# Non-relativistic limit

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

## Non-relativistic limit of NLKG

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

utt / c2 − Δu + c2u + f(u) = 0.

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

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

In the non-relativistic limit this becomes two paraboloids

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

and so one expects u to resolve as

u = exp(ic2t)v + + exp( − ic2t)v
ut = ic2exp(ic2t)v + ic2exp(ic2t)v

where v + , v solve some suitable NLS.

A special case arises if one assumes (utic2u) to be small at time zero (say o(c) in some Sobolev norm). Then one expects 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 v + and v to evolve by a coupled NLS; see MasNa2002.

Heuristically, the frequency $\ll c$ portion of the evolution should evolve in a Schrodinger-type manner, while the frequency $\gg c$ 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 H1 hypotheses on the initial data BecMauSb-p. The asymptotic relation between the MKG-CG fields f , A, A0 and the Schrodinger-Poisson fields u, v^+, v^- are

A0˜u
f˜exp(ic2t)v + + exp( − ic2t)v
ft˜iMexp(ic2)v + iMexp( − ic2t)v

where $M = \sqrt{c^4 - c^2 \Delta}$ (a variant of c2).