Is Measuring the One-Way Speed of Light Possible Near a Black Hole?

[FranzS]

Hi PF,
as far as my modest knowledge goes, measuring the one-way speed of light is impossible / non-sensical (?).
Thus I'm not here to actually propose a possible method for that purpose, but I just would like the savvy among you to point out the fallacy in my reasoning.

That said, I was thinking about an ideal situation where one could find himself right on the event horizon of a (non-spinning, non-charged?) black hole and could emit light tangentially to the event horizon surface. Ideally, wouldn't light travel around the black hole following a geodesic on the event horizon surface and eventually come back to where it started? In case it would, can this be considered as a one-way path?

Sorry in advance if I'm posing a stupid question, I'm here to learn. Thanks!

 

[PeterDonis]

measuring the one-way speed of light is impossible / non-sensical (?).

No, it's not impossible/non-sensical. It's just frame-dependent; in other words, what you're measuring isn't actually a property of light, it's a property of whatever reference frame you have chosen.

one could find himself right on the event horizon of a (non-spinning, non-charged?) black hole

You can, but only for an instant; you can't stay there. You must fall inward. Why? Because to stay at the horizon, you would have to move radially outward at the speed of light. And you can't.

and could emit light tangentially to the event horizon surface.

"Tangentially" is ambiguous here. If you think of the horizon (or more precisely the cross section of it that you are "at" at the instant when you, falling inward, are just crossing the horizon) as a 2-sphere, "tangentially" could mean emitting light tangentially to that 2-sphere. But such light would not stay at the horizon; it would fall inward (but not as fast as you do).

Or "tangentially" could mean emitting light tangentially to the horizon in a spacetime sense--but this "tangentially" is radially outward in a spatial sense, i.e., not tangential to the 2-sphere in the sense given above. Light emitted in this way will stay at the horizon, but (a) not in the way you are thinking (see below), and (b) you won't stay at the horizon, so you will never see that light again in any case.

wouldn't light travel around the black hole following a geodesic on the event horizon surface and eventually come back to where it started?

No. The horizon is a surface in spacetime; light emitted radially outward at the horizon travels in a null direction that works like traveling into the future, the way light emitted radially outward from an ordinary 2-sphere does. The difference is that, due to the curvature of spacetime, this radially outgoing light stays at the same radius, i.e., if you imagine the entire 2-sphere cross section of the horizon at an instant, and imagine a 2-sphere of light emitted radially outward from all around that 2-sphere, the 2-sphere of light will maintain a constant radius (more precisely a constant surface area, from which we compute a "radius"), even though it is moving radially outward. But no light ray ever returns to a point in spacetime that it was at before, and no light ray goes around the 2-sphere; each light ray stays at the same point on the 2-sphere that it was emitted from.

 

[Dale]

Ideally, wouldn't light travel around the black hole following a geodesic on the event horizon surface and eventually come back to where it started?

What you want is the photon sphere, not the event horizon.

This is a so called two-way measurement, the same as if you had placed a bunch of mirrors or a fiber optic cable to route the light in a circle.

 

[FactChecker]

Replace "one-way speed of light" with "one-direction speed of light" and you will see that your proposal is not what they wanted.

 

[PeterDonis]

Replace "one-way speed of light" with "one-direction speed of light" and you will see that your proposal is not what they wanted.

The OP specifically described light going around in a circle. So the photon sphere scenario @Dale described is what the OP said they wanted. (The OP just didn't realize that it needs to be done at the photon sphere, not the horizon, and that it does not qualify as a one-way speed of light measurement.)

Also, light that goes around a photon sphere does go in one direction only. So the scenario described meets your implied criterion for a "one-direction speed of light" measurement. It just doesn't qualify as a "one-way" measurement as that term is used in the literature. (We have had a number of previous threads on this topic in which references to that literature have been given, quite a few of them by @Dale.)

 

[FactChecker]

The OP specifically described light going around in a circle. So the photon sphere scenario @Dale described is what the OP said they wanted.

Is that what people really want when they say "one-way speed of light"?

 

[PeterDonis]

Is that what people really want when they say "one-way speed of light"?

"What people really want" when they ask questions without fully realizing the actual implications according to GR (or indeed any physical theory) often does not exist. The best we can do is to describe what the actual implications are and wait for the OP to ask follow up questions.

 

[Dale]

Replace "one-way speed of light" with "one-direction speed of light" and you will see that your proposal is not what they wanted.

The terminology isn’t great, but at this point we are pretty much stuck with it. “One way” is a path with a distinct beginning and end and “two way” is a closed loop path.

What the OP is describing is a closed loop. So it is a two-way measurement according to the unfortunate standard terminology.

Is that what people really want when they say "one-way speed of light"?

I think that we have to use the standard terminology. Part of the problem is that people have heard that there is some problem with measuring the one way speed of light. But they don’t know what distinguishes a one way measurement from a two way one. We have to teach that, at a minimum, so that they can understand the other stuff they are reading.

 

[Ibix]

The definition is that a one-way speed measure has the start and end points in different places, and a two-way measure has the start and end in the same place. The physical distinction is that a two-way measure allows you to use one clock that is local to the start and end of the track for the timing. On the other hand, a one-way measure involves two clocks, or moving a clock, or watching a distant clock, or something of that nature - and handling the effects of this action ends up requiring an assumption of the one-way speed, and you then "measure" this assumed value.

As already pointed out, circular orbits around black holes are available at the "photon sphere", which is 1.5 times the Schwarzschild radius for a Schwarzschild black hole. They would be a two-way measure in this sense because it's a closed light path.

 

[pervect]

What we call "the speed of light" could also be called "the limiting speed of mater" or "the ultimate speed". If you have a powerful particle accelerator, you can accelerate pieces of matter (electrons, protons, ions, whatever you like, really) as hard as you can, i.e. put as much energy into the particles as you can - but no matter how much energy you put into the particles, their speed will never reach the speed of light, though it will get closer and closer to the speed of light the more energy you put into it.

This has been done experimentally. It's a routine phenomenon in high energy physics, , but Bertozzi also wrote a peer-reviewed paper on it and produced a video for educational purposes. See for instance and https://ui.adsabs.harvard.edu/abs/1964AmJPh..32..551B/abstract.

The speed of light being the same in both directions is the same thing, then, as the ultimate speed of a particle being the same in "both directions". Or really, being the same any direction you pick. This later formulation, the idea that the ultimate speed is the same in any direcction, is known as "isotropy".

While it is not mathematically inconsistent to say that there is no isotropy, that some direction in space is favored over others, it's unnatural and compilciated. Newtonian mechanics certainly also has isotropy as a feature - the familiar Newtonian equations do not depend on the direction of travel. It'd be much more complicated and annoying to work with if it did. It's also rather rare (though not impossible) for a failure of isotropy to have any experimental consequences. That's an important, but somewhat advanced topic - perhaps I could write more on it if there is any interest, but I'll avoid it unless prompted.

 

[Ibix]

the particle simply crosses the horizon in finite time and is inevitably drawn inward

Not quite. As Peter pointed out, there's one exception - a light ray emitted at the horizon going radially outward will "hover" at the horizon. And, unfortunately given the OP's choice of wording, in the 4d sense this path is tangent to the horizon. This is easiest to see on a Kruskal diagram where light in the plane of the diagram travels on 45° sloped paths and the horizon itself is a 45° sloped path, so light emitted on the horizon will stay there because it is on the horizon and parallel to it - i.e. tangential.

Because of this, a traditional one-way speed measure (i.e. a clock and a laser a known distance away from another synchronised clock with a photodetector) dropped laser-end-first into a black hole could, with careful timing, have both its emission and reception events on the horizon and a light path tangential to the horizon. This still doesn't solve the issues with one way speed measures because the horizon isn't a place in the strict meaning of the word and you can't have clocks that stay there.

The OP almost certainly had something more like the photon sphere orbits in mind, which is why they were brought up, but a one-way speed experiment with a light path tangent (in the 4d sense) to the horizon is possible and fits the OP's wording pretty closely.