Photon Velocity and Pythagorean Theorem

[phys-learner]

If two photons are traveling at right angles to each other with velocity vectors of c then what is the velocity vector of the hypotenuse of the right triangle and does this velocity triangle form an equilateral triangle with three angles of 90 degrees if c^2 + c^2 = c^2

 

[Doc Al]

I'm not sure what you mean by "velocity vector of the hypotenuse", but you can certainly calculate the separation rate of the photons according to lab frame observers. That separation rate will be the hypotenuse of a right triangle with sides equal to c, thus will equal √2 c.

 

[phys-learner]

I'm not sure what you mean by "velocity vector of the hypotenuse", but you can certainly calculate the separation rate of the photons according to lab frame observers. That separation rate will be the hypotenuse of a right triangle with sides equal to c, thus will equal √2 c.

How can two photons have a separation rate of √2 c ~= 1.414 c , which is greater than c?

Wouldn't the separation rate be c since two photons separating at 180 degrees = (c+c)/√(1+(c/c)^2) = c ?

 

[Fredrik]

No, that's how you would calculate the velocity of one of the particles in the other's rest frame if they're moving in opposite directions in your frame, but that doesn't make sense here since a photon doesn't have a rest frame. If something is moving relative to you in the x direction with speed u<c and something else is moving relative to you in the -x direction with speed v<c, then the speed of the second object in the first object's rest frame is (u+v)/(1+uv/c²), but the time derivative of their distance from each other in your frame as a function of time can be close to 2c.

If they're moving in the x and y directions in your frame, as you described in #1, then their coordinates at time t are (t,0,ct) and (t,ct,0) respectively. So the distance between them in the x-y plane at time t is just $\sqrt{(ct)^2+(ct^2)}=\sqrt 2 ct$, and the derivative of that with respect to t is $\sqrt 2 c$. Nothing weird about that.

Looks like Frederik beat me to it.

Not quite. I was still editing when I saw your post. I still hadn't explained the $\sqrt 2$.

 

[Hootenanny]

How can two photons have a separation rate of √2 c ~= 1.414 c , which is greater than c?

Why would you say they can't.

Wouldn't the separation rate be c since two photons separating at 180 degrees = (c+c)/√(1+(c/c)^2) = c ?

You said that in your original post they are traveling perpendicular (90^o) to each other, not anti-parallel (180^o) to each other.

Furthermore, if you are attempting to apply the additive velocity formula for SR, you are applying it incorrectly. This is irrelevant in any case, since the formula is singular for velocities equal to c.

Edit: Looks like Frederik beat me to it.

 

[lightarrow]

If two photons are traveling at right angles to each other with velocity vectors of c then what is the velocity vector of the hypotenuse of the right triangle and does this velocity triangle form an equilateral triangle with three angles of 90 degrees if c^2 + c^2 = c^2

If you have two objects which travels at right angles with speeds modulus v1 and v2, then one sees (measures) the other's speed as:
V = Sqrt[v1^2 + v2^2 - (v1^2*v2^2)/c^2].
You couldn't use that equation for photons because you can't find a ref frame traveling exactly at c, but making the limit for v1, or v2 or both -->c you still get c from it.

 

[phys-learner]

Why would you say they can't.

Two photons traveling relative to each other - cannot have a relative separation velocity greater than c, whatever angle they may be relative to each other, or can they?

 

[lightarrow]

Two photons traveling relative to each other - cannot have a relative separation velocity greater than c, whatever angle they may be relative to each other, or can they?

That's correct (not exactly for photons because you can't compute it in that case, but in the limit v-->c).

 

[Hootenanny]

Two photons traveling relative to each other - cannot have a relative separation velocity greater than c, whatever angle they may be relative to each other, or can they?

Ahh, there you said it: relative to each other. Notice that Doc Al emphasised the rate of separation of the photons as viewed in the lab, this rate of separation is as observed by the lab, not by the photons themselves.

Do you see the difference?

 

[Dale]

To avoid any problems with the fact that photons, by definition, don't have a rest frame, it is easier to talk about protons in the LHC. You wind up with two beams of protons going in opposite directions, one 0.99999999 c clockwise and the other 0.99999999 c counter clockwise. In the lab frame therefore their closing rate is just under 2 c, but in either beam's frame the velocity of the other beam is just under 1 c.

 

[phys-learner]

Two photonic life forms having the ability to travel at c, grab a third photonic stretch creature with the ability to stretch at rate c, ...take off at right angles to each other but it seems that the photonic stretch creature cannot have a stretch rate faster than c ... relative to the others?

 

[atyy]

Beautiful! Bravo, bravissimo!

 

[atyy]

Yes, I think you are right. Photons don't interact with each other. They interact with electrons and protons and other charged particles. So your photonic creatures exist, but they cannot grab each other.

In fact, if you have a bunch of photons of different momenta all located at a particular spot at a particular time, they would all go out radially from that spot with the same speed, which obviously wouldn't happen if two photons were able to grab each other.

 

[Doc Al]

Two photonic life forms having the ability to travel at c, grab a third photonic stretch creature with the ability to stretch at rate c, ...take off at right angles to each other but it seems that the photonic stretch creature cannot have a stretch rate faster than c ... relative to the others?

This is pure fantasy now. Massive objects cannot travel at speed c, and it's meaningless to talk about speeds relative to photons (or anything else traveling at speed c). If you're willing to ignore such issues, don't be surprised when nonsensical results appear.

If two photons travel at right angles with respect to the lab frame (not each other!), then they will separate at a rate of √2 c as seen by lab observers (the same observers who see the photons moving at right angles to each other). Just to be clear, this means: Photon A moves east, photon B moves north (in the lab frame). Lab observers measure the distance between photons A and B as a function of time. That distance increases at a rate of √2 c. Note that nothing moves faster than light in this example.

Now if you would like to talk about speeds relative to a moving object, then let's give that object a physically meaningful speed. Here's an example. Say a spaceship moves east at speed 0.99c with respect to the lab frame. That spaceship emits a photon traveling north with respect to the ship. What's the speed and direction of the photon according to lab frame observers? If you do the calculation, it turns out that the speed of the photon is c (of course!) but its direction is not north but about 8.11° north of east.