Asymptotic Symmetries

Prerequisites: General Relativity, Noether's Theorems. Full path (22 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

[S]ymmetries of gravitational systems, that is, changes of coordinates that can be applied to space-time and that leave invariant the large-distance behaviour of the gravitational field. They are known as asymptotic symmetries and can be thought of as a generalization of Poincaré symmetry for systems endowed with a weak gravitational field. In other words, these symmetries are those one would observe by looking at a gravitational system “from far away”B. Oblak, BMS Particles in Three Dimensions, Springer Theses

• http://jakobschwichtenberg.com/surprising-symmetries/

Concrete

A warm-up in electromagnetism. A gauge transformation $A_\mu \to A_\mu + \partial_\mu \varepsilon$ does not change $F_{\mu\nu}$. By Noether's second theorem the charge associated with a local symmetry is a surface integral. For a gauge parameter $\varepsilon(x)$ the charge on a time slice is (Heaviside-Lorentz units) $$Q_\varepsilon = \oint_{S^2_\infty} \varepsilon\, \vec E \cdot d\vec S ,$$ evaluated on a sphere at spatial infinity. If $\varepsilon \to 0$ at infinity, then $Q_\varepsilon = 0$. Such transformations only relate different descriptions of the same physical state and are true redundancies. If $\varepsilon$ approaches a constant, $Q_\varepsilon$ is that constant times the total electric charge. If $\varepsilon$ approaches a function $\varepsilon(\theta,\varphi)$ of the angles, we get an infinite family of charges, one for each function on the sphere. These "large" gauge transformations change the physical state and are called asymptotic symmetries.

Gravity: the BMS group. In general relativity diffeomorphisms play the role of gauge transformations. For asymptotically flat spacetimes one works near future null infinity $\mathscr I^+$ in retarded Bondi coordinates $(u, r, \theta, \varphi)$, where $u = t - r$ at large $r$ and the metric approaches $$ds^2 = -du^2 - 2\,du\,dr + r^2 d\Omega^2 + \frac{2 G m_B}{r} du^2 + \dots$$ (signature $(-,+,+,+)$, $c = 1$). The function $m_B(u,\theta,\varphi)$ is the Bondi mass aspect, normalised so that $m_B = M$ for the Schwarzschild solution. Further terms, falling off in $r$, encode gravitational radiation.

One then asks which diffeomorphisms preserve this form of the metric and these fall-off conditions. Besides the Lorentz transformations one finds the supertranslations, which at leading order act on $\mathscr I^+$ as $$u \to u + f(\theta,\varphi),$$ with an arbitrary smooth function $f$ on the sphere. If we expand $f$ in spherical harmonics, the $\ell = 0$ and $\ell = 1$ components are the four ordinary spacetime translations (one time translation, three space translations). All components with $\ell \ge 2$ are new. Each supertranslation has a charge, built from the integral of $f\, m_B$ over a cut $u = \text{const}$ of $\mathscr I^+$ plus terms involving the radiation data. For $f = 1$ it reduces to the Bondi mass $$M_B(u) = \frac{1}{4\pi}\oint d\Omega\; m_B(u,\theta,\varphi),$$ which decreases when gravitational waves carry energy to $\mathscr I^+$.

The group of supertranslations together with the Lorentz group is the Bondi-van der Burg-Metzner-Sachs (BMS) group. It is infinite-dimensional and contains the Poincaré group as a subgroup, but the Poincaré group is not a normal subgroup, so there is no canonical way to single it out. A practical consequence: two spacetimes that differ by a supertranslation are physically distinct. The passage of gravitational radiation can shift the asymptotic metric by a supertranslation, which is the gravitational memory effect, a permanent relative displacement of free test masses after a burst of waves has passed.

Abstract

General definition. Consider a gauge theory with gauge transformations $\delta_\xi$ and a choice of boundary conditions on the fields near some asymptotic boundary. Let $\mathcal A$ be the group of gauge transformations that preserve the boundary conditions, and let $\mathcal T \subset \mathcal A$ be the normal subgroup of those whose canonical generators vanish on-shell (their Noether charges are zero for every solution). The asymptotic symmetry group is $$G_{\text{asym}} = \mathcal A / \mathcal T .$$ Elements of $\mathcal T$ are redundancies of the description. Elements of $G_{\text{asym}}$ have surface charges $Q_\xi$ and act nontrivially on the phase space. The charges realize the group by Poisson brackets, $\{Q_\xi, Q_\eta\} = Q_{[\xi,\eta]} + K(\xi,\eta)$, possibly with a central extension $K$. The result depends on the boundary conditions, which are part of the definition of the theory.

Asymptotically flat gravity. In four dimensions the BMS group of $\mathscr I^+$ is the semidirect product $$\mathrm{BMS}_4 = \mathcal S \rtimes SO^+(3,1),$$ where the abelian normal subgroup $\mathcal S$ of supertranslations consists of functions $f$ on the celestial sphere, and the Lorentz group acts on the sphere $S^2 \simeq \mathbb{CP}^1$ by Möbius transformations, $SO^+(3,1) \simeq PSL(2,\mathbb C)$. The ordinary translations form the unique four-dimensional normal subgroup of $\mathrm{BMS}_4$. Proposed extensions replace the global conformal group by local conformal transformations of the sphere (superrotations) or by all diffeomorphisms of $S^2$.

Strominger and collaborators showed that the Ward identities of supertranslation charges, with a suitable matching between past and future null infinity, are equivalent to Weinberg's soft graviton theorem. The same structure links large $U(1)$ gauge transformations to the soft photon theorem, and both link to memory effects.

Anti-de Sitter space in three dimensions. For three-dimensional gravity with negative cosmological constant $\Lambda = -1/\ell^2$ and Brown-Henneaux boundary conditions, the asymptotic symmetry algebra is two copies of the Virasoro algebra, $$[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}m(m^2-1)\,\delta_{m+n,0}, \qquad c = \frac{3\ell}{2G},$$ and the same for $\bar L_n$. The finite-dimensional isometry group $SO(2,2)$ of $\mathrm{AdS}_3$ is generated by $L_{0,\pm1}$ and $\bar L_{0,\pm 1}$. This central charge is an early precursor of the AdS/CFT correspondence.

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