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Why can there be no instant changes in energy density in GR? And exactly how smooth do they need to be?

Also, how does this make sense in things like quantum tunneling or pair production? Aren't those instantaneous changes?

Flamethrower
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1 Answers1

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In principle it doesn't have to. In textbooks one usually looks at smooth (or at least $C^2$) solutions, in practice mathematicians are interested in all sorts of generalizations and weaker notions of solutions.

This is the field of so-called low regularity metrics. The basic idea is that you give up the (fairly restrictive) condition of everywhere continuous derivatives (i.e. you extend your search beyond the $C^k$ class) and instead consider solutions (matter fields, metrics, curvatures, etc..) in a distributional or weak sense only, that is you only want your field equations to hold under an integral, thus "blurring out" all the unpleasantries that arise when you try to take derivatives of discontinuous functions. For more details, see Bruhat's "GR and the Einstein Equations" or Ringström's "The Cauchy problem in GR" books.

To connect it to your question, luckily GR is a hyperbolic theory, so non-smooth phenomena like

  • "energy density suddenly vanishing at some time" (like the disappearance of a BH's mass in the linked question)

and

  • "energy density ending in a sharp cutoff" (like for instance in some stellar models where the density abruptly jumps at the star's surface)

are qualitatively similar. The latter has more obvious applications and thus has an extensive literature, see e.g. [KE93], [BM17], [DN02].

Such instantaneous (whether in the time or space direction) densities don't generically create horizons.

[KE93] : Kind & Ehlers, "Initial-boundary value problem for the spherically symmetric Einstein equations for a perfect fluid", 1993

[BM17]: Brito & Mena, "Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure", 2017

[DN02]: Dain & Nagy, "Initial data for fluid bodies in general relativity", 2002

Quasi-Logical
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