Curved Space-Time and the Speed of Light

[Dale]

I find no problem in applying this idea of "unique parallel transport"to my concepts.

Excellent. Then please do so rigorously, i.e. In a manifestly covariant way. Perhaps when you do so you will find a problem.

 

[nismaratwork]

On the Uniqueness of Parallel Transport

The concept of parallel concept can be made unique by choosing a curve between a pair of points such that the inner product of any pair of vectors get preserved along the curve.

[Mathematically such a curve is one for which the components of the covariant derivatives of g(mu,nu) are zero]

This is a commonly accepted practice in relation to methods of General Relativity and has been discussed quite elaborately [from the mathematical point of view]by Robert M. Wald in the text "General Relativity" in Chapter Three[Curvature],section 3.1[Derivative Operators and Parallel Transport]

In the introduction to this chapter(ie chapter three) the author remarks "We will show in section 3.1 that a given metric(of any signature), there is a unique definition of parallel transport which presreves the the inner product of all pairs of vectors. The existence of a metric gives rise to a unique notion of parallel transport and ,thus, to an intrinsic notion of the curvature of the manifold."
Such a definition of uniqueness in relation to parallel transport ensures the the uniqueness of derivative operator for a given system of metrics.

I find no problem in applying this idea of "unique parallel transport"to my concepts.

I say this with all due respect, but you last sentence sounds a bit mad to me. If you can do that, then do it. I've read this thread from the beginning, and you seem to be saying the same thing in a hundred different ways. I think the keyword here is "[YOUR] concepts" which sounds fun, but don't reflect physics as I understand them, or as anyone else here does it seems.

You're saying outright that DaleSpam is wrong, and that in a curved space you find something meaningful by comparing distant speeds? I'm sorry, but that makes no sense at all to me, can you show some work to support this?!

 

[yuiop]

... then the observations can be logically explained by the speed of light slowing down deeper in the gravitational field. ...

Indeed. In that respect a quite a parallel view is expressed in the Lorentz ether theory for flat spacetime.

However, and here is a key difference, in the extreme case, passed the event horizon, the interpretation under GR must be different because of...

Do you agree that if we dismiss the light slowing down notion, that the only alternative is to accept that a single stationary rod extending from r1 to r2, really does have two separate physical lengths at the same time?

 

[starthaus]

Do you agree that if we dismiss the light slowing down notion, that the only alternative is to accept that a single stationary rod extending from r1 to r2, really does have two separate physical lengths at the same time?

1. Light doesn't "slow down", the speed of light measured in a small vicinity is always c.
2. The rod does not "have two separate physical lengths", it is just that the local observer measures a different length from the length measured by a distant observer.

1. and 2. above can be proven really easily in three lines of simple computations.

 

[Passionflower]

Do you agree that if we dismiss the light slowing down notion, that the only alternative is to accept that a single stationary rod extending from r1 to r2, really does have two separate physical lengths at the same time?

Well, you can't have your cake and eat it too. If you assume that light slows down in a gravitational field then, at the event horizon, it must completely stop. Then how fast do you think it goes passed the event horizon? I know you can do fancy math tricks there but we are talking physics right?

Could you explain the case where you think you have two separate physical lengths at the same time?

1. Light doesn't "slow down", the speed of light measured in a small vicinity is always c.

Well obviously, as the gravitational effect in a small vicinity is always zero and the clock is running slower so it will not be able to detect the assumed slowing down of light.

 

[yuiop]

1. Light doesn't "slow down", the speed of light measured in a small vicinity is always c.
2. The rod does not "have two separate physical lengths", it is just that the local observer measures a different length from the length measured by a distant observer.

1. and 2. above can be proven really easily in three lines of simple computations.

If we agree that the stationary rod extending from r1 to r2 has some physical proper length, how do we explain that the two observers at either end obtain different radar lengths?

If you explain that by gravitational time dilation, then that is in effect admitting that clocks lower down run slower. If we admit that clocks lower down run slower and yet they measure the local speed of light to be the same as local speed higher up, then the logical conclusion is that the speed of light must in some real sense be slower lower down.

P.S. @2. I am not talking about a distant observer. I am talking about 2 local observers. One at the top end of the the vertical rod and one at the lower end.

 

[starthaus]

.

P.S. @2. I am not talking about a distant observer. I am talking about 2 local observers. One at the top end of the the vertical rod and one at the lower end.

I am talking about a distant observer vs. local observer.

 

[yuiop]

Could you explain the case where you think you have two separate physical lengths at the same time?

If two observers at either end of the vertical rod measure the radar length to be different, then there are 2 possible explanations:

1)The rod has two separate physical lengths at the same time (very unlikely).
2)The clock of the observer at the lower end is physically running slower than the clock of the observer at the top end. (more likely).

If we accept proposition (2) and note that the observer lower down with the slower clock measures the local speed of light to be the same as the local speed of light measured by the observer at the top, then it logically follows that the real speed of light lower down must be slower than higher up. I am not talking about what is measured, but how a sentient being would explain what is measured.

It is not difficult to prove that clocks lower down physically run slower than clocks higher up. All we have to do is start with two synchonised clocks higher up, send one to a lower level and hold it stationary for a while and then send the second clock down. When they are alongside each other the clock that been down the longest will have "aged" less than the other clock. A sort of gravitational twins experiment.

 

[Passionflower]

If we agree that the stationary rod extending from r1 to r2 has some physical proper length, how do we explain that the two observers at either end obtain different radar lengths?

If you explain that by gravitational time dilation, then that is in effect admitting that clocks lower down run slower. If we admit that clocks lower down run slower and yet they measure the local speed of light to be the same as local speed higher up, then the logical conclusion is that the speed of light must in some real sense be slower lower down.

P.S. @2. I am not talking about a distant observer. I am talking about 2 local observers. One at the top end of the the vertical rod and one at the lower end.

Again I do not think there is anything wrong in having the position that the speed of light slows down in a gravitational field (it is, by analogy, preferring LET over SR), but as soon as you do that you cannot defend the interior solution of the Schwarzschild metric on physical grounds. In the modern interpretation of GR there is not such thing as "one physical proper length". To try to put it in words: the length of something is an operational function not an intrinsic property.

 

[yuiop]

Again I do not think there is anything wrong in having the position that the speed of light slows down in a gravitational field (it is, by analogy, preferring LET over SR), but as soon as you do that you cannot defend the interior solution of the Schwarzschild metric on physical grounds. In the modern interpretation of GR there is not such thing as "one physical proper length". To try to put it in words: the length of something is an operational function not an intrinsic property.

.. which is basically saying what I said earlier. If we dismiss the notion of light slowing down in a real sense deeper in a gravitational field we are forced to accept that a single vertical rod has two (or more) physical proper lengths at the same time.

P.S. I think most people here know I prefer the more physical interpretaion of LET over SR, even though they are the same mathematically and even though LET is very unfashionable these days. As for what happens exactly at the event horizon and below, I am not certain. It is very difficult to analyse. The place to start is outside the EH where we have experimental data to confirm the theory.

 

[Passionflower]

.. which is basically saying what I said earlier. If we dismiss the notion of light slowing down in a real sense deeper in a gravitational field we are forced to accept that a single vertical rod has two (or more) physical proper lengths at the same time.

No, it is not the same as what I am saying.

 

[yuiop]

No, it is not the same as what I am saying.

Ok, I will ponder some more on the difference in meaning between:

a)... there is no such thing as "one physical proper length".

and

b) has two (or more) physical proper lengths at the same time.

 

[Passionflower]

a)... there is no such thing as "one physical proper length".

This is what I wrote:

In the modern interpretation of GR there is not such thing as "one physical proper length". To try to put it in words: the length of something is an operational function not an intrinsic property.

 

[Mentz114]

If 'proper length' is loosely defined as the length measured by an observer in their locally Minkowski frame, then there's an infinity of them. And it's not a very useful concept for this very reason. Also in regions of strong curvature the locally flat regions are getting too small.

It seems more meaningful to use operational definitions - like Einsteins clocks and rulers, to which we can add radar and lasers.

 

[Dale]

If 'proper length' is loosely defined as the length measured by an observer in their locally Minkowski frame, then there's an infinity of them. And it's not a very useful concept for this very reason.

I agree with your sentiments here. Also, the discussion here was previously about "physical length" (the "proper" came later and is wrong). "Physical length" is an even sloppier and less useful concept.

The "physical length" described here seems to be the length measured by radar, and even in flat spacetime one object has an infinite number of "physical lengths" depending on the relative speed of the observer. Why should it be the least bit surprising or worrisome that in curved spacetime one object has different "physical lengths" depending on the relative position of the observer?

 

[AWA]

-it is well known that coordinate speed of light is not constant in GR (it is in SR)

From a layperson point of view. Have I misread that GR is generalization of SR? Can GR contradict SR in its basic postulate about c? Am I wrong to understand from the quote that there is some contradiction between SR and GR?

By reading this thread I get the idea that GR seems to be easier to apply locally and that to get to physical plausible velocities in different (non local) frames of reference one has to recur to SR. Is this reasonable at all?

 

[Mentz114]

From a layperson point of view. Have I misread that GR is generalization of SR? Can GR contradict SR in its basic postulate about c? Am I wrong to understand from the quote that there is some contradiction between SR and GR?

By reading this thread I get the idea that GR seems to be easier to apply locally and that to get to physical plausible velocities in different (non local) frames of reference one has to recur to SR. Is this reasonable at all?

GR is a generalization of SR but it should be interpreted from local frames. In these frames the speed of light is always measured to be the same c. It is difficult to define operationally the speed of light in another frame, and it would be coordinate dependent making it a not-very-useful concept.

 

[Dale]

From a layperson point of view. Have I misread that GR is generalization of SR? Can GR contradict SR in its basic postulate about c? Am I wrong to understand from the quote that there is some contradiction between SR and GR?

Even in SR the postulate refers only to inertial frames and it is possible to have non-inertial frames where the coordinate velocity is not c.

By reading this thread I get the idea that GR seems to be easier to apply locally and that to get to physical plausible velocities in different (non local) frames of reference one has to recur to SR. Is this reasonable at all?

No, the point is that you simply cannot get physically plausible velocities non-locally. If you naively try to use SR then your results will be wrong if the curvature is significant.

 

[AWA]

No, the point is that you simply cannot get physically plausible velocities non-locally. If you naively try to use SR then your results will be wrong if the curvature is significant.

Ok, I understand this to mean that in ths non-inertial frames where coordinate velocity can be higher than c, in their own frame of reference the limit is still c, thus it makes no sense to ask if their "physical" velocity is higher than c? does this mean that accelerated frames like for example gravitational fields wih curved spacetime, have properties that are in a way only local, and here is where GR enters with its solutions?

 

[Dale]

Ok, I understand this to mean that in ths non-inertial frames where coordinate velocity can be higher than c, in their own frame of reference the limit is still c, thus it makes no sense to ask if their "physical" velocity is higher than c?

I don't know what you are trying to say here.

does this mean that accelerated frames like for example gravitational fields wih curved spacetime, have properties that are in a way only local, and here is where GR enters with its solutions?

Yes.

 

[AWA]

I don't know what you are trying to say here.

You said:"it is possible to have non-inertial frames where the coordinate velocity is not c." and : "the point is that you simply cannot get physically plausible velocities non-locally".
I was just trying to rephrase it my way. But not succesfully I guess.

 

[yuiop]

If 'proper length' is loosely defined as the length measured by an observer in their locally Minkowski frame, then there's an infinity of them. And it's not a very useful concept for this very reason. ...

There is an infinity of coordinate lengths of an object because it depends on the relative velocity of the observer, but "proper length" is usually defined as the length measured by an observer at rest with the object being measured and this is a single value that all observers can agree on. In SR the proper length of a non-accelerating object can be measured using short measuring rods laid end to end (ruler length) or by timing light signals (radar length).

It seems more meaningful to use operational definitions - like Einsteins clocks and rulers, to which we can add radar and lasers.

While the proper length measured by radar is equivalent to ruler length for an inertially moving object, the radar length of an accelerating object even in flat spacetime can vary depending upon which end you take the radar measurement from. The ruler measurement is not dependent on the location of the observer as long as the rulers are at rest with the object being measured. Ruler length remains a true indication of the proper length of an object even when it is accelerating.

I agree with your sentiments here. Also, the discussion here was previously about "physical length" (the "proper" came later and is wrong). "Physical length" is an even sloppier and less useful concept.

The "physical length" described here seems to be the length measured by radar, and even in flat spacetime one object has an infinite number of "physical lengths" depending on the relative speed of the observer. Why should it be the least bit surprising or worrisome that in curved spacetime one object has different "physical lengths" depending on the relative position of the observer?

What I am looking for is "intrinsic length". This is the length that an object must have even when it is a gravitational field and different observers at different ends of the object get different radar lengths. I guess the nearest you can get to the intrinsic length is the ruler length using infinitesimal measuring rods at rest with the object being measured. Physical length is not usually defined clearly in the textbooks, but usually equated with proper length. However, I guess the coordinate length is the nearest you can get to a definition of physical length because to any given observer the coordinate length appears to be the "physically real" length of the object in every way that he can measure it, but other observers will have a different opinion of what the physical length of the object is depending on their state of motion. What I am trying to get at, is that in a gravitational field an object might have an infinite amount of radar lengths depending on where you take the measurement my intuition is that the object has a single "intrinsic length" rather than a sort of quantum mechanical fuzzy superimposition of an infinite amount of observer dependent lengths.

So earlier when I said physical length, it was a bit sloppy and perhaps I should have said proper length or ruler length.

Just kicking some ideas around here

 

[Austin0]

While the proper length measured by radar is equivalent to ruler length for an inertially moving object, the radar length of an accelerating object even in flat spacetime can vary depending upon which end you take the radar measurement from. The ruler measurement is not dependent on the location of the observer as long as the rulers are at rest with the object being measured. Ruler length remains a true indication of the proper length of an object even when it is accelerating.

Just kicking some ideas around here

Hi kev Just a couple of questions. Assuming constant proper acceleration; What is the basis of the difference of measured radar lengths in an accelerating frame depending on which end the signals are sent from?
A) SImply the result of the acceleration during the propagation of the signals
or
B) The assumption of a dilation differential between the clocks in the front and back?


A related question stemming from your posting in another thread regarding radar ranging of the radial distance [ length] between two points at different G potentials.
In that case you stated ; based on the assumption of invariant c that the distance "up" would be different than the distance "down" due to the greater dilation of the clocks at the lower potential. This is clear enough.
My question is: does the assumption of a constant c mean that it is not possible to actually measure the speed of a light signal in this situation. It seems reasonable that this would be the case. That even if you could momentarily synch the respective clocks that the difference in periodicity between them would neccessarily result in different measured speeds up and down. But in case there are other factors I may be missing I thought I would check.
I.e. Is it possible to measure the radial speed of light up and down the well ? And if so how??
Thanks