Black Hole Waterfall Analogy & Light Speed

[Rene Dekker]

TL;DR Summary

The "waterfall analogy" visualises that space flows towards mass, and that the effective speed of light slows down when going against that flow. Is that correct?

Recently I have seen a number of General Relativity visualisations that show spacetime flowing towards any mass, similar to water flowing into a sink hole. ScienceClic's video is an example. That model is also used in the "waterfall model" to explain the event horizon of a black hole, as the point where space flows faster than the speed of light, so that light does not manage to swim against it anymore.

But if that model is correct, doesn't that have consequences for the measured speed of light? If space flows towards a black hole that way, then it should also flow towards the Earth. And if the effective light speed is lower when going against the flow of space, then it should also be lower here on Earth. That is, the upwards light speed should differ from the downwards light speed.

Is that model correct, and should the light speed theoretically differ in upwards and downwards direction?
Is that difference measurable?

Suppose that model is not correct, and light does not loose speed when moving away from a mass, but looses only energy. Then how is the event horizon of a black hole explained? Shouldn't it differ from a hard boundary in that case, but depend on how far from the black hole the light is generated?

 

[Orodruin]

Summary:: The "waterfall analogy" visualises that space flows towards mass, and that the effective speed of light slows down when going against that flow. Is that correct?

Is that model correct, and should the light speed theoretically differ in upwards and downwards direction?

No. Any popularized model has limitations and will always be an imprecise analogy of the full theory.

 

[PeroK]

Suppose that model is not correct, and light does not loose speed when moving away from a mass, but looses only energy. Then how is the event horizon of a black hole explained? Shouldn't it differ from a hard boundary in that case, but depend on how far from the black hole the light is generated?

Any textbook on GR will explain

a) The measurements of the frequency of light at different positions relative to a large mass.

b) The behaviour of null (light-like) geodesics in relation to the event horizon of a black hole. In particular, that light signals may cross the horizon in only direction (inwards).

 

[Rene Dekker]

Any textbook on GR will explain

a) The measurements of the frequency of light at different positions relative to a large mass.

b) The behaviour of null (light-like) geodesics in relation to the event horizon of a black hole. In particular, that light signals may cross the horizon in only direction (inwards).

Would it be correct to say that we, from our position far away from a black hole, would deduce that the speed of the light near a black hole must be low wrt. us? But that somebody who is there locally would still measure it as c ?

 

[PeroK]

Would it be correct to say that we, from our position far away from a black hole, would deduce that the speed of the light near a black hole must be low wrt. us? But that somebody who is there locally would still measure it as c ?

Strictly speaking, in a curved spacetime a speed only makes sense when measured locally. When you say "the speed of light near a black hole wrt us", then the "wrt us" is ambiguous. There is no unique way to define or measure the speed of an object except locally.

 

[Rene Dekker]

Strictly speaking, in a curved spacetime a speed only makes sense when measured locally. When you say "the speed of light near a black hole wrt us", then the "wrt us" is ambiguous. There is no unique way to define or measure the speed of an object except locally.

Suppose two stars are circling around each other with their rotational axis pointing straight at us. Then we can use our knowledge of how far away the stars are, and simple triangulation methods, to calculate a speed for the stars. If we take such a speed calculation as the definition of "speed wrt. us", then would it be fair to say that the speed of light moving away from a black hole would be judged to be less than c by us?

 

[Nugatory]

The "waterfall analogy" visualises that space flows towards mass, and that the effective speed of light slows down when going against that flow. Is that correct?

It's an analogy so pretty much by definition it is not correct. Analogies can be useful for understanding some aspects of a problem, but they're analogies not the real thing so they will break down if pushed too far. The waterfall analogy is somewhat helpful for understanding the red shift of outwards-directed light, and doesn't get in the way of visualizing anything that happens locally, but is seriously misleading when it comes to thinking about the speeds of any distant objects.
The basic problem is that it that it assumes such speeds, including that of the infalling "water" can be defined in a non-arbitrary way. They can't, and you will see why if you try constructing a thought experiment that would permit us to measure the speed of a non-local light signal. So...

Would it be correct to say that we, from our position far away from a black hole, would deduce that the speed of the light near a black hole must be low wrt us?

Depending on which measurements we perform and especially how we define "at the same" across the entire area, we can deduce just about any speed we please. But

But that somebody who is there locally would still measure it as c ?

Yes. Relative speeds are only unambiguously defined locally, and then the speed of light is always $c$.

 

[PeroK]

simple triangulation methods

... only make sense in flat (Euclidean) space. Think of gravitational lensing. Is a distant star really in two places at once if we see its light from two different directions? Think of the experiment that first corroborated GR with the bending of light around the Sun. Where, precislely, is the star according to your simple triangulation methods?

You can, of course, unambiguously deduce or estimate the local proper distance between two points and the local proper time; and, thereby, deduce what a local measurement of the speed (of light) would yield.

 

[Rene Dekker]

... only make sense in flat (Euclidean) space. Think of gravitational lensing. Is a distant star really in two places at once if we see its light from two different directions? Think of the experiment that first corroborated GR with the bending of light around the Sun. Where, precislely, is the star according to your simple triangulation methods?

I understand the limitations of such methods. But thinking about such remote measurements helps me visualise what is going on. We do calculations of remote speeds all the time. For example, to calculate the orbital speed of a satellite. We don't go to the satellite to measure it locally, we calculate it from measurements made on Earth. And yes, I understand that the spacetime disturbances around a black hole are much bigger than for a satellite around the Earth, but still wonder whether a similar reasoning can be used to think about the speed of light moving away from (or towards, of course) a black hole.

 

[PeroK]

but still wonder whether a similar reasoning can be used to think about the speed of light moving away from (or towards, of course) a black hole.

The theory of GR postulates that the locally measured speed of light is a universal constant. Other measurements of the speed of light lead to coordinate dependent numbers and, therefore, are not physically meaningful.

I saw this earlier:

I am late to the party, and won't comment on the good philosophical discussion, but like to comment on your original question. I have always found it easier to understand and accept length contraction, by viewing it as a rotation instead.

We are all familiar with normal rotations in 3D space, and don't think much about the effects that it has. Imagine a rod is lying some distance away on the its side, perpendicular to your viewing direction, such that you see the full length of the rod. If I now rotate the rod, then it appears to shrink in size to you. We normally don't think about it that way, because our brain compensates for that immediately. But if you would take photographs before and after the rotation, and compare those, then the rod appears to have shortened. After the rotation you would also see one of its short ends, and one end would be further away than the other.

Something similar happens with length contraction with a faster moving rod. It is kind of like a rotation in time. The rod appears shorter, you see one of its short ends, and one end would be "further away" in time than the other. If you look at the mathematics of it, then it is also more like a rotation than a contraction.

That's a similar idea. Measuring the speed of a light in some coordinates and getting a coordinate dependent answer says nothing physically meaningful about that particular light.

 

[Orodruin]

But thinking about such remote measurements helps me visualise what is going on.

... but not correctly. If anything it creates a false sense of understanding.

 

[PeterDonis]

We do calculations of remote speeds all the time.

That's because the spacetime geometry in the scenarios in question is such that the errors in such calculations are small enough to be acceptable. That is not the case for calculations of "remote speeds" of objects close to a black hole's horizon.

 

[Rene Dekker]

That's a similar idea. Measuring the speed of a light in some coordinates and getting a coordinate dependent answer says nothing physically meaningful about that particular light.

I understand that. But it may give a meaningful idea about how that light appears to move, to a far away observer. Just like the length contraction example, which gives you a meaningful idea how a moving spaceship looks like for a stationary observer.

Thanks everybody for the answers, I get the main idea. The speed of light is always equal to c when measured locally. And the "waterfall analogy" can give some insights into how the cogwheels of GR work, but cannot be used to draw any conclusions from.

 

[PeterDonis]

But it may give a meaningful idea about how that light appears to move, to a far away observer. Just like the length contraction example, which gives you a meaningful idea how a moving spaceship looks like for a stationary observer.

Actually, length contraction does not give you a meaningful idea how a moving spaceship actually looks to a stationary observer. Look up "Penrose-Terrell rotation".

Similarly, looking at the coordinate speed of light near a black hole does not give you a meaningful idea of what a faraway observer actually sees. You need to look at the spacetime trajectories of the light rays.

 

[Rene Dekker]

Actually, length contraction does not give you a meaningful idea how a moving spaceship actually looks to a stationary observer. Look up "Penrose-Terrell rotation".

The "Penrose-Terrell rotation" is actually what I meant, I should have said that instead of "length contraction". The rotation gives you a meaningful idea how a moving spaceship would look like.

I had the idea that a similar kind of projection for black holes could "show" that light moving away from a black hole would appear to move more slowly, according to a far away observer. But you all have made it clear that that is not a particular fruitful way of looking at things.

 

[PeterDonis]

The rotation gives you a meaningful idea how a moving spaceship would look like.

Yes, but the calculations that show you why Penrose-Terrell rotation happens are more complicated than just "length contraction". Similarly, the calculations that show you how an object near a black hole's horizon actually appears to a faraway observer are more complicated than just "coordinate speed of light is slower".