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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.
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 :
When deciding whether to use general relativity for perturbation theory, note that Newtonian physics is only applicable in some cases such as for scales smaller than the Hubble horizon, where spacetime is sufficiently flat, and for which speeds are non-relativistic.
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 .
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, i = 1, 2, 3, are the components of the magnetic vector potential that may all explicitly depend on and .
Non-relativistic quantum electrodynamics (NRQED) is a low energy approximation of quantum electrodynamics which describes the interaction of (non-relativistic, i.e. moving at speeds much smaller than the speed of light) spin one-half particles (e.g., electrons) with the quantized electromagnetic field.
Thus it is necessary to check that relativistic invariance is not lost. Alternatively, the Feynman integral approach is available for quantizing relativistic fields, and is manifestly invariant. For non-relativistic field theories, such as those used in condensed matter physics, Lorentz invariance is not an issue.