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

[PeterDonis]

The transition from the outside (Flamm's paraboloid) to the inside (it also looks like a paraboloid) should be smooth.

Yes, it's smooth. I'm not sure where you are getting "looks like a paraboloid" from. My understanding is that the spatial geometry of the interior solution is a 3-sphere.

There are inflection points on the surface of the object

If you mean an inflection point of the "slope" of the spatial geometry in an embedding diagram, yes.

 

[A.T.]

The transition from the outside (Flamm's paraboloid) to the inside (it also looks like a paraboloid) ...

The interior geometry is rather spherical:
https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Visualization

... should be smooth.

To some degree of continuity. The curvature switches from negative to positive.

 

[vanhees71]

But that's not a "black-hole solution" but the original Schwarzschild solution for a static incompressible fluid of constant density.

 

[PeterDonis]

But that's not a "black-hole solution" but the original Schwarzschild solution for a static incompressible fluid of constant density.

The OP said they were interested in both.

 

[Bosko]

Take any static spacetime. The metric in such a spacetime may be written in static coordinates $t$ and $x^i$ as
$$
ds^2 = f(x)\, dt^2 - d\Sigma^2
= f(x)\, dt^2 - g_{ij} dx^i dx^j
$$
where $d\Sigma^2$ is the (Riemannian) metric on the spatial slices defining simultaneous static time.

By the time translation symmetry, it follows that
$$
E = g(\partial_t, \dot \gamma) = f(x)\dot t
$$
is constant for any spacetime geodesic $\gamma$. The spatial parts of the spacetime geodesic equations take the form (assuming I did the math right)
$$
\bar\nabla_{\dot x} \dot x^j = \frac{E^2}2 g^{jk}\partial_k(1/f)
$$
with $\bar\nabla$ representing the affine connection on the spatial slices. For the worldline to correspond to a geodesic when projected on the spatial slices, this should at the very least satisfy the non-affinely parametrised geodesic equations on the spatial slice. This would require the RHS to be proportional to $\dot x^j$, which is not the case.

For which observer does the speed of light slows down entering a stronger gravitational field?

Let's imagine...

Let two rays of light emitted at point B arrive at (detector) point A at the same time.
Does $L_A=L_B$ for both observers?

For example, $t_A=20 ns, L_A=6 m$ and $t_B=10 ns, L_B=3 m$.
If $L_{AB}$ is removed from both paths, we get $2L_A=2L_B$ and $L_A=L_B$.
(Speed of light = 30 cm, or foot, per nano second)

The speed of light appears to be constant for any observer regardless of the gravitational field.
I'm interested in whether it slows down as approaches the photon sphere (or the event horizon).

 

[PeterDonis]

Let two rays of light emitted at point B arrive at (detector) point A at the same time.

How is that even possible? Two different rays would be emitted from point B at different times, and that means they would arrive at point A at different times.

 

[PeterDonis]

Does $L_A=L_B$ for both observers?

Yes. The time taken for light to travel is what is different for the two observers. The simplest way to test that is for each observer to send a round-trip light signal that reflects off a mirror at the other observer's location, and then compare the round-trip travel times by the two observers' clocks.

 

[Bosko]

How is that even possible? Two different rays would be emitted from point B at different times, and that means they would arrive at point A at different times.

Both ray are emitted from the same source and at same time from point B ( see image)

 

[PeterDonis]

Both ray are emitted from the same source and at same time from point B ( see image)

Then what's the point of having two rays? The two rays aren't telling you anything that one ray wouldn't tell you.

 

[Bosko]

Then what's the point of having two rays? The two rays aren't telling you anything that one ray wouldn't tell you.

By adjusting the LA, you can set them to arrive at the same time.
Then $L_A=L_B$ for both observers.

 

[PeterDonis]

By adjusting the LA, you can set them to arrive at the same time.

You can't "adjust" the LA; both rays leave from the same place, point B, and arrive at the same place, point A. That means they both travel the same distance. Whether that same distance covered by both rays has the same numerical as measured by both observers is a separate question that your scenario does not give any information about at all.

Then $L_A=L_B$ for both observers.

That is true, but your scenario has, as far as I can tell, nothing to do with this true statement.

 

[PeroK]

For which observer does the speed of light slows down entering a stronger gravitational field?

Let's imagine...
View attachment 337779
Let two rays of light emitted at point B arrive at (detector) point A at the same time.
Does $L_A=L_B$ for both observers?

For example, $t_A=20 ns, L_A=6 m$ and $t_B=10 ns, L_B=3 m$.
If $L_{AB}$ is removed from both paths, we get $2L_A=2L_B$ and $L_A=L_B$.
(Speed of light = 30 cm, or foot, per nano second)

The speed of light appears to be constant for any observer regardless of the gravitational field.
I'm interested in whether it slows down as approaches the photon sphere (or the event horizon).

It seems to me there is a lot wrong with this. Especially what you mean by certain terms. What does length mean (for an observer)? How is the length of a path through space defined for a given observer?

How is the speed of light defined in curved spacetime?

Ultimately, it seems like a simplistic analysis that doesn't take account of how theese quantities are defined.

 

[Bosko]

It seems to me there is a lot wrong with this. Especially what you mean by certain terms. What does length mean (for an observer)? How is the length of a path through space defined for a given observer?

The initial assumption is a constant speed of light for a local observer.
It measures some local distances. ( for example using https://www.fluke.com/en/product/building-infrastructure/laser-distance-meters/417d)

How is the speed of light defined in curved spacetime?

The definition is self-explanatory to the local observer. (Distance 30 cm that light travels for a time unit of 1 nanosecond)

Ultimately, it seems like a simplistic analysis that doesn't take account of how theese quantities are defined.

I analyse how two observers can agree on the measured distances.
I am interested in questions:
Is the speed of light the same for any observer regardless of the gravitational field?
Does light travel along the (locally) shortest path between two points?

 

[pervect]

The initial assumption is a constant speed of light for a local observer.
It measures some local distances. ( for example using https://www.fluke.com/en/product/building-infrastructure/laser-distance-meters/417d)

The definition is self-explanatory to the local observer. (Distance 30 cm that light travels for a time unit of 1 nanosecond)

I analyse how two observers can agree on the measured distances.
I am interested in questions:
Is the speed of light the same for any observer regardless of the gravitational field?
Does light travel along the (locally) shortest path between two points?

You are talking about the local speed of light as if it is something that were measured. With modern SI definitions, the speed of light is no longer something that is measured. Since it's clear that you're not (cannot be) using the SI defintion of distance, , to give some meaningful answer to your question, we'd need to know what you are basing your concept of "measuring the speed of light" on.

Fundamentally, it's not clear what you're asking, because it's not clear how you are regarding light as something whose speed needs to be measured. I used to assume that people asking this question were using one particular old definition of the meter based on a standard "prototype" meter bar. However, I quickly found that actually people had no idea of what they thought a meter was - it was some internal mental concept to them, and they never thought about the process of how it was standardized so it could be communicated to others and realized in practice. There's quite a bit of history on the topic of the issue of measurement, starting with the "treaty of the metre" and the origin of the BIPM, "the international organization established by the Metre Convention". Where in this history you are is unclear, it's only clear that you're not using the modern ideas.

As I mentioned previously, the general process of defining distance is based on reducing a 4 dimensional space-time manifold to a three dimensional spatial manifold. It's reasonably clear how this is done when the space-time geometry is static (as on an elevator accelerating with a constant acceleration or with the Schwarzschild geometry), but to actually answer your question in a more general case, you'd need to define how you are reducing the 4-d space-time manifold to a 3-d space only manifold.

As I recall, you gave a sensible (though not very rigorous) answer to this question when I asked it earlier, so it's not clear what went wrong with the communication process in that you're asking the same question again. It seems worthwhile to repeat myself (at greater length) once, but not more. If this attempt doesn't work this time, I'm going to assume it's a lost cause :(.

The answer to your second question, "does light always travel on a path of shortest distance", with distance being defined by the above approach of reducing a 4d manifold to a 3d manifold, is in general no. I've proposed some specific examples, but I haven't talked about them at length. I'd suggest you take some time to think about the issue - it could be productive for you describe a particular 4d geometry (by giving us a metric), then defining the induced 3d geometry (most clearly done by giving us the 3d line element).

To give an example in practice, Einstein's elevator can be described with the Rindler metric. Using geometric units, the 4d line element is:

$$-c^2 z^2 dt^2 + dx^2 + dy^2 + dz^2$$

This represents an elevator accelerating in the "z" direction. Not that the origin of the coordinates is at z=1. There are ways to reformumulate this so that the origin is at z=0, but it'd be confusing to introduce two examples so I'll use this one, unless you think it'd be helpful to use a different one.

The induced 3d metric is given by the line element

$$dx^2 + dy^2 + dz^2$$

Given this framework, we can then answer the question. Does light follow a straight line path in the 3d spatial submanifold? The answer is no, it does not. Physically, if we shine a light beam along the line "x=constant", it appears to drop as the elevator accelerates, following a path that is roughly (but not exactly) parabolic.

We can also answer the question: is the coordinate speed of light constant? The answer is again no, the coordinate speed of light depends on the "height" , given by the coordinate z. So the coordinate speed of light is equal to c only at z=1.

If you have an alternative example, we can discuss it if you give the needed information - which is specifying the 4d manifold with a line element, describing the process by which you reduce the 4d manifold to a 3d manifold, and then, as a double check, giving us the line element for the induced line element.

If you wish to discuss speed, and you want to discuss something other than coordinate speed, you'll need to tell us what notion of "speed" you are using.

 

[PeterDonis]

I analyse how two observers can agree on the measured distances.

I don't see that you've actually done any analysis. You just posed a scenario that makes no sense and is irrelevant to what you are asking about.

Is the speed of light the same for any observer regardless of the gravitational field?

It depends on what you mean by "the speed of light". The light cone structure of spacetime is invariant. But the "speed of light" defined by non-local measurements, such as two observers at different altitudes in a gravitational field exchanging light signals, might not be.

Does light travel along the (locally) shortest path between two points?

Not in the sense you mean. As has already been said in this thread (more than once, IIRC), the projection of a null (or timelike) geodesic of spacetime into "space", i.e., a spacelike surface of constant coordinate time in some relevant coordinate chart, will not be a geodesic of the spacelike surface.

 

[Bosko]

It depends on what you mean by "the speed of light". The light cone structure of spacetime is invariant. But the "speed of light" defined by non-local measurements, such as two observers at different altitudes in a gravitational field exchanging light signals, might not be.

Are those two observers are static?
How they exchange light signals to get different results?

 

[Bosko]

...
To give an example in practice, Einstein's elevator can be described with the Rindler metric. Using geometric units, the 4d line element is:

$$-c^2 z^2 dt^2 + dx^2 + dy^2 + dz^2$$

This represents an elevator accelerating in the "z" direction. Not that the origin of the coordinates is at z=1. There are ways to reformumulate this so that the origin is at z=0, but it'd be confusing to introduce two examples so I'll use this one, unless you think it'd be helpful to use a different one.

The induced 3d metric is given by the line element

$$dx^2 + dy^2 + dz^2$$

If light follow the null geodesic of 4D interval defined as $-c^2 z^2 dt^2 + dx^2 + dy^2 + dz^2$
$dx^2 + dy^2 + dz^2=c^2 z^2 dt^2$

$\frac{dx^2 + dy^2 + dz^2}{ z^2}=c^2 dt^2$

3D spacelike "differential" interval should be the left side

$\frac{dx^2 + dy^2 + dz^2}{ z^2}$

Given this framework, we can then answer the question. Does light follow a straight line path in the 3d spatial submanifold? The answer is no, it does not. Physically, if we shine a light beam along the line "x=constant", it appears to drop as the elevator accelerates, following a path that is roughly (but not exactly) parabolic.

Is that path the sortest?

 

[Ibix]

Are those two observers are static?
How they exchange light signals to get different results?

You stand at the top of a building and I'll stand at the bottom. We use a ruler to measure the distance between us and we will agree that value - we must, because we use the same ruler at the same time in the same way.

Now we bounce radar pulses off each other. We do not agree the return time due to gravitational time dilation.

We agree distance travelled but not time taken. This we do not agree speed.

 

[Bosko]

We agree distance travelled but not time taken. This we do not agree speed.

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.

Let's imagine that time ticking 1/10 slower on my clock then on yours ...
For me the radar pulse goes up and back in 600 nano seconds ...
and I am calculating 300 nano seconds times 1 feet (30 cm) = 300 feet (100 meters )

For you the radar pulse goes down and back in 660 nano seconds ...
and you are calculating 330 nano seconds times 1 feet (30 cm) = 330 feet (110 meters )

Any speeds we measure are the same but time and distance measured by me have to be multiplied by 1,1 to agree with your measurement

 

[PeterDonis]

Are those two observers are static?

Yes.

How they exchange light signals to get different results?

Their clocks will measure different round-trip travel times for the light signals.

 

[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.