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Velocity transformation 
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#1
Jan1308, 08:55 AM

P: 23

Let observer O' move respect to observer O with velocity V. Suppose, that O sees an object M moving with velocity v. A standard vector formula for velocity v' of the object M seen by observer O' is attached.
My question is the following: suppose that the velocity v and v' are given. What is the relative velocity V of the two observers? Of course in one dimension the answer is very easy :) Thanks for your help! 


#2
Jan1308, 10:02 AM

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PF Gold
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Just consider M's frame.



#3
Jan1308, 10:40 AM

P: 23

It is not always possible to attribute a real reference frame to M. The simplest example  M could be a particle moving with velocity of light, or a mexican wave on a stadion, which can move with arbitrary velocity v, not necessarily slower than c.



#4
Jan1408, 07:31 PM

P: 23

Velocity transformation
Unfortunatelly, as it usually is, a "simple" special relativity problem turns out to be too difficult for specialists in general relativity. This always made me wonder how surprisingly difficult special relativity is. And it is always so easy when it comes to storytelling about "what is time", "black holes", etc..



#5
Jan1408, 07:51 PM

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To do it 'properly' you've got to use the fact that: dx/dt = v^{x} dy/dt = v^{y} dz/dt = v^{z} dx'(x,t)/dt'(x,t) = v'^{x'} dy'/dt'(x,t) = dy/dt'(x,t) = v'^{y'} dz'/dt'(x,t) = dz/dt'(x,t) = v'^{z'} with the form of x'(x,t) and t'(x,t) being given the usual equations for a Lorentz boost. There's the problem that if you allow supeluminal velocities you cannot get away from the fact that v or v' may be infinite. 


#6
Jan1408, 07:58 PM

P: 23

Unfortunatelly when you consider 3 systems of coordinates, they cannot all be made "parallel" in general. So the transormations involed are not bare boosts, but space rotations also. This complicates the problem very much, because the rotations are parametrized by the velocity that you would like to find.



#7
Jan1508, 01:34 AM

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P: 4,087

Particles with mass are never seen travel at c from any inertial frame, so the question is meaningless. All frames of reference in SR must be related to matter. 


#8
Jan1508, 05:11 AM

P: 23

Mentz114,
First of all the problem is difficult to solve even for all velocities below c. I can understand that you don't see it, but once you actually try to solve it, maybe your eyes will open. Of course I will be extremely happy if you prove me wrong and show me the right formula, but somehow I have second thoughts about it. Other thing  "fictional situations". I understand that you never heard of photons? When you derive the formula for velocity transformation you only assume that V relates two inertial frames. v and v' can also describe a nonuniformly moving object. Who told you that they must describe inertial frames? Jcsd, I think you are right. I believe it is possible to start with a less general problem and to attribute inertial frame to M and then generalize it, however, as I said, the set of equations one gets seems algebraically hard to solve. 


#9
Jan1508, 05:37 AM

PF Gold
P: 4,087

I don't know what point you are trying to make with this post. Your question about the relative velocities has been answered. You have a fundemental misunderstanding if you think 1. Particles with mass can reach light speed 2. Lorentz transformations apply between accelerating frames. 


#10
Jan1508, 06:50 AM

P: 23

Suppose that in one inertial frame a photon moves with velocity v (of course v=c). What is the velocity of the photon v' (v'=c) in a frame that moves with velocity V? The answer is simple and given by formula from the first post.
My question is reverse  given two velocities of photon v and v' in two inertial frames derive the formula for the relative velocity of the frames V. And generalize to arbitrary case, not only photons. Now you got it? Do you see now, that it is not always possible to atribute an inertial system to the observed object? 


#11
Jan1508, 06:58 AM

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#12
Jan1508, 07:03 AM

P: 23

jcsd you are right, unfortunatelly it seems that considering the other frame allows one only to find the v as a function of v' and V (and the solution is trivial, you simply change the sign of V), and what I am looking for is V as a function of v and v'. This function is not symmetric, hence the difficulties.



#13
Jan1508, 08:22 AM

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P: 4,087

Dragan:
Given that your observers O and O' are inertial, if either sees M with velocity c, then the other will also, and M is not an inertial frame. If all three frames are inertial, this is a standard problem. 


#14
Jan1508, 08:58 AM

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P: 1,883

If v<c, jcsd already gave you the answer. Try again to understand what he's doing. 


#15
Jan1508, 09:04 AM

P: 23

Please show me the right formula for V (assuming v<c, v'<c) and your PayPal account info and I promise to send you a $100 gift once I see the right answer before tomorrow. 


#16
Jan1508, 09:14 AM

P: 23

But the formula for velocity transform "works" not only for M being material particles or photons, but anything at all. It could be also superluminal phase velocity of a wave and the velocity transomation is still valid and gives proper answer. My question is to reverse it. At least till tomorrow's evening :) 


#17
Jan1508, 04:10 PM

PF Gold
P: 4,087

dragan:
Here's the standard textbook answer from Prof. J. Baez. The general flow of the expanding universe can carry a frame outside our lightcone for instance, in which case Lorentz transformations do not apply and the formulae above become meaningless. M 


#18
Jan1508, 04:31 PM

P: 23

Thanks for the formula. There is no derivation given, but it is quite simple: one considers velocity fourvectors of B and C, calculates the product, which is invariant. Then one calculates the same product in an inertial frame attributed to B and equals the two. The formula for w^2 follows easily.
The only problem is that the formula gives only the magnitude, not the whole vector w (in my notation it was V)  so it is not the complete solution that I am looking for. Thanks for your help anyway! This is anyhow only the special case when one can attribute an inertial frame of reference with the observed object. 


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