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Old Apr 4th 2010, 05:45 PM   #1
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Codirectional 3-vectors

From Rindler prob 5.14.

We shall say that three particles move codirectionally if their three velocities are parallel in some inertial frame. Prove that the necessary and sufficient condition for this to be the case is that the 4-velocities U,V,W of these particles be linearly dependant. [Hint component specialisation].

I have found a reasonable proof that the condition is necessary. But I cannot show that it is sufficient.

Any ideas?

I tried orientating (specialising) the spacial axes so that the z component of all three (3 and 4) vectors is zero. Then the direction of each 3-vector is given by the ratio of its x' and y' components. I can then find the velocity v that makes u and v parallel. I hoped to then show that w would also be parallel, but the algebra seems to be unreasonably complicated.
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Old Apr 5th 2010, 05:14 PM   #2
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"three velocities are parallel in some inertial frame".
Direct x axis along this direction. These 3 velocities vectors have only x component,
y and z are zeros. This is S frame.
Transfer these 3 velocities to inertial frame S' moving with v along S frame.
Examine new components.
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Old Apr 6th 2010, 02:48 AM   #3
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I think I have solved it for myself! Starting from nothing more than the linear dependence of U,V,W.

Since they are linearly dependant we can write:

W=a*U+b*V where a and b are constants. Being a 4-vector equation this is true in any frame.

If we transform to the rest frame of U then in this frame w is parallel to v by W=a*U+b*V, u=0,0,0. Now we rotate the spatial axes so that x is in the same direction as w (and hence v).

Finally performing a standard LT with x axis defined in this way, does not change the direction of v or w and causes u to transform from its rest frame into a frame in which it is parallel to x and hence v and w.
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Old Apr 7th 2010, 03:32 PM   #4
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I am glad all is ok.
I wanted to make LT 3 times forward and 3 times backwards, I not finished yet,
good luck.
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