jimmy-neutron-decay.html

Alright jumbo, let's set the scene. This one scroll will be simple and concise. We simply have a neutron moving at a speed 0.885v, that is 88.5% of the speed of light along the x-axis. Suddenly and spontaneously, it disintegrates into two particles, a proton and an electron, don't start asking me about the neutrino, I'm just telling you what's happening. 

We're interested in a few things, the velocity of the particles (and naturally the momentum and the energy) in two reference frames, the neutron's and the planet's. We'll also try to see if mass is conserved (surprising). 

A good place to start is to write up the mom energy four-vector of the electron in the frame of the neutron. Trying my best to keep this as physics-y as possible without too much math but it feels awkward without the maths, but this is sooo important. 

By now you should know the difference between for example E' and E. Using a bit of linear algebra we obtain nice equations

These equations tell us that Energy and momentum aren't two separate quantities, they are closely related, we can somewhat think of them the same way we think of time and space, the subscript rel is just relative. 

Starting with newtonian momentum mV where V is a four vector we can easily show that the momentum of an object is given by: 

Good stuff eh? In an identical manner we get the same expression for the electron, swapping out the p's with e's. For the neutron's it's a different story, because we are in its frame of reference meaning it has a velocity of 0, meaning that it will be identical to the expression above, except m_p is -_n and v = 0. 

We know that momentum-energy is conserved in combination. That is, the neutron disintegrating will have the same momentum-energy as the sum of the momentum energy of the electron and the proton. You know all about momentum in newtonian physics from your school, it's kind of the same except momentum isn't conserved on its own, it's the combination of momentum and energy. 

Solving this equation is just algebraic manipulation, but the result is what matters. We get this expression

and several others.