Asymptotic Structure of Electromagnetism and Gravity in The

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux Introduction Asymptotic structure of electromagnetism and The problem in a nutshell - Electromagnetism gravity in the asymptotically ﬂat case New boundary conditions

Gravity

Conclusions and Marc Henneaux comments

Geometry and Duality Workshop, AEI, Potsdam - 6 December 2019

1 / 33 Asymptotic symmetries play a central role in holographic duality. This is familiar from the AdS/CFT context, where the asymptotic symmetry group (which is inﬁnite-dimensional in the case of AdS3 gravity) is the group of rigid symmetries of the dual boundary theory. What is the situation in the asymptotically ﬂat context ? This will be the subject of this talk.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

2 / 33 This is familiar from the AdS/CFT context, where the asymptotic symmetry group (which is inﬁnite-dimensional in the case of AdS3 gravity) is the group of rigid symmetries of the dual boundary theory. What is the situation in the asymptotically ﬂat context ? This will be the subject of this talk.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction Asymptotic symmetries play a central role in holographic duality. The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

2 / 33 What is the situation in the asymptotically ﬂat context ? This will be the subject of this talk.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction Asymptotic symmetries play a central role in holographic duality. The problem in a nutshell - This is familiar from the AdS/CFT context, where the asymptotic Electromagnetism symmetry group (which is inﬁnite-dimensional in the case of New boundary conditions AdS3 gravity) is the group of rigid symmetries of the dual Gravity boundary theory. Conclusions and comments

2 / 33 This will be the subject of this talk.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

2 / 33 Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

2 / 33 The BMS (Bondi-Metzner-Sachs) group was shown long ago to be the group of asymptotic symmetries of gravity in the asymptotically flat context. It is infinite-dimensional and contains the Poincaré group as a subgroup, which is the group of isometries (exact symmetries) of the background Minkowski space. This is another instance where the group of asymptotic symmetries is strictly larger than the group of exact symmetries of the most symmetric solution and even infinite-dimensional !

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

3 / 33 It is inﬁnite-dimensional and contains the Poincaré group as a subgroup, which is the group of isometries (exact symmetries) of the background Minkowski space. This is another instance where the group of asymptotic symmetries is strictly larger than the group of exact symmetries of the most symmetric solution and even inﬁnite-dimensional !

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case The BMS (Bondi-Metzner-Sachs) group was shown long ago to be Marc Henneaux the group of asymptotic symmetries of gravity in the Introduction asymptotically ﬂat context. The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

3 / 33 which is the group of isometries (exact symmetries) of the background Minkowski space. This is another instance where the group of asymptotic symmetries is strictly larger than the group of exact symmetries of the most symmetric solution and even inﬁnite-dimensional !

Introduction

New boundary subgroup, conditions

Gravity

Conclusions and comments

3 / 33 This is another instance where the group of asymptotic symmetries is strictly larger than the group of exact symmetries of the most symmetric solution and even inﬁnite-dimensional !

Introduction

New boundary subgroup, conditions

Gravity which is the group of isometries (exact symmetries) of the

Conclusions and background Minkowski space. comments

3 / 33 and even inﬁnite-dimensional !

Introduction

New boundary subgroup, conditions

Gravity which is the group of isometries (exact symmetries) of the

Conclusions and background Minkowski space. comments This is another instance where the group of asymptotic symmetries is strictly larger than the group of exact symmetries of the most symmetric solution

3 / 33 Introduction

New boundary subgroup, conditions

Gravity which is the group of isometries (exact symmetries) of the

3 / 33 This remarkable result was first received with embarrassment because the meaning of the enlargement was not understood. Furthermore, there was a tension between studies at null infinity and at spatial infinity : while the BMS group naturally emerges at null infinity, previous analyses of asymptotically flat spaces at spatial infinity did not exhibit any sign of the BMS group.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

4 / 33 because the meaning of the enlargement was not understood. Furthermore, there was a tension between studies at null infinity and at spatial infinity : while the BMS group naturally emerges at null infinity, previous analyses of asymptotically flat spaces at spatial infinity did not exhibit any sign of the BMS group.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction This remarkable result was ﬁrst received with embarrassment

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

4 / 33 Furthermore, there was a tension between studies at null infinity and at spatial infinity : while the BMS group naturally emerges at null infinity, previous analyses of asymptotically flat spaces at spatial infinity did not exhibit any sign of the BMS group.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction This remarkable result was ﬁrst received with embarrassment

The problem in a nutshell - because the meaning of the enlargement was not understood. Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

4 / 33 while the BMS group naturally emerges at null infinity, previous analyses of asymptotically flat spaces at spatial infinity did not exhibit any sign of the BMS group.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction This remarkable result was ﬁrst received with embarrassment

Conclusions and comments

4 / 33 previous analyses of asymptotically ﬂat spaces at spatial inﬁnity did not exhibit any sign of the BMS group.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction This remarkable result was ﬁrst received with embarrassment

The problem in a nutshell - because the meaning of the enlargement was not understood. Electromagnetism Furthermore, there was a tension between studies at null infinity New boundary conditions and at spatial infinity : Gravity while the BMS group naturally emerges at null infinity, Conclusions and comments

4 / 33 Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction This remarkable result was ﬁrst received with embarrassment

4 / 33 For that reason, it was sometimes stated that the BMS group was intrinsically connected with gravitational radiation and could only be seen in formulations adapted to null inﬁnity. But if a transformation is a symmetry of a theory, it should be visible in any formulation ! So : which picture should be trusted ?

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

5 / 33 But if a transformation is a symmetry of a theory, it should be visible in any formulation ! So : which picture should be trusted ?

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a For that reason, it was sometimes stated that the BMS group was nutshell - Electromagnetism intrinsically connected with gravitational radiation and could

New boundary only be seen in formulations adapted to null inﬁnity. conditions

Gravity

Conclusions and comments

5 / 33 So : which picture should be trusted ?

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a For that reason, it was sometimes stated that the BMS group was nutshell - Electromagnetism intrinsically connected with gravitational radiation and could

New boundary only be seen in formulations adapted to null inﬁnity. conditions But if a transformation is a symmetry of a theory, it should be Gravity

Conclusions and visible in any formulation ! comments

5 / 33 Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a For that reason, it was sometimes stated that the BMS group was nutshell - Electromagnetism intrinsically connected with gravitational radiation and could

New boundary only be seen in formulations adapted to null inﬁnity. conditions But if a transformation is a symmetry of a theory, it should be Gravity

Conclusions and visible in any formulation ! comments So : which picture should be trusted ?

5 / 33 This state of affairs considerably changed recently. First, remarkable work showed that the extra charges could be associated with the soft graviton theorems through the corresponding Ward identities. This gives “physical existence” to the BMS group. The enlargement is a gift ! Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

6 / 33 First, remarkable work showed that the extra charges could be associated with the soft graviton theorems through the corresponding Ward identities. This gives “physical existence” to the BMS group. The enlargement is a gift ! Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case This state of affairs considerably changed recently. Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

6 / 33 through the corresponding Ward identities. This gives “physical existence” to the BMS group. The enlargement is a gift ! Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case This state of affairs considerably changed recently. Marc Henneaux First, remarkable work showed that the extra charges could be Introduction associated with the soft graviton theorems The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

6 / 33 This gives “physical existence” to the BMS group. The enlargement is a gift ! Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

Introduction

New boundary conditions

Gravity

Conclusions and comments

6 / 33 The enlargement is a gift ! Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

Introduction

Gravity

Conclusions and comments

6 / 33 Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

Introduction

Gravity The enlargement is a gift !

Conclusions and comments

6 / 33 through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

Introduction

Gravity The enlargement is a gift !

Conclusions and comments Second, the tension between null inﬁnity and spatial inﬁnity has been resolved

6 / 33 Analyses at null inﬁnity and at spatial inﬁnity completely agree !

Introduction

Gravity The enlargement is a gift !

Conclusions and comments Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity.

6 / 33 Introduction

Gravity The enlargement is a gift !

Conclusions and comments Second, the tension between null infinity and spatial infinity has been resolved through a reconsideration of the boundary conditions at spatial infinity. Analyses at null infinity and at spatial infinity completely agree !

6 / 33 The purpose of this talk is to explain how the infinite-dimensional BMS group appears at spatial infinity. The analysis will be carried on spacelike hypersurfaces that are asymptotically flat hyperplanes, using Hamiltonian methods. It completely solves the difficulties mentioned above. (Work done in collaboration with Cédric Troessaert.) The analysis also provides an opportunity to develop general considerations on asymptotic symmetries. One bonus of investigating the asymptotic properties at spatial infinity is that a sufficiently smooth null infinity, with the implicitly imposed peeling properties of the Weyl tensor, might not exist ! (This is a very delicate dynamical question, see Friedrich 2018.) Does the BMS symmetry depends on the existence of a sufficiently smooth null infinity ?

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

7 / 33 The analysis will be carried on spacelike hypersurfaces that are asymptotically flat hyperplanes, using Hamiltonian methods. It completely solves the difficulties mentioned above. (Work done in collaboration with Cédric Troessaert.) The analysis also provides an opportunity to develop general considerations on asymptotic symmetries. One bonus of investigating the asymptotic properties at spatial infinity is that a sufficiently smooth null infinity, with the implicitly imposed peeling properties of the Weyl tensor, might not exist ! (This is a very delicate dynamical question, see Friedrich 2018.) Does the BMS symmetry depends on the existence of a sufficiently smooth null infinity ?

Introduction

Asymptotic structure of electromagnetism The purpose of this talk is to explain how the and gravity in the asymptotically infinite-dimensional BMS group appears at spatial infinity. flat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

7 / 33 (Work done in collaboration with Cédric Troessaert.) The analysis also provides an opportunity to develop general considerations on asymptotic symmetries. One bonus of investigating the asymptotic properties at spatial infinity is that a sufficiently smooth null infinity, with the implicitly imposed peeling properties of the Weyl tensor, might not exist ! (This is a very delicate dynamical question, see Friedrich 2018.) Does the BMS symmetry depends on the existence of a sufficiently smooth null infinity ?

Introduction

Introduction asymptotically ﬂat hyperplanes, using Hamiltonian methods. It

The problem in a completely solves the difﬁculties mentioned above. nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

7 / 33 The analysis also provides an opportunity to develop general considerations on asymptotic symmetries. One bonus of investigating the asymptotic properties at spatial infinity is that a sufficiently smooth null infinity, with the implicitly imposed peeling properties of the Weyl tensor, might not exist ! (This is a very delicate dynamical question, see Friedrich 2018.) Does the BMS symmetry depends on the existence of a sufficiently smooth null infinity ?

Introduction

Introduction asymptotically ﬂat hyperplanes, using Hamiltonian methods. It

The problem in a completely solves the difﬁculties mentioned above. nutshell - Electromagnetism (Work done in collaboration with Cédric Troessaert.) New boundary conditions

Gravity

Conclusions and comments

7 / 33 One bonus of investigating the asymptotic properties at spatial infinity is that a sufficiently smooth null infinity, with the implicitly imposed peeling properties of the Weyl tensor, might not exist ! (This is a very delicate dynamical question, see Friedrich 2018.) Does the BMS symmetry depends on the existence of a sufficiently smooth null infinity ?

Introduction

Introduction asymptotically ﬂat hyperplanes, using Hamiltonian methods. It

7 / 33 is that a sufficiently smooth null infinity, with the implicitly imposed peeling properties of the Weyl tensor, might not exist ! (This is a very delicate dynamical question, see Friedrich 2018.) Does the BMS symmetry depends on the existence of a sufficiently smooth null infinity ?

Introduction

Introduction asymptotically ﬂat hyperplanes, using Hamiltonian methods. It

7 / 33 Does the BMS symmetry depends on the existence of a sufﬁciently smooth null inﬁnity ?

Introduction

Introduction asymptotically ﬂat hyperplanes, using Hamiltonian methods. It

The problem in a completely solves the difficulties mentioned above. nutshell - Electromagnetism (Work done in collaboration with Cédric Troessaert.) New boundary conditions The analysis also provides an opportunity to develop general Gravity considerations on asymptotic symmetries. Conclusions and comments One bonus of investigating the asymptotic properties at spatial infinity is that a sufficiently smooth null infinity, with the implicitly imposed peeling properties of the Weyl tensor, might not exist ! (This is a very delicate dynamical question, see Friedrich 2018.)

7 / 33 Introduction

Introduction asymptotically ﬂat hyperplanes, using Hamiltonian methods. It

7 / 33 A central role in the analysis will be played by the gravitational action which reads, in Hamiltonian form,

Z ½Z ¾ ij i 3 ³ ij i grav grav´ S[gij,π ,N,N ] dt d x π ∂t gij N H NH B = − i − − ∞

where B is a boundary term at inﬁnity and where ∞

grav 1 ij 1 2 grav j H pgR (π πij π ) 0, H 2 jπ 0. = − + pg − 2 ≈ i = − ∇ i ≈

(Dirac, Arnowitt-Deser-Misner, Regge-Teitelboim)

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

8 / 33 which reads, in Hamiltonian form,

Z ½Z ¾ ij i 3 ³ ij i grav grav´ S[gij,π ,N,N ] dt d x π ∂t gij N H NH B = − i − − ∞

where B is a boundary term at inﬁnity and where ∞

grav 1 ij 1 2 grav j H pgR (π πij π ) 0, H 2 jπ 0. = − + pg − 2 ≈ i = − ∇ i ≈

(Dirac, Arnowitt-Deser-Misner, Regge-Teitelboim)

Introduction

Asymptotic structure of electromagnetism A central role in the analysis will be played by the gravitational and gravity in the asymptotically action ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

8 / 33 Z ½Z ¾ ij i 3 ³ ij i grav grav´ S[gij,π ,N,N ] dt d x π ∂t gij N H NH B = − i − − ∞

where B is a boundary term at inﬁnity and where ∞

grav 1 ij 1 2 grav j H pgR (π πij π ) 0, H 2 jπ 0. = − + pg − 2 ≈ i = − ∇ i ≈

(Dirac, Arnowitt-Deser-Misner, Regge-Teitelboim)

Introduction

Asymptotic structure of electromagnetism A central role in the analysis will be played by the gravitational and gravity in the asymptotically action ﬂat case Marc Henneaux which reads, in Hamiltonian form,

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

8 / 33 where B is a boundary term at inﬁnity and where ∞

grav 1 ij 1 2 grav j H pgR (π πij π ) 0, H 2 jπ 0. = − + pg − 2 ≈ i = − ∇ i ≈

(Dirac, Arnowitt-Deser-Misner, Regge-Teitelboim)

Introduction

The problem in a nutshell - Z ½Z ¾ Electromagnetism ij i 3 ³ ij i grav grav´ S[gij,π ,N,N ] dt d x π ∂t gij N H NH B New boundary = − i − − ∞ conditions

Gravity

Conclusions and comments

8 / 33 grav 1 ij 1 2 grav j H pgR (π πij π ) 0, H 2 jπ 0. = − + pg − 2 ≈ i = − ∇ i ≈

(Dirac, Arnowitt-Deser-Misner, Regge-Teitelboim)

Introduction

The problem in a nutshell - Z ½Z ¾ Electromagnetism ij i 3 ³ ij i grav grav´ S[gij,π ,N,N ] dt d x π ∂t gij N H NH B New boundary = − i − − ∞ conditions

Gravity

Conclusions and comments where B is a boundary term at inﬁnity and where ∞

8 / 33 (Dirac, Arnowitt-Deser-Misner, Regge-Teitelboim)

Introduction

The problem in a nutshell - Z ½Z ¾ Electromagnetism ij i 3 ³ ij i grav grav´ S[gij,π ,N,N ] dt d x π ∂t gij N H NH B New boundary = − i − − ∞ conditions

Gravity

Conclusions and comments where B is a boundary term at inﬁnity and where ∞

grav 1 ij 1 2 grav j H pgR (π πij π ) 0, H 2 jπ 0. = − + pg − 2 ≈ i = − ∇ i ≈

8 / 33 Introduction

Introduction

The problem in a nutshell - Z ½Z ¾ Electromagnetism ij i 3 ³ ij i grav grav´ S[gij,π ,N,N ] dt d x π ∂t gij N H NH B New boundary = − i − − ∞ conditions

Gravity

Conclusions and comments where B is a boundary term at inﬁnity and where ∞

grav 1 ij 1 2 grav j H pgR (π πij π ) 0, H 2 jπ 0. = − + pg − 2 ≈ i = − ∇ i ≈

(Dirac, Arnowitt-Deser-Misner, Regge-Teitelboim)

8 / 33 ﬁnite and invariant under all (asymptotic) Poincaré symmetries, which are thus canonical transformations.

We shall insist that the boundary conditions make the action :

This puts strong and interesting restrictions.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

9 / 33 ﬁnite and invariant under all (asymptotic) Poincaré symmetries, which are thus canonical transformations. This puts strong and interesting restrictions.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction The problem in a We shall insist that the boundary conditions make the action : nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

9 / 33 and invariant under all (asymptotic) Poincaré symmetries, which are thus canonical transformations. This puts strong and interesting restrictions.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction The problem in a We shall insist that the boundary conditions make the action : nutshell - Electromagnetism ﬁnite New boundary conditions

Gravity

Conclusions and comments

9 / 33 This puts strong and interesting restrictions.

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction The problem in a We shall insist that the boundary conditions make the action : nutshell - Electromagnetism ﬁnite New boundary conditions and invariant under all (asymptotic) Poincaré symmetries, which

Gravity are thus canonical transformations.

Conclusions and comments

9 / 33 Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Gravity are thus canonical transformations. Conclusions and This puts strong and interesting restrictions. comments

9 / 33 The same tension arises for electromagnetism, where the existing boundary conditions at spatial infinity do not yield the infinite-dimensional angle-dependent u(1) symmetry present at null infinity. We shall focus on this more familiar and simpler case first. The action is in that case

Z ½Z Z µ ¶¾ i 3 i 3 1 i 1 ij SH [Ai,π ,At ] dt d xπ ∂t Ai d x π πi F Fij At G = − 2 + 4 +

where G ∂ πk 0 (Gauss’ law). = − k ≈

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

10 / 33 We shall focus on this more familiar and simpler case ﬁrst. The action is in that case

Z ½Z Z µ ¶¾ i 3 i 3 1 i 1 ij SH [Ai,π ,At ] dt d xπ ∂t Ai d x π πi F Fij At G = − 2 + 4 +

where G ∂ πk 0 (Gauss’ law). = − k ≈

Introduction

Asymptotic structure of electromagnetism and gravity in the asymptotically flat case The same tension arises for electromagnetism, where the existing Marc Henneaux boundary conditions at spatial infinity do not yield the Introduction infinite-dimensional angle-dependent u(1) symmetry present at The problem in a nutshell - null infinity. Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

10 / 33 The action is in that case

Z ½Z Z µ ¶¾ i 3 i 3 1 i 1 ij SH [Ai,π ,At ] dt d xπ ∂t Ai d x π πi F Fij At G = − 2 + 4 +

where G ∂ πk 0 (Gauss’ law). = − k ≈

Introduction

New boundary We shall focus on this more familiar and simpler case ﬁrst. conditions

Gravity

Conclusions and comments

10 / 33 Z ½Z Z µ ¶¾ i 3 i 3 1 i 1 ij SH [Ai,π ,At ] dt d xπ ∂t Ai d x π πi F Fij At G = − 2 + 4 +

where G ∂ πk 0 (Gauss’ law). = − k ≈

Introduction

New boundary We shall focus on this more familiar and simpler case ﬁrst. conditions The action is in that case Gravity

Conclusions and comments

10 / 33 where G ∂ πk 0 (Gauss’ law). = − k ≈

Introduction

New boundary We shall focus on this more familiar and simpler case ﬁrst. conditions The action is in that case Gravity

Conclusions and comments Z ½Z Z µ ¶¾ i 3 i 3 1 i 1 ij SH [Ai,π ,At ] dt d xπ ∂t Ai d x π πi F Fij At G = − 2 + 4 +

10 / 33 Introduction

New boundary We shall focus on this more familiar and simpler case ﬁrst. conditions The action is in that case Gravity

Conclusions and comments Z ½Z Z µ ¶¾ i 3 i 3 1 i 1 ij SH [Ai,π ,At ] dt d xπ ∂t Ai d x π πi F Fij At G = − 2 + 4 +

where G ∂ πk 0 (Gauss’ law). = − k ≈

10 / 33 k The canonical variables are Ak and π (electric ﬁeld). The standard boundary conditions are

a(1)(n) a(2)(n) k k 2 A o(r− ) k = r + r2 + for the spatial components of the connexion and

pk (n) pk (n) k (1) (2) 3 π o(r− ) = r2 + r3 + for the electric ﬁeld. These conditions imply

f (1)(n) f (2)(n) ik ik 3 F o(r− ) ik = r2 + r3 + for the magnetic ﬁeld.

Maxwell theory in 4D

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

11 / 33 The standard boundary conditions are

a(1)(n) a(2)(n) k k 2 A o(r− ) k = r + r2 + for the spatial components of the connexion and

pk (n) pk (n) k (1) (2) 3 π o(r− ) = r2 + r3 + for the electric ﬁeld. These conditions imply

f (1)(n) f (2)(n) ik ik 3 F o(r− ) ik = r2 + r3 + for the magnetic ﬁeld.

Maxwell theory in 4D

Asymptotic structure of elec- k tromagnetism The canonical variables are Ak and π (electric ﬁeld). and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

11 / 33 pk (n) pk (n) k (1) (2) 3 π o(r− ) = r2 + r3 + for the electric ﬁeld. These conditions imply

f (1)(n) f (2)(n) ik ik 3 F o(r− ) ik = r2 + r3 + for the magnetic ﬁeld.

Maxwell theory in 4D

Asymptotic structure of elec- k tromagnetism The canonical variables are Ak and π (electric ﬁeld). and gravity in the asymptotically The standard boundary conditions are ﬂat case Marc Henneaux a(1)(n) a(2)(n) Introduction k k 2 Ak 2 o(r− ) The problem in a = r + r + nutshell - Electromagnetism for the spatial components of the connexion and New boundary conditions

Gravity

Conclusions and comments

11 / 33 These conditions imply

f (1)(n) f (2)(n) ik ik 3 F o(r− ) ik = r2 + r3 + for the magnetic ﬁeld.

Maxwell theory in 4D

11 / 33 Maxwell theory in 4D

f (1)(n) f (2)(n) ik ik 3 F o(r− ) ik = r2 + r3 + for the magnetic ﬁeld.

11 / 33 Problems : R 3 k (1) The kinetic term d xπ (x)dAk(x) is in general (logarithmically) divergent. (2) The generator of boosts ¡ R d3xr(E2 B2)¢ is in general ∼ + (logarithmically) divergent. One needs to strengthen the boundary conditions.

Maxwell theory in 4D

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

12 / 33 R 3 k (1) The kinetic term d xπ (x)dAk(x) is in general (logarithmically) divergent. (2) The generator of boosts ¡ R d3xr(E2 B2)¢ is in general ∼ + (logarithmically) divergent. One needs to strengthen the boundary conditions.

Maxwell theory in 4D

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction Problems : The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

12 / 33 (2) The generator of boosts ¡ R d3xr(E2 B2)¢ is in general ∼ + (logarithmically) divergent. One needs to strengthen the boundary conditions.

Maxwell theory in 4D

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction Problems : The problem in a nutshell - R 3 k Electromagnetism (1) The kinetic term d xπ (x)dAk(x) is in general

New boundary (logarithmically) divergent. conditions

Gravity

Conclusions and comments

12 / 33 One needs to strengthen the boundary conditions.

Maxwell theory in 4D

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction Problems : The problem in a nutshell - R 3 k Electromagnetism (1) The kinetic term d xπ (x)dAk(x) is in general

New boundary (logarithmically) divergent. conditions (2) The generator of boosts ¡ R d3xr(E2 B2)¢ is in general Gravity ∼ + Conclusions and (logarithmically) divergent. comments

12 / 33 Maxwell theory in 4D

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction Problems : The problem in a nutshell - R 3 k Electromagnetism (1) The kinetic term d xπ (x)dAk(x) is in general

New boundary (logarithmically) divergent. conditions (2) The generator of boosts ¡ R d3xr(E2 B2)¢ is in general Gravity ∼ + Conclusions and (logarithmically) divergent. comments One needs to strengthen the boundary conditions.

12 / 33 k One imposes that the first coefficients in Ak and π have definite parity properties :

a(1)( n) a(1)(n), pk ( n) pk (n). k − = k (1) − = − (1) It follows that f (1)( n) f (1)(n) ik − = − ik

These conditions eliminate the above divergences. These conditions are also invariant under Lorentz transformations because the leading orders of Lorentz parameters are parity-odd (boosts and rotations).

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

13 / 33 a(1)( n) a(1)(n), pk ( n) pk (n). k − = k (1) − = − (1) It follows that f (1)( n) f (1)(n) ik − = − ik

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the k asymptotically One imposes that the first coefficients in Ak and π have definite flat case Marc Henneaux parity properties :

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

13 / 33 It follows that f (1)( n) f (1)(n) ik − = − ik

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the k asymptotically One imposes that the first coefficients in Ak and π have definite flat case Marc Henneaux parity properties :

Introduction (1) (1) k k The problem in a ak ( n) ak (n), p(1)( n) p(1)(n). nutshell - − = − = − Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

13 / 33 These conditions eliminate the above divergences. These conditions are also invariant under Lorentz transformations because the leading orders of Lorentz parameters are parity-odd (boosts and rotations).

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the k asymptotically One imposes that the first coefficients in Ak and π have definite flat case Marc Henneaux parity properties :

Introduction (1) (1) k k The problem in a ak ( n) ak (n), p(1)( n) p(1)(n). nutshell - − = − = − Electromagnetism It follows that New boundary (1) (1) conditions fik ( n) fik (n) Gravity − = −

Conclusions and comments

13 / 33 These conditions are also invariant under Lorentz transformations because the leading orders of Lorentz parameters are parity-odd (boosts and rotations).

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the k asymptotically One imposes that the first coefficients in Ak and π have definite flat case Marc Henneaux parity properties :

Conclusions and comments These conditions eliminate the above divergences.

13 / 33 because the leading orders of Lorentz parameters are parity-odd (boosts and rotations).

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the k asymptotically One imposes that the first coefficients in Ak and π have definite flat case Marc Henneaux parity properties :

Conclusions and comments These conditions eliminate the above divergences. These conditions are also invariant under Lorentz transformations

13 / 33 Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the k asymptotically One imposes that the first coefficients in Ak and π have definite flat case Marc Henneaux parity properties :

Conclusions and comments These conditions eliminate the above divergences. These conditions are also invariant under Lorentz transformations because the leading orders of Lorentz parameters are parity-odd (boosts and rotations).

13 / 33 These parity conditions are thus fulfilled by the Liénard-Wichiert potentials (up to a gauge transformation). They are also fulfilled by the electromagnetic field of a magnetic monopole (up to a gauge transformation). From the point of view of containing the known solutions, these boundary conditions seem therefore acceptable.

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

14 / 33 They are also fulﬁlled by the electromagnetic ﬁeld of a magnetic monopole (up to a gauge transformation). From the point of view of containing the known solutions, these boundary conditions seem therefore acceptable.

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction These parity conditions are thus fulﬁlled by the Liénard-Wichiert The problem in a nutshell - potentials (up to a gauge transformation). Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

14 / 33 From the point of view of containing the known solutions, these boundary conditions seem therefore acceptable.

Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction These parity conditions are thus fulfilled by the Liénard-Wichiert The problem in a nutshell - potentials (up to a gauge transformation). Electromagnetism New boundary They are also fulfilled by the electromagnetic field of a magnetic conditions monopole (up to a gauge transformation). Gravity

Conclusions and comments

14 / 33 Maxwell theory in 4D - Standard parity conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Conclusions and From the point of view of containing the known solutions, these comments boundary conditions seem therefore acceptable.

14 / 33 We now turn to the computation of the asymptotic symmetries. The physical asymptotic symmetry algebra is the quotient of all the gauge transformations preserving the boundary conditions by the ideal of “trivial" asymptotic symmetries, which have zero charges for all conﬁgurations obeying the boundary conditions. We must therefore both determine the asymptotic gauge transformations that preserve the boundary conditions and compute the value of their generators.

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

15 / 33 The physical asymptotic symmetry algebra is the quotient of all the gauge transformations preserving the boundary conditions by the ideal of “trivial" asymptotic symmetries, which have zero charges for all conﬁgurations obeying the boundary conditions. We must therefore both determine the asymptotic gauge transformations that preserve the boundary conditions and compute the value of their generators.

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction We now turn to the computation of the asymptotic symmetries.

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

15 / 33 We must therefore both determine the asymptotic gauge transformations that preserve the boundary conditions and compute the value of their generators.

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction We now turn to the computation of the asymptotic symmetries. The problem in a The physical asymptotic symmetry algebra is the quotient of all nutshell - Electromagnetism the gauge transformations preserving the boundary conditions New boundary conditions by the ideal of “trivial" asymptotic symmetries, which have zero

Gravity charges for all conﬁgurations obeying the boundary conditions.

Conclusions and comments

15 / 33 Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Gravity charges for all conﬁgurations obeying the boundary conditions. Conclusions and We must therefore both determine the asymptotic gauge comments transformations that preserve the boundary conditions and compute the value of their generators.

15 / 33 The boundary conditions are invariant under gauge transformations δ A ∂ ², ² k = k where ² behaves asymptotically as

² ² λ(n) o(r0). = 0 + +

Here ²0 is a constant and λ(n) is parity-odd,

λ( n) λ(n). − = −

This seems to be an inﬁnite-dimensional global symmetry with angle-dependent gauge transformations...... but is this the case ?

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

16 / 33 where ² behaves asymptotically as

² ² λ(n) o(r0). = 0 + +

Here ²0 is a constant and λ(n) is parity-odd,

λ( n) λ(n). − = −

This seems to be an inﬁnite-dimensional global symmetry with angle-dependent gauge transformations...... but is this the case ?

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the The boundary conditions are invariant under gauge asymptotically ﬂat case transformations δ²Ak ∂k², Marc Henneaux =

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

16 / 33 Here ²0 is a constant and λ(n) is parity-odd,

λ( n) λ(n). − = −

This seems to be an inﬁnite-dimensional global symmetry with angle-dependent gauge transformations...... but is this the case ?

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the The boundary conditions are invariant under gauge asymptotically ﬂat case transformations δ²Ak ∂k², Marc Henneaux = where ² behaves asymptotically as Introduction

The problem in a 0 nutshell - ² ²0 λ(n) o(r ). Electromagnetism = + +

New boundary conditions

Gravity

Conclusions and comments

16 / 33 This seems to be an inﬁnite-dimensional global symmetry with angle-dependent gauge transformations...... but is this the case ?

Asymptotic symmetries and charges

The problem in a 0 nutshell - ² ²0 λ(n) o(r ). Electromagnetism = + +

New boundary conditions

Gravity Here ²0 is a constant and λ(n) is parity-odd, Conclusions and comments λ( n) λ(n). − = −

16 / 33 ... but is this the case ?

Asymptotic symmetries and charges

The problem in a 0 nutshell - ² ²0 λ(n) o(r ). Electromagnetism = + +

New boundary conditions

Gravity Here ²0 is a constant and λ(n) is parity-odd, Conclusions and comments λ( n) λ(n). − = −

This seems to be an inﬁnite-dimensional global symmetry with angle-dependent gauge transformations...

16 / 33 Asymptotic symmetries and charges

The problem in a 0 nutshell - ² ²0 λ(n) o(r ). Electromagnetism = + +

New boundary conditions

Gravity Here ²0 is a constant and λ(n) is parity-odd, Conclusions and comments λ( n) λ(n). − = −

This seems to be an inﬁnite-dimensional global symmetry with angle-dependent gauge transformations...... but is this the case ?

16 / 33 The generator of gauge transformations is Z 3 ³ k ´ G[²] d x²(x) π ,k Q[² ] = − + ∞

(Gauss’ law + surface term) with

Z I k 2 2 r Q[² ] ²π d Sk d x²π ∞ = S2 = ∞ (in polar coordinates ; barred quantites = leading orders). Since πr is even, the charges G[²] reduce on-shell to

I G[²] d2x² πr. ≈ even

The odd part ²odd of the gauge parameter ² gives a zero contribution to the charges G[²].

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

17 / 33 Z I k 2 2 r Q[² ] ²π d Sk d x²π ∞ = S2 = ∞ (in polar coordinates ; barred quantites = leading orders). Since πr is even, the charges G[²] reduce on-shell to

I G[²] d2x² πr. ≈ even

The odd part ²odd of the gauge parameter ² gives a zero contribution to the charges G[²].

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism The generator of gauge transformations is and gravity in the asymptotically Z ﬂat case 3 ³ k ´ G[²] d x²(x) π ,k Q[² ] Marc Henneaux = − + ∞

Introduction

The problem in a (Gauss’ law + surface term) with nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

17 / 33 Since πr is even, the charges G[²] reduce on-shell to

I G[²] d2x² πr. ≈ even

The odd part ²odd of the gauge parameter ² gives a zero contribution to the charges G[²].

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism The generator of gauge transformations is and gravity in the asymptotically Z ﬂat case 3 ³ k ´ G[²] d x²(x) π ,k Q[² ] Marc Henneaux = − + ∞

Introduction

The problem in a (Gauss’ law + surface term) with nutshell - Electromagnetism Z I New boundary k 2 2 r conditions Q[² ] ²π d Sk d x²π ∞ = S2 = Gravity ∞

Conclusions and (in polar coordinates ; barred quantites = leading orders). comments

17 / 33 I G[²] d2x² πr. ≈ even

The odd part ²odd of the gauge parameter ² gives a zero contribution to the charges G[²].

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism The generator of gauge transformations is and gravity in the asymptotically Z ﬂat case 3 ³ k ´ G[²] d x²(x) π ,k Q[² ] Marc Henneaux = − + ∞

Introduction

The problem in a (Gauss’ law + surface term) with nutshell - Electromagnetism Z I New boundary k 2 2 r conditions Q[² ] ²π d Sk d x²π ∞ = S2 = Gravity ∞

Conclusions and (in polar coordinates ; barred quantites = leading orders). comments Since πr is even, the charges G[²] reduce on-shell to

17 / 33 The odd part ²odd of the gauge parameter ² gives a zero contribution to the charges G[²].

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism The generator of gauge transformations is and gravity in the asymptotically Z ﬂat case 3 ³ k ´ G[²] d x²(x) π ,k Q[² ] Marc Henneaux = − + ∞

Introduction

The problem in a (Gauss’ law + surface term) with nutshell - Electromagnetism Z I New boundary k 2 2 r conditions Q[² ] ²π d Sk d x²π ∞ = S2 = Gravity ∞

Conclusions and (in polar coordinates ; barred quantites = leading orders). comments Since πr is even, the charges G[²] reduce on-shell to

I G[²] d2x² πr. ≈ even

17 / 33 Asymptotic symmetries and charges

Asymptotic structure of electromagnetism The generator of gauge transformations is and gravity in the asymptotically Z ﬂat case 3 ³ k ´ G[²] d x²(x) π ,k Q[² ] Marc Henneaux = − + ∞

Introduction

The problem in a (Gauss’ law + surface term) with nutshell - Electromagnetism Z I New boundary k 2 2 r conditions Q[² ] ²π d Sk d x²π ∞ = S2 = Gravity ∞

Conclusions and (in polar coordinates ; barred quantites = leading orders). comments Since πr is even, the charges G[²] reduce on-shell to

I G[²] d2x² πr. ≈ even

The odd part ²odd of the gauge parameter ² gives a zero contribution to the charges G[²].

17 / 33 It follows that the gauge transformations with ²odd are proper gauge transformations that do not change the physical state of the system.

By contrast, the gauge transformations with ²even are improper gauge transformations that do change the physical state of the system.

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

18 / 33 By contrast, the gauge transformations with ²even are improper gauge transformations that do change the physical state of the system.

Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a It follows that the gauge transformations with ²odd are proper nutshell - Electromagnetism gauge transformations that do not change the physical state of

New boundary the system. conditions

Gravity

Conclusions and comments

18 / 33 Asymptotic symmetries and charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a It follows that the gauge transformations with ²odd are proper nutshell - Electromagnetism gauge transformations that do not change the physical state of

New boundary the system. conditions

Gravity By contrast, the gauge transformations with ²even are improper

Conclusions and gauge transformations that do change the physical state of the comments system.

18 / 33 Therefore, since ² ² λ (n), all charges are in fact zero except = 0 + odd the charge associated with ²0, Z k G[λodd(n)] Q[λodd(n)] λodd(n)π dSk 0 = = S = ∞ so that the quotient algebra is just u(1), with generator

Z k G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

19 / 33 Z k G[λodd(n)] Q[λodd(n)] λodd(n)π dSk 0 = = S = ∞ so that the quotient algebra is just u(1), with generator

Z k G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

19 / 33 so that the quotient algebra is just u(1), with generator

Z k G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Asymptotic structure of electromagnetism Therefore, since ² ²0 λodd(n), all charges are in fact zero except and gravity in the = + asymptotically the charge associated with ²0, ﬂat case Marc Henneaux Z Introduction k G[λodd(n)] Q[λodd(n)] λodd(n)π dSk 0 The problem in a = = S = nutshell - ∞ Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

19 / 33 with generator

Z k G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Gravity

Conclusions and comments

19 / 33 Z k G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Gravity

Conclusions and comments

19 / 33 The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Gravity Z Conclusions and k comments G[²0] Q[²0] ²0 π dSk. = = S ∞

19 / 33 Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Gravity Z Conclusions and k comments G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional.

19 / 33 Can one devise a “more liberal” strengthening of the original boundary conditions ?

Charges

Gravity Z Conclusions and k comments G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ?

19 / 33 Charges

Gravity Z Conclusions and k comments G[²0] Q[²0] ²0 π dSk. = = S ∞

The symmetry is one-dimensional. Is this due to a bad choice of boundary conditions ? Can one devise a “more liberal” strengthening of the original boundary conditions ?

19 / 33 The answer is afﬁrmative... and surprisingly simple ! The new boundary conditions that do not suffer from the above drawback are simply the above ones... up to a gradient !

That is, one demands that the leading order of Ak be even up to ∂kΦ for some Φ (of order 1)

a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1) “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

Idea

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

20 / 33 and surprisingly simple ! The new boundary conditions that do not suffer from the above drawback are simply the above ones... up to a gradient !

That is, one demands that the leading order of Ak be even up to ∂kΦ for some Φ (of order 1)

a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1) “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

Idea

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case The answer is afﬁrmative... Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

20 / 33 The new boundary conditions that do not suffer from the above drawback are simply the above ones... up to a gradient !

That is, one demands that the leading order of Ak be even up to ∂kΦ for some Φ (of order 1)

a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1) “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

Idea

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case The answer is afﬁrmative... Marc Henneaux and surprisingly simple ! Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

20 / 33 up to a gradient !

That is, one demands that the leading order of Ak be even up to ∂kΦ for some Φ (of order 1)

a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1) “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

Idea

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case The answer is afﬁrmative... Marc Henneaux and surprisingly simple ! Introduction The new boundary conditions that do not suffer from the above The problem in a nutshell - drawback are simply the above ones... Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

20 / 33 That is, one demands that the leading order of Ak be even up to ∂kΦ for some Φ (of order 1)

a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1) “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

Idea

Gravity

Conclusions and comments

20 / 33 a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1) “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

Idea

Gravity That is, one demands that the leading order of Ak be even up to Conclusions and ∂kΦ for some Φ (of order 1) comments

20 / 33 “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

Idea

Gravity That is, one demands that the leading order of Ak be even up to Conclusions and ∂kΦ for some Φ (of order 1) comments

a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1)

20 / 33 Idea

Gravity That is, one demands that the leading order of Ak be even up to Conclusions and ∂kΦ for some Φ (of order 1) comments

a(1)( n) a(1)(n) (∂ Φ)(1) (n), pk ( n) pk (n). k − = k + k (1) − = − (1) “Parity-twisted boundary conditions” with a twist given by a gauge transformation.

20 / 33 One might think that one could set Φ 0 by a gauge = transformation... but this is generically an improper gauge transformation with a non-vanishing charge. It does change the physical state of the system and it would be wrong to take the quotient by such transformations.

Twist is given by an improper gauge transformations

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

21 / 33 but this is generically an improper gauge transformation with a non-vanishing charge. It does change the physical state of the system and it would be wrong to take the quotient by such transformations.

Twist is given by an improper gauge transformations

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction One might think that one could set Φ 0 by a gauge = The problem in a transformation... nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

21 / 33 with a non-vanishing charge. It does change the physical state of the system and it would be wrong to take the quotient by such transformations.

Twist is given by an improper gauge transformations

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Gravity

Conclusions and comments

21 / 33 It does change the physical state of the system and it would be wrong to take the quotient by such transformations.

Twist is given by an improper gauge transformations

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Conclusions and comments

21 / 33 and it would be wrong to take the quotient by such transformations.

Twist is given by an improper gauge transformations

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction One might think that one could set Φ 0 by a gauge = The problem in a transformation... nutshell - Electromagnetism but this is generically an improper gauge transformation New boundary conditions with a non-vanishing charge. Gravity It does change the physical state of the system Conclusions and comments

21 / 33 Twist is given by an improper gauge transformations

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

21 / 33 Are these conditions acceptable ? Is the symplectic form ﬁnite ? Yes, provided one imposes that the constraint holds (off-shell) at inﬁnity one order faster than anticipated,

i 4 ∂ π O(r− ). i =

This obviously does not eliminate any solution (for which ∂ πi 0) and i = R d3xπi A˙ R d3xπi ∂ Φ˙ Finite R d3x∂ πi Φ˙ Finite i ∼ i + ∼ − i + is indeed ﬁnite.

Symplectic structure

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

22 / 33 Is the symplectic form ﬁnite ? Yes, provided one imposes that the constraint holds (off-shell) at inﬁnity one order faster than anticipated,

i 4 ∂ π O(r− ). i =

This obviously does not eliminate any solution (for which ∂ πi 0) and i = R d3xπi A˙ R d3xπi ∂ Φ˙ Finite R d3x∂ πi Φ˙ Finite i ∼ i + ∼ − i + is indeed ﬁnite.

Symplectic structure

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Are these conditions acceptable ? Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

22 / 33 Yes, provided one imposes that the constraint holds (off-shell) at inﬁnity one order faster than anticipated,

i 4 ∂ π O(r− ). i =

This obviously does not eliminate any solution (for which ∂ πi 0) and i = R d3xπi A˙ R d3xπi ∂ Φ˙ Finite R d3x∂ πi Φ˙ Finite i ∼ i + ∼ − i + is indeed ﬁnite.

Symplectic structure

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Are these conditions acceptable ? Marc Henneaux Is the symplectic form ﬁnite ? Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

22 / 33 i 4 ∂ π O(r− ). i =

This obviously does not eliminate any solution (for which ∂ πi 0) and i = R d3xπi A˙ R d3xπi ∂ Φ˙ Finite R d3x∂ πi Φ˙ Finite i ∼ i + ∼ − i + is indeed ﬁnite.

Symplectic structure

Asymptotic structure of electromagnetism and gravity in the asymptotically flat case Are these conditions acceptable ? Marc Henneaux Is the symplectic form finite ? Introduction Yes, provided one imposes that the constraint holds (off-shell) at The problem in a nutshell - infinity one order faster than anticipated, Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

22 / 33 This obviously does not eliminate any solution (for which ∂ πi 0) and i = R d3xπi A˙ R d3xπi ∂ Φ˙ Finite R d3x∂ πi Φ˙ Finite i ∼ i + ∼ − i + is indeed ﬁnite.

Symplectic structure

New boundary conditions i 4 ∂iπ O(r− ). Gravity = Conclusions and comments

22 / 33 R d3xπi A˙ R d3xπi ∂ Φ˙ Finite R d3x∂ πi Φ˙ Finite i ∼ i + ∼ − i + is indeed ﬁnite.

Symplectic structure

New boundary conditions i 4 ∂iπ O(r− ). Gravity = Conclusions and comments This obviously does not eliminate any solution (for which ∂ πi 0) and i =

22 / 33 is indeed ﬁnite.

Symplectic structure

New boundary conditions i 4 ∂iπ O(r− ). Gravity = Conclusions and comments This obviously does not eliminate any solution (for which ∂ πi 0) and i = R d3xπi A˙ R d3xπi ∂ Φ˙ Finite R d3x∂ πi Φ˙ Finite i ∼ i + ∼ − i +

22 / 33 Symplectic structure

22 / 33 First, we have only “half” of the asymptotic angle-dependent u(1)

It is clear that these boundary conditions are invariant under gauge transformations with both even and odd parts. We have thus succeeded in giving boundary conditions that have a non trivial inﬁnite-dimensional symmetry. However, they suffer from two difﬁculties.

These problems can be cured and one ﬁnds in the end complete agreement with the null inﬁnity analysis (see MH + C. Troessaert).

Difﬁculties with boosts

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

23 / 33 First, we have only “half” of the asymptotic angle-dependent u(1)

We have thus succeeded in giving boundary conditions that have a non trivial inﬁnite-dimensional symmetry. However, they suffer from two difﬁculties.

These problems can be cured and one ﬁnds in the end complete agreement with the null inﬁnity analysis (see MH + C. Troessaert).

Difﬁculties with boosts

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

23 / 33 First, we have only “half” of the asymptotic angle-dependent u(1)

However, they suffer from two difﬁculties.

These problems can be cured and one ﬁnds in the end complete agreement with the null inﬁnity analysis (see MH + C. Troessaert).

Difﬁculties with boosts

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case It is clear that these boundary conditions are invariant under Marc Henneaux gauge transformations with both even and odd parts. Introduction We have thus succeeded in giving boundary conditions that have The problem in a nutshell - a non trivial inﬁnite-dimensional symmetry. Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

23 / 33 First, we have only “half” of the asymptotic angle-dependent u(1)

transformations, namely those described by an even ²even. Where is the other half ? Second, as soon as one includes in the potential an odd component, the boosts cease to be canonical transformations. These problems can be cured and one ﬁnds in the end complete agreement with the null inﬁnity analysis (see MH + C. Troessaert).

Difﬁculties with boosts

New boundary However, they suffer from two difﬁculties. conditions

Gravity

Conclusions and comments

23 / 33 Second, as soon as one includes in the potential an odd component, the boosts cease to be canonical transformations. These problems can be cured and one ﬁnds in the end complete agreement with the null inﬁnity analysis (see MH + C. Troessaert).

Difﬁculties with boosts

23 / 33 These problems can be cured and one ﬁnds in the end complete agreement with the null inﬁnity analysis (see MH + C. Troessaert).

Difﬁculties with boosts

New boundary However, they suffer from two difﬁculties. conditions First, we have only “half” of the asymptotic angle-dependent u(1) Gravity transformations, namely those described by an even ²even. Where is Conclusions and comments the other half ? Second, as soon as one includes in the potential an odd component, the boosts cease to be canonical transformations.

23 / 33 Difﬁculties with boosts

23 / 33 Similar considerations apply to gravity. ij The canonical variables are gij and π . The usually assumed fall-off is (in cartesian coordinates)

1 ij 2 g δ O(r− ), π O(r− ). ij = ij + =

As in the case of electromagnetism, this generically leads to a logarithmic divergence in the symplectic structure (kinetic term)

Z d3xπij g˙ lnr. ij ∼

Gravity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

24 / 33 ij The canonical variables are gij and π . The usually assumed fall-off is (in cartesian coordinates)

1 ij 2 g δ O(r− ), π O(r− ). ij = ij + =

As in the case of electromagnetism, this generically leads to a logarithmic divergence in the symplectic structure (kinetic term)

Z d3xπij g˙ lnr. ij ∼

Gravity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Similar considerations apply to gravity. Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

24 / 33 The usually assumed fall-off is (in cartesian coordinates)

1 ij 2 g δ O(r− ), π O(r− ). ij = ij + =

As in the case of electromagnetism, this generically leads to a logarithmic divergence in the symplectic structure (kinetic term)

Z d3xπij g˙ lnr. ij ∼

Gravity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Similar considerations apply to gravity. Marc Henneaux ij The canonical variables are gij and π . Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

24 / 33 1 ij 2 g δ O(r− ), π O(r− ). ij = ij + =

As in the case of electromagnetism, this generically leads to a logarithmic divergence in the symplectic structure (kinetic term)

Z d3xπij g˙ lnr. ij ∼

Gravity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Similar considerations apply to gravity. Marc Henneaux ij The canonical variables are gij and π . Introduction The usually assumed fall-off is (in cartesian coordinates) The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

24 / 33 As in the case of electromagnetism, this generically leads to a logarithmic divergence in the symplectic structure (kinetic term)

Z d3xπij g˙ lnr. ij ∼

Gravity

Conclusions and comments

24 / 33 Z d3xπij g˙ lnr. ij ∼

Gravity

Conclusions and comments As in the case of electromagnetism, this generically leads to a logarithmic divergence in the symplectic structure (kinetic term)

24 / 33 Gravity

Gravity

Conclusions and comments As in the case of electromagnetism, this generically leads to a logarithmic divergence in the symplectic structure (kinetic term)

Z d3xπij g˙ lnr. ij ∼

24 / 33 One way to cure this would be to impose that the leading terms of the metric and its conjugate momentum have opposite parity properties under the antipodal map,

k hij(n ) 1 k k hij gij δij O( ), hij( n ) hij(n ) ≡ − = r + r2 − = and πij(nk) 1 πij O( ), πij( nk) πij(nk). = r2 + r3 − = − . But these strict parity conditions leave no room for the BMS group. The asymptotic symmetry reduces to the Poincaré group. Some form of parity conditions are needed, however, in order to have a ﬁnite symplectic structure.

Parity conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

25 / 33 k hij(n ) 1 k k hij gij δij O( ), hij( n ) hij(n ) ≡ − = r + r2 − = and πij(nk) 1 πij O( ), πij( nk) πij(nk). = r2 + r3 − = − . But these strict parity conditions leave no room for the BMS group. The asymptotic symmetry reduces to the Poincaré group. Some form of parity conditions are needed, however, in order to have a ﬁnite symplectic structure.

Parity conditions

Asymptotic structure of electromagnetism One way to cure this would be to impose that the leading terms of and gravity in the asymptotically the metric and its conjugate momentum have opposite parity ﬂat case Marc Henneaux properties under the antipodal map,

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

25 / 33 But these strict parity conditions leave no room for the BMS group. The asymptotic symmetry reduces to the Poincaré group. Some form of parity conditions are needed, however, in order to have a ﬁnite symplectic structure.

Parity conditions

25 / 33 The asymptotic symmetry reduces to the Poincaré group. Some form of parity conditions are needed, however, in order to have a ﬁnite symplectic structure.

Parity conditions

25 / 33 Some form of parity conditions are needed, however, in order to have a ﬁnite symplectic structure.

Parity conditions

Introduction k The problem in a hij(n ) 1 k k nutshell - hij gij δij O( ), hij( n ) hij(n ) Electromagnetism ≡ − = r + r2 − = New boundary conditions and ij k Gravity ij π (n ) 1 ij k ij k π 2 O( 3 ), π ( n ) π (n ). Conclusions and = r + r − = − comments . But these strict parity conditions leave no room for the BMS group. The asymptotic symmetry reduces to the Poincaré group.

25 / 33 Parity conditions

25 / 33 One must allow a “parity-twisted component” in the leading orders of the asymptotic metric and momenta in order to get a non trivial action for the BMS symmetry. This parity-twisted component takes the form of a gauge transformation (rewritten in Hamiltonian form). It is physically illegitimate to impose strict parity conditions as this requires improper gauge transformations, which one cannot use in gauge fixing. Do these relaxed parity conditions involving a twist lead to a consistent description (finite symplectic form, well-defined generators) ?

Parity-twisted boundary conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

26 / 33 in order to get a non trivial action for the BMS symmetry. This parity-twisted component takes the form of a gauge transformation (rewritten in Hamiltonian form). It is physically illegitimate to impose strict parity conditions as this requires improper gauge transformations, which one cannot use in gauge fixing. Do these relaxed parity conditions involving a twist lead to a consistent description (finite symplectic form, well-defined generators) ?

Parity-twisted boundary conditions

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

26 / 33 This parity-twisted component takes the form of a gauge transformation (rewritten in Hamiltonian form). It is physically illegitimate to impose strict parity conditions as this requires improper gauge transformations, which one cannot use in gauge fixing. Do these relaxed parity conditions involving a twist lead to a consistent description (finite symplectic form, well-defined generators) ?

Parity-twisted boundary conditions

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case One must allow a “parity-twisted component” in the leading Marc Henneaux orders of the asymptotic metric and momenta Introduction in order to get a non trivial action for the BMS symmetry. The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

26 / 33 It is physically illegitimate to impose strict parity conditions as this requires improper gauge transformations, which one cannot use in gauge fixing. Do these relaxed parity conditions involving a twist lead to a consistent description (finite symplectic form, well-defined generators) ?

Parity-twisted boundary conditions

New boundary transformation (rewritten in Hamiltonian form). conditions

Gravity

Conclusions and comments

26 / 33 which one cannot use in gauge fixing. Do these relaxed parity conditions involving a twist lead to a consistent description (finite symplectic form, well-defined generators) ?

Parity-twisted boundary conditions

New boundary transformation (rewritten in Hamiltonian form). conditions

Gravity It is physically illegitimate to impose strict parity conditions as

Conclusions and this requires improper gauge transformations, comments

26 / 33 Do these relaxed parity conditions involving a twist lead to a consistent description (ﬁnite symplectic form, well-deﬁned generators) ?

Parity-twisted boundary conditions

New boundary transformation (rewritten in Hamiltonian form). conditions

Gravity It is physically illegitimate to impose strict parity conditions as

Conclusions and this requires improper gauge transformations, comments which one cannot use in gauge ﬁxing.

26 / 33 (ﬁnite symplectic form, well-deﬁned generators) ?

Parity-twisted boundary conditions

New boundary transformation (rewritten in Hamiltonian form). conditions

Gravity It is physically illegitimate to impose strict parity conditions as

Conclusions and this requires improper gauge transformations, comments which one cannot use in gauge ﬁxing. Do these relaxed parity conditions involving a twist lead to a consistent description

26 / 33 Parity-twisted boundary conditions

New boundary transformation (rewritten in Hamiltonian form). conditions

Gravity It is physically illegitimate to impose strict parity conditions as

Conclusions and this requires improper gauge transformations, comments which one cannot use in gauge fixing. Do these relaxed parity conditions involving a twist lead to a consistent description (finite symplectic form, well-defined generators) ?

26 / 33 The procedure turns out to be very similar to what was found for electromagnetism. (1) The symplectic form is finite provided one requires that the constraints hold faster at infinity than dictated by the decrease of the canonical variables. (2) The (asymptotic) boosts also raise difficulties. These are solved by imposing h 0 to leading order. rA =

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux The answer is afﬁrmative and requires some work (even though Introduction the idea is elementary). The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

27 / 33 (1) The symplectic form is finite provided one requires that the constraints hold faster at infinity than dictated by the decrease of the canonical variables. (2) The (asymptotic) boosts also raise difficulties. These are solved by imposing h 0 to leading order. rA =

BMS group at spatial inﬁnity

New boundary electromagnetism. conditions

Gravity

Conclusions and comments

27 / 33 (2) The (asymptotic) boosts also raise difﬁculties. These are solved by imposing h 0 to leading order. rA =

BMS group at spatial inﬁnity

27 / 33 BMS group at spatial inﬁnity

New boundary electromagnetism. conditions (1) The symplectic form is finite provided one requires that the Gravity constraints hold faster at infinity than dictated by the decrease of Conclusions and comments the canonical variables. (2) The (asymptotic) boosts also raise difficulties. These are solved by imposing h 0 to leading order. rA =

27 / 33 Once this is done, one ﬁnds that the asymptotic symmetries are given by hypersurface deformations that behave asymptotically as

i ¡ 1¢ ξ b x T(n) O r− = i + + i i j ¡ 1¢ ξ b x W (n) O r− , b b , W (n) ∂ (rW(n)). = j + i + ij = − ji i = i

where T is even and W is odd. i i j The terms bix and b jx describe respectively boosts and spatial rotations. The zero mode of T and the ﬁrst spherical harmonic component of W describe translations.

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

28 / 33 i ¡ 1¢ ξ b x T(n) O r− = i + + i i j ¡ 1¢ ξ b x W (n) O r− , b b , W (n) ∂ (rW(n)). = j + i + ij = − ji i = i

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the Once this is done, one ﬁnds that the asymptotic symmetries are asymptotically ﬂat case given by hypersurface deformations that behave asymptotically Marc Henneaux as Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

28 / 33 where T is even and W is odd. i i j The terms bix and b jx describe respectively boosts and spatial rotations. The zero mode of T and the ﬁrst spherical harmonic component of W describe translations.

BMS group at spatial inﬁnity

The problem in a nutshell - Electromagnetism i ¡ 1¢ ξ bix T(n) O r− New boundary = + + conditions i i j ¡ 1¢ ξ b jx Wi(n) O r− , bij bji, Wi(n) ∂i(rW(n)). Gravity = + + = − =

Conclusions and comments

28 / 33 i i j The terms bix and b jx describe respectively boosts and spatial rotations. The zero mode of T and the ﬁrst spherical harmonic component of W describe translations.

BMS group at spatial inﬁnity

The problem in a nutshell - Electromagnetism i ¡ 1¢ ξ bix T(n) O r− New boundary = + + conditions i i j ¡ 1¢ ξ b jx Wi(n) O r− , bij bji, Wi(n) ∂i(rW(n)). Gravity = + + = − =

Conclusions and comments where T is even and W is odd.

28 / 33 The zero mode of T and the ﬁrst spherical harmonic component of W describe translations.

BMS group at spatial inﬁnity

The problem in a nutshell - Electromagnetism i ¡ 1¢ ξ bix T(n) O r− New boundary = + + conditions i i j ¡ 1¢ ξ b jx Wi(n) O r− , bij bji, Wi(n) ∂i(rW(n)). Gravity = + + = − =

Conclusions and comments where T is even and W is odd. i i j The terms bix and b jx describe respectively boosts and spatial rotations.

28 / 33 BMS group at spatial inﬁnity

The problem in a nutshell - Electromagnetism i ¡ 1¢ ξ bix T(n) O r− New boundary = + + conditions i i j ¡ 1¢ ξ b jx Wi(n) O r− , bij bji, Wi(n) ∂i(rW(n)). Gravity = + + = − =

Conclusions and comments where T is even and W is odd. i i j The terms bix and b jx describe respectively boosts and spatial rotations. The zero mode of T and the ﬁrst spherical harmonic component of W describe translations.

28 / 33 The higher spherical harmonics desribe general supertranslations. In fact, the even function T and the odd function W combine to form a single arbitrary function of the angles, as in the null inﬁnity description of the supertranslations. The symmetries are canonical transformations with generators R ¡ ¢ P [g ,πij] d3x ξH ξiH B [g ,πij] ξ ij = + i + ξ ij ij where Bξ[gij,π ] is a surface term, the explicit form of which can be found in MH and C. Troessaert. The algebra of the generators can be easily veriﬁed to be the BMS algebra.

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

29 / 33 In fact, the even function T and the odd function W combine to form a single arbitrary function of the angles, as in the null inﬁnity description of the supertranslations. The symmetries are canonical transformations with generators R ¡ ¢ P [g ,πij] d3x ξH ξiH B [g ,πij] ξ ij = + i + ξ ij ij where Bξ[gij,π ] is a surface term, the explicit form of which can be found in MH and C. Troessaert. The algebra of the generators can be easily veriﬁed to be the BMS algebra.

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case The higher spherical harmonics desribe general Marc Henneaux supertranslations. Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

29 / 33 The symmetries are canonical transformations with generators R ¡ ¢ P [g ,πij] d3x ξH ξiH B [g ,πij] ξ ij = + i + ξ ij ij where Bξ[gij,π ] is a surface term, the explicit form of which can be found in MH and C. Troessaert. The algebra of the generators can be easily veriﬁed to be the BMS algebra.

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case The higher spherical harmonics desribe general Marc Henneaux supertranslations. Introduction In fact, the even function T and the odd function W combine to The problem in a nutshell - form a single arbitrary function of the angles, as in the null Electromagnetism

New boundary inﬁnity description of the supertranslations. conditions

Gravity

Conclusions and comments

29 / 33 R ¡ ¢ P [g ,πij] d3x ξH ξiH B [g ,πij] ξ ij = + i + ξ ij ij where Bξ[gij,π ] is a surface term, the explicit form of which can be found in MH and C. Troessaert. The algebra of the generators can be easily veriﬁed to be the BMS algebra.

BMS group at spatial inﬁnity

New boundary inﬁnity description of the supertranslations. conditions The symmetries are canonical transformations with generators Gravity

Conclusions and comments

29 / 33 The algebra of the generators can be easily veriﬁed to be the BMS algebra.

BMS group at spatial inﬁnity

New boundary inﬁnity description of the supertranslations. conditions The symmetries are canonical transformations with generators Gravity ij R 3 ¡ i ¢ ij Conclusions and Pξ[gij,π ] d x ξH ξ Hi Bξ[gij,π ] comments = + + ij where Bξ[gij,π ] is a surface term, the explicit form of which can be found in MH and C. Troessaert.

29 / 33 BMS group at spatial inﬁnity

29 / 33 There is complete agreement with the null infinity results (when the initial data leads to fields that are sufficiently smooth at null infinity). In particular, the “matching conditions” of Strominger, which involve the antipodal map, are in fact a consequence of the boundary conditions at spatial infinity (and leading singularities are absent).

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

30 / 33 In particular, the “matching conditions” of Strominger, which involve the antipodal map, are in fact a consequence of the boundary conditions at spatial inﬁnity (and leading singularities are absent).

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Gravity

Conclusions and comments

30 / 33 (and leading singularities are absent).

BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction There is complete agreement with the null infinity results (when The problem in a nutshell - the initial data leads to fields that are sufficiently smooth at null Electromagnetism infinity). New boundary conditions In particular, the “matching conditions” of Strominger, which Gravity involve the antipodal map, are in fact a consequence of the Conclusions and comments boundary conditions at spatial infinity

30 / 33 BMS group at spatial inﬁnity

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

30 / 33 References

Asymptotic structure of electromagnetism and gravity in the asymptotically flat case M. Henneaux and C. Troessaert, “Asymptotic symmetries of Marc Henneaux electromagnetism at spatial infinity,” JHEP 1805 (2018) 137 Introduction [arXiv :1803.10194 [hep-th]] ; The problem in a nutshell - Electromagnetism M. Henneaux and C. Troessaert, “Hamiltonian structure and New boundary conditions asymptotic symmetries of the Einstein-Maxwell system at spatial Gravity infinity,” JHEP 1807 (2018) 171 [arXiv :1805.11288 [gr-qc]] ; Conclusions and comments M. Henneaux and C. Troessaert, “The asymptotic structure of gravity at spatial infinity in four spacetime dimensions,” arXiv :1904.04495 [hep-th], review article dedicated to Andrei Slavnov on the occasion of his 80th birthday.

31 / 33 This is clear whether one deals with null infinity or spatial infinity. In order to reveal the action of the angle-dependent u(1) symmetry (electromagnetism) or of the BMS group (gravity) at spatial infinity, one needs to include a twist in the parity conditions imposed in standard treatments. This twist has the form of an improper gauge transformation that changes the physical state and hence cannot be set to zero. Once it is included, one finds a perfect match with the results obtained at null infinity.

Conclusions and comments

Asymptotic structure of electromagnetism The asymptotic symmetries of electromagnetism and gravity in and gravity in the asymptotically the asymptotically flat context are infinite-dimensional and flat case Marc Henneaux contain the finite dimensional symmetries of the background as

Introduction a subgroup.

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

32 / 33 In order to reveal the action of the angle-dependent u(1) symmetry (electromagnetism) or of the BMS group (gravity) at spatial infinity, one needs to include a twist in the parity conditions imposed in standard treatments. This twist has the form of an improper gauge transformation that changes the physical state and hence cannot be set to zero. Once it is included, one finds a perfect match with the results obtained at null infinity.

Conclusions and comments

Introduction a subgroup. The problem in a This is clear whether one deals with null inﬁnity or spatial nutshell - Electromagnetism inﬁnity. New boundary conditions

Gravity

Conclusions and comments

32 / 33 one needs to include a twist in the parity conditions imposed in standard treatments. This twist has the form of an improper gauge transformation that changes the physical state and hence cannot be set to zero. Once it is included, one ﬁnds a perfect match with the results obtained at null inﬁnity.

Conclusions and comments

Introduction a subgroup. The problem in a This is clear whether one deals with null inﬁnity or spatial nutshell - Electromagnetism inﬁnity. New boundary conditions In order to reveal the action of the angle-dependent u(1)

Gravity symmetry (electromagnetism) or of the BMS group (gravity) at Conclusions and spatial inﬁnity, comments

32 / 33 This twist has the form of an improper gauge transformation that changes the physical state and hence cannot be set to zero. Once it is included, one ﬁnds a perfect match with the results obtained at null inﬁnity.

Conclusions and comments

Gravity symmetry (electromagnetism) or of the BMS group (gravity) at Conclusions and spatial inﬁnity, comments one needs to include a twist in the parity conditions imposed in standard treatments.

32 / 33 Once it is included, one ﬁnds a perfect match with the results obtained at null inﬁnity.

Conclusions and comments

Gravity symmetry (electromagnetism) or of the BMS group (gravity) at Conclusions and spatial inﬁnity, comments one needs to include a twist in the parity conditions imposed in standard treatments. This twist has the form of an improper gauge transformation that changes the physical state and hence cannot be set to zero.

32 / 33 Conclusions and comments

32 / 33 It is because this twist was set to zero in earlier treatments that there was no sign of the angle-dependent u(1) symmetry or of the BMS symmetry in previous treatments of spatial inﬁnity. [Note : other boundary conditions are possible.] Another context where an inﬁnite-dimensional asymptotic symmetry algebra emerges is 2 1 gravity with negative + cosmological constant... and this is a bonus ! More work is still needed (and is going on) to understand more completely the physical implications.

THANK YOU !

Conclusions and comments

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case Marc Henneaux

Introduction

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

33 / 33 [Note : other boundary conditions are possible.] Another context where an inﬁnite-dimensional asymptotic symmetry algebra emerges is 2 1 gravity with negative + cosmological constant... and this is a bonus ! More work is still needed (and is going on) to understand more completely the physical implications.

THANK YOU !

Conclusions and comments

Asymptotic structure of electromagnetism and gravity in the asymptotically ﬂat case It is because this twist was set to zero in earlier treatments that Marc Henneaux there was no sign of the angle-dependent u(1) symmetry or of the

Introduction BMS symmetry in previous treatments of spatial inﬁnity.

The problem in a nutshell - Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

33 / 33 Another context where an inﬁnite-dimensional asymptotic symmetry algebra emerges is 2 1 gravity with negative + cosmological constant... and this is a bonus ! More work is still needed (and is going on) to understand more completely the physical implications.

THANK YOU !

Conclusions and comments

Introduction BMS symmetry in previous treatments of spatial inﬁnity.

The problem in a nutshell - [Note : other boundary conditions are possible.] Electromagnetism

New boundary conditions

Gravity

Conclusions and comments

33 / 33 and this is a bonus ! More work is still needed (and is going on) to understand more completely the physical implications.

THANK YOU !

Conclusions and comments

Introduction BMS symmetry in previous treatments of spatial inﬁnity.

The problem in a nutshell - [Note : other boundary conditions are possible.] Electromagnetism Another context where an inﬁnite-dimensional asymptotic New boundary conditions symmetry algebra emerges is 2 1 gravity with negative + Gravity cosmological constant... Conclusions and comments

33 / 33 More work is still needed (and is going on) to understand more completely the physical implications.

THANK YOU !

Conclusions and comments

Introduction BMS symmetry in previous treatments of spatial inﬁnity.

33 / 33 THANK YOU !

Conclusions and comments

Introduction BMS symmetry in previous treatments of spatial inﬁnity.

33 / 33 Conclusions and comments

Introduction BMS symmetry in previous treatments of spatial inﬁnity.

THANK YOU !

33 / 33