Fourvectors in RelativityIn the literature of relativity, spacetime coordinates and the energy/momentum of a particle are often expressed in fourvector form. They are defined so that the length of a fourvector is invariant under a coordinate transformation. This invariance is associated with physical ideas. The invariance of the spacetime fourvector is associated with the fact that the speed of light is a constant. The invariance of the energymomentum fourvector is associated with the fact that the rest mass of a particle is invariant under coordinate transformations.

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Lorentz Tranformation of FourvectorsThe Lorentztransformation of both spacetime and momentumenergy fourvectors can be expressed in matrix form.
The spacetime Lorentz transformation produces the result: and the energymomentum 4vector transforms as 
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Fourvector Sum for MomentumEnergyTwo momentumenergy fourvectors can be summed to form a fourvector. The length of this fourvector is an invariant The momenta of two particles in a collision can then be transformed into the zeromomentum frame for analysis, a significant advantage for highenergy collisions. For the two particles, you can determine the length of the momentumenergy 4vector, which is an invariant under Lorentz transformation. The practical advantage of this for high energy collisions is that it allows you to calculate the momentum of each particle in the zeromomentum frame. One approach to this for a two particle system involves adding the momenta and energy for the two particles: Transforming this to the zeromomentum frame While this gives the form of the necessary transformation, we don't know the values for β and γ necessary to achieve the zeromomentum condition. That's where the invariance of the length of the energymomentum 4vector is of value. Now evaluating the length of the momentumenergy 4vector from the experimental information we have in the laboratory frame gives the quantity s above. Since s can be evaluated from laboratory information, we can concentrate on the expression for s in the zeromomentum frame. where p*c has been used to represent the value of each momentum since they are constrained to be equal. This expression can be used to determine p*c, and that value can be compared to the original value of the momentum for one of the particles to determine the values of for β and γ necessary for the transformation.

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