# Non-relativistic limit

The non-relativistic limit of a relativistic equation (which thus involves the speed of light 'c') denotes the limit when ${\displaystyle c\to \infty }$. It is the opposite of the vanishing dispersion limit.

## Non-relativistic limit of NLKG

By inserting a parameter ${\displaystyle c}$ (the speed of light), one can rewrite the nonlinear Klein-Gordon equation as

${\displaystyle u_{tt}/c^{2}-\Delta u+c^{2}u+f(u)=0}$.

One can then ask for what happens in the non-relativistic limit ${\displaystyle c\rightarrow \infty }$ (keeping the initial position fixed, and dealing with the initial velocity appropriately). In Fourier space, ${\displaystyle u}$ should be localized near the double hyperboloid

${\displaystyle t=\pm c{\sqrt {c^{2}+x^{2}}}}$.

In the non-relativistic limit this becomes two paraboloids

${\displaystyle t=\pm (c^{2}+x^{2}/2)}$

and so one expects ${\displaystyle u}$ to resolve as

${\displaystyle u=exp(ic^{2}t)v_{+}+exp(-ic^{2}t)v_{-}}$
${\displaystyle u_{t}=ic^{2}exp(ic^{2}t)v_{+}-ic^{2}exp(ic^{2}t)v_{-}}$

where ${\displaystyle v_{+}}$, ${\displaystyle v_{-}}$ solve some suitable NLS.

A special case arises if one assumes ${\displaystyle (u_{t}-ic^{2}u)}$ to be small at time zero (say ${\displaystyle o(c)}$ in some Sobolev norm). Then one expects ${\displaystyle v_{-}}$ to vanish and to get a scalar NLS. Many results of this nature exist, see Mac-p, Nj1990, Ts1984, MacNaOz-p, Na-p. In more general situations one expects ${\displaystyle v_{+}}$ and ${\displaystyle v_{-}}$ to evolve by a coupled NLS; see MasNa2002.

Heuristically, the frequency ${\displaystyle \ll c}$ portion of the evolution should evolve in a Schrodinger-type manner, while the frequency ${\displaystyle \gg c}$ portion of the evolution should evolve in a wave-type manner. (This is consistent with physical intuition, since frequency is proportional to momentum, and hence (in the nonrelativistic regime) to velocity).

A similar non-relativistic limit result holds for the Maxwell-Klein-Gordon system (in the Coulomb gauge), where the limiting equation is a coupled Schrodinger-Poisson system under reasonable ${\displaystyle H^{1}}$ hypotheses on the initial data BecMauSb-p. The asymptotic relation between the MKG-CG fields ${\displaystyle f}$ , ${\displaystyle A}$, ${\displaystyle A_{0}}$ and the Schrodinger-Poisson fields u, v^+, v^- are

${\displaystyle A_{0}\sim u}$
${\displaystyle f\sim exp(ic^{2}t)v^{+}+exp(-ic^{2}t)v^{-}}$
${\displaystyle f_{t}\sim iMexp(ic^{2})v^{+}-iMexp(-ic^{2}t)v^{-}}$

where ${\displaystyle M={\sqrt {c^{4}-c^{2}\Delta }}}$ (a variant of ${\displaystyle c^{2}}$).