Does Light Travel the Shortest Path in Curved Space-Time Around a Neutron Star?

[Ibix]

We should agree on the electro-magnetic radiation speed but not agree on the distance - the height of the building.

Ok. But in that case the two observers don't share a notion of space, so light didn't follow the same spatial path for them.

 

[PeterDonis]

We should agree on the electro-magnetic radiation speed but not agree on the distance - the height of the building.
The ruler is the same but we can't agree about its length.

In the case of two static observers at different heights (top and bottom of a building), you are incorrect. The observers agree on the height of the building but not the time taken for light to travel between them.

Any speeds we measure are the same

No, they are not. Each observer measures the same local speed of light, using measurements made just at their height; but the two observers are at different heights, so any measurement involving light traveling between both of them is not local and neither observer can assume that the speed of light over the entire trip is $c$. And in fact their observations will tell them explicitly that it is not: they each measure different times for light to travel the same distance.

 

[PeterDonis]

in that case the two observers don't share a notion of space

I'm not sure what case is being discussed here. There is no case where two static observers at different heights don't agree on the height difference between them.

 

[Ibix]

I'm not sure what case is being discussed here. There is no case where two static observers at different heights don't agree on the height difference between them.

On a Kruskal diagram, the two observers are hyperbolae in region I, and the ruler fills the region between them. What you'd normally call the distance between the observers is measured along a straight radial line from the origin. But I can pick any spacelike curve joining the intersections of the observers and the radial line and call it my spatial slice, and hence get a different distance.

There's a lot of work to do to derive this distance from the raw ruler measure, but it's allowed. I can even Kruskal boost the diagram and repeat, to prove I have a time-independent notion of space. I should be able to get an arbitrarily short distance by a nearly-null spacelike path, so I should always be able to get equal speeds. Obviously, the spacelike projection of null worldlines depends on my choice of space.

Clearly nobody sane would actually do this.

 

[Bosko]

In the case of two static observers at different heights (top and bottom of a building), you are incorrect. The observers agree on the height of the building but not the time taken for light to travel between them.

How they agree about measurement method?
By using the same coordinate system and positions in it ?

No, they are not. Each observer measures the same local speed of light, using measurements made just at their height; but the two observers are at different heights, so any measurement involving light traveling between both of them is not local and neither observer can assume that the speed of light over the entire trip is $c$. And in fact their observations will tell them explicitly that it is not: they each measure different times for light to travel the same distance.

The same local speed of light means ... ( let's A be on top and B on the bottom of the building )
$(\Delta x_A)^2=(\Delta t_A)^2$
$(\Delta x_B)^2=(\Delta t_B)^2$

If clock A ticks 2 nano seconds while clock B ticks 1 nano second ...
Light will travel 2 feet ( 60 cm) on A (top) and 1 foot on B (bottom)

 

[Bosko]

Ok. But in that case the two observers don't share a notion of space, so light didn't follow the same spatial path for them.

The light follow the same spatial path but length of that path is different for different oservers

If clock A ticks 2 nano seconds while clock B ticks 1 nano second ...
They will agree on any speed $\frac{\Delta x}{\Delta t}$ but not on distances and time passed
Any distance for observer B (bottom) is 2 times smaller then the same distance for observer A.

 

[PeterDonis]

I can pick any spacelike curve joining the intersections of the observers and the radial line and call it my spatial slice, and hence get a different distance.

Yes, agreed, but I think that is beyond the scope of this thread. The "natural" distance is the one along the spacelike path that is orthogonal to both observers' worldlines, and that is the one that I was describing when I said both observers would agree on it.

 

[PeterDonis]

The light follow the same spatial path but length of that path is different for different oservers

Please show your work. Either you are wrong (which is what I suspect), or you are using some different notion of "distance" (which @Ibix correctly says is possible, but that doesn't mean you can just wave your hands about it, you need to actually do the math and show it to us).

At this point you either need to show us math or this thread will be closed.

 

[PeterDonis]

If clock A ticks 2 nano seconds while clock B ticks 1 nano second ...
They will agree on any speed $\frac{\Delta x}{\Delta t}$ but not on distances and time passed
Any distance for observer B (bottom) is 2 times smaller then the same distance for observer A.

Here you are simply wrong. Even in the kind of alternate case that @Ibix was talking about, these statements will not be true.

 

[Bosko]

Please show your work. Either you are wrong (which is what I suspect), or you are using some different notion of "distance" (which @Ibix correctly says is possible, but that doesn't mean you can just wave your hands about it, you need to actually do the math and show it to us).

I am analysing the geometry around a massive spherical object (neutron stars, black holes)

The existing Schwarzschild and alternative coordinates do not look good to me. That's why I don't use them.
Is there any experiment that confirms a different speed of light in a strong gravitational field for any observer?

The mathematics I am working on has many pages and is not complete. And it's not just math. I am trying to use C++ and finite element method ...
That's way off topic. Maybe for a topic outside of the standard models.

This topic is - the basic principles on which I want to build further analysis.

If light travels at the same speed for, at least, every static observer, and if it follows from Fermat's principle that a ray of light moves locally along the shortest path then:

The gravitational field becomes weaker inside the photon sphere.
That is the motivation

At this point you either need to show us math or this thread will be closed.

That's fine, close it or move it to "Beyond the Standard Models"

 

[Bosko]

Here you are simply wrong. Even in the kind of alternate case that @Ibix was talking about, these statements will not be true.

Is there experimental evidence for this?
Depending on the selected coordinate system: Schwarzschild, Kruskal - Szekeres, Harmonic, Isotropic, Lemaître, Gullstrand – Pulneve (not sure), Eddington-Frinkelstein,... you can get various values
Why are there so many different ones for the same physical object, I mean, the same spherically symmetrical space-time geometry.

 

[PeterDonis]

I am analysing the geometry around a massive spherical object (neutron stars, black holes)

Which is fine, but it is also a very basic topic in GR, which is covered in every GR textbook, and there is a wealth of material available to help you. It doesn't seem like you are familiar with any of that or taking any advantage of it to help you.

The existing Schwarzschild and alternative coordinates do not look good to me. That's why I don't use them.

This doesn't seem like very good judgment to me. See further comments below.

Is there any experiment that confirms a different speed of light in a strong gravitational field for any observer?

You have already had the relevant experimental results described to you: two static observers at different heights will measure the same distance between them but their clocks will measure different elapsed times for light to travel between them. This experiment has been done: look up the Pound-Rebka and Pound-Snider experiments. Same distance + different elapsed times = different calculated speeds for the light.

The mathematics I am working on

Is off limits for discussion here ("here" in this case means "anywhere on PF") unless and until you publish it in a peer-reviewed paper.

This topic is - the basic principles on which I want to build further analysis.

If you have not read any textbooks, and are not using any of the standard tools that have served physicists well for decades in this problem domain, I think you are not going down a good path as far as "basic principles" is concerned.

If light travels at the same speed for, at least, every static observer

"Light travels at the same speed" is only true for local measurements, i.e., measurements confined to a single local inertial frame. The measurements described above, in the experiments referred to above, were not local measurements. You have already been told this.

if it follows from Fermat's principle that a ray of light moves locally along the shortest path

Fermat's principle does not say that light moves along the shortest spatial path. It says that light travels the path of least time. But to make use of this in the context of General Relativity, you need to be very careful.

The gravitational field becomes weaker inside the photon sphere.

This is wrong. It doesn't.

That's fine, close it or move it to "Beyond the Standard Models"

I'll take the first option. Thread closed.

 

[Nugatory]

Why are there so many different ones for the same physical object, I mean, the same spherically symmetrical space-time geometry.

For the same reason that we have cartesian $x,y$ and polar $r,\theta$ coordinates (as well as more outre ones, like log and log-log scales) for the same simple flat Euclidean geometry: depending on the problem at hand, one coordinate system will be easier to work with than another.

Schwarzschild coordinates are generally most convenient when we want to know how things look around the black hole. I find Kruskal coordinates best when I’m trying to reason about objects falling through the horizon; and so forth.

Coordinates are tools - and I own easily a dozen tools all designed for turning 10mm hex-head nuts/bolts.