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Total energy is the sum of rest energy = and relativistic kinetic energy: = = + Invariant mass is mass measured in a center-of-momentum frame. For bodies or systems with zero momentum, it simplifies to the mass–energy equation E 0 = m 0 c 2 {\displaystyle E_{0}=m_{0}c^{2}} , where total energy in this case is equal to rest energy.
The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Q k. Normal coordinates decouple the classical vibrational motions of the molecule and thus give an easy route to ...
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
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. In that case θ = 0, and cos θ = 1, which gives:
Kinetic energy in special relativity and Newtonian mechanics. Relativistic kinetic energy increases to infinity when approaching the speed of light, thus no massive body can reach this speed. Tests of relativistic energy and momentum are aimed at measuring the relativistic expressions for energy, momentum, and mass.
The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side: ε = Mc 2 and ε 0 = mc 2, with ε being the total energy (rest energy plus kinetic energy) of a moving material point, ε 0 its rest energy, M the relativistic mass, and m the invariant mass.
Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass: = = = + | | = where = is the metric tensor of special relativity with metric signature for definiteness chosen to be (–1, 1, 1, 1).
Spinors find several important applications in relativity. Their use as a method of analysing spacetimes using tetrads, in particular, in the Newman–Penrose formalism is important. Another appealing feature of spinors in general relativity is the condensed way in which some tensor equations may be written using the spinor formalism.