How do the concepts of 'constant c' and 'locality' work in GR?

[arindamsinha]

Drawing on a number of threads in this forum, and my very limited understanding of relativity, I am intrigued by the following thoughts:

If I have understood correctly, in GR, c is supposed to be only a 'local global constant', and not exactly a Universal constant.

Meaning that, two distant observer will each measure c 'locally' to be a given constant value (299,792,458 m/s as per most sources), and this is valid everywhere in the Universe 'locally'. In addition, both observers can conclude that the c at the other observers location is different from his local c (e.g. c is comparatively lower in the vicinity of a large mass, than farther away).

My questions are:
- How is 'local' or 'locality' defined (i.e. where is the boundary between local and non-local)?
- If c can be observed to have different values at various distances from an observer as per GR, does SR have any validity in the real 'gravitational' Universe?
- How can c then be considered an 'ultimate speed limit of the Universe', since it can vary at different locations even for a single observer?

 

[bcrowell]

The main point to understand is that GR doesn't have any meaningful way to describe the velocity of an object that is distant from the observer.

You seem to have gotten the impression that c is variable. It isn't. In the SI, it has a defined value. In the units normally used be relativists, it equals 1. 1 doesn't have a varying value.

In addition, both observers can conclude that the c at the other observers location is different from his local c (e.g. c is comparatively lower in the vicinity of a large mass, than farther away).

No, each observer can conclude that the other observer is measuring the ordinary, local value of c for phenomena close to him.

- How is 'local' or 'locality' defined (i.e. where is the boundary between local and non-local)?

There is no sharp boundary. You can, for example, quantify curvature effects by saying that they scale like a certain power of the distance.

- If c can be observed to have different values at various distances from an observer as per GR, does SR have any validity in the real 'gravitational' Universe?

SR is valid locally, as an approximation. The error in the approximation scales like a certain power of the distance.

- How can c then be considered an 'ultimate speed limit of the Universe', since it can vary at different locations even for a single observer?

It's a local speed limit.

 

[arindamsinha]

Ben,

Thank for the response.

I have seen in some references that light travels slower in the vicinity of a large mass that farther away. Doesn't that mean that the c differs at different locations?

Or am I missing the whole concept?

 

[tom.stoer]

I have seen in some references that light travels slower in the vicinity of a large mass that farther away. Doesn't that mean that the c differs at different locations?

That's exactly what Ben means with

The main point to understand is that GR doesn't have any meaningful way to describe the velocity of an object that is distant from the observer.

The so-called Shapiro delay can be cast in a form where it seems as if c - as seen by a distant observer - is slower, i.e. c < 1, near a large mass. But at each point a local observer would not see any variation of c.

 

[bcrowell]

The so-called Shapiro delay can be cast in a form where it seems as if c - as seen by a distant observer - is slower, i.e. c < 1, near a large mass. But at each point a local observer would not see any variation of c.

Yeah, vast amounts of harm are done by these popularizations that uncritically describe observers as perceiving things at a distance. One big example is descriptions of matter falling into a black hole, as "seen" by an observer at infinity. Another is when people talk about cosmological expansion and assume uncritically that it makes sense to describe the rate of increase of proper length as a velocity "seen" by a particular observer (which leads, for example, to the incorrect belief that this velocity can't be >c for observable galaxies).

 

[arindamsinha]

Ben and Tom,

If I understand what you are saying, GR should only be applied locally, and the local c to everybody is the known constant. GR should not be used to explain or predict what a distant observer might see. Is this understanding correct?

 

[bcrowell]

If I understand what you are saying, GR should only be applied locally, and the local c to everybody is the known constant. GR should not be used to explain or predict what a distant observer might see. Is this understanding correct?

No, that would be much too broad. When I look at a cosmologically distant galaxy, I see something. What I see is red-shifted, for example. GR predicts correctly what I see.

If you tell us something about your background in math and physics, we might be able to point you to a book to read that would help you get going more on understanding the subject deeply. But you can't just interpret everything in terms of sound-bites.

 

[arindamsinha]

If you tell us something about your background in math and physics, we might be able to point you to a book to read that would help you get going more on understanding the subject deeply. But you can't just interpret everything in terms of sound-bites.

I did my bachelor's degree in engineering where I had some advanced level courses in physics and mathematics, though my major was in computer science. This of course was 20+ years ago. Since then physics has been more of a hobby than anything else.

So I suppose I can handle reasonably advanced calculus, but not tensor calculus etc.

If there are some really good books you can suggest, that will be helpful. Thanks in advance.

PS: I have gone through some of the papers and books by Einstein, though cannot claim to have studied them in great depth, or understood everything they say.

 

[HomogenousCow]

I am asking some similar questions in my thread, https://www.physicsforums.com/showthread.php?t=638937

I too find the this subject very confusing, I know this is just because my lack of understanding on the subject, but many threads seem to contradict each other in how frames are dealt with in GR.

 

[ImaLooser]

I did my bachelor's degree in engineering where I had some advanced level courses in physics and mathematics, though my major was in computer science. This of course was 20+ years ago. Since then physics has been more of a hobby than anything else.

So I suppose I can handle reasonably advanced calculus, but not tensor calculus etc.

If there are some really good books you can suggest, that will be helpful. Thanks in advance.

PS: I have gone through some of the papers and books by Einstein, though cannot claim to have studied them in great depth, or understood everything they say.

I'd recommend Reflections on Relativity by Kevin Brown. It might be perfect for you. Rigorous but not a boring textbook.

 

[arindamsinha]

I'd recommend Reflections on Relativity by Kevin Brown. It might be perfect for you. Rigorous but not a boring textbook.

Thanks a lot