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These equations were numerically integrated to find t t , x x , u t u^{t} and u x u^{x} of the particles using the inbuilt Ordinary Differential Equation (ODE) solver ode23s in Matlab. Only one pair of these variables ( t t and u t u^{t} , or x x and u x u^{x} ) needed to be solved for, however both were found and the normalisation of the 4 velocity was used as a check on the accuracy of the integration. In cases where u t u^{t} and u x u^{x} diverge, the equations were numerically integrated to find x and d ​ x d ​ t {dx\over dt} using ode45

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