first order gravity on the light front

First order gravity

on the light front

Sergei Alexandrov

Laboratoire Charles Coulomb Montpellier

work in progress with Simone Speziale

Light Front Light cone coordinates:


Main features:

• Triviality of the vacuum

– energy

– momentum


Main features:


– energy

– momentum

• Non-trivial physics of zero modes


Main features:


– energy

– momentum


• Importance of boundary conditions at


Main features:


– energy

– momentum

linear in velocities

• Presence of second class constraints


• Importance of boundary conditions at

Null surfaces are natural in gravity (Penrose,…)

Gravity on the light front

null vectors

• Sachs(1962) – constraint free formulation

• conformal metrics on

• intrinsic geometry of

• extrinsic curvature of

• Reisenberger – symplectic structure on

the constraint free data

null foliation



null vectors

• Torre(1986) – canonical formulation in the metric formalism

• Goldberg,Robinson,Soteriou(1991) – canonical formulation in the complex Ashtekar variables

• Inverno,Vickers(1991) – canonical formulation in the complex Ashtekar variables adapted to the double null foliation

Constraint algebra

becomes a Lie algebra







null foliation



null vectors

• Torre(1986) – canonical formulation in the metric formalism

• Goldberg,Robinson,Soteriou(1991) – canonical formulation in the complex Ashtekar variables

• Inverno,Vickers(1991) – canonical formulation in the complex Ashtekar variables adapted to the double null foliation

Constraint algebra

becomes a Lie algebra

• Veneziano et al. (recent) – light-cone averaging in cosmology







null foliation

Motivation

Can the light front formulation be useful in quantum gravity (black holes, spin foams…)?

Motivation

– was not analyzed yet • One needs the real first order formulation on the light front


Motivation

Exact path integral (still to be defined)



• Can one find constraint free data in the first order formulation? (preferably without using double null foliation)

Motivation



• The issue of zero modes in gravity was not studied yet



Motivation



• The issue of zero modes in gravity was not studied yet



• In the first order formalism the null condition can be controlled

by fields in the tangent space

Technical motivation

3+1 decomposition of the tetrad

Used in various approaches to quantum gravity (covariant LQG, spin foams…)




lapse

shift

spatial metric



determines the nature of the foliation

spacelike spacelike lightlike timelike


lapse

shift

spatial metric





light front formulation


What happens at ?

lapse

shift

spatial metric



Perform canonical analysis for the real first order

formulation of general relativity on a lightlike foliation



light front formulation


What happens at ?

lapse

shift

spatial metric

1. Canonical formulation of field theories on the light front

2. A review of the canonical structure of first order gravity

3. Canonical analysis of first order gravity on the light front

4. The issue of zero modes

Plan of the talk

Massless scalar field in 2d

Solution:


Solution:

Primary constraint Hamiltonian

Light front formulation


Solution:


Stability condition:

is of second class



Solution:



is of second class

zero mode

first class

Identification:



Solution:



is of second class

zero mode

first class

Identification:

• the phase space is one-dimensional • the lost dimension is encoded in the Lagrange multiplier

Conclusions:


Massive theories

One generates the same constraint but different Hamiltonian

Massive theories


inhomogeneous equation


Massive theories




The existence of the zero mode contradicts to the natural boundary

conditions

Massive theories


In massive theories the light front constraints do not have first class zero modes




conditions

Massive theories


In massive theories the light front constraints do not have first class zero modes




conditions

In higher dimensions:

behave like massive 2d case

Dimensionality of the phase space

On the light front

dim. phase space = num. of deg. of freedom


On the light front


Fourier decompositions


On the light front


Second class constraint

Symplectic structure is non-degenerate

Dirac bracket

Fourier decompositions

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables:




Linear simplicity constraints





Hamiltonian is a linear combination of constraints






3 6






3 6

secondary constraints






3 6


1st class

2d class






3 6


dim. of phase space = 2×18 – 2(3+3+1)-(3+9+6)=4

1st class

2d class

Cartan equations

Cartan equations

Cartan equations

Cartan equations

do not contain time

derivatives and Lagrange multipliers

12 constraints

Cartan equations

Cartan equations

do not contain time


12 constraints

for fixed do not contain

time derivatives

Cartan equations

Cartan equations

do not contain time


12 constraints


time derivatives

fix 3 components of

( – 2d class)

Cartan equations

Cartan equations

do not contain time


12 constraints


time derivatives

fix 3 components of

( – 2d class)

fix 2 components of and the lapse

Cartan equations

Cartan equations

do not contain time


12 constraints


time derivatives

We expect that the Hamiltonian constraint becomes second class

fix 3 components of

( – 2d class)

fix 2 components of and the lapse

Hamiltonian analysis on the light front

Light front condition








where





2 7


where





2 7


where


equation fixing the lapse +

Tertiary constraints The crucial observation:

has 2 null eigenvectors

induced metric

on the foliation

and



induced metric

on the foliation

and

There are two

tertiary constraints

Projector on the

null eigenvectors



induced metric

on the foliation

and

Stabilization procedure stops due to

There are two

tertiary constraints

Projector on the

null eigenvectors

Summary

List of constraints:

Gauss

preserving

Gauss

rotating

spatial

diffeos Hamiltonian

primary

simplicity secondary simplicity

tertiary

Summary


Gauss

preserving

4

Gauss

rotating

spatial

diffeos Hamiltonian

primary


tertiary

First class Second class

3 2 1 9 6 2

2 1 4 2

2 4

Summary


Gauss

preserving

4

Gauss

rotating

spatial

diffeos Hamiltonian

primary


tertiary


3 2 1 2

Lie algebra

2 1 4 2

2 4

Summary


Gauss

preserving

4

Gauss

rotating

spatial

diffeos Hamiltonian

primary


tertiary


3 2 1 2

dim. of phase space = 2×18 – 2(4+3)-(2+1+9+6+2)=2

as it should be on the light front

Lie algebra

Zero modes

The zero modes of constraints are

determined by equations fixing Lagrange multipliers

Zero modes



Potential first class constraints:

Zero modes



Potential first class constraints:

homogeneous equations

Conjecture

A zero mode can exist only if the corresponding Lagrange multiplier satisfies

a homogeneous equation

One may expect only two zero modes

In our case there are two homogenous equations for and

Conjecture





Gravity behaves like

2d massless theory

exist

Conjecture





Gravity behaves like

2d massless theory

exist

Initial data on one null hypersurface fix

the solution

do not exist

Open problems

• Do the zero modes in gravity really exist? If yes, what is the geometric meaning of the zero modes? • What are the appropriate boundary conditions along ? • How do singularities appear in this formalism?

• Can one solve (at least formally) all constraints?

• What is the right symplectic structure (Dirac bracket)?

• Can this formulation be applied to quantum gravity problems?

first order gravity on the light front

Documents