Can gravitational energy be localized in the case of plane waves?

[Kostik]

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?

 

[anuttarasammyak]

In the beginnig of the section, Dirac starts from (16.4) which is weak gravity case where second order terms are disregarded. We should remind that his conclusion is under this approximation.

 

[Kostik]

OK, but the equivalence principle still applies. If gravitational energy and momentum density and fluxes constitute a tensor, then in a locally inertial coordinate system all entries vanish. Hence such a tensorial quantity must vanish in all coordinate systems … a contradiction.

 

[renormalize]

How can Dirac be right, in view of the equivalence principle? What exactly is he saying here?

Dirac is simply saying that weak gravitational waves propagating in a single direction $l_{\mu}$ have an energy-momentum pseudo-tensor $\tau_{\mu\nu}$ that is covariant under the restricted coordinate transformations $x^{\mu\prime}=x^{\mu}+b^{\mu}\left(l\cdot x\right)$. The pseudo-tensor is therefore non-vanishing in any such restricted coordinate system and hence localizable by Dirac's criterion. But note that $\tau_{\mu\nu}$ is quadratic in the first derivatives of the metric $g_{\mu\nu ,\sigma}$ or, in other words, this pseudo-tensor is quadratic in the Christoffel symbols $\Gamma{}^{\alpha}{}_{\mu\nu}$. Since $\tau_{\mu\nu}$ is non-zero in any restricted system, then the Christoffel symbols must satisfy $\Gamma{}^{\alpha}{}_{\mu\nu}\neq 0$ everywhere in any such system. Consequently, these restricted systems never constitute an inertial (free-falling) frame, for which we must have $\Gamma{}^{\alpha}{}_{\mu\nu}=0$ at some point. So the equivalence principle doesn't come into play: in the context of gravitational waves travelling in one direction, Dirac is restricting attention to a narrow class of non-inertial coordinate systems.

 

[Kostik]

Dirac is simply saying that weak gravitational waves propagating in a single direction $l_{\mu}$ have an energy-momentum pseudo-tensor $\tau_{\mu\nu}$ that is covariant under the restricted coordinate transformations $x^{\mu\prime}=x^{\mu}+b^{\mu}\left(l\cdot x\right)$. The pseudo-tensor is therefore non-vanishing in any such restricted coordinate system and hence localizable by Dirac's criterion. But note that $\tau_{\mu\nu}$ is quadratic in the first derivatives of the metric $g_{\mu\nu ,\sigma}$ or, in other words, this pseudo-tensor is quadratic in the Christoffel symbols $\Gamma{}^{\alpha}{}_{\mu\nu}$. Since $\tau_{\mu\nu}$ is non-zero in any restricted system, then the Christoffel symbols must satisfy $\Gamma{}^{\alpha}{}_{\mu\nu}\neq 0$ everywhere in any such system. Consequently, these restricted systems never constitute an inertial (free-falling) frame, for which we must have $\Gamma{}^{\alpha}{}_{\mu\nu}=0$ at some point. So the equivalence principle doesn't come into play: in the context of gravitational waves travelling in one direction, Dirac is restricting attention to a narrow class of non-inertial coordinate systems.

Yes, by Dirac (34.2) the energy pseudo-tensor is positive definite and never zero. But from Dirac (33.9) the pseudotensor is $\Gamma^\nu_{\alpha\beta}$ multiplied by another factor. Hence, $\Gamma$ never vanishes.

So, yes, the psuedotensor is covariant within this restricted class of coordinate transformations with the property that if $g_{\mu\nu}=g_{\mu\nu}(l_\sigma x^\sigma)$ then the transformed metric $g'_{\mu\nu}$ is likewise a function of $l_\sigma x^\sigma$. This category includes transformations of the kind $x^\mu \rightarrow x^\mu + b^\mu(l_\sigma x^\sigma)$ with $|b^\mu_{\,\,,\sigma}| \ll 1$ (not hard to show this).

Still, does this show that gravitational wave energy is truly localized? Clearly, in another coordinate system (such as a locally inertial one) which doesn't fall into the class mentioned, the pseudotensor is not coordinate invariant. So does an observer in that frame of reference agree that energy is localized?

 

[PeterDonis]

does this show that gravitational wave energy is truly localized?

You answer that question yourself:

Clearly, in another coordinate system (such as a locally inertial one) which doesn't fall into the class mentioned, the pseudotensor is not coordinate invariant.

And in fact it is perfectly possible to find a coordinate chart in which the pseudotensor vanishes. What does that tell you?

 

[Kostik]

I must conclude that Dirac chose his words poorly in stating that "with the restriction that we have waves moving only in one direction, gravitational energy can be localized."

At a particular $x$, a locally inertial coordinate system is one determined by an observer in free fall. (It's not unique, since the observer may have started from rest at any other position or time.) $x^0$ in this system is the observer's proper time, and $x^1, x^2, x^3$ are a system of Cartesian coordinates in the observer's reference frame with the observer at the origin.

In such a coordinate system, there is no gravitational field. Hence, the action of the gravitational wave cannot accelerate the observer. It can only cause two separated points (observers) to move relative to one another by altering the space between them.

 

[anuttarasammyak]

We have succeeded to observe gravitational wave with LIGO. I think that no direct energy-momentum observation of gravitational waves but something related to energy-momentum of gravitational wave which Dirac explained in Chap 33 and 34, was observed. The detection signal generation at LIGO detector is the 4-vector EVENT, with When and Where , which is shared with any FR including co-moving FR where $\Gamma$ is zero. Here, don't we need any conciliation between the features of EVENT and Non-Localizability ?

 

[Kostik]

The detection signal generation at LIGO detector is the 4-vector EVENT, with When and Where , which is shared with any FR including co-moving FR where $\Gamma$ is zero. Here, don't we need any conciliation between the features of EVENT and Non-Localizability ?

I have no idea what you’re saying.