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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 relativistic mass of an object is given by the relativistic energy divided by c 2. [10] Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly synonymous; the only difference between them is the units.
Relativistic kinetic energy: The relativistic kinetic energy relation takes the slightly modified form: = = As is a function of , the non-relativistic limit gives / =, as expected from Newtonian considerations.
Derivation of the relativistic Doppler shift If an object emits a beam of light or radiation, the frequency, wavelength, and energy of that light or radiation will look different to a moving observer than to one at rest with respect to the emitter.
In special relativity, four-momentum (also called momentum–energy or momenergy [1]) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions ; similarly four-momentum is a four-vector in spacetime .
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
The relativistic expressions for E and p obey the relativistic energy–momentum relation: [12] = where the m is the rest mass, or the invariant mass for systems, and E is the total energy. The equation is also valid for photons, which have m = 0 : E 2 − ( p c ) 2 = 0 {\displaystyle E^{2}-(pc)^{2}=0} and therefore E = p c {\displaystyle E=pc}
So relativistic energy and momentum significantly increase with speed, thus the speed of light cannot be reached by massive particles. In some relativity textbooks, the so-called "relativistic mass" = is used as well. However, this concept is considered disadvantageous by many authors, instead the expressions of relativistic energy and momentum ...