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veeman88
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1. Hi everyone, I'm a final year in university and have got a "Relativity and Electrodynamics" exam coming up. I'm going through past papers and can't seem to work out how to tackle this problem, any help would be much appreciated.
Suppose in some inertial frame S a photon has 4-momentum components:
[p^[mu]] = [E, E, 0, 0]
There is a special class of Lorentz Transformations called the "little group of p"
which leaves the components of p unchanged (see example below). You are to find
one sequence of at least one pure boost and at least one pure rotation whose product
is not a pure rotation in the y-z-plane, but is in the little group of p.
(i) Start your sequence with a pure boost followed by a pure rotation to re-align
the reference frame axes. Determine the rotation angle as a function of the
boost speed β. {13}
(ii) Finalise the sequence by stating and justifying a third and last step. Apply this
last transformation. {5}
(iii) Derive the condition on velocities involved in your sequence. {4}
I know it's a bit of a tough one but anyone who is good at this stuff could really help me out.
Thanks.
2. An example for a transformation belonging to the little group would be a pure
rotation through an angle [THETA] in the y-z-plane:
[ 1 , 0 , 0 , 0 ] [E] [E]
[ 0 , 1 , 0 , 0 ] [E] = [E]
[ 0 , 0 , cos[theta] , -sin[theta] ] [0] [0]
[ 0 , 0 , sin[theta] , cos[theta] ] [0] [0]
3. I don't know where to start.
Suppose in some inertial frame S a photon has 4-momentum components:
[p^[mu]] = [E, E, 0, 0]
There is a special class of Lorentz Transformations called the "little group of p"
which leaves the components of p unchanged (see example below). You are to find
one sequence of at least one pure boost and at least one pure rotation whose product
is not a pure rotation in the y-z-plane, but is in the little group of p.
(i) Start your sequence with a pure boost followed by a pure rotation to re-align
the reference frame axes. Determine the rotation angle as a function of the
boost speed β. {13}
(ii) Finalise the sequence by stating and justifying a third and last step. Apply this
last transformation. {5}
(iii) Derive the condition on velocities involved in your sequence. {4}
I know it's a bit of a tough one but anyone who is good at this stuff could really help me out.
Thanks.
2. An example for a transformation belonging to the little group would be a pure
rotation through an angle [THETA] in the y-z-plane:
[ 1 , 0 , 0 , 0 ] [E] [E]
[ 0 , 1 , 0 , 0 ] [E] = [E]
[ 0 , 0 , cos[theta] , -sin[theta] ] [0] [0]
[ 0 , 0 , sin[theta] , cos[theta] ] [0] [0]
3. I don't know where to start.