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I have just recently begun delving into SR, and there are a few questions I would like clarified if possible, with your assistance. NB These questions were discussed during some of the lectures and are NOT HW assignments.

1)If a particle of rest mass m breaks down, whilst at rest, into a particle of mass m' and a photon, what would be the energies of the particle with mass m' and the photon created in the process?:

I understand that the four-momentum of the original particle of mass m is (m,0) and those of m' and the photon are (E,p) and (ε,εn^). I also understand that since the center of mass reference frame of the original particle is also that of the products (m' and the photon), the overall four-momentum must be conserved and the following equality must hold:

P_{tot}= (m,0)=(E+ε,p+n^).

What is not clear to me is why would that entail E + p = m? I don't quite understand this equality. What does it designate? What is its meaning?

2)In reference frame O two photons with frequencies [itex]\nu[/itex]_{1}and[itex]\nu[/itex]_{2}are moving in the positive and negative x^ directions, respectively. What would be the CM of the two photon's velocity wrt O?

The four-momentum of the first photon may be written thus: h[itex]\nu[/itex]_{1}(1,n^)

Similarly for the second: h[itex]\nu[/itex]_{2}(1,-n^).

Now, the total four-momentum must be conserved: P = h([itex]\nu[/itex]_{1}+ [itex]\nu[/itex]_{2}, ([itex]\nu[/itex]_{1}- [itex]\nu[/itex]_{2})n^).

What I don't quite understand is how applying Boost in the n^ direction at a velocity u would yield:

h[itex]\gamma[/itex]([itex]\nu[/itex]_{1}- [itex]\nu[/itex]_{2}-u([itex]\nu[/itex]_{1}+ [itex]\nu[/itex]_{2})).

Could someone please explain this delicate point to me step-by-step?

3)Suppose P^{μ}= (M + h[itex]\nu[/itex],h[itex]\nu[/itex]n^). Why does that squared yield: M^{2}+ 0 + 2h[itex]\nu[/itex]M? What does the zero denote?

Also, what is the difference between the following notations: P_{μ}and P^{μ}?

I'd truly appreciate any insightful remarks.

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# Special Relativity, four-momenta questions.

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