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The definition of the non-relativistic gravitational fields provides the answer to this question, and thereby describes the image of the metric tensor in Newtonian physics. These fields are not strictly non-relativistic. Rather, they apply to the non-relativistic (or post-Newtonian) limit of GR.
The classic example of a non-relativistic spacetime is the spacetime of Galileo and Newton. It is the spacetime of everyday "common sense". [1] Galilean/Newtonian spacetime assumes that space is Euclidean (i.e. "flat"), and that time has a constant rate of passage that is independent of the state of motion of an observer, or indeed of anything external.
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
These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of the Galilean group. In contrast to the inertial frame, a non-inertial frame of reference is one in which fictitious forces must be invoked to explain observations. An example is an observational ...
As in classical mechanics, in special relativity the relative velocity | is the velocity of an object or observer B in the rest frame of another object or observer A. However, unlike the case of classical mechanics, in Special Relativity, it is generally not the case that
A free particle with mass in non-relativistic quantum mechanics is described by the free Schrödinger equation: (,) = (,) where ψ is the wavefunction of the particle at position r and time t . The solution for a particle with momentum p or wave vector k , at angular frequency ω or energy E , is given by a complex plane wave :
In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative to the frame until acted upon by external forces. In such a frame the laws of nature can be observed ...
In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units): = ˙ + ˙, where q is the electric charge of the particle, φ is the electric scalar potential, and the A i are the components of the magnetic vector potential that may all explicitly depend on and .