# Relative speed of two photons

• MeAndMyLucidLife
In summary: Velocity is distance over time, but there is no time, so there can't be a velocity. Therefore, there can't be a reference frame. This is why the concept of a reference frame moving at the speed of light is nonsensical.I'm not sure I agree with this. It seems to me that any reference frame moving at c would have a time coordinate, it would just have a very different and bizarre metric.I'm not sure I agree with this. It seems to me that any reference frame moving at c would have a time coordinate, it would just have a very different and bizarre metric.In a frame moving at the speed of light,

## Homework Statement

What is the speed of a photon with respect to another photon if:
1. the two photons are going in the same direction.
2. they are going in opposite direction?
2. The attempt at a solution
I think the answer to the first question should be zero and to the second one be 2xC; C⇒speed of light.
But at the same time i think that the speed of anything can't exceed C, thus my second answer should be wrong. I also feel the we cannot have an inertial frame which is traveling at the speed C as then the Newtonian laws of motion will fail.
I don't know much about Relativity, and am pretty confident about some flaws in my approach.
Thank You:)

MeAndMyLucidLife said:

## Homework Statement

What is the speed of a photon with respect to another photon if:
1. the two photons are going in the same direction.
2. they are going in opposite direction?
2. The attempt at a solution
I think the answer to the first question should be zero and to the second one be 2xC; C⇒speed of light.
But at the same time i think that the speed of anything can't exceed C, thus my second answer should be wrong. I also feel the we cannot have an inertial frame which is traveling at the speed C as then the Newtonian laws of motion will fail.
I don't know much about Relativity, and am pretty confident about some flaws in my approach.
Thank You:)

The question is ambiguous. You are right that there is no frame of reference for a photon. But, by "speed" the question may mean "separation speed", being the speed at which two objects (or photons) converge or diverge in a given reference frame.

MeAndMyLucidLife said:

## Homework Statement

What is the speed of a photon with respect to another photon if:
1. the two photons are going in the same direction.
2. they are going in opposite direction?
2. The attempt at a solution
I think the answer to the first question should be zero and to the second one be 2xC; C⇒speed of light.
But at the same time i think that the speed of anything can't exceed C, thus my second answer should be wrong. I also feel the we cannot have an inertial frame which is traveling at the speed C as then the Newtonian laws of motion will fail.
I don't know much about Relativity, and am pretty confident about some flaws in my approach.
Thank You:)
This is a tricky question. A photon moves at the speed of light, and the speed of light is invariant, meaning it is the same in all reference frames. So in the reference frame of photon A, what is the speed of photon B?

PeroK said:
The question is ambiguous. You are right that there is no frame of reference for a photon. But, by "speed" the question may mean "separation speed", being the speed at which two objects (or photons) converge or diverge in a given reference frame.

"Separation speed" can exceed ##c##, even for non-photons.

In some inertial frame O, say particle A is moving with velocity ##v_A =(.75 c, 0,0)## while B is moving with velocity ##v_B = (-.75 c, 0, 0).## In frame O the separation speed is ##1.5 c##, but the separation speed of B as seen from the rest frame of A is less than ##c##; ditto for A as seen by B.

For photons there are no "rest frames". An observer cannot ride along on photon A and then make measurements for photon B.

tnich said:
So in the reference frame of photon A, what is the speed of photon B?
There is no "reference frame of a photon".

David Lewis and MeAndMyLucidLife
What if you replaced photon with "2 protons with speeds of say .999999 c"?
It seems that is what the question implies ( reference frames moving at speed c).

J Hann said:
What if you replaced photon with "2 protons with speeds of say .999999 c"?
It seems that is what the question implies ( reference frames moving at speed c).
That is a sensible question with a very interesting and important answer, but the answer and the thought process used to find that answer tell us nothing about the situation when the speed is 1.0c instead of .99999999...c

Nugatory said:
That is a sensible question with a very interesting and important answer, but the answer and the thought process used to find that answer tell us nothing about the situation when the speed is 1.0c instead of .99999999...c

On the other hand: ##1.0c = 0.999 \dots c##.

vela and SammyS
PeroK said:
On the other hand: ##1.0c = 0.999 \dots c##.
That is true, and I'm guilty of (knowingly) perpetuating a sloppiness from the post I was replying to. Getting to a rigorous statement is beyond the scope of this homework thread though.

MeAndMyLucidLife and PeroK
J Hann said:
What if you replaced photon with "2 protons with speeds of say .999999 c"?
It seems that is what the question implies ( reference frames moving at speed c).
Reference frames do not "move at speed c". First of all, all motion is relative. Second, no matter how close to c a reference frame moves relative to another, it is still not anywhere close to be "moving at c".

Lots of people in this thread have pointed out that reference frames can't move at the speed of light, but I'm not sure anybody has pointed out why that is the case.

The fundamental problem is time.

It is certainly possible to create something that looks like a hypothetical rest frame that moves at the speed of light. This is easy to do since we don't need to think about mass. But if you look at the properties of that hypothetical rest frame you have a problem: there is no time coordinate. You can see this by examining time dilation in the limit as ##v -> c##, which shows that no matter how much time passes in an inertial frame, no time ever passes in this hypothetical speed-of-light frame. That lack of time means that most of the questions we normally associate with a reference frame become nonsensical. For instance, velocity can't be measured because time doesn't pass, so questions about "relative velocity" are pointless.

So this problem cannot be answered because the "reference frame" of a photon isn't a proper reference frame at all (because it has no time coordinate).

Orodruin, Delta2, MeAndMyLucidLife and 1 other person
kimbyd said:
[...] S this problem cannot be answered because the "reference frame" of a photon isn't a proper reference frame at all (because it has no time coordinate).
Note, however, that the original problem statement did not mention the phrase "reference frame".

@MeAndMyLucidLife:
As stated, the original problem can be answered sensibly in terms of limits. I.e., take the usual addition rule for Lorentz boosts in a given direction (with ##|v| < c##), and take a limit as ##v## approaches ##c##.

E.g., consider bodies "A,B" moving at colinear velocities ##-v## and ##v## respectively relative to "O". The velocity ## v_{OA}## of O relative to A is ##+v##, and the velocity ##v_{BO}## of B relative to O is also ##+v##. Hence the velocity ##v_{BA}(v)## of B relative to A in this scenario is $$v_{BA}(v) ~=~ \frac{v + v}{1 + v^2/c^2} ~=~ \frac{2v}{1 + v^2/c^2}~.$$ Now take the limit: $$\lim_{v\to c^-} v_{BA}(v) ~=~ \lim_{v\to c^-} \frac{2v}{1 + v^2/c^2} ~=~ c ~.$$ More rigorously, one should talk about ##\epsilon##'s and ##\delta##'s, but the above is sufficient "at the usual level of rigor in theoretical physics".

Briefly, this approach is valid because these transformations depend on the underlying Lie group theory (wherein the group is a well-behaved differentiable manifold). ##|v|=c## happens to be a fixed point in the group, but we can still talk about sensibly in terms of limits because of the underlying theory.

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MeAndMyLucidLife
strangerep said:
Note, however, that the original problem statement did not mention the phrase "reference frame".

@MeAndMyLucidLife:
As stated, the original problem can be answered sensibly in terms of limits. I.e., take the usual addition rule for Lorentz boosts in a given direction (with ##|v| < c##), and take a limit as ##v## approaches ##c##.

E.g., consider bodies "A,B" moving at colinear velocities ##-v## and ##v## respectively relative to "O". The velocity ## v_{OA}## of O relative to A is ##+v##, and the velocity ##v_{BO}## of B relative to O is also ##+v##. Hence the velocity ##v_{BA}(v)## of B relative to A in this scenario is $$v_{BA}(v) ~=~ \frac{v + v}{1 + v^2/c^2} ~=~ \frac{2v}{1 + v^2/c^2}~.$$ Now take the limit: $$\lim_{v\to c^-} v_{BA}(v) ~=~ \lim_{v\to c} \frac{2v}{1 + v^2/c^2} ~=~ c ~.$$ More rigorously, one should talk about ##\epsilon##'s and ##\delta##'s, but the above is sufficient "at the usual level of rigor in theoretical physics".

Briefly, this approach is valid because these transformations depend on the underlying Lie group theory (wherein the group is a well-behaved differentiable manifold). ##|v|=c## happens to be a fixed point in the group, but we can still talk about sensibly in terms of limits because of the underlying theory.
Nope, this doesn't work. You've only avoided talking about a reference frame, rather than actually avoiding it altogether. It still has the same exact problem of time I mentioned.

The only way to make the problem correct is to consider the "velocity of photon A relative to photon B" in a reference frame that is neither A's nor B's. In that case the answer is pretty trivial: just add or subtract their velocities as you would any two objects given their relative direction of motion. This estimated velocity wouldn't be anything real per se, but it would be the time rate of change of distance between the two photons as measured by an observer.

kimbyd said:
Nope, this doesn't work. You've only avoided talking about a reference frame, rather than actually avoiding it altogether.
Of course I've avoided talking about a "reference frame of a photon". That's my point: by re-expressing the statement in terms of limits, things become better defined. I avoided the ill-defined concept of "reference frame of a photon" by using something else that is well-defined, just as in ordinary differential calculus, one replaces the ill-defined division by ##h=0## by the limit of a well-defined ratio.

All the answers are pretty convincing!

Things which I learned are:
We can't assume an inertial frame with respect to a photon!
So we cannot take any inertial frame into consideration.
I think this is because time and space are not defined for a photon.

So, What should be the proper answer statement for the question in which two photons are traveling in opposite directions?

And if both the photons are traveling in the same direction, then can't we see the motion with respect to either of the photon?
Assume the light wave: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > ; where ~ denotes a single photon.

Now considering any two photons from this wave train, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ >

Now these two photons will always be moving along the same line of path and with equal velocities ie. C.
So now, can't we say that the relative speed between them is ZERO?

Thank You:-)

MeAndMyLucidLife said:
All the answers are pretty convincing!

Things which I learned are:
We can't assume an inertial frame with respect to a photon!
So we cannot take any inertial frame into consideration.
I think this is because time and space are not defined for a photon.

So, What should be the proper answer statement for the question in which two photons are traveling in opposite directions?

And if both the photons are traveling in the same direction, then can't we see the motion with respect to either of the photon?
Assume the light wave: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > ; where ~ denotes a single photon.

Now considering any two photons from this wave train, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ >

Now these two photons will always be moving along the same line of path and with equal velocities ie. C.
So now, can't we say that the relative speed between them is ZERO?

Thank You:-)

"Relative" speed has a precise meaning, which is the speed of one object in the reference frame of the other. As speed is distance/time and neither is defined for a photon, then relative speed of any object is not defined for a photon.

"Separation speed" is defined as above and hence is defined within any inertial frame for two photons, and is either 0 or 2c in these cases.

Cutter Ketch and MeAndMyLucidLife
PS in physics you must be careful not to confuse everyday meanings with precise physics meanings. E.g. energy and power.

This also applies to words like "relative", which has a precise meaning in this case.

MeAndMyLucidLife
MeAndMyLucidLife said:
We can't assume an inertial frame with respect to a photon!
So we cannot take any inertial frame into consideration.
I think this is because time and space are not defined for a photon.
That last statement is incorrect. Rather, it's because a photon has no rest frame, as others have explained.

So, What should be the proper answer statement for the question in which two photons are traveling in opposite directions?
[...]
these two photons will always be moving along the same line of path and with equal velocities ie. C.
So now, can't we say that the relative speed between them is ZERO?
Keep in mind that this is a homework thread. Therefore we're not supposed to simply give you the answers, but rather to offer hints to help you reach the answer for yourself.

Indeed, I think I might have given too much of a hint in my post #12, which already contains both the answers ,if you study and apply it. I.e., don't guess, and don't try to reason "intuitively". Use the math of special relativity, in this case the velocity addition formula.

MeAndMyLucidLife
PeroK said:
"Relative" speed has a precise meaning, which is the speed of one object in the reference frame of the other. As speed is distance/time and neither is defined for a photon, then relative speed of any object is not defined for a photon.

"Separation speed" is defined as above and hence is defined within any inertial frame for two photons, and is either 0 or 2c in these cases.

Now i got it!
Thanks for making my concepts and terminology clearer...
Thanks a lot!

strangerep said:
Of course I've avoided talking about a "reference frame of a photon". That's my point: by re-expressing the statement in terms of limits, things become better defined. I avoided the ill-defined concept of "reference frame of a photon" by using something else that is well-defined, just as in ordinary differential calculus, one replaces the ill-defined division by ##h=0## by the limit of a well-defined ratio.
The limit of the thing is not always the thing at the limit. Especially when the thing at the limit is undefined.

strangerep said:
Of course I've avoided talking about a "reference frame of a photon". That's my point: by re-expressing the statement in terms of limits, things become better defined. I avoided the ill-defined concept of "reference frame of a photon" by using something else that is well-defined, just as in ordinary differential calculus, one replaces the ill-defined division by ##h=0## by the limit of a well-defined ratio.
It still doesn't work because it leads to an inconsistency in the case where the two photons are traveling in the same direction.

kimbyd said:
It still doesn't work because it leads to an inconsistency in the case where the two photons are traveling in the same direction.
What inconsistency?

jbriggs444 said:
The limit of the thing is not always the thing at the limit. Especially when the thing at the limit is undefined.
I know that. That's why I was re-interpreting the original question in terms of approaching a limit rather than "at the limit point".

strangerep said:
What inconsistency?
The observed speed of the other photon would be zero. It's not possible for photons to have a speed anything other than c, for any observer.

Really, there are only two ways to interpret this problem:
2. It doesn't mean that we should take the difference in speed in a photon's reference frame at all, but by an outside observer.

Edit:
Also, you are using a reference frame of a photon. You can't do that, because there's no time coordinate, and thus no way to measure velocity at all. You avoid the word reference frame, but that doesn't change the math you're using.

kimbyd said:
[...] Also, you are using a reference frame of a photon.

No I'm not -- I'm using a sequence of ordinary reference frames.

[Aside: I'm getting the feeling you're not reading what I actually wrote, so I leave this conversation here.]

MeAndMyLucidLife said:

## Homework Statement

What is the speed of a photon with respect to another photon if:
1. the two photons are going in the same direction.
2. they are going in opposite direction?
2. The attempt at a solution
I think the answer to the first question should be zero and to the second one be 2xC; C⇒speed of light.
Measurement of the photon relative speed depends on where and when the photons were created and where and when they were detected, as measured in an inertial observer's frame. The photons must exist at the same time in the observer's reference frame: i.e. the observer must observe detection of each of the photons to occur after the creation of the other photon. So your answer would be correct if you were asking what the relative speed would be as measured in an inertial observer's frame of reference in which the detection of each of the photons occurred after the creation of the other as measured in that inertial reference frame.

If the reference frame in which these events are observed is one in which the detection of one photon occurs before the creation of the other, then the relative speed has no meaning because according to the observer's measurements, the two photons do not exist at the same time.

If you are asking what is the speed of separation of the two photons from the imaginary "frame of reference" of one of the photons, the relative speed, it seems to me, would have no meaning for the same reason: the detection event for each photon could never occur after the creation event of the other as measured in that (imaginary) frame.

AM

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MeAndMyLucidLife
Andrew Mason said:
Measurement of the photon relative speed depends on where and when the photons were created and where and when they were detected.
Learned something new!
Thanks a Lot

strangerep said:
What inconsistency?
consider the limit of v and v in the same direction, as v approaches c; and the limit of v and c in the same direction as v approaches c. For either limit to be physically meaningful, they ought to be the same. Yet the former is zero and latter is c.

I strongly agree with @kimbyd here.

PAllen said:
consider the limit of v and v in the same direction, as v approaches c; and the limit of v and c in the same direction as v approaches c. For either limit to be physically meaningful, they ought to be the same. Yet the former is zero and latter is c.

I strongly agree with @kimbyd here.
More generally, take each v as function of some parameter such that as parameter goes to infinitiy, v goes to c. Then, by choice of functions, you can get your pick of -c, 0, or c for the limit as the parameter goes to infinity, even though each v is clearly approaching c.

## 1. What is the relative speed of two photons?

The relative speed of two photons is always equal to the speed of light, which is approximately 299,792,458 meters per second in a vacuum. This means that no matter the direction or relative motion of the photons, they will always move at the same speed.

## 2. Can two photons have different relative speeds?

No, according to the theory of relativity, the speed of light is a constant and cannot be changed by the relative velocity of the observer. Therefore, two photons will always have the same relative speed regardless of the observer's reference frame.

## 3. How does the relative speed of two photons affect their energy?

The energy of a photon is directly proportional to its frequency, which is determined by its speed. Since the relative speed of two photons is constant, their energy will also be equal.

## 4. Is there a limit to the relative speed of two photons?

According to the theory of relativity, the speed of light is the maximum speed at which any object can travel. Therefore, the relative speed of two photons cannot exceed the speed of light.

## 5. Can the relative speed of two photons be measured?

Yes, the relative speed of two photons can be measured using advanced techniques such as interferometry. This involves measuring the interference patterns created when two photons of different frequencies are superimposed, which can be used to calculate their relative speed.