In 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. Note that this differs from the ordinary scalar product of vectors because of the minus sign. That minus sign is necessary for the property of invariance of the length of the 4vectors. The length of a 4vector is invariant, being the same in every inertial frame. This invariance is associated with the constancy of the speed of light. This expression can be seen to be the equation of a sphere, with light propagating outward from the origin at speed c in all directions so that the radius of the sphere at time t is ct. The length of this 4vector is the rest energy of the particle. The invariance is associated with the fact that the rest mass is the same in any inertial frame of reference. The Lorentztransformation of both spacetime and momentumenergy fourvectors can be expressed in matrix form. 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: 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. Source.
Fourvectors in Relativity
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