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I am reading Strominger's lecture notes "The infrared structure of gravity and gauge theory" (https://arxiv.org/abs/1703.05448). In chapter two, while trying to derive an expression about the conserved charges $Q_{\varepsilon}$, the author assumes some that the behavior of the fields at $\mathcal{I}^+$. Specifically, he assumes that there are no long range magnetic fields at the future of $\mathcal{I}^+$, which he denotes as $\mathcal{I}^+_+$ and neither on its past, which he denotes as $\mathcal{I}^+_-$. Later, he claims that if we wish to have finite energy, the electromagnetic potential better be pure gauge. Why is that? I have also seen the stack exchange question (Electromagnetic radiation flux through null infinity) but I can not connect the arguments in the link with the gauge field being pure gauge. Also, I can not fully understand why the flux is given by the following formula in the link: $\int_{\mathcal{I}^+}*(T_{\mu\nu}X^{\nu}dx^{\mu})$

I have read the original paper, which I think Strominger bases its lecture notes regarding this part (https://arxiv.org/abs/1407.3789), and the author choose configurations which revert to the vacuum at $\mathcal{I}^+_+$ and hence $F_{ur}|_{\mathcal{I}^+_+}=0$ and $F_{uz}|_{\mathcal{I}^+_+}=0$. Is there a physical intuition related to this choice? Can someone elaborate on our motivation for choosing those boundary conditions?

Any help will be appreciated.

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The energy flux through a cut $[u_0,u_1]$ of ${\cal I}^+$ and charge flux through $i^+$ is given by $$ E_{[u_0,u_1]} \sim \int_{u_0}^{u_1} du \int d^2 z | F_{uz} |^2 , \qquad Q \sim \int_{{\cal I}^+_+} d^2 z r^2 F_{ur} $$ Since the authors want the system to revert to pure vacuum in the far future, the energy flux must vanish as $u_1,u_0\to\infty$ and the charge flux must also vanish. This implies $$ F_{uz} |_{{\cal I}^+_+} = 0 , \qquad F_{ur} |_{{\cal I}^+_+} = 0. $$ These conditions only make physical sense in the absence of massive charged particles. In later papers (should also be in the lecture notes), the effect of massive states has been studied and these conditions have been relaxed somewhat.

The two conditions together with the Bianchi identities implies that the full field strength vanishes in the far future (and past) from which we surmise that the gauge field is pure gauge in that region.

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  • $\begingroup$ okay thanks Prahar. Can you elaborate a bit on why the absence of massive charged particles provides some physical sense to these conditions?? I mean, I know that charged massive particles can not travel to $i^+$, since they always follow timelike trajectories. But what is the exact relation between $i^+$ and $\mathcal{I}^+_+$?? From the arguments used so far, it seems to me that they are one and the same? Is this true? and why so? $\endgroup$
    – schris38
    May 28, 2022 at 20:28
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    $\begingroup$ $\partial i^+ = {\cal I}^+_+$ $\endgroup$
    – Prahar
    May 28, 2022 at 20:32
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    $\begingroup$ $i^+$ is a single point in the conformal compactification. That is not a true representation of the asymptotic structure of a spacetime. $i^+$ and $i^-$ are 3-dimensional Euclidean hypersurfaces. ${\cal I}^+$ and ${\cal I}^-$ are 3-dimensional null hypersurfaces and $i^0$ is a 3-dimensional timelike hypersurface. $\endgroup$
    – Prahar
    Oct 12, 2023 at 12:32
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    $\begingroup$ @GRrocks - ya, that is correct. Conformal compactifications preserve conformal structures. This is why it is useful for resolving null boundaries but fails to accurately represent the non-null boundaries. The surfaces are perfectly well-defined in advanced and retarded coordinates. The problem arises when you compactify the space by multiplying the metric by a Weyl factor that is badly behaved on the non-null boundaries. $\endgroup$
    – Prahar
    Oct 12, 2023 at 20:58
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    $\begingroup$ @Prahar got it, thanks! I'm starting to learn celestial holography and somehow this basic point about the asymptotic structure puzzled me because GR books don't really stress the point you made above. $\endgroup$
    – GRrocks
    Oct 12, 2023 at 21:13

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