Curved Space-time and Relative Velocity

[Dale]

Can you give me more detail on just what is involved in the Einstein Convention?

Here is the original paper by Einstein:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

The simultaneity convention is explained in section 1. Just use the same convention for a non-inertial observer and you have Dolby and Gull's radar time.

I have to say I doubt the wisdom of that technique. It works fine in an inertial frame, but it shouldn't be used while you are accelerating. By the time the signal comes back to you you will not have the same lines of simultaneity as when you sent the signal.

The point is that you have to define your lines of simultaneity by adopting some convention. Any convention you pick is fine, so why not use the same convention that you use for inertial frames?

 

[PAllen]

I have to say I doubt the wisdom of that technique. It works fine in an inertial frame, but it shouldn't be used while you are accelerating. By the time the signal comes back to you you will not have the same lines of simultaneity as when you sent the signal.

The papers you mention in discussion with Dalespam all agree that there is no such thing as objective lines of simultaneity. An infinite number of consistent definitions are possible for a single oberver (one of the papers even parameterizes an infinite family of valid definitions of planes of simultaneity). The Lorentz transform embodies one possible definition for inertial frames.

Given this, feel free to have doubts about Einstein's convention, but note it is the one he used throughout his papers on special relativity.

I also have a problem with it, in that it requires the ability extend an observer's world line to the prior light cone of a distant event. Even where this is possible, I find it inelegant.

Say I was trying to determine what the y-coordinate of an object were on a graph as I was rotating. I figure out what the y-coordinate is, and a moment later, after I've rotated 30 degrees, I find what the y-coordinate is again. Would it be valid in ANY way for me to just take the average of those two y-coordinates, and claim it as the "radar y-coordinate?"

The aim of this convention is simply to provide one meaningful answer to what time on an oberver's timeline corresonds to some distent event. The recipe (where achievable) is simple and intuitive: find a pair of null lines connecting some point t1 on oberver's world line to the distant event, and another connecting the event to a later point t2 on the observer's world line; assume the distant event occurred at (t1+t2)/2. This can be done unambiguously in SR (may have a couple of solutions, I think, in weird GR geometries), doesn't care about how wild or rotating the observer's state of motion is, and can be computed in any frame of reference and get the same result (for simultaneity by this definition for a specified observer (world line)).

Not sure exactly what you're asking about a strongly curved model, but to get inflation, you just apply a Lorentz Transformation around some event later than the Big Bang event in Minkowski space. The Big Bang gets moved further into the past, and voila... inflation.

So far as I know, it is impossible to impose a global minkowski coordinate system on a general solution in GR. Also, so far as I understand it, it is not unusual to have a GR solution where event e1 is in the prior light cone of e2, but no event in the prior light cone of e2 is in the prior light cone of e1. In such a situations, the Einstein convention is impossible to apply, as is any global minkowski coordinates. However, my definition provides a consistent simultaneity definition for such a case.

 

[Ich]

Just for the record: there is IMHO a most general definition of "as minkowski as possible" coordinates. Operationally, it's a chain of observers, starting at the prime observer, each at rest (two-way doppler=0) and synchronized wrt its neighbours. Time coordinate is the proper time of the prime observer, space coordinate is the distance measured along the chain.
This definition reproduces not only Minkowski distance, but also Rindler distance in the case of an accelerating prime observer.
Mathematically, we're talking about geodesics orthogonal to the prime observer's worldline.

 

[PAllen]

Just for the record: there is IMHO a most general definition of "as minkowski as possible" coordinates. Operationally, it's a chain of observers, starting at the prime observer, each at rest (two-way doppler=0) and synchronized wrt its neighbours. Time coordinate is the proper time of the prime observer, space coordinate is the distance measured along the chain.
This definition reproduces not only Minkowski distance, but also Rindler distance in the case of an accelerating prime observer.
Mathematically, we're talking about geodesics orthogonal to the prime observer's worldline.

Try applying it to the complete Schwarzschild geometry. Two way doppler doesn't exist for events separated by the event horizon. Yet an event outside the horizon can be in the prior lightcone of an event inside the horizon, but not vice versa. I am thinking about sufficiently general notions of simultaneity such that if e1 gets signal from e2, no other information is needed to define a plausible sense of when e2 occurred from the point of view of e1.

Also, note that the rindler metric includes horizons across which two way doppler is impossible.

 

[Ich]

Also, note that the rindler metric includes horizons across which two way doppler is impossible.

Of course. Two way doppler establishes staticity as far as possible. In static coordinates there are sometimes horizons, and static coordinates necessarily reflect their existence. It's not a bug, it's a feature.

I am thinking about sufficiently general notions of simultaneity such that if e1 gets signal from e2, no other information is needed to define a plausible sense of when e2 occurred from the point of view of e1.

Yes, that's rather a revised luminosity distance, if the emitter's luminosity is known. Quite a messy calculation, and vulnerable (think of gravitational lenses), but there's always a price to pay.

 

[Dale]

It's not a bug, it's a feature.

Hehe, only if it is in the documentation!

 

[Ich]

Hehe, only if it is in the documentation!

Of course, page #303 first sentence. My lawyer reads it as: the cutomer has been warned explicitly that using this program will most certainly lead to apocalypse, so in case something goes wrong it is the customer's own fault.
That's how it is in software business.

 

[JDoolin]

Here is the original paper by Einstein:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

The simultaneity convention is explained in section 1. Just use the same convention for a non-inertial observer and you have Dolby and Gull's radar time.

The point is that you have to define your lines of simultaneity by adopting some convention. Any convention you pick is fine, so why not use the same convention that you use for inertial frames?

It depends on how you define "the same convention."

To me, "the same convention" would be to use the line of simultaneity of the momentarily comoving inertial frame. i.e. "the same convention" is the one that yields the "same result."

You're wanting to use the same technique for accelerated observers, but you will get an entirely different result.

For one thing, simultaneity should be based solely on "now." It should not be averaged out between some time in the future depending on what accelerations you plan to make in the future, and some time in the past based on your acceleration history.

 

[JDoolin]

Just for the record: there is IMHO a most general definition of "as minkowski as possible" coordinates. Operationally, it's a chain of observers, starting at the prime observer, each at rest (two-way doppler=0) and synchronized wrt its neighbours. Time coordinate is the proper time of the prime observer, space coordinate is the distance measured along the chain.
This definition reproduces not only Minkowski distance, but also Rindler distance in the case of an accelerating prime observer.
Mathematically, we're talking about geodesics orthogonal to the prime observer's worldline.

I don't get this at all. Minkowski coordinates are x, y, z, t. It's the Cartesian Coordinate system plus time. Operationally it's this way, that way, the other way, and waiting.

 

[JDoolin]

While distant simultaneity is a matter of convention, I prefer choices that rely on some operational definition. The Einstein convention (equiv. radar time) is a particularly intuitive operational definition. However, one issue I have with it in cosmological (GR) context is that it requires that one be able (at minimum) to extend an observer's worldline back to the past light cone of distant event. In cosmology, for a very distant object, this is simply impossible (before the big bang anyone?)

Ah, now I understand what you meant. Because for an accelerating observer, the Einstein convention/Radar time, calculating simultaneity requires you to use both your future motion and your past motion, it can require you to use motion from before you even existed; before your particles had burst asunder from stars. And even before that, before the universe existed.

So each particle in your body has a different calculation of radar time for the most distant events.

 

[PAllen]

I don't get this at all. Minkowski coordinates are x, y, z, t. It's the Cartesian Coordinate system plus time. Operationally it's this way, that way, the other way, and waiting.

An observatory took a picture of a comet crashing into Jupiter. Tell me exactly how you assign x,y,z,t to this? You must use several operational definitions to achieve this. Even a magic 200 million mile tape measure is an operational definition. You could say, completely arbitrarily (and validly), that the event of my taking the picture is (0,0,0,1) and the event of the collision is (1,1,1,0). But then, to form a metric (or supply the 'c' in a Lorentz transform) you need operational definitions to relate these coordinates to observable invariants.

 

[Dale]

It depends on how you define "the same convention."

To me, "the same convention" would be to use the line of simultaneity of the momentarily comoving inertial frame. i.e. "the same convention" is the one that yields the "same result."

The Einstein synchronization convention is an experimental procedure that can be used to determine if events were simultaneous. Different observers disagree on the result of this procedure. That is the whole point of the relativity of simultaneity.

Defining the convention by the result is rather inappropriate in this case and doesn't even work for inertial frames.

Ah, now I understand what you meant. Because for an accelerating observer, the Einstein convention/Radar time, calculating simultaneity requires you to use both your future motion and your past motion

The Einstein synchronization convention requires you to use both your future and past motion for an inertial observer too. It is just that inertial motion is particularly easy to describe.

JDoolin, whether it is the same convention or not, if you have a strong preference for one convention over another for some personal reason (or even for no reason at all) that is perfectly fine. You don't have to justify your preference to me nor to anyone else. What is not fine is for you to attempt to elevate your personal preference to the status a physical requirement. No coordinate system has that status. Do you understand that now?

 

[JDoolin]

An observatory took a picture of a comet crashing into Jupiter. Tell me exactly how you assign x,y,z,t to this? You must use several operational definitions to achieve this. Even a magic 200 million mile tape measure is an operational definition. You could say, completely arbitrarily (and validly), that the event of my taking the picture is (0,0,0,1) and the event of the collision is (1,1,1,0). But then, to form a metric (or supply the 'c' in a Lorentz transform) you need operational definitions to relate these coordinates to observable invariants.

One needs two eyes with a high enough resolution to see the event, spaced far enough apart that the parallax can be measured at that resolution. In order to find the value of c, techniques from Ole Romer, James Bradley, Louis Fizeau would work. If we had good enough clocks and cameras, even Galileo's method would work.

Once you have the speed of light, and parallax measurements of the distance, you can calculate the time the event happened by dividing the distance by the speed of light.

 

[PAllen]

One needs two eyes with a high enough resolution to see the event, spaced far enough apart that the parallax can be measured at that resolution. In order to find the value of c, techniques from Ole Romer, James Bradley, Louis Fizeau would work. If we had good enough clocks and cameras, even Galileo's method would work.

Once you have the speed of light, and parallax measurements of the distance, you can calculate the time the event happened by dividing the distance by the speed of light.

Two way speed of light can be objectively defined for one observer. One way speed of light being constant (even for an inertial observer) is an additional assumption that cannot be directly verified, and is one of many equally possible conventions (read all the papers you listed above carefully). Assuming one way speed of light is constant is exactly Einstein's convention. There is no way for one observer to measure one way speed of light. You need two separated observers, with distant clock synchronization established.

Parallax requires distant simultaneity between two separate observers (your eyes or even opposite ends of the Earth are no good for astronomic events). So you are back to a conventions about distant simultaneity to actually measure parallax - which all the papers under discussion here agree is impossible to define objectively.

Trying to be explicit about the operational definitions behind real measurements is exactly what leads to relativity, and to clarifying which parts of it are fundamental features of the universe and which parts are possibly useful conventions. This is also part of what leads to the quantum revolution.

 

[JDoolin]

The Einstein synchronization convention is an experimental procedure that can be used to determine if events were simultaneous. Different observers disagree on the result of this procedure. That is the whole point of the relativity of simultaneity.

Defining the convention by the result is rather inappropriate in this case and doesn't even work for inertial frames.

The Einstein synchronization convention requires you to use both your future and past motion for an inertial observer too. It is just that inertial motion is particularly easy to describe.

JDoolin, whether it is the same convention or not, if you have a strong preference for one convention over another for some personal reason (or even for no reason at all) that is perfectly fine. You don't have to justify your preference to me nor to anyone else. What is not fine is for you to attempt to elevate your personal preference to the status a physical requirement. No coordinate system has that status. Do you understand that now?

We have opinions about the two conventions based on facts; either facts we have wrong, or facts we have right. Even if we differ in opinion, we should agree about the facts.


You are saying that the whole point of relativity is that different observer's disagree on the results of this procedure. I don't know if you mean it this way, but it sounds like you are saying that there is some arbitrary or random way in which the results disagree, (perhaps that different observers view things differently based on the opinions of the observer, or how they weigh the information; or based on whether they decide to use cartesian or spherical coordinates, or whether they measure in feet or meters.) This is NOT the point of Special Relativity.

The point of Special Relativity is to describe exactly how and why different inertial observer's disagree on the results of the procedure. And the result of of that description is the Lorentz Transformation equations.

Changing from one inertial reference frame to another using the Lorentz Transformations is a NON-rigid transformation. In Cartesian space, the objects contract or uncontract. Events move apart or closer together.

It is my "opinion" that the easiest way to determine the observations of Barbara, at any given time, is to apply the Lorentz Transformations until we are looking at the reference frame in which Barbara is at currently at rest. (Then to do further calculation to account for the finite speed of light) It is my opinion that Tom Fontenot's procedure to use the Momentarily Comoving Inertial Reference Frame (MCIRF) to calculate the Current Age of Distant Objects" (CADO), is a better method for describing simultaneity for accelerating observers than the Einstein Convention.

Another fact is that the Lorentz Transformation already provides a one-to-one mapping of the events.

Another fact is that performing the Lorentz Transformation has no effect on the clock-values that you would get using radar time.

I leave it to you to describe your opinion.

 

[Dale]

It is my "opinion" that the easiest way to determine the observations of Barbara, at any given time, is to apply the Lorentz Transformations until we are looking at the reference frame in which Barbara is at currently at rest. (Then to do further calculation to account for the finite speed of light) It is my opinion that Tom Fontenot's procedure to use the Momentarily Comoving Inertial Reference Frame (MCIRF) to calculate the Current Age of Distant Objects" (CADO), is a better method for describing simultaneity for accelerating observers than the Einstein Convention.

Is it your opinion that that is the only correct way to determine the observations of Barbara at any given time?

Assuming that your answer to the above is that it is not your opinion that it is the only way, then the only potential disagreement we have is here:

Another fact is that the Lorentz Transformation already provides a one-to-one mapping of the events.

This may only require clarification and we may actually agree. Are you referring here to a single inertial reference frame, or are you referring to a non-inertial reference frame formed by stitching together Barbara's MCIF's?

 

[JDoolin]

Is it your opinion that that is the only correct way to determine the observations of Barbara at any given time?

Let me be explicit.

Pick an arbitrary event on Barbara's worldline. Construct an inverted light-cone down from that event into the past. This lightcone represents the locus of events that Barbara will see at that instant. This lightcone will correctly tell you what events Barbara sees at that instant, but will not tell you the correct distances.

If you want to know the correct distances, the procedure I would follow would be to perform the Lorentz Transformation so that the tangent of Barbara's world-line at that instant is "vertical" and her lines of simultaneity are perpendicular to the tangent world-line. Then the x, y, and z coordinates of the events in the inverted light cone will be the correct distances that Barbara sees.

There may be mathematical methods to produce the same results, so I can't say this is the only way, but if you find another way to determine the observations, it should produce the same results.

Assuming that your answer to the above is that it is not your opinion that it is the only way, then the only potential disagreement we have is here:This may only require clarification and we may actually agree. Are you referring here to a single inertial reference frame, or are you referring to a non-inertial reference frame formed by stitching together Barbara's MCIF's?

Actually, I think possibly, we've been meaning different things by "one-to-one"

When I'm saying the Lorentz Transformation produces a one-to-one mapping of events, I mean for every event in Barbara's current comoving inertial reference frame, the same event happens in every other inertial reference frame.

I think, possibly what you have been calling the one-to-one mapping is: for every event that happens to Barbara (a continuous locus of events forming her curving worldline) there is an event that happens to Alex (a continuous locus of events forming his straight worldline). So by using the radar time, the Einstein convention, you can construct a one-to-one mapping between the time on Barbara's clock and the time on Alex's clock.

Is your main concern, then, mapping time to time, or is your main concern mapping event to event?

Edit: Just to be clear, The MCIRF and CADO (Current Age of Distant Objects) definitely do not create a 1-to-1 mapping of t_Barbara to t_Alex.

 

[JDoolin]

Two way speed of light can be objectively defined for one observer. One way speed of light being constant (even for an inertial observer) is an additional assumption that cannot be directly verified, and is one of many equally possible conventions (read all the papers you listed above carefully). Assuming one way speed of light is constant is exactly Einstein's convention. There is no way for one observer to measure one way speed of light. You need two separated observers, with distant clock synchronization established.

Parallax requires distant simultaneity between two separate observers (your eyes or even opposite ends of the Earth are no good for astronomic events). So you are back to a conventions about distant simultaneity to actually measure parallax - which all the papers under discussion here agree is impossible to define objectively.

Trying to be explicit about the operational definitions behind real measurements is exactly what leads to relativity, and to clarifying which parts of it are fundamental features of the universe and which parts are possibly useful conventions. This is also part of what leads to the quantum revolution.

I'm not sure if I'm entirely following your argument.

Yes, I mean to make the assumption that the one-way speed of light is the same as the two-way speed of light. I believe that assumption is implicit in the Lorentz Transformations.

I agree with your point that we do not have the resources to put two comoving cameras in positions far away from each other so that they can get parallax for astronomical distances. But in your earlier post you proposed a 200 million mile meter stick. Instead I would prefer a pair of eyes with perfect (infinite density) resolution. Then I would have no trouble perceiving the parallax of Jupiter.

Obviously in real life, I can't have the infinite resolution pair of eyes. But I can easily put my brain in the center of those eyes, and have the two images processed simultaneously. I have an operational definition of simultaneous; that the events that reached my brain simultaneously are simultaneous* in my brain's frame of reference.

Even if my eyes can't make out parallax at any great distance, the mathematical form for calculating parallax still gives a finite value; a value smaller than the resolution power of my rods and cones, of course, but it is still an unambiguous determination of the distance.

*Edit: (reached my eyes simultaneously)

 

[Dale]

There may be mathematical methods to produce the same results, so I can't say this is the only way, but if you find another way to determine the observations, it should produce the same results.

OK, we agree here then.

Actually, I think possibly, we've been meaning different things by "one-to-one"

When I'm saying the Lorentz Transformation produces a one-to-one mapping of events, I mean for every event in Barbara's current comoving inertial reference frame, the same event happens in every other inertial reference frame.

Yes, we have been meaning different things, thanks for the clarification.

The mapping I am referring to is the mapping from points in the manifold (events in spacetime) to coordinates (ordered 4-tuples of real numbers). For any mathematically valid coordinate system this mapping must be 1-to-1 otherwise the mapping will not be invertible and you will not be able to transform coordinate systems.

When we speak of an observer's "perspective" in relativity we are referring to this mapping between points and numbers. For an inertial observer there is a standard mapping given by the Einstein/radar convention, and it is 1-to-1. For a non-inertial observer there is no standard mapping, and we have to be a little more explicit in defining it. The radar approach gives a strange mapping (bent lines of simultaneity) that is 1-to-1, and the many MCIF's approach gives a less strange (straight lines of simultaneity) mapping that is not 1-to-1.

 

[JDoolin]

OK, we agree here then.

Yes, we have been meaning different things, thanks for the clarification.

The mapping I am referring to is the mapping from points in the manifold (events in spacetime) to coordinates (ordered 4-tuples of real numbers). For any mathematically valid coordinate system this mapping must be 1-to-1 otherwise the mapping will not be invertible and you will not be able to transform coordinate systems.

When we speak of an observer's "perspective" in relativity we are referring to this mapping between points and numbers. For an inertial observer there is a standard mapping given by the Einstein/radar convention, and it is 1-to-1. For a non-inertial observer there is no standard mapping, and we have to be a little more explicit in defining it. The radar approach gives a strange mapping (bent lines of simultaneity) that is 1-to-1, and the many MCIF's approach gives a less strange (straight lines of simultaneity) mapping that is not 1-to-1.

I'm not sure if I fully grasp the idea of a manifold, but are there two other possible one-to-one mappings that we are neglecting? Does a manifold have to be about distant simultaneity, or could it be a manifold mapping mapping causality? In determining "radar time" you use a signal from Barbara to Alice, and a signal back from Alice to Barbara. These two signals both represent unambiguous one-to-one mappings between Alice's world-line and Barbara's world-line. If we extended these out to inverted lightcones from each observer's positions, and allowed the lightcones to stack as the observers progressed forward through time, would these stacks of lightcones form manifolds?

Is there a way we could have a separate manifold for each observer representing distant "observations" instead of distant "simultaneity?" ...or am I misunderstanding the manifol idea?

 

[JDoolin]

Causality; the idea that events from the future cannot affect the past.

Radar simultaneity relies on the mathematical tautology that two one-to-one functions added together will also be one-to-one. This method does not rely in any way on the results of Special Relativity. It only relies only on causality; that the future cannot affect the past.

The MCIRF/CADO method relies on the assumption that the results of special relativity are valid; that the one-way and two-way speeds of light are the same; that the Lorentz Transformations map between intertial reference frames. Within this framework "simultaneity" is already defined as a function of the observer's velocity; a line of constant t, or a line of constant t' in spacetime represents a line of simultaneity.

The two methods have an equal number of assumptions, I think.

(1) They both require the causality requirement to be met.

(2a) Radar time also requires that the definition of simultaneity be one-to-one.

(2b) MCIRF/CADO requires that the Lorentz Transformations apply and uses lines of constant t to define simultaneity.

 

[Dale]

I'm not sure if I fully grasp the idea of a manifold, but are there two other possible one-to-one mappings that we are neglecting? Does a manifold have to be about distant simultaneity, or could it be a manifold mapping mapping causality? In determining "radar time" you use a signal from Barbara to Alice, and a signal back from Alice to Barbara. These two signals both represent unambiguous one-to-one mappings between Alice's world-line and Barbara's world-line. If we extended these out to inverted lightcones from each observer's positions, and allowed the lightcones to stack as the observers progressed forward through time, would these stacks of lightcones form manifolds?

Is there a way we could have a separate manifold for each observer representing distant "observations" instead of distant "simultaneity?" ...or am I misunderstanding the manifol idea?

A manifold is a very fundamental topological space. A plane, the surface of a sphere, and the surface of a torus are all examples of 2D manifolds. Manifolds, by themselves, don't have any notion of angles, relative velocities, durations, distances, or causality. They are topological spaces rather than geometrical spaces.

In order to add notions of angles, relative velocities, durations, distances, and causality a manifold can be equipped with a metric. The metric defines these coordinate-independent geometrical relationships in the manifold. The purpose of expressing the laws of physics in terms of tensors is to identify these underlying geometrical relationships that do not depend on the coordinate system.

If we wish to use notions like simultaneity or co-location then we need to equip the manifold with a coordinate system. A coordinate system is not necessary for these geometric relationships, but it makes working with the math a lot easier. When you express the metric in terms of your coordinates then you provide the key connection beteween your coordinate system and the underlying geometry.

I hopes this helps clarify the differences between manifolds, metrics, and coordinates.

 

[Dale]

(2a) Radar time also requires that the definition of simultaneity be one-to-one.

This isn't a requirement of radar time. It is a requirement of any coordinate system on any manifold. If a mapping isn't 1-to-1 then it isn't a coordinate system by definition.

 

[Anamitra]

I would request the audience to consider[reconsider] the links below:
https://www.physicsforums.com/showpost.php?p=2871702&postcount=136
https://www.physicsforums.com/showpost.php?p=2872317&postcount=137

In the first link there are two examples and in the second one there are two partial answers.I find it important to bring the matter into further consideration with the participation of the audience.

 

[Dale]

I would request the audience to consider[reconsider] the links below:
https://www.physicsforums.com/showpost.php?p=2871702&postcount=136
https://www.physicsforums.com/showpost.php?p=2872317&postcount=137

In the first link there are two examples and in the second one there are two partial answers.I find it important to bring the matter into further consideration with the participation of the audience.

What aspect do you want to consider further?

 

[Anamitra]

Of course, if you smooth out the corners then you are taking a different path so you will, in general, get a different result. However, if the path is only changed very slightly and the Christoffel symbols vary only slightly over that change then the final result will differ only slightly. Since a sphere is so symmetric I wouldn't expect a large difference without a large change in the path, but I would have to work it out for myself to be sure.

On rounding off the edges the direction of the parallel transported vector does not change or it changes by a very small amount/insignificant amount.If the corners are sharp there is a significant change in the orientation of the vector notwithstanding the fact that the nature of the surface included does not change in a significant manner.This seems quite peculiar and I would request some clarification from the side of the audience.In case Dalespam has done some calculations he is requested to present them. But this is not mandatory

[This is in relation to the second link in Thread #325]

 

[Dale]

On rounding off the edges the direction of the parallel transported vector does not change or it changes by a very small amount/insignificant amount.If the corners are sharp there is a significant change in the orientation of the vector notwithstanding the fact that the nature of the surface included does not change in a significant manner.

Do you have the metric for the two spaces in question? I do not.

 

[Anamitra]

The metric relating to the surface does not change at all,so far its form is concerned. We are simply changing the curves --that too slightly --to investigate how the large change in the direction of the parallel-transported vector corresponds to the fact the of surface enclosed has changed by a small amount.The metric does not change in form.

 

[Dale]

I don't know the form of the metric for the space you are talking about here. Do you?

 

[Anamitra]

Just think of an ordinary sphere--you don't need to consider a space-time sphere to under the particular aspect of the problem being considered.

Metric:
{ds}^{2}{=}{r}^{2}{(}{{d}{\theta}}^{2}{+}{{sin}{\theta}}^{2}{{d}{\phi}}^{2}{)}

We simply do not have two spaces here as referred to in thread #327