On the Physical Separation of Time

[Anamitra]

g(00) is not a function of time in a direct or an explicit manner[for stationary fields].But as we move along the curve of integration ,g(00) changes from point to point. So we may construct a relationship between t and r for the purpose of integration.

 

[JesseM]

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.

I understand that your "physical time" is different from proper time, I don't understand it's "importance" though. Do you agree that all calculations that physicists use to synchronize GPS clocks are done using coordinate time and proper time, not your notion of "physical time"?

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.

It isn't "clear" to me. Could you please just give a direct answer to my question from post #30?

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.

Please, no reference to "intervals" unless you specify what specific particular pair of physical events (for example, the event of a ground clock sending a signal and the event of a satellite receiving that signal) you are taking an interval between. Perhaps you are talking about an "interval" between two successive events of a signal being sent from the ground, and comparing with an "interval" between two successive events of a satellite receiving a signal?

 

[JesseM]

g(00) is not a function of time in a direct or an explicit manner[for stationary fields].But as we move along the curve of integration ,g(00) changes from point to point. So we may construct a relationship between t and r for the purpose of integration.

You still haven't given a direct answer to my question--was the equation I wrote down in post #35 the same as what you had in mind for "physical time", yes or no?

 

[Anamitra]

I understand that your "physical time" is different from proper time, I don't understand it's "importance" though. Do you agree that all calculations that physicists use to synchronize GPS clocks are done using coordinate time and proper time, not your notion of "physical time"?

With out the notion of physical time you cannot have clocks running at different rates at different points.
Coordinate separation [temporal] cannot produce this effect, typical of General Relativity.
Proper time is not related to this issue.

It isn't "clear" to me. Could you please just give a direct answer to my question from post #30?

In #30 the Wikipedia reference relates to PROPER TIME and not PHYSICAL time

Please, no reference to "intervals" unless you specify what specific particular pair of physical events (for example, the event of a ground clock sending a signal and the event of a satellite receiving that signal) you are taking an interval between. Perhaps you are talking about an "interval" between two successive events of a signal being sent from the ground, and comparing with an "interval" between two successive events of a satellite receiving a signal?

I would request you not to complicate your own thinking.Just think of a pair of events occurring in curved spacetime [at finite separation]. How do you calculate the physical time difference?

I am not meandering with my responses. I am trying to get the point to you which you are unwilling to accept

 

[Anamitra]

The equation you wrote in #35 can be meaningful only if the path is specified.I have clearly indicated that[you may consider #1 and some others also] , and we can get different results for different curves--and that is the crux of the problem.

 

[JesseM]

With out the notion of physical time you cannot have clocks running at different rates at different points.

What do you mean "cannot have"? Do you actually mean that you think you'll get some different predictions about local events (like the times that a particular clock receives signals from another clock) if you don't make use of "physical time" in your calculations? (please give a direct yes or no answer to this question) Or would you agree with me that all predictions about local coordinate-invariant facts are the same regardless of what method we use to calculate things, but you think "physical time" is necessary if we want to define some non-coordinate-invariant notion (i.e. a coordinate-dependent notion) of the "rate" that clocks run at different points? The normal way of defining the "rate" of a clock in a coordinate-dependent way is just to look at $$d\tau/dt$$, the rate proper time is increasing relative to coordinate time. Certainly it is true in Schwarzschild coordinates that $$d\tau/dt$$ will be smaller for a clock hovering at a lower radius than for a clock hovering at a greater radius, which is what physicists say when they talk about low-altitude clocks "running slower" than high-altitude clocks.

In #30 the Wikipedia reference relates to PROPER TIME and not PHYSICAL time

I didn't ask you to address the entirety of post #30, I asked you to address this particular question from post #30:

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.

There is no "wikipedia reference" in this question. Please address this question, specifically.

I would request you not to complicate your own thinking.Just think of a pair of events occurring in curved spacetime [at finite separation]. How do you calculate the physical time difference?

This isn't helping me to understand what you meant with your comment that prompted my question above, namely:

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

Communication is going to be impossible if each time you make a statement which I find confusing and I ask you a question about it, you avoid answering the question and just make some new confusing statements. So please, let's straighten out what you meant by distinguishing "one set of intervals" associated with "transmission" and "another set" associated with "reception" before moving on to other issues. What is the exact nature of the "intervals" associated with transmission? What events are you calculating the intervals between?

 

[JesseM]

The equation you wrote in #35 can be meaningful only if the path is specified.I have clearly indicated that[you may consider #1 and some others also] , and we can get different results for different curves--and that is the problem.]

That doesn't answer the question of whether #35 actually captures what you meant when you wrote the equation in post #1. Yes or no? If no, does that mean you meant #1 to possibly have an interpretation where the value of the integral would not depend on the choice of path?

 

[Anamitra]

For events occurring at a fixed point[spatial point] proper time difference is the same as physical time difference. But for events occurring a pair of distant points in curved spacetime physical time difference and proper time difference are not the same. To get the proper time one has to travel from one point to the other between the events with the clock in his hand.But I am standing at a third point and I am not ready to move---I am in a laboratory.I want to have an estimate of the time difference.Coordinate time difference would not help me----it may have units different from time in certain types of metrics.What should I do in such a situation?

 

[Dickfore]

Let's say point A has (spatial) coordinates $x^{i}$ and point B is infinitesimally close with coordinates $x^{i} + dx^{i}$. We shine a light signal from B to A at time $x^{0}$ according to B. The signal travels towards A, reflects and reaches B again. Since it is traveling along a null geodesic, the equation for the light beam is:

$$g_{0 0} (dx^{0})^{2} + 2 g_{0 k} dx^{k} \, (dx^{0}) + g_{i k} dx^{i} dx^{k} = 0$$

This is a quadratic equation w.r.t. $dx^{0}$ and it has two solutions:

$$(dx_{0})_{1/2} = \frac{-g_{0 i} dx^{i} \mp \sqrt{(g_{0 i} g_{0 k} - g_{0 0} g_{i k}) dx^{i} dx^{k}}}{g_{0 0}}$$

corresponding to the time coordinates $x^{0} + (dx^{0})_{1}$ and $x^{0} + (dx^{0})_{2}$ according to A when the emission and reception of the light beam at B took place. The midpoint of the two:

$$x^{0} + \frac{(dx^{0})_{1} + (dx^{0})_{2}}{2} = x^{0} + g_{; i} dx^{i}, \; g_{; i} \equiv - g_{0 i}/g_{0 0}$$

is, by the operational definition of synchonization, synchronous to the event with time coordinate $x^{0}$ at A when the reflection took place. Thus, the synchronization of the clocks at A and B requires an offset by an amount:

$$d(\Delta x^{0}) = g_{; i} \, dx^{i}$$

If the points are separated by a finite amount, then we need to integrate:

$$\Delta x^{0} = \int{g_{; i} dx^{i}}$$

Notice that this integral is along a spatial curve. Also, it still depends parametrically on $x^{0}$. If you require path independence of this integral, it means that the integral:

$$\oint{g_{; i} dx^{i}} = 0$$

should be zero along any closed spatial curve. For this, it is necessary and sufficient that the integrand is a gradient:

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

But, the mixed second derivatives of the function $\psi$ need to be equal, which means:

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

Substituting the expression for $g_{; i}$ and performing the differentiation leads to:

$$g_{0 0} \left(\frac{\partial g_{0 i}}{\partial x^{k}} - \frac{\partial g_{0 k}}{\partial x^{i}}\right) - \left(g_{0 i} \frac{\partial g_{0 0}}{\partial x^{k}} - g_{0 k} \frac{\partial g_{0 0}}{\partial x^{i}}\right) = 0$$

I am still not sure how to express this condition in terms of the Christoffel symbols.

 

[JesseM]

For events occurring at a fixed point[spatial point] proper time difference is the same as physical time difference. But for events occurring a pair of distant points in curved spacetime physical time difference and proper time difference are not the same. To get the proper time one has to travel from one point to the other between the events with the clock in his hand.But I am standing at a third point and I am not ready to move---I am in a laboratory.I want to have an estimate of the time difference.Coordinate time difference would not help me----it may have units different from time in certain types of metrics.What should I do in such a situation?

Are you claiming that there must be a single objective truth about "the time difference"? Or are you just looking for some definition that will allow you to define a "time difference" between an arbitrary pair of events in curved spacetime, without any notion that this definition is physically preferred over other possible ways we might define "time difference"?

In the latter case, my understanding is that as long as a spacetime is globally hyperbolic it should be possible to "foliate" it into a series of spacelike surfaces, so you could always build a coordinate system where each spacelike surface is a surface of constant t-coordinate, and then I would guess it should then be possible to define the coordinate system in such a way that all curves of constant position coordinate would be timelike curves. In this case, coordinate time difference between two events should always have units of time. The only spacetimes that aren't "globally hyperbolic" are ones with weird properties, like spacetimes containing closed timelike curves (i.e spacetimes where it is possible to 'travel backwards in time' and revisit events in your own past light cone). In the case of a nonrotating uncharged black hole, if you choose Kruskal-Szekeres coordinates it will be true that any curve of constant position-coordinate is a purely timelike curve, so the difference in coordinate time between any two events in these coordinates should have units of time, even if one event is outside the event horizon and the other is inside.

 

[Anamitra]

Response to #44
The initial and the final points are the same spatiallyin the procedure given by Dickfore [at least in stationary fields]and so the spatial separations $${{\Delta}{x}}_{i}{=}{0}$$
So the quadratic equation should undergo a drastic modification.

We consider each term [spatial]=$${g}_{ij}{{\Delta}{x}}_{i}{{\Delta}{x}}_{j}$$
For varying fields the value of g(ij) may change with time. But what about the spatial elements $${{\Delta}{x}}_{i}$$ if these terms are considered individually?

[May I refer to the picture/Diagram given in Landau and Lifgarbagez["The Classical Theory of Fields" Chapter--10,Section83] for the visualization of the procedure given by Dickfore in the initial part of the treatment]

 

[Anamitra]

Section 83 in the previous post should be replaced by Section84, Figure 18

 

[Anamitra]

The first equation in #44 could represent the travel of light in either direction from A to B or from B to A.So the two roots can represent the two times and the process has been worked out by the mirror.But if we consider the total travel from B to A and than back to B we still have a null geodesic[if a sharp bend/reflecting point in the path is given due consideration].Now the spatial elements work out to zero value and at the same time ds =0. Therefore dt=0.

What would be the case like if we make the sharp bend smooth without increasing the path in an appreciable manner?One does not have to consider a sharp bend in such a situation.

 

[Anamitra]

We consider the travel of a light ray between the spatial points A and B[A light ray traveling from B to A to report an event at B]

Physical Element[Spatial], $${dL}{=}{\sqrt{{g}_{11}{dx1}^{2}{+}{g}_{22}{dx2}^{2}{+}{g}_{33}{dx3}^{2}}$$
Now,
$${ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}$$
Implies,
$${ds}^{2}{=}{g}_{00}{dt}^{2}{-}{dL}^{2}$$
For a null geodesic,
$${dL}{=}{\sqrt{{g}_{00}}{dt}$$

Time of travel of the light ray,

=$${\int dL}$$ ,noting c=1 in the natural units.

=$${\int {\sqrt{{g}_{00}}{dt}$$
So physical time as considered in #1 is simply the time of travel of the light ray between the pair of points. In curved spacetime we may have several null geodesics connecting a pair of points[gravitational lensing]. The same event may appear to be occurring at different locations----why not at different instants of time?
[Stationary fields are being considered]

 

[Anamitra]

Distance along the x-axis:$${\int \sqrt{{g}_{11}}{dx}$$
Distance along the y-axis:$${\int \sqrt{{g}_{22}}{dy}$$
Distance along the z-axis:$${\int \sqrt{{g}_{33}}{dz}$$

Analogously,distance along the time-axis should be:$${\int \sqrt{{g}_{00}}{dt}$$

This is to be interpreted as the time taken by a light ray to travel between the points so far as the general nature of the metrics is concerned.

 

[Anamitra]

In #49 we have considered a pair of spatial points and have connected them by a light ray passing between them . But what about fixed space-time points? They may not lie on the natural path of a light ray,ie a null geodesic.
Well, in such a case we may think of some sort of a mirror arrangement/mirror combination that could work out the concept of physical time in a practical way. Perhaps, this could solve the problem.

 

[JesseM]

Distance along the x-axis:$${\int \sqrt{{g}_{11}}{dx}$$

It's a little misleading to call this "distance along the x-axis". If we take two events, consider a path between them, and evaluate $${\int \sqrt{{g}_{11}}{dx}$$ along that path, then the result will not necessarily be equal to the coordinate distance between the events along the x-axis (at least not if my interpretation of what it means to integrate the metric along a path in post #33 was correct, you never answered my question about whether that matched your own notion). Of course you can always take this integral to be a definition of your own invented phrase "physical distance" along the x-axis.

 

[Anamitra]

I #50 I have talked of physical distances and not coordinate distances.These physical always have the dimension of length.

 

[JesseM]

I #50 I have talked of physical distances and not coordinate distances.

But "physical distance" is just an arbitrary phrase you made up to describe that integral, which has no meaning apart from your definition--correct? If so you should be careful to use the full phrase "physical distance" and not just something more vague like "distance along the x-axis".

These physical always have the dimension of length.

Well, coordinate distance does too as long as you use a coordinate system where surfaces of constant time are everywhere spacelike.

 

[Anamitra]

But "physical distance" is just an arbitrary phrase you made up to describe that integral, which has no meaning apart from your definition--correct? If so you should be careful to use the full phrase "physical distance" and not just something more vague like "distance along the x-axis".

Just think of two points an ordinary sphere[three dimensional-spatial]

$${\int {{d}{\theta}}}$$ gives an angular distance, when integration is performed between the limits $${(}{\theta}_{1}{,}{\theta}_{2}{)}$$
While the integral,
$${\int {r}{d}{\theta}}{=}{\int{\sqrt{{g}_{\theta\theta}}{d}{\theta}}$$
Conforms to the idea of distance[it has the dimension of length] along the curve r=constant and phi=constant

"Physical distance" is not an arbitrary phase. It has a well defined physical concept behind it.

I would request the mentors to comment on the issue.

 

[JesseM]

Just think of two points an ordinary sphere[three dimensional-spatial]

$${\int {{d}{\theta}}}$$ gives an angular distance, when integration is performed between the limits $${(}{\theta}_{1}{,}{\theta}_{2}{)}$$
While the integral,
$${\int {r}{d}{\theta}}{=}{\int{\sqrt{{g}_{\theta\theta}}{d}{\theta}}$$
Conforms to the idea of distance[it has the dimension of length] along the curve r=constant and phi=constant

And what if you pick some different curve with varying phi and/or r? In that case this integral will not necessarily correspond to the geometric length of the curve, do you agree?

"Physical distance" is not an arbitrary phase. It has a well defined physical concept behind it.

Can you explain what that "well defined physical concept" actually is?

 

[Anamitra]

And what if you pick some different curve with varying phi and/or r? In that case this integral will not necessarily correspond to the geometric length of the curve, do you agree?

I am measuring the distance along the axis defined by r=const and phi=constant[only theta changes along this direction]

 

[JesseM]

I am measuring the distance along the axis defined by r=const and phi=constant[only theta changes along this direction]

This seems like an artificial restriction since there is no mathematical reason that the integral couldn't be evaluated along paths where r and/or phi are allowed to vary. Besides, when it comes to defining "physical distance" and "physical time" in spacetime, you didn't mention any requirement that all the other coordinates besides the one being integrated must be constant, so if the sphere is supposed to be an analogy to what you're talking about in spacetime, imposing such requirements in one case but not the other seems to make the analogy break down.

Finally, if you can't actually explain what the "well defined physical concept" behind your notion of "physical distance" is supposed to be, then it follows that you don't actually have any well-defined concept in mind that you used to derive the equation.

 

[Anamitra]

We can always calculate the physical distance along an axis.Let us consider the points$${(}{r}{,}{\theta}{,}{\phi}{)}$$ and $${(}{r}{,}{\theta}^{'}{,}{\phi}^{'}{)}$$

We may calculate the distance between them along r=const,phi=$${\phi}$$,[taken const]
or r=constant ,phi=$${\phi}^{'}$$, [taken constant]
between the aforesaid coordinates using the second integral in #55

We may also calculate the distance between them along r=const,theta=$${\theta}$$
[taken constant]
or r=constant ,theta=$${\theta}^{'}$$, [taken constant]

between the aforesaid points using the integral:

$${\int{ \sqrt{g}_{\phi\phi}}{d}{\phi}$$
For the two parallel lines[of latitude lying between the meridians concerned] we get different values. This is quite natural
[I am referring to a common,3D spherical surface,obviously a spatial one]

 

[Anamitra]

If one wants to calculate the physical distance along some path connecting $${(}{r}{,}{\theta}{,}{\phi}{)}$$ and $${(}{r}{,}{\theta}^{'}{,}{\phi}^{'}{)}$$

He/she can use the integral $${\int{dL}}{=}{\int{\sqrt{{g}_{rr}{dr}^{2}{+}{g}_{\theta\theta}{{d}{\theta}}^{2}{+}{g}_{\phi\phi}{d}{{\phi}}^{2}}}$$
Along the specified path lying on the surface of the sphere

 

[JesseM]

If one wants to calculate the physical distance along some path connecting $${(}{r}{,}{\theta}{,}{\phi}{)}$$ and $${(}{r}{,}{\theta}^{'}{,}{\phi}^{'}{)}$$

He/she can use the integral $${\int{dL}}{=}{\int{\sqrt{{g}_{rr}{dr}^{2}{+}{g}_{\theta\theta}{{d}{\theta}}^{2}{+}{g}_{\phi\phi}{d}{{\phi}}^{2}}}$$
Along the specified path lying on the surface of the sphere

Obviously you can get the length along a path using the whole metric, and similarly in spacetime you can get the coordinate-invariant proper time along a timelike path, or the coordinate-invariant proper distance along a spacelike path, by integrating the whole spacetime metric along that path, i.e. $$\int{\sqrt{{g}_{tt}{dt}^{2}{+}{g}_{xx}{{dx}^{2}{+}{g}_{yy}{dy}^{2}}{+}{g}_{zz}{dz}^{2}}$$. And obviously if you pick a path of constant r and phi, so dr = dphi = 0 all along the path, then the total spherical integral in 3D space which you wrote above reduces to $$\int \sqrt{ g_{\theta\theta} \, d\theta^2 } = \int \sqrt{g_{\theta\theta}} \, d\theta$$, and similarly if we pick a spacelike path through spacetime with constant t, y, and z coordinate, then the integral in that case would reduce to $$\int \sqrt{ g_{xx} \, dx^2 } = \int \sqrt{g_{11}} \, dx$$ which is what you wrote down earlier. But when you wrote down this integral, you just said it was the "distance along the x-axis", you didn't specify that it applies only to paths where the x-coordinate varies while the t, y, and z coordinates are constant. Did you mean to imply that restriction? (please answer this question yes or no) I didn't think you were implying this, since you seemed to say earlier that your notion of "physical time" could be applied to arbitrary paths, not just paths of constant position coordinate (i.e. t varying while x, y, z stay constant). And if you did intend for your integral to define "distance along the x-axis" for arbitrary paths, not just paths where 3 coordinates are held constant, then the analogy to distance in 3D space doesn't work, because the integral $$\int \sqrt{g_{\theta\theta}} d\theta$$ doesn't define "distance" for arbitrary paths in space, it only defines distance along paths where r and phi are constant.

 

[Anamitra]

But when you wrote down this integral, you just said it was the "distance along the x-axis", you didn't specify that it applies only to paths where the x-coordinate varies while the t, y, and z coordinates are constant. Did you mean to imply that restriction? (please answer this question yes or no) I didn't think you were implying this, since you seemed to say earlier that your notion of "physical time" could be applied to arbitrary paths, not just paths of constant position coordinate (i.e. t varying while x, y, z stay constant). And if you did intend for your integral to define "distance along the x-axis" for arbitrary paths, not just paths where 3 coordinates are held constant, then the analogy to distance in 3D space doesn't work, because the integral $$\int \sqrt{g_{\theta\theta}} d\theta$$ doesn't define "distance" for arbitrary paths in space, it only defines distance along paths where r and phi are constant.

By x-axis we mean paths along which the other coordinates do not change.
Similarly for the time axis we have to consider a fixed spatial location. Interval is given by the integral: $${\int {\sqrt{g}_{00}}{dt}}$$

But what am I to understand by the time interval between events occurring at distant points A and B if I am working in a laboratory at A.?Proper time would not help me since I am not ready to run between the events with a clock in my hand. Coordinate time difference may not have the unit of time itself.

The time interval in such a situation may be conveniently defined by the integral:$${\int {\sqrt{g}_{00}}{dt}}$$
and this may be a path dependent quantity.I have explained this in #49

In fact time interval between a pair of events is meaningful only if signals can be passed between them.You may consider [as an example]a pair of events A and B which have a spacelike separation. If you are stationed at A the event B[at a spacelike separation] exits hypothetically--its there in your imagination or you can have it in the papers of mathematics![since you can never have any information from it]If you are in a closed isolated room you can only speculate what is happening in the outside world --you can never observe them.Quite meaningless to say --when did such events occur?

If signals can be passed between a pair of events[considering a finite separation] you may define the time difference by the line integral I have given.The value may be path dependent--that should not usher in any type of contradiction.

 

[JesseM]

By x-axis we mean paths along which the other coordinates do not change.
Similarly for the time axis we have to consider a fixed spatial location. Interval is given by the integral: $${\int {\sqrt{g}_{00}}{dt}}$$

But what am I to understand by the time interval between events occurring at distant points A and B if I am working in a laboratory at A.?Proper time would not help me since I am not ready to run between the events with a clock in my hand. Coordinate time difference may not have the unit of time itself.

The time interval in such a situation may be conveniently defined by the integral:$${\int {\sqrt{g}_{00}}{dt}}$$
and this may be a path dependent quantity.

So you are trying to evaluate this integral along paths where the x,y,z coordinates are not constant, correct? If so, I think you can see why the analogy with "distance" in 3D space doesn't work--evaluating this integral along paths of non-constant x,y,z is analogous in 3D to evaluating the integral $$\int \sqrt{g_{\theta\theta}} \, d\theta$$ along paths where r and phi are not constant, and this integral does not give what we ordinarily define to be the "distance" along such a path in 3D space. So, your argument in post #55 doesn't work as a justification for why you think it makes sense to label the integral $${\int {\sqrt{g}_{xx}}{dx}}$$ as a "distance" even when evaluating along a path where the other coordinates aren't held constant. So, you still haven't provided any meaningful justification for your statement in post #55:

"Physical distance" is not an arbitrary phase. It has a well defined physical concept behind it.

I would request the mentors to comment on the issue.

If you just wanted to say that "physical distance" was an arbitrary label you came up with for integrals like $${\int {\sqrt{g}_{xx}}{dx}}$$ that would be fine with me, but there is no reason it is "natural" to label it this way given our prior notions of "distance".

 

[JesseM]

But what am I to understand by the time interval between events occurring at distant points A and B if I am working in a laboratory at A.?Proper time would not help me since I am not ready to run between the events with a clock in my hand. Coordinate time difference may not have the unit of time itself.

Ordinarily a physicist will choose a coordinate system where paths of constant position coordinate are actually timelike and paths of constant time coordinate are actually spacelike, and in this case the coordinate time difference will have units of time. It's true that in the case of Schwarzschild coordinates the t-coordinate becomes spacelike inside the horizon, but I would think (and someone more experienced in how these coordinate systems are normally used can correct me if I'm wrong) that Schwarzschild coordinates would mostly be used to analyze events outside the horizon, whereas if a physicist wanted to analyze events inside the horizon it would be more common to use a coordinate system where the t-coordinate is still timelike inside the horizon, like "free-fall coordinates" or Kruskal-Szekeres coordinates illustrated near the bottom of this page.

In any case, it seems to me that if you do choose to use a coordinate system where the t-coordinate is spacelike rather than timelike, then in that case the integral $$\int \sqrt{g_{tt}} \, dt$$ will not actually have units of time! If we define the metric in such a way that ds^2 is positive for timelike intervals and negative for spacelike intervals, then this means that if you integrate the metric along a path (i.e. evaluate the integral $$\int \sqrt{ds^2}$$ along the path) and get a real number then the path's length in spacetime has "units" of time, while if you get an imaginary number then the path's length in spacetime has "units" of distance. So, let's consider a simple example of a coordinate system where the t-coordinate is spacelike: let's suppose in Minkowski spacetime we use a coordinate system where the "t-axis" was identical to the "x-axis" of an ordinary SR inertial frame, and likewise the "x-axis" of this coordinate system was identical to the "t-axis" of the SR inertial frame, and both coordinate systems had identical y and z axes. In this case the metric for this new coordinate system would have $$g_{tt} = g_{yy} = g_{zz} =-1/c^2$$ and $$g_{xx} = 1$$. If we picked a path of constant x,y,z coordinates and varying t-coordinate and evaluated $$\int \sqrt{g_{tt}} \, dt$$, then the result will be an imaginary number, which means the answer has units of distance, not time (since this coordinate system is just an SR inertial frame with the x and t axes switched, then we should get the same answer as if we evaluated the integral $$\int \sqrt{g_{xx}} \, dx$$ using the Minkowski metric, where $$g_{xx} = -1/c^2$$).

In fact time interval between a pair of events is meaningful only if signals can be passed between them.You may consider [as an example]a pair of events A and B which have a spacelike separation. If you are stationed at A the event B[at a spacelike separation] exits hypothetically--its there in your imagination or you can have it in the papers of mathematics![since you can never have any information from it]If you are in a closed isolated room you can only speculate what is happening in the outside world --you can never observe them.Quite meaningless to say --when did such events occur?

But even if the events are spacelike-separated so that no signal can pass between the events themselves, signals about each event might later reach an observer in the overlap of each event's future light cone, so this observer can learn the coordinates of each event and calculate things such as the coordinate time between them.

 

[Anamitra]

One can consider a very simple case: An event occurs at point[spatial] B and is reported at A[spatial point].

Events:1)Transmission of signal from B
2)Reception of the same signal at A
This signal can reach A by several paths. So we are considering a number of events at A--A1,A2,A3,...They have the same spatial coordinates but different temporal coordinates. For each pair [Event at B which includes the time part, Ai] ,the time interval is=Time taken by the light ray along the path concerned=$${\int {\sqrt{{g}_{00}}dt}}$$
[please see #49 for the calculations]

Each set [event at B, Ai] may have a different value for the time interval [depending on the path of travel of the light ray]but this interval is being correctly measured/depicted by the integral[ which is nothing but the time of travel of the light ray between the events]

Each event is ,of course, observed at a different instant of time as I have said in #49.

The real problem lies in the fact that if a pair of events[space-time points] are specified, it may not always be possible to connect them by a light ray--they may not lie lie on a null geodesic.In such a situation we may use mirrors to guide the ray from one point to the other.This may be possible in a huge number of cases.

 

[Anamitra]

A pair of events may not lie on a null geodesic.[One may consider a spacelike separation by way of example]But we can always change the temporal coordinate at the reception point to enable a connection by a null geodesic.[ I believe ,this has been suggested by Jesse herself--please do correct me if I am mistaken!]This may help us.
But for a certain period of time we remain ignorant of the information at the reception center

 

[JesseM]

But we can always change the temporal coordinate at the reception point to enable a connection by a null geodesic.[ I believe ,this has been suggested by Jesse herself--please do correct me if I am mistaken!]

By "change the temporal coordinate", do you mean pick a different coordinate system which assigns a different time coordinate to one event, or actually pick a different event in spacetime with the same position coordinate but a different time coordinate, or something else? Either way, I never suggested anything like this (and by the way, I am a he rather than a she).

 

[Anamitra]

This is in relation to #67
A pair of events (t,x,y,z) and (t',x',y',z') with spatial separation are considered.We cannot connect them by a signal.But at the spatial location (x,y,z) we have the event (t1,x,y,z) at a later point of time.
Now the events (t1,x,y,z) and (t',x',y',z') may be connected by a null geodesic to get information from the second point. But one has to wait for the coordinate interval (t1-t) or the physical interval g(00)(t1-t) to get the information--which may be a hundred years or more.

[Change in temporal coordinate:t1-t]

The coordinate system is not being changed.

 

[ApplePion]

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.

I'm not so sure that there actually is a problem with your delta t being dependent on path.

 

[Anamitra]

The integral in the last posting is not path dependent----it simply represents the time of travel of a light ray between a pair of points along some specified null geodesic. This matter has been explained in #49,#65

The issue may be recast in the following manner:

Suppose you are standing in your laboratory and a pair of events occur in some distant galaxies in curved spacetime.What do you understand by the time difference between the two events?
[Your laboratory is also in a region of sufficient spactime curvature]
Since you in your laboratory, you cannot travel between the events along a spacetime path.Proper time will not help you.

Coordinate time should not be used since the speed of light in vacuum changes if coordinate time is considered.You may take the example of the Schwarzschild Geometry.

Physical time could have been a useful concept if it was independent of path.But it is not.
In most cases the spacetime points do not correspond to conjugate points--they cannot be connected by several null geodesics. Physical time gives us a unique picture in many cases.