# I would like to see the math of something

If someone would be so kind as to show me the math of why a reference frame for a photon wouldn't work?

"Problem #1: In the frame of reference of some photon in that beam, what is the velocity of some other photon in the beam? We have a slight problem here with the second postulate of special relativity, which says that the local speed of light is the exact same value, c, in all reference frames.

Problem #2: Another aspect of special relativity is that one can transform from any one inertial frame to another using the Poincare transform. Try going to/from the photon frame of reference using this transformation. There will be a slight problem with dividing by zero / multiplying by infinity here."

This was quoted from the user D H in a thread I came across. I would like to see the math involed with is, along with any other equations that back up why photons have a frame of reference wouldn't make sense.

Thank you

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## Answers and Replies

Bacle2
Science Advisor
You may have better luck posting this in the/a physics forum.

HallsofIvy
Science Advisor
Homework Helper
I'm moving this thread to the "Special and General Relativity" forum.

"Math?" Why does there have to be math for everything?

The speed of light is c in all inertial reference frames, by postulate. An inertial frame in which a photon is at rest violates this postulate. Done.

Yes, I know that ZikZak, you're on a science forum, of course we are going to want see math done to back up reasoning...

Now again, I ask if someone would do me the favor of showing me the formula or equations in which 'C' is entered and then the equation comes out with a divison by zero or an answer of infinity or something of the like.

I wish to see this just to better my understanding of the math behind the reasoning. After all math is "the language of physics."

Matterwave
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Gold Member
Reasoning is reasoning and math is math. Math is important in physics, but if you become so dependent on the math that you won't accept physical reasoning, then I think you've gone too far.

But ok, the Lorentz boost in the x-direction is defined by:

$$t'=\gamma (t-\frac{vx}{c^2})$$
$$x'=\gamma (x-vt)$$
$$y'=y$$
$$z'=z$$

We have defined:
$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

If you plug in v=c (boosting to the frame of the photon), you get gamma is a division by 0 which is a non-nonsensical answer.

You know, you put a lot of trust into equations :)

How about experiments?

We have countless experiments showing us that 'light' doesn't care about what 'speed' you think you are doing, relative some other frame of reference. It always travel at 'c' measured locally. That's the 'frame of reference' for a 'photon'. To deny this you will need to prove it otherwise, not theoretically but by experiment.

You also will need to explain all other experiments telling us that 'c' is a constant.

(Eh, not you matter wave:)

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I didn't mean to come off the way I may have. I just always see people talking about light and referencing that a frame of reference for it doesn't work. I understand that reasoning, I do. But for me at least, when I see the math, it becomes even more clear becasue I can picture in my head doing the math instead of picturing in my head a photon traveling at C.

So thank you.

atyy
Science Advisor
The "frame" of a photon can be defined using "light cone" coordinates.

If you have a Lorentz inertial frame whose coordinates are (t,x,y,z), the light cone coordinates are (a,b,c,d), with

a=(t+x)/√2
b=(t-x)/√2
c=y
d=z.

That is not a Lorentz transformation (you can check by using Matterwave's equations in post #6). Since (a,b,c,d) results from a non-Lorentz transformation applied to the Lorentz inertial coordinates (t,x,y,z), the light cone coordinates do not form a Lorentz inertial frame.

ghwellsjr
Science Advisor
Gold Member
You know, you put a lot of trust into equations :)

How about experiments?

We have countless experiments showing us that 'light' doesn't care about what 'speed' you think you are doing, relative some other frame of reference. It always travel at 'c' measured locally. That's the 'frame of reference' for a 'photon'. To deny this you will need to prove it otherwise, not theoretically but by experiment.

You also will need to explain all other experiments telling us that 'c' is a constant.

(Eh, not you matter wave:)
The 'c' that is measured locally is always a round-trip "average speed" for light which has nothing to do with any 'frame of reference', only that the measurement takes place under conditions of the apparatus being inertial, that is, not accelerating. This kind of measurement is covered by Einstein's first postulate, the principle of relativity.

But then Einstein has a second postulate which states that the unmeasurable and unknowable one-way speed of light is also 'c' which is a mathematical statement and the basis for his purely mathematical definition for a Frame of Reference which includes the concept of space-time and four-dimensional events. He also derives the mathematical Lorentz Transform to connect events from one Frame of Reference to another FoR moving at some speed, v, with respect to the first one.

So Einstein's mathematical definition of a FoR and the Lorentz Transform that won't allow for a FoR for anything traveling at v=c with respect to any other FoR which includes photons.

There cannot be any experimental proof for this since the foundation of Einstein's Theory of Special Relativity is purely mathematical. If you want to prove that a photon travels at 'c' or that light propagates at 'c', you're going to need a different theory based on experiments, not mathematics, and I don't think you're going to find one.

Don't get you there ghwellsjr?

Light has only one 'speed' locally, in any experiment, as far as I know. The only way you ever will measure anything is locally. And it doesn't matter how you measure your 'motion' relative some other frame for this.

Are you telling me that this is wrong?

(ignoring it travelling through 'mediums' as glass and water etc for this)

ghwellsjr
Science Advisor
Gold Member
When you make a round-trip measurement of the speed of light, you have one timing device located at the source of the light and a mirror some measured distance away. All you know is total time it takes for the light to get from the source to the mirror and back to the source where the timer is located. You cannot know what time the light hit the mirror and therefore you cannot know in that experiment how fast the light was traveling in each direction. Einstein's solution is to make the two time intervals equal (mathematically). That's his second postulate. Now you can define time on a remote clock with respect to a local clock and from that you can build the mathematical concept of a Frame of Reference.

What if the final answer in the eqution in post 6 came out to be an actual number with no divison by zero (meaning the object wasn't moving at C), what does that actually mean? Like if the asnwer comes out to .8, what is that telling you?

robphy
Science Advisor
Homework Helper
Gold Member
Problem #2: Another aspect of special relativity is that one can transform from any one inertial frame to another using the Poincare transform. Try going to/from the photon frame of reference using this transformation. There will be a slight problem with dividing by zero / multiplying by infinity here."

The lorentz (boost) transformation must preserve the square-norms of 4-vectors.
So, a timelike-vector (associated with a typical inertial frame) [with square-norm 1] cannot be transformed to a lightlike-vector [with square-norm 0]... and vice versa.

Fredrik
Staff Emeritus
Science Advisor
Gold Member
"Problem #1: In the frame of reference of some photon in that beam, what is the velocity of some other photon in the beam? We have a slight problem here with the second postulate of special relativity, which says that the local speed of light is the exact same value, c, in all reference frames.
You seem to be overlooking the fact that if there is such a thing as an inertial reference frame of the photon, then that photon would have to have both speed 0 and speed c in it. There is an immediate contradiction even if you don't consider any other photons, so the problem is anything but "slight". (The speed must be 0 because it's assumed to be the reference frame in which the photon is at rest, and the speed must be c because it's assumed to be an inertial frame. The contradiction means that if a photon has a reference frame, it's not an inertial frame). Of course, the same thing can be said about any other photon in the beam.

Problem #2: Another aspect of special relativity is that one can transform from any one inertial frame to another using the Poincare transform. Try going to/from the photon frame of reference using this transformation. There will be a slight problem with dividing by zero / multiplying by infinity here."
This is what matterwave was showing you. However, rather than concluding that there's a division by zero problem, you should note that Poincaré transformations don't apply to "the reference frame of the photon", because it can't be defined as an inertial frame. That last part is proved by what I said in the reply to problem #1.

What if the final answer in the eqution in post 6 came out to be an actual number with no divison by zero (meaning the object wasn't moving at C), what does that actually mean? Like if the asnwer comes out to .8, what is that telling you?
It means that v=0.6c. The other equalities tell you that if I'm moving with speed 0.6c relative to you, I would be assigning coordinates (t',x',y',z') to the event you would be assigning coordinates (t,x,y,z). This is assuming that we both use the inertial coordinate systems that are associated with our motions in a standard way. I would say that the reason why "the rest frame of the photon" doesn't make sense (even if we would allow it to be a non-inertial coordinate system) is that this standard way of associating coordinate systems with non-accelerating objects doesn't work for photons. Some of the details are explained in this post.

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Fredrik
Staff Emeritus
Science Advisor
Gold Member
The "frame" of a photon can be defined using "light cone" coordinates.
You can pick any coordinate system with a time axis that coincides with the photon's world line and call it "the frame of the photon" if you want to. But I don't know a reason why the light cone coordinates should be preferred over any of the others.

atyy
Science Advisor
You can pick any coordinate system with a time axis that coincides with the photon's world line and call it "the frame of the photon" if you want to. But I don't know a reason why the light cone coordinates should be preferred over any of the others.

Yes. The aim was just to pick a reasonable definition (this one that is actually useful) and show that it is not a Lorentz inertial frame.

Good information in this thread, thanks. Though I would like to point out, Fredrik, that part of my OP was quoted from the user D H in another thread. It isn't my words.

You seem to be overlooking the fact that if there is such a thing as an inertial reference frame of the photon, then that photon would have to have both speed 0 and speed c in it. There is an immediate contradiction even if you don't consider any other photons, so the problem is anything but "slight". (The speed must be 0 because it's assumed to be the reference frame in which the photon is at rest, and the speed must be c because it's assumed to be an inertial frame. The contradiction means that if a photon has a reference frame, it's not an inertial frame). Of course, the same thing can be said about any other photon in the beam.

Wait, are you saying in the bolded above that if we were looking for a reference frame of a photon, then we don't neccesarily need to have a speed of 0?

Fredrik
Staff Emeritus
Science Advisor
Gold Member
Wait, are you saying in the bolded above that if we were looking for a reference frame of a photon, then we don't neccesarily need to have a speed of 0?
No, the "reference frame of" something is always a frame or a coordinate system such that the "something" has speed 0. What I'm saying is that such a frame must be non-inertial, because in any inertial frame, the speed of light is c≠0

(I also said that there's no non-inertial frame with properties that singles it out as "the" reference frame of the photon).

When you make a round-trip measurement of the speed of light, you have one timing device located at the source of the light and a mirror some measured distance away. All you know is total time it takes for the light to get from the source to the mirror and back to the source where the timer is located. You cannot know what time the light hit the mirror and therefore you cannot know in that experiment how fast the light was traveling in each direction. Einstein's solution is to make the two time intervals equal (mathematically). That's his second postulate. Now you can define time on a remote clock with respect to a local clock and from that you can build the mathematical concept of a Frame of Reference.

Sure, but why citing me for that? Use the Michelson–Morley experiment, find a speed. It will be 'c'. That's your experimental definition. Now accelerate a spaceship to some speed relative Earth, then move uniformly. Do the same experiment, you will again find 'c'.

That's what I meant saying that 'c' doesn't care about your moving relative some other 'frame of reference'. It will always come out as 'c', and that's also where the Lorentz contraction comes in, to explain the fact that it never varies, all as I see it.

ghwellsjr
Science Advisor
Gold Member
Sure, but why citing me for that? Use the Michelson–Morley experiment, find a speed. It will be 'c'. That's your experimental definition. Now accelerate a spaceship to some speed relative Earth, then move uniformly. Do the same experiment, you will again find 'c'.

That's what I meant saying that 'c' doesn't care about your moving relative some other 'frame of reference'. It will always come out as 'c', and that's also where the Lorentz contraction comes in, to explain the fact that it never varies, all as I see it.
MMX showed that the two-way speed of light was a constant when measured along different directions and at different speeds for the apparatus, as you pointed out. (Although it didn't measure the value 'c', other experiments did that.) It didn't measure the one-way speed of light, nor did it show that it was a constant in all directions. Neither has any other experiment, nor can any experiment do so. But in your post,
You know, you put a lot of trust into equations :)

How about experiments?

We have countless experiments showing us that 'light' doesn't care about what 'speed' you think you are doing, relative some other frame of reference. It always travel at 'c' measured locally. That's the 'frame of reference' for a 'photon'. To deny this you will need to prove it otherwise, not theoretically but by experiment.

You also will need to explain all other experiments telling us that 'c' is a constant.

(Eh, not you matter wave:)
you equated the locally measured speed of light with the speed that light travels (or propagates at) and also with a 'frame of reference' for a 'photon'. But we cannot measure how light propagates or the speed of a photon (since it always travels in one direction). My previous post was to make this clear and especially to point out that we cannot do any experiment to show that any light or photons travel at c, even when we have done an experiment to show that the "average" round trip speed of light is c. Also, I want to make clear that Einstein's second postulate was about the unmeasurable one-way speed of light.

So my question for you is what did you mean by this statement?
"To deny this you will need to prove it otherwise, not theoretically but by experiment."​

Sorry,this isn't clicking fully with me yet guys, but why exactly can't the one way speed of light be measured?

ghwellsjr
Science Advisor
Gold Member
Sorry,this isn't clicking fully with me yet guys, but why exactly can't the one way speed of light be measured?
Because we do not have anything faster than the speed of light to communicate to us when the light arrived at a distant location. We cannot see the light once it has left us so we cannot track its progress. Only by having the light reflect off of distant objects can we tell that it has arrived at those objects but it will take some time for the image of the reflection to travel back toward us.

And we cannot just move another clock from our present location to a distant location and expect it to have the same time on it as our local clock because we know that if we bring it back it will have less time on it and we really don't know what happened to the time while it is in transit in either direction.

Okay, I see your thinking now. I wrote "We have countless experiments showing us that 'light' doesn't care about what 'speed' you think you are doing, relative some other frame of reference."

Should have put it "relative 'you comparing it to' some other frame of reference."

That's what I meant when I wrote it :)
Ah well..