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Kostik
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- TL;DR Summary
- Dirac says, "With the restriction that we have waves moving only in one direction, gravitational energy can be localized." Yet this seems to contradict the equivalence principle.
Reading Dirac's "General Theory of Relativity", Chap. 33 "Gravitational waves". He shows that in a weak gravitational field (##g_{\mu\nu}## approximately constant), using harmonic coordinates, we have a wave equation ##g^{\mu\nu}g_{\rho\sigma,\mu\nu}\approx \eta^{\mu\nu}g_{\rho\sigma,\mu\nu}=0##.
He also shows that the Einstein energy-momentum pseudotensor for a plane wave with wave vector ##l_\mu## is $$16\pi {t_\mu}^\nu = \frac{1}{2}\left(u_{\alpha\beta}u^{\alpha\beta} - \frac{1}{2}u^2\right) l_\mu l^\nu$$ where ##u_{\mu\nu}## is the derivative of the function ##g_{\mu\nu}## of the single variable ##l_\sigma x^\sigma##, and ##u=u^\mu_\mu##.
Dirac concludes:
"We have a result for ##{t_\mu}^\nu## that looks like a tensor. This means that ##{t_\mu}^\nu## transforms like a tensor under those transformations that preserve the character of the field consisting only of waves moving in the direction ##l_\sigma##, so that the ##g_{\mu\nu}## remain functions of the single variable ##l_\sigma x^\sigma##. Such transformations must consist only in the introduction of coordinate waves moving in the direction ##l_\sigma##, of the form $$x^{\mu'}=x^\mu+b^\mu$$ where ##b_\mu## is a function of ##l_\sigma x^\sigma##. With the restriction that we have waves moving only in one direction, gravitational energy can be localized."
It is usually understood that the equivalence principle makes it impossible to localize gravitational energy-momentum. If there were a gravitational field energy-momentum tensor, it would vanish in locally inertial coordinates, hence it would vanish everywhere. That's why there are only pseudotensors of various kinds that satisfy ##\left( [t^{\mu\nu}+T^{\mu\nu}]\sqrt{-g}\right)_{,\nu}=0##.
How can Dirac be right, in view of the equivalence principle? What exactly is he saying here?
He also shows that the Einstein energy-momentum pseudotensor for a plane wave with wave vector ##l_\mu## is $$16\pi {t_\mu}^\nu = \frac{1}{2}\left(u_{\alpha\beta}u^{\alpha\beta} - \frac{1}{2}u^2\right) l_\mu l^\nu$$ where ##u_{\mu\nu}## is the derivative of the function ##g_{\mu\nu}## of the single variable ##l_\sigma x^\sigma##, and ##u=u^\mu_\mu##.
Dirac concludes:
"We have a result for ##{t_\mu}^\nu## that looks like a tensor. This means that ##{t_\mu}^\nu## transforms like a tensor under those transformations that preserve the character of the field consisting only of waves moving in the direction ##l_\sigma##, so that the ##g_{\mu\nu}## remain functions of the single variable ##l_\sigma x^\sigma##. Such transformations must consist only in the introduction of coordinate waves moving in the direction ##l_\sigma##, of the form $$x^{\mu'}=x^\mu+b^\mu$$ where ##b_\mu## is a function of ##l_\sigma x^\sigma##. With the restriction that we have waves moving only in one direction, gravitational energy can be localized."
It is usually understood that the equivalence principle makes it impossible to localize gravitational energy-momentum. If there were a gravitational field energy-momentum tensor, it would vanish in locally inertial coordinates, hence it would vanish everywhere. That's why there are only pseudotensors of various kinds that satisfy ##\left( [t^{\mu\nu}+T^{\mu\nu}]\sqrt{-g}\right)_{,\nu}=0##.
How can Dirac be right, in view of the equivalence principle? What exactly is he saying here?