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The equation of state for ordinary non-relativistic 'matter' (e.g. cold dust) is =, which means that its energy density decreases as =, where is a volume.In an expanding universe, the total energy of non-relativistic matter remains constant, with its density decreasing as the volume increases.
Actions are typically written using Lagrangians, formulas which depend upon the object's state at a particular point in space and/or time. In non-relativistic mechanics, for example, a point particle's Lagrangian is the difference between kinetic and potential energy: =.
The non-relativistic equation for the energy of a free particle is ... Applying the formula for the stress–energy tensor to the Lagrangian density (the quantity ...
This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis.
The Larmor formula can only be used for non-relativistic particles, which limits its usefulness. The Liénard-Wiechert potential is a more comprehensive formula that must be employed for particles travelling at relativistic speeds. In certain situations, more intricate calculations including numerical techniques or perturbation theory could be ...
The non-relativistic version was found by Hans Bethe in 1930; the relativistic version (shown below) was found by him in 1932. [2] The most probable energy loss differs from the mean energy loss and is described by the Landau-Vavilov distribution. [3]
Paul Dirac incorporated special relativity and quantum mechanics into a single formulation that simplifies to the Schrödinger equation in the non-relativistic limit. This is the Dirac equation, which contains a single derivative in both space and time.
For example, for a speed of 10 km/s (22,000 mph) the correction to the non-relativistic kinetic energy is 0.0417 J/kg (on a non-relativistic kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a non-relativistic kinetic energy of 5 GJ/kg). The relativistic relation between kinetic energy and momentum is given by