kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero. Newtonian momenta, calculated as , fail to behave properly under Lorentzian transformation. The linear transformation of velocities is replaced by the highly nonlinear so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. This second option was what he chose. The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass. Beyond the basics The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding Introduction to curved spacetime. Rapidity Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas. This nonlinearity is an artifact of our choice of parameters. We have previously noted that in an spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other. The natural functions for expressing these relationships are the hyperbolic analogs of the trigonometric functions. Fig. 4-1a shows a unit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that a is interpreted, not as the angle between the ray and the , but as twice the area of the sector swept out by the ray from the . (Numerically, the angle and measures for the unit circle are identical.)