symmetries & charges
at spatial infinity

Kartik Prabhu Brown University, Providence, RI

with Ibrahim Shehzad [arXiv:1912.04305]


Asymptotic boundaries \(\mathscr I^-\), \(\mathscr I^+\) and \(i^0\)

  1. \(\mathscr I^\pm\) have infinite-number of symmetries; BMS group with Lorentz and supertranslations
  2. Infinite-number of charges called supermomenta; not conserved but have flux due to radiation

Is there any relation between symmetries and charges on \(\mathscr I^-\) and \(\mathscr I^+\)?

Global conservation (?)

  1. relate BMS symmetry on \(\mathscr I^-\) to another one on \(\mathscr I^+\)?
  2. incoming flux on \(\mathscr I^-\) = outgoing flux on \(\mathscr I^+\)?

Infinite-number of global conservation laws; constraints on classical scattering

Quantum S-matrix, soft theorems, information loss (?)

Global conservation (known results)

(not a complete list)
  1. Bondi \(4\)-momentum in full GR. Ashtekar, Magnon-Ashtekar [1979]
  2. Angular momentum in stationary spacetimes full GR. Ashtekar, Streubel [1979]
  3. Minkowski background: EM Campiglia, Eyheralde [2017], linearised gravity Troessaert [2017], Grgin index (twistor theory literature; very old result)
  4. General spacetimes: EM Prabhu [2018], supertranslations Prabhu [2019]

Lorentz charges on general spacetimes?


  1. Solve the scattering problem through the interior of the spacetime
  2. Somehow use spatial infinity \(i^0\)

Both methods are do-able in Minkowski spacetime

  1. try solve exactly given incoming data (Kirchoff integrals; assuming analyticity)
  2. \(i^0\) is analytic so can “pass through” spatial infinity

\(i^0\) is a funny place!

Ashtekar-Hansen structure at \(i^0\):

  • \(\nabla_a\Omega = 0\); \(\nabla_a \nabla_b\Omega = 2 g_{ab}\) (cannot use Bondi frame)
  • if \(mass \neq 0\): \(\partial_a g_{bc}\) is direction-dependent

Very low differential structure at \(i^0\):

  • The manifold is \(C^{>1}\)
  • Metric \(g_{ab}\) is \(C^{>0}\)

Blowup \(i^0\) — space of directions: \(\mathscr H\)

\(\mathscr H\): hyperboloid of spatial directions \(\vec\eta\) in \(Ti^0\) (a.k.a. deSitter)

  • Weyl tensor \(C_{abcd}\) gives electric and magnetic fields on \(\mathscr H\):
    \(\mathbf E_{ab} = \lim\limits_{\to i^0} \Omega^{1/2}C_{acbd}\eta^c \eta^d\)
    \(\mathbf B_{ab} = \lim\limits_{\to i^0} \Omega^{1/2} *C_{acbd}\eta^c \eta^d\)
  • Einstein equation:
    \(\mathbf D_{[a} \mathbf E_{b]c} = \mathbf D_{[a} \mathbf B_{b]c} = 0\)

Asymptotic symmetries

diffeos \(\xi^a\) in the physical spacetime which preserve the asymptotic structure at \(i^0\)

  1. preserve \(i^0\) : \(\xi^a\vert_{i^0} = 0\)
  2. preserve \(C^{>1}\)-structure : \(\Omega^{-1/2}\xi^a\) is direction-dependent
  3. preserve the metric and directions: \((\mathbf f, \mathbf X^a)\)
    supertranslation: \(\mathbf f \in C^\infty(\mathscr H)\)
    Lorentz: \(\mathbf X^a\) is a Killing field on \(\mathscr H\)

Lagrangian and symplectic current

Lagrangian \(L\) as a \(4\)-form in physical spacetime

  • \(\delta L = \mathcal E^{ab} \delta g_{ab} + d \theta(\delta g)\)
  • \(\mathcal E^{ab} = 0\) are equations of motion. \(\theta(\delta g)\) symplectic potential
  • \(\omega(\delta_1 g, \delta_2 g) = \delta_1 \theta(\delta_2 g) - \delta_2 \theta(\delta_1 g) \) symplectic current
  • \(\omega(\delta g, £_\xi g) = d [\delta Q_\xi - \xi \cdot \theta(\delta g)] + EOM\) Hamiltonian flow of a diffeo is a boundary term
  • The boundary term defines a perturbed charge

Symplectic current: limit to \(i^0\)

asymptotic conditions on metric perturbations: \(\Omega^{3/2}\omega\) has a direction-dependent limit

pullback to \(\mathscr H\): \(\underleftarrow{\mathbf \omega} = \mathbf \varepsilon_3 ( \delta_1 \mathbf E \delta_2 \mathbf K - \delta_2 \mathbf E \delta_1 \mathbf K)\)

\(\mathbf E\) and \(\mathbf K_{ab}\) are potentials for the Weyl tensor: \[ \mathbf E_{ab} = -\tfrac{1}{4}~ (\mathbf D_a \mathbf D_b \mathbf E + \mathbf h_{ab} \mathbf E) \quad \mathbf B_{ab} = - \tfrac{1}{4}~ \mathbf \varepsilon_{cda} \mathbf D^c \mathbf K^d{}_b \]

Symplectic current: Symmetries

  • Can be written as a total derivative (computation to show this) \(\underleftarrow{\mathbf \omega} (\delta g, \delta_{(\mathbf f, \mathbf X)}g) = \mathbf \varepsilon_3 \mathbf D^a \mathbf Q_a(\delta g, (\mathbf f, \mathbf X)) + EOM\)
  • Use \(\mathbf Q_a\) to define a charge on cross-sections of \(\mathscr H\)

Charge: Supertranslations

  • \(\mathbf Q_a = \delta (\mathbf E \mathbf D_a \mathbf f - \mathbf f \mathbf D_a \mathbf E)\)
  • Supermomentum: \(\mathcal Q[\mathbf f; S] = \int\limits_S \mathbf \varepsilon_2 \mathbf u^a (\mathbf E \mathbf D_a \mathbf f - \mathbf f \mathbf D_a \mathbf E) \)
  • Can be matched up to the supermomentum on \(\mathscr I^\pm\)

Charge: Lorentz (perturbed)

  • \(\mathbf Q_a = \delta \mathbf W_{ab} {}^\star \mathbf X^b - \tfrac{1}{8} \delta \mathbf E \mathbf K \mathbf X_{a}\)
  • \(\mathbf{W}_{ab} = \mathbf{\beta}_{ab} + \tfrac{1}{8}~ \mathbf{\varepsilon}_{cd(a} \mathbf D^{c} \mathbf{E} \mathbf K^d{}_{b)} - \tfrac{1}{16}~ \mathbf{\varepsilon}_{abc}\mathbf{K}\mathbf D^{c}\mathbf{E}\)
  • \(\mathbf \beta_{ab}\) is the subleading magnetic Weyl tensor
  • Note that \(\mathbf Q_a\) is not \(\delta (something)\). This can be fixed using the Wald-Zoupas prescription (whole another talk!!)

Charge: Lorentz

  • Do the Wald-Zoupas stuffs
  • \(\mathcal Q[\mathbf{X}^{a};S] = \int_S\mathbf{\varepsilon}_{2}~ \mathbf u^{a} [ \mathbf{W}_{ab} {}^\star{\mathbf{X}}^{b} - \tfrac{1}{8} \mathbf{K} \mathbf{E} \mathbf{X}_{a}]\)
  • Same as the charge found by Ashtekar-Hansen when \(\mathbf K_{ab} = 0\)
  • Same as the charge found by Compère-Dehouck when \(\mathbf K = 0\)

In the oven

  • Does the Lorentz charge at \(i^0\) match the ones from \(\mathscr I^\pm\)?
  • implications for scattering and conservation laws in quantum theory?
  • Since \(\mathscr H\) is the same as deSitter, is there any relation to deSitter holography?


  1. KP and I. Shehzad, Asymptotic symmetries and charges at spatial infinity in general relativity, [arXiv:1912.04305].
  2. KP, Conservation of asymptotic charges from past to future null infinity: Maxwell fields, JHEP 10 113 (, [arXiv:1808.07863].
  3. KP, Conservation of asymptotic charges from past to future null infinity: Supermomentum in general relativity, JHEP 03 148 (, [arXiv:1902.08200].