Rest frame in the context of Coordinate system

In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstein's formula: ${\displaystyle E=mc^{2}}$. In a reference frame where the system is moving, its relativistic energy and relativistic mass (instead of rest mass) obey the same formula.

The formula defines the energy (E) of a particle in its rest frame as the product of mass (m) with the speed of light squared (c). Because the speed of light is a large number in everyday units (approximately 300000 km/s or 186000 mi/s), the formula implies that a small amount of mass corresponds to an enormous amount of energy.

Proper length or rest length is the length of an object in the object's rest frame.

The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously. But in the theory of relativity, the notion of simultaneity is dependent on the observer.

Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald contraction (after Hendrik Lorentz and George Francis FitzGerald) and is usually only noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light relative to the observer.

The relative velocity of an object B with respect to an observer A, denoted ${\displaystyle \mathbf {v} _{B\mid A}}$ (also ${\displaystyle \mathbf {v} _{BA}}$ or ${\displaystyle \mathbf {v} _{B\operatorname {rel} A}}$), is the velocity vector of B measured in the rest frame of A.The relative speed is the vector norm of the relative velocity, ${\displaystyle v_{B\mid A}=\|\mathbf {v} _{B\mid A}\|}$.

Peculiar motion or peculiar velocity refers to the velocity of an object relative to a rest frame—usually a frame in which the average velocity of some objects is zero.