Speed of light and acceleration in general relativity

[Rerry]

Mentor note: this discussion was split out of a different thread.

The speed of light in a vacuum is constant, but what I would like some information regarding is Black Holes. Does a Black Hole increase the speed of a light photon as it is being pulled into the Event Horizon?

 

[jbriggs444]

The speed of light in a vacuum is constant, but what I would like some information regarding is Black Holes. Does a Black Hole increase the speed of a light photon as it is being pulled into the Event Horizon?

In order to model a black hole, one must be using general relativity. But in general relativity, gravity is not a pull. It is curved space-time. Objects that are freely falling under the influence of gravity are not accelerated. They are following "straight line" trajectories -- geodesics.

If one calculated the speed of light against a locally inertial reference frame (i.e. a reference frame that is freely falling), the result would always be c. That is to say that the speed of light is unchanged by the presence of a black hole. [In general relativity the curvature of space-time can be ignored and the rules of special relativity and inertial frames applied as long as one considers a sufficiently small local region]

If one calculates the speed of light using a global frame of reference then the speed of light can be determined to be greater than c. But the speed you calculate depends on the coordinate system you choose. There is no such thing as a global inertial frame in general relativity, no preferred notion of simultaneity at a distance and, accordingly, no unambiguously correct way to specify how the speed of light should be calculated.

 

[Jon Richfield]

Light travels c in a vacuum. According to special relativity, something traveling at c is impossible to accelerate.

What on Earth (or in space) is all this about the constancy of speed (such as c) implying constancy of velocity (v)?
Denial of that idea can't be just my ignorance; others already have pointed it out, apparently casting their pearls futilely.
That there are frames of reference according to which a given photon does not change its velocity is irrelevant to the extent that it might be true in any given context.

Almost everywhere in our universe there are gravitational fields such that to an extent (slight in slight fields) any photon soon changes its velocity relative to nearly all observers in the universe. (Always recognising that from the photon's point of view, everything (if anything) happens instantly!)

If at time 0 the speed (NB!) of a photon in space is c in the direction (X0,Y0,Z0) and at time 1 (relevantly different from time 0) the speed still is c, but in the direction (X1,Y1,Z1) (not = (X0,Y0,Z0)), that is change in velocity. Change in velocity is acceleration whether the speed changes or not and whether the sign of the change is positive or not. The change of a photon's velocity could happen, not merely in our local space, but in intergalactic space far from the madding mirror, just coasting through gravitational gradients, but that duzzint make no neverminds -- any non-zero change in velocity still is acceleration.
Otherwise what IS acceleration? If I swing a cat at constant speed, are you arguing that the thongs are not accelerating because a fly sitting on a thong notices no change in airspeed?
If moving through an arc at constant speed is not acceleration, then acceleration aint what it used to be!

 

[jbriggs444]

Otherwise what IS acceleration? If I swing a cat at constant speed, are you arguing that the thongs are not accelerating because a fly sitting on a thong notices no change in airspeed?

If you spin in a circle while your cat sits on the couch, is your cat accelerating? It is in that sense of the word acceleration that light accelerates as it moves near gravitating objects.

 

[A.T.]

If moving through an arc at constant speed is not acceleration,

That is "coordinate acceleration" (dV/dt), which is frame dependent. But there is also the frame invariant "proper acceleration (what an accelerometer measures)" .

 

[Jon Richfield]

If you spin in a circle while your cat sits on the couch, is your cat accelerating? It is in that sense of the word acceleration that light accelerates as it moves near gravitating objects.

Try working that out in GR, and the tensor values will explain why your static scourge on the couch (no one spoke of felines! That was why I spoke of a fly on the thong) experiences no stresses, whereas your ears do.
Note that just as "centrifugal forces" will tell who is "spinning" with reference to "the fixed stars", similarly, the change of velocity (ahem! ) and therefore momentum, colour etc, will tell which photon passed through a gravitational field. All of that in any form demands acceleration whether there is a change of speed or not. Ask those fixed stars if you doubt me!

 

[Jon Richfield]

If you spin in a circle while your cat sits on the couch, is your cat accelerating? It is in that sense of the word acceleration that light accelerates as it moves near gravitating objects.

Oh, and it emphatically is NOT in that sense of the word acceleration that light accelerates in gravitational fields. If you don't believe me, try measuring the red shift before and after the acceleration. The stationary cat experiences no change in red shift, while the swung cat does.

 

[jbriggs444]

Oh, and it emphatically is NOT in that sense of the word acceleration that light accelerates in gravitational fields.

It most certainly is.

If you don't believe me, try measuring the red shift before and after the acceleration. The stationary cat experiences no change in red shift, while the swung cat does.

Please keep up. You are the one arguing that the cat on the couch is accelerating.

 

[Jon Richfield]

It most certainly is.

Please keep up. You are the one arguing that the cat on the couch is accelerating.

Nope! You are the one arguing that there is no change in the momentum or velocity of either scourge or your ears or in the red shift of the gravitationally refracted light!

Nice try though!

 

[Jon Richfield]

That is "coordinate acceleration" (dV/dt), which is frame dependent. But there is also the frame invariant "proper acceleration (what an accelerometer measures)" .

Both are actual physical acceleration either in GR or Sr or Newtonian physics.
In particular both will affect for example the energy, momentum, or red shift that an unaccelerated observer will experience (measure) on encountering such an accelerated photon.

 

[jbriggs444]

Nope! You are the one arguing that there is no change in the momentum or velocity of either scourge or your ears or in the red shift of the gravitationally refracted light!

Please do not jump ahead.

Before we can address the question of light, we must address the question of that cat on the couch. Is it accelerating or is it not? You have held that any change of velocity involves an acceleration. In coordinates in which you are stationary, the cat is moving around you in a circle with a constant speed and a changing velocity. May we take it that your position is that the cat is accelerating? Or may we take it that your position is that changes in velocity do not always involve accelerations?

 

[ZapperZ]

This has gone very far beyond the scope and understanding of the OP (remember him/her?). At what point did people become oblivious to helping that person at a level that he/she can understand?

Zz.

 

[Jon Richfield]

This has gone very far beyond the scope and understanding of the OP (remember him/her?). At what point did people become oblivious to helping that person at a level that he/she can understand?

Zz.

Your point is not clear.
The question was whether light can accelerate in a vacuum.
Several persons have explained that it can because acceleration involves a change in velocity, not necessarily in speed, and gave examples of how and why.
Some others have argued, patently fallaciously, that because light cannot change in speed it cannot accelerate.
One of them apparently even argues that one cannot distinguish between accelerations of different entities (swung cats and all that).
Surely it is no favour to the OP to let such assertions go unchallenged?

 

[mfb]

I split the threads, the original one is here. Please respect the level indicators - details of general relativity are certainly not [B].

 

[jbriggs444]

Your point is not clear.
[...]
One of them apparently even argues that one cannot distinguish between accelerations of different entities (swung cats and all that).
Surely it is no favour to the OP to let such assertions go unchallenged?

Thank you, mfb for splitting the threads.
Jon, no one is saying that one cannot distinguish between coordinate acceleration (a sitting cat) and proper acceleration (a swung cat).

I am trying to get you to clarify which you are speaking of when you say that light experiences changes in velocity.

 

[PeterDonis]

Always recognising that from the photon's point of view, everything (if anything) happens instantly!

This is not correct; the photon has no "point of view" in this sense.

 

[DrStupid]

What on Earth (or in space) is all this about the constancy of speed (such as c) implying constancy of velocity (v)?

It's just a fact, that the velocity of light in vacuum can't change without changing it's speed. At first view light deflection in a gravitational field seems to be an example for acceleration with constant speed. I also made the mistake to exclude the Shapiro delay in the attempt to keep it simple. But that's an oversimplification because there is no light deflection without Shapiro delay.

 

[pervect]

Mentor note: this discussion was split out of a different thread.

The speed of light in a vacuum is constant, but what I would like some information regarding is Black Holes. Does a Black Hole increase the speed of a light photon as it is being pulled into the Event Horizon?

There is some ambiguity in what one might mean when one talks about "the speed of light". To simplify it as much as possible, consider a light beam right at the event horizon moving outwards. In one sense, which I'll call a coordinate sense, the "speed" of that light is zero, because the beam of light is staying right at the event horizon. In another sense, for instance if one measured the speed of light via an instrument falling through the black hole (see https://en.wikipedia.org/wiki/Fizeau–Foucault_apparatus for a detailed description of such an instrument) the instrument would not detect any change in the speed of light as the instrument was dropped into the black hole.

So "the speed of light" is a constant in the later sense, the coordinate-independent sense of what an instrument would measure, while it is not necessarily a constant in terms of any given coordinate system (such as Schwarzschild coordinates - there's nothing necessarily fixed about the "speed" of light defined as a ratio of the distance coordinate divided by the time coordinate.).

 

[Jon Richfield]

It's just a fact, that the velocity of light in vacuum can't change without changing it's speed. At first view light deflection in a gravitational field seems to be an example for acceleration with constant speed. I also made the mistake to exclude the Shapiro delay in the attempt to keep it simple. But that's an oversimplification because there is no light deflection without Shapiro delay.

How does that affect the question, which was about whether light can be accelerated, not about whether its speed can be changed? It is not clear to me whether in terms of GR, Shapiro delay involves a conceptual change to the speed of light or not, and I for one did not address the question of constant speed at all.
I still don't.
All I was considering was "can light be accelerated", and clearly it can be and routinely is.
Remember that gravitationally deflected light can be distinguished from undeflected light by where and when it arrives, by its red/blue shift and resultant energy etc, in ways that are closely analogous to the effects of the acceleration of massive matter.

It quacks like a duck say I, and I quack back, saying: "Duck!"

 

[DrStupid]

How does that affect the question, which was about whether light can be accelerated, not about whether its speed can be changed?

Depending on the coordinates the speed of light in vacuum is either constant or not and light is accelerated if and only if it's speed changes.

 

[Jon Richfield]

This is not correct; the photon has no "point of view" in this sense.

Says who?
Consider:
two electrons and two positrons and some independent observers are buzzing about their respective trajectories. (you might like to consider more massive bodies and antimatter bodies according to taste, but I want to keep things simple.)

One electron and positron fuse and form a gamma photon. For some convenient (longish) period that situation holds. Then suitable circumstances cause the gamma to split into an electron and positron.
From the point of view of the "reconstituted" particles the elapsed time is from start of experiment to fusion, plus from splitting to end of experiment. (zero subjective time in photon, whether accelerated or not). For the other two bodies and any other observers. the elapsed time is the full period from start to end.
The degree to which all observers agree depends on their various frames of reference of course, but all of them apart from the gamma pair agree that there was a non-zero interval that to the gamma pair was of duration zero.
To the gamma, the event of fusion and the event of splitting were simultaneous.
From the point of view of the other observers, they were not.
(Unless the gamma pair kept a clock outside to refer back to, which I think you will agree stretches the complexity of the assumptions and interpretations a bit much!)
That is about as meaningful a point of view as an electron or positron could have, as long as one doesn't insist on being anthropomorphic about it

 

[DrStupid]

Says who?

Einstein. A "point of view of a photon" would imply a rest frame of the photon. But as the photon would be at rest in its own rest frame (per definition) this would be a violation of the second postulate. Therefore such a frame of reference does not exist.

 

[Jon Richfield]

Thank you, mfb for splitting the threads.
Jon, no one is saying that one cannot distinguish between coordinate acceleration (a sitting cat) and proper acceleration (a swung cat).

I am trying to get you to clarify which you are speaking of when you say that light experiences changes in velocity.

I already said in effect and have said again:
If you fire off a photon under circumstances that affect neither its wavelength (energy) nor its direction, then we can all agree that it has not been accelerated.
And similarly, if you fire off a bullet under analogous circumstances (free fall and all that) then we can all agree that it has not been accelerated either.
This has certain measurable and calculable implications for where, when and how energetically your bullet and photon arrive.
OK.
BUT
Stick a gravitational gradient in anywhere along the path so as to affect the path, and it will have other calculable implications for where, when and how energetically your bullet and photon arrive, whether it affects their speed or not, and these affect the momentum, direction and all that, all of them being absolutely dependent on the energies involved and so on.
All of it being fully subject to everyday (and GR) interpretations of acceleration, whether coordinate acceleration or any other.
The energetic implications remain the same as in F=MA.

Hmmm... You might find it helpful to reconsider your cat analogy in terms of the difference between a swung photon and an unswung photon. Or something like that!

 

[jbriggs444]

I already said in effect and have said again:
If you fire off a photon under circumstances that affect neither its wavelength (energy) nor its direction, then we can all agree that it has not been accelerated.
And similarly, if you fire off a bullet under analogous circumstances (free fall and all that) then we can all agree that it has not been accelerated either.
This has certain measurable and calculable implications for where, when and how energetically your bullet and photon arrive.

As I understand this, you are envisioning the firing of a photon in flat space-time. Similarly for the bullet. And observing that the photon arrives at its destination with the same energy with which it was launched. Similarly for the bullet. You appear to be implicitly assuming a global inertial frame of reference for all of this.

That's fine. One could quibble but I will not. Yes we can agree on that much.

BUT
Stick a gravitational gradient in anywhere along the path so as to affect the path, and it will have other calculable implications for where, when and how energetically your bullet and photon arrive, whether it affects their speed or not, and these affect the momentum, direction and all that, all of them being absolutely dependent on the energies involved and so on.
All of it being fully subject to everyday (and GR) interpretations of acceleration, whether coordinate acceleration or any other.

The velocity and energy at arrival will depend on your choice of coordinates. Similarly, whether any coordinate acceleration has occurred en route will depend on your choice of coordinates. Whether any proper acceleration has occurred will not depend on your choice of coordinates

One can have a trajectory with zero proper acceleration but a non-zero change of velocity or of energy. You just have to use a non-inertial frame. In general relativity there is little choice. There is no such thing as a global inertial frame.

 

[weirdoguy]

Says who?

It's a basic fact in relativity and it's been discussed here multiple times...

 

[PeterDonis]

Says who?

You are skating perilously close to a warning here. The fact that you cannot construct an inertial frame in which a photon is at rest is standard special relativity, known for decades.

To the gamma, the event of fusion and the event of splitting were simultaneous.

Wrong. These events are null separated. Null separated events cannot be simultaneous; only spacelike separated events can be simultaneous (and only if you choose the appropriate coordinates).

 

[PeterDonis]

Everyone, please remember that this is an "A" level thread. That means you are expected to have the background knowledge of a physics graduate student. It also means you should not be using imprecise ordinary language; you should be using precise math, or at least precise ordinary language that translates directly into precise math. Words like "speed", "acceleration", and "point of view" are not precise. Even "frame of reference" is not really precise (if you don't understand why not, you should not be posting in an "A" level thread).

(Yes, that means the OP question in itself is not precise. The way to fix that is to propose a precise interpretation, and then answer that. See my next post for some ways of doing so.)

 

[PeterDonis]

Does a Black Hole increase the speed of a light photon as it is being pulled into the Event Horizon?

Per my previous post, we need to make this question precise in order to answer it. Unfortunately there is more than one way of doing that. Here are some of the ways:

(1) As long as the photon is above the horizon, there will be "static" observers (observers who are "hovering" at a constant altititude above the horizon) who can measure the photon's speed as it passes them. All of these observers will measure the photon's speed to be $c$ as it passes them. So in this sense, the answer to the question is "no".

(2) We can also consider infalling observers, who are free-falling into the hole, and who therefore can observe the photon passing them even if they are at or below the horizon. All of these observers will also measure the photon's speed to be $c$ as it passes them. So in this sense also, the answer to the question is "no".

(Note: what will be different for these various observers is the frequency they will measure the photon to have. If we assume that the photon is emitted with some known reference frequency, the static observers will measure it to be blueshifted when it passes them, and the blueshift will be larger the closer the observers are to the horizon. The infalling observers will measure the photon to be redshifted, and the redshift will be larger the further the observers have fallen when the photon passes them.)

(3) If we adopt Schwarzschild coordinates in the region outside the horizon, then the coordinate speed of the photon does change as it falls towards the horizon; but the change is a decrease, not an increase. In the limit as the photon approaches the horizon, its coordinate speed in these coordinates goes to zero.

(4) If we adopt Painleve coordinates, which can cover all the way down through the horizon and on to $r = 0$, then the coordinate speed of the photon does increase as it falls. So on this particular choice of coordinates, the answer to the question is "yes".