t))} \, dt$$. Is this what the more detailed form of your integral would be? or do you want the r in (1 - r_s/r) to be a constant rather than a function r(t)? (and if it's a constant, what constant value should it take? After all the two events we're interested in may have different r-coordinates)

r changes as we move along the curve from one point to the other during the process of integration.In fact we can have several such curves which is a basic feature of the problem[especially, in relation to general type of metrics which may be complicated functions of the coordinate variables even in the stationary case. We are excluding the explicit dependence of the metric,g(00) on coordinate time].

 

[JesseM]

r changes as we move along the curve from one point to the other during the process of integration.

So is this equation correct?

$$\int_{t_0}^{t_1} \sqrt{(1 - r_s / r(t))} \, dt$$

If so, you agree that different curves between the same events could have different functions for r(t) and so you could get different answers by evaluating it along different curves?