Centripetal Acceleration of Photons?

[Comeback City]

This question came to mind from the thread...

https://www.physicsforums.com/threads/can-a-photon-be-accelerated.901484/

Knowing...
1) Strong gravitational fields create strong curves in spacetime
2) Light traveling through strong gravitational fields get curved along the spacetime
3) Centripetal acceleration occurs at a constant speed, and light travels at constant speed c

Can it be concluded that photons traveling through strong gravitational fields undergo centripetal acceleration?

Also, do photons have relativistic mass? If so, could it be said that photons also experience centripetal force using the relativistic mass? Or is relativistic mass too obscure to be used for centripetal force?

 

[PeterDonis]

Centripetal acceleration occurs at a constant speed

What do you mean by this? What causes "centripetal acceleration" and how is it defined?

do photons have relativistic mass?

"Relativistic mass" is just a synonym for "energy", and photons have energy, so yes.

could it be said that photons also experience centripetal force using the relativistic mass?

What do you mean by "centripetal force"? What causes it and how is it defined?

Are you aware that in GR, gravity is not a force?

 

[Comeback City]

What do you mean by this? What causes "centripetal acceleration" and how is it defined?

Centripetal acceleration is defined as the acceleration on a constantly-moving body caused by the constantly-changing direction in circular motion. The equation for it is...
a_c=v²/r
Where v is velocity and r is radius.

What do you mean by "centripetal force"? What causes it and how is it defined?

Centripetal force is the force on a moving body that is in uniform motion around a circular path, and is caused by the centripetal acceleration. The equation for it is...
F_c=mv²/r
Where v is velocity, r is radius, and m is mass.

Are you aware that in GR, gravity is not a force?

Yes, I never said it was.

 

[A.T.]

Gravity in GR can result in centripetal coordinate acceleration, but not via a centripetal force.

 

[Comeback City]

Gravity in GR can result in centripetal coordinate acceleration, but not via a centripetal force.

I figured that Centripetal force would not be relevant in this situation, but wasn't sure because of the relativistic mass possibility. And when the light is traveling through the curved spacetime, the speed will still remain exactly at c, correct?

 

[A.T.]

And when the light is traveling through the curved spacetime, the speed will still remain exactly at c, correct?

In the local free falling frame. Not in the global coordinates.

 

[Comeback City]

In the local free falling frame. Not in the global coordinates.

Would it change by a lot in the global coordinates? Or would that depend on the degree of the curvature?

 

[A.T.]

Would it change by a lot in the global coordinates? Or would that depend on the degree of the curvature?

https://en.wikipedia.org/wiki/Shapiro_delay.

 

[PeterDonis]

Centripetal acceleration is defined as the acceleration on a constantly-moving body caused by the constantly-changing direction in circular motion.

In GR, this is called "coordinate acceleration"; you can make it go away by choosing appropriate coordinates. Anything that depends on coordinates, in GR, can't appear in a physical law; physical laws have to be based on invariants, things that don't change when you change coordinates.

Centripetal force is the force on a moving body that is in uniform motion around a circular path, and is caused by the centripetal acceleration.

Same comment here. In a gravitational field, a moving body can be on a uniform circular path (in space), but be in free fall, feeling no force. But by your definition, there would be a "centripetal force" on it (which would in Newtonian terms be the gravity of the body it is orbiting). In GR, this definition of "force" is not used; "force" has to be something that is actually felt. A body in free fall has zero force on it in GR.

I never said it was.

You did implicitly by defining "centripetal force" as you did. See above.

Also, your definitions above, since they explicitly require circular motion, don't apply to light passing through a gravitational field (unless it happens to be at exactly the right altitude above a black hole to be at the "photon sphere", where light can orbit in a circle around the hole).

 

[Comeback City]

Also, your definitions above, since they explicitly require circular motion, don't apply to light passing through a gravitational field (unless it happens to be at exactly the right altitude above a black hole to be at the "photon sphere", where light can orbit in a circle around the hole).

So you are saying that there is no centripetal acceleration of photons except for the photon sphere?

 

[PeterDonis]

So you are saying that there is no centripetal acceleration of photons except for the photon sphere?

I'm saying that, since your definition specified circular motion, it only applies to things that are moving in circles. Photons only do that at the photon sphere around a black hole. But that's a problem with your definition, not with photons.

Also, even if we fix your definition so it applies more generally, it's still the wrong concept to use, because, as I said before, it depends on the coordinates you choose.

 

[Comeback City]

I'm saying that, since your definition specified circular motion, it only applies to things that are moving in circles. Photons only do that at the photon sphere around a black hole. But that's a problem with your definition, not with photons.

Also, even if we fix your definition so it applies more generally, it's still the wrong concept to use, because, as I said before, it depends on the coordinates you choose.

Well I'm sorry that my definitions have problems. I don't think it requires perfect circular motion to have centripetal acceleration, but we would need an expert of circular motion to resolve that; doesn't seem like either of us are experts of it.

 

[PeterDonis]

I don't think it requires perfect circular motion to have centripetal acceleration

I don't either, but that's the way you defined it. Would you like to try to fix your definition?

doesn't seem like either of us are experts of it

Perhaps it wasn't clear that I already know that it doesn't require circular motion, and that your definition was too restricted. I was hoping you would figure that out if I gave some hints.

 

[pervect]

This question came to mind from the thread...

https://www.physicsforums.com/threads/can-a-photon-be-accelerated.901484/

Knowing...
1) Strong gravitational fields create strong curves in spacetime
2) Light traveling through strong gravitational fields get curved along the spacetime
3) Centripetal acceleration occurs at a constant speed, and light travels at constant speed c

Can it be concluded that photons traveling through strong gravitational fields undergo centripetal acceleration?

Also, do photons have relativistic mass? If so, could it be said that photons also experience centripetal force using the relativistic mass? Or is relativistic mass too obscure to be used for centripetal force?

Photons do have energy, and energy is equivalent to relativistic mass, so it can be said that photons have relativistic mass.

However, applying Newtonian formula to photons isn't going to give any insight into the answers that GR gives for the paths of photons. Before discussing the details, I'll just mention that one of the classical tests of GR is the fact that light deflects twice the amount it does in Newtonian gravity. It's problematic to say that this extra deflection is due to a force. I have a personal way I look at it, but since I don't have a good reference that looks at it the way I do, it might be better if I don't mention my personal interpretation as to why this happens, and stick to what's written about it in the literature. And if there is a treatment in the literature that uses "forces" at all, I haven't seen it. (Which isn't quite the same thing as saying it doesn't exist, of course.).

If we remove the Newtonian conceptual framework, and ask "can photons move in a circular orbit around a heavy massive object", the answer is yes, they can. It's called the photon sphere <<wiki link>>.

To provide a hint of the conceptual framework that GR uses, without getting into a whole lot of detail, I'll just say that photons travel along worldlines (paths) in GR that are known as geodesics, and that there is a differential equation, called the geodesic equation, that describes these paths. "Forces" are not needed to write these equations, and as I mentioned before I'm not aware of any good treatment of the issue that uses the concept of "force".

On a related note, it's worth pointing out that massive particles not subject to any external forces other than gravity also travel along geodesic paths. Therefore the same formula that work for ultra-relativistic massive particles work (in the apporopriate relativistic limit) to give nearly the same path as photons - i.e. if you accelerate massive particles fast enough, so they are traveling almost at the speed of light, and said massive particles do not experience any force other than "gravity", they'll travel along nearly the same path that massless photons do.

I don't think it's problematical to say, even without a technical reference, that the extra deflection that light undergoes can be attributed to it's velocity, and not to something special about light, giving the preceding observation. It's probably wrong in detail to think of this extra deflection as being due to a "velocity dependent force", but if you can't get around thinking about the issue in terms of "forces" as being the only thing that could cause deflection, it will at least steer you towards the actual observed behavior of photons.

One paper that might be useful (if you can get a hold of it) is Olson & Guarino's paper "Measuring the active gravitational mass of a moving object". It's oriented towards describing the deflection of ultra-relativistic massive particles, but, as I mentioned, ultra-relativistic massive particles in the appropriate limit move along nearly the same paths as photons do.

 

[Comeback City]

I don't either, but that's the way you defined it. Would you like to try to fix your definition?

Well I suppose I would say that rather than perfect circular motion, it would simply be a "curved" motion. Is that accurate?

Perhaps it wasn't clear that I already know that it doesn't require circular motion, and that your definition was too restricted. I was hoping you would figure that out if I gave some hints.

I was confused why you were basing your decision off my definition.

 

[Comeback City]

I have a personal way I look at it, but since I don't have a good reference that looks at it the way I do, it might be better if I don't mention my personal interpretation as to why this happens, and stick to what's written about it in the literature.

Indeed they are pretty strict here about personal beliefs/theories. If you want to message me directly, I would be glad to hear you out.

On a related note, it's worth pointing out that massive particles not subject to any external forces other than gravity also travel along geodesic paths. Therefore the same formula that work for ultra-relativistic massive particles work (in the apporopriate relativistic limit) to give nearly the same path as photons - i.e. if you accelerate massive particles fast enough, so they are traveling almost at the speed of light, and said massive particles do not experience any force other than "gravity", they'll travel along nearly the same path that massless photons do.

Just for clarification purposes, when you say massive particles, are you just referring to elementary particles that have mass (in order to differentiate from massless particles), are do you mean particles with a significantly large mass?

 

[PeterDonis]

I suppose I would say that rather than perfect circular motion, it would simply be a "curved" motion. Is that accurate?

Yes, if "curved" means "curved in a particular chosen set of coordinates". But this kind of "curved" depends on your choice of coordinates, so it can't appear in any of the laws of physics.

The other kind of "curved", the kind that can appear in the laws of physics, is "nonzero proper acceleration". In other words, the way to tell whether the path of a particular object through spacetime is curved is to attach an accelerometer to it and see what it reads. If it reads nonzero, the path is curved; if it reads zero, the path is straight. By this definition, the path of a light ray in free space is straight, even if it is passing near a gravitating mass. (We can't actually attach an accelerometer to a light ray, but we can make other measurements which are equivalent to doing that.) The apparent "curvature" in coordinates centered on the mass is due to the curvature of spacetime, not of the light's path.

 

[Comeback City]

Yes, if "curved" means "curved in a particular chosen set of coordinates". But this kind of "curved" depends on your choice of coordinates, so it can't appear in any of the laws of physics.

How do you choose coordinates in a gravitational field?

 

[PeterDonis]

How do you choose coordinates in a gravitational field?

Coordinates are always chosen: they are labels we put on events to keep track of them. Nothing in spacetime comes pre-labeled with coordinates.

 

[Comeback City]

Coordinates are always chosen: they are labels we put on events to keep track of them. Nothing in spacetime comes pre-labeled with coordinates.

Oh so is it just a "frame of reference" type thing?

 

[PeterDonis]

Oh so is it just a "frame of reference" type thing?

Yes, that's what coordinates are. (At least, the terms are usually used synonymously; technically a "frame of reference" is not quite the same as a coordinate chart, but for most purposes the differences don't matter.)

 

[Comeback City]

Yes, that's what coordinates are. (At least, the terms are usually used synonymously; technically a "frame of reference" is not quite the same as a coordinate chart, but for most purposes the differences don't matter.)

Alright thanks!

 

[pervect]

Indeed they are pretty strict here about personal beliefs/theories. If you want to message me directly, I would be glad to hear you out.

Just for clarification purposes, when you say massive particles, are you just referring to elementary particles that have mass (in order to differentiate from massless particles), are do you mean particles with a significantly large mass?

I was implicitly thinking elementary particles, but the math works the same for, say, ping-ping balls, or even baseballs. The technical term for what I was thinking about when I wrote that is "test particles". I was also thinking about said test particles being deflected by an object that's massive enough to cause significant - or at least measurable - deflection of light. So to set the scale, I was thinking of said test particles (baseballs, say) being deflected by objects of at least a stellar mass.

If you start to consider two large masses, say a pair of black holes orbiting each other, you can't use the test particle assumptions that I was thinking of when I wrote the post, and you can no longer say that the masses follow geodesics. The test-particle approximation breaks down in this case. The limit as to when an object stops being a "test particle" is a bit vague, but a good guideline is if the object has significant gravity of it's own, it's not a test particle.

I believe what's important for the math is that the mass ratio between the test particle and the central mass be "high enough". I don't have a firm mathematical handle on how high is "high enough", but as a practical guide to my thoughts, a baseball orbiting a black hole would be well approximated by this test particle approximation, a binary inspiral of two black holes (such as those observed by Ligo) would, as I mentioned, not be well-approximated.

Of course, it'd be difficult with current technology to get a baseball up to reasonable fraction of the speed of light, but there are no theoretical difficulties with predicting what would happen according to GR if you managed to accelerate a baseball to this speed.

 

[Comeback City]

I was implicitly thinking elementary particles, but the math works the same for, say, ping-ping balls, or even baseballs. The technical term for what I was thinking about when I wrote that is "test particles". I was also thinking about said test particles being deflected by an object that's massive enough to cause significant - or at least measurable - deflection of light. So to set the scale, I was thinking of said test particles (baseballs, say) being deflected by objects of at least a stellar mass.

If you start to consider two large masses, say a pair of black holes orbiting each other, you can't use the test particle assumptions that I was thinking of when I wrote the post, and you can no longer say that the masses follow geodesics. The test-particle approximation breaks down in this case. The limit as to when an object stops being a "test particle" is a bit vague, but a good guideline is if the object has significant gravity of it's own, it's not a test particle.

I believe what's important for the math is that the mass ratio between the test particle and the central mass be "high enough". I don't have a firm mathematical handle on how high is "high enough", but as a practical guide to my thoughts, a baseball orbiting a black hole would be well approximated by this test particle approximation, a binary inspiral of two black holes (such as those observed by Ligo) would, as I mentioned, not be well-approximated.

Of course, it'd be difficult with current technology to get a baseball up to reasonable fraction of the speed of light, but there are no theoretical difficulties with predicting what would happen according to GR if you managed to accelerate a baseball to this speed.

Thanks for the explanation! I am going to try to read that paper when I get a chance also.

 

[pervect]

Oh so is it just a "frame of reference" type thing?

Pretty much. If you try to imagine building a frame of reference on the surface of the globe, you'll see some of the difficulties that you have in applying the concept to curved spaces. For instance, if you try to make a mosaic of square rods, it will work for small areas, but if you try to cover the entire globe with such a framework, it won't work.

 

[stoomart]

Well I suppose I would say that rather than perfect circular motion, it would simply be a "curved" motion. Is that accurate?

As a layman, I like a good visual for this kind of thing if possible: from Wikipedia's Schwarzschild geodesics article:

https://en.m.wikipedia.org/wiki/Schwarzschild_geodesics#Bending_of_light_by_gravity

 

[rude man]

Also, do photons have relativistic mass?

They have ONLY relativistic mass. Rest mass is zero.