Relativistic Mechanics

relativistic mechanics, science concerned with the motion of bodies whose relative velocities approach the speed of light c, or whose kinetic energies are comparable with the product of their masses m and the square of the velocity of light, or mc². Such bodies are said to be relativistic, and when their motion is studied, it is necessary to take into account Einstein’s special theory of relativity. As long as gravitational effects can be ignored, which is true so long as gravitational potential energy differences are small compared with mc², the effects of Einstein’s general theory of relativity may be safely ignored.

The bodies concerned may be sufficiently small that one may ignore their internal structure and size and regard them as point particles, in which case one speaks of relativistic point-particle mechanics; or one may need to take into account their internal structure, in which case one speaks of relativistic continuum mechanics. This article is concerned only with relativistic point-particle mechanics. It is also assumed that quantum mechanical effects are unimportant, otherwise relativistic quantum mechanics or relativistic quantum field theory—the latter theory being a quantum mechanical extension of relativistic continuum mechanics—would have to be considered. The condition that allows quantum effects to be safely ignored is that the sizes and separations of the bodies concerned are larger than their Compton wavelengths. (The Compton wavelength of a body of mass m is given by h/mc, where h is Planck’s constant.) Despite these restrictions, there are nevertheless a number of situations in nature where relativistic mechanics is applicable. For example, it is essential to take into account the effects of relativity when calculating the motion of elementary particles accelerated to higher energies in particle accelerators, such as those at CERN (European Organization for Nuclear Research) near Geneva or at Fermilab (Fermi National Accelerator Laboratory) near Chicago. Moreover, such particles are caused to collide, thus creating further particles; although this creation process can only be understood through quantum mechanics, once the particles are well separated, they are subject to the laws of special relativity.

Similar remarks apply to cosmic rays that reach Earth from outer space. In some cases, these have energies as high as 10²⁰ electron volts (eV). An electron of that energy has a velocity that differs from that of light by about 1 part in 10²⁸, as can be seen from the relativistic relation between energy and velocity, which will be given later. For a proton of the same energy, the velocity would differ from that of light by about 1 part in 10²². At a more mundane level, relativistic mechanics must be used to calculate the energies of electrons or positrons emitted by the decay of radioactive nuclei. Astrophysicists need to use relativistic mechanics when dealing with the energy sources of stars, the energy released in supernova explosions, and the motion of electrons moving in the atmospheres of pulsars or when considering the hot big bang. At temperatures in the very early universe above 10¹⁰ kelvins (K), at which typical thermal energies kT (where k is Boltzmann’s constant and T is temperature) are comparable with the rest mass energy of the electron, the primordial plasma must have been relativistic. Relativistic mechanics also must be considered when dealing with satellite navigational systems used, for example, by the military, such as the Global Positioning System (GPS). In this case, however, it is the purely kinematic effect on the rate of clocks on board the satellites (i.e., time dilation) that is important rather than the dynamic effects of relativity on the motion of the satellites themselves.