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I'm having trouble with what strikes me as an inconsistency within conservation of energy, conservation of momentum, and the four-momentum invariant equation (E^{2}-p^{2}c^{2}= m^{2}c^{4}). For the sake of this question, I'll be using non-relativistic mass--i.e. mass is the same in all reference frames and p=γmv and E = γmc^{2}.

Here is the situation I'm imagining: a particle of mass M moves in the +x direction with some velocity when suddenly it decays into two particles of mass (2/5)M moving in any ol' direction. By the conservation of momentum and energy, we know that some of our mass before the decay has been converted to kinetic energy. That's all okay.

The issue: since energy and momentum are conserved, combinations of them are conserved. For problem solving, a convenient combination is E_{i}^{2}-p_{i}^{2}c^{2}= E_{f}^{2}-p_{f}^{2}c^{2}. But if we use the four-momentum invariant equation, we can substitute in m_{i}^{2}c^{4}and m_{f}^{2}c^{4}respectively. This leaves us with a problem; in the example, the M_{f}is less than the M_{i}, but we just derived that the final mass energy squared is equal to the initial mass energy squared. These statements are contradictory!

So where does the physics/math go awry?

Any and all help is appreciated!

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# Four-Momentum Invariant and Conservation Laws Yielding Contradictory Results

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