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

• Non-trivial physics of zero modes • Importance of boundary conditions at • Presence of second class constraints

linear in velocities Gravity on the light front

• Sachs(1962) – constraint free formulation null vectors

• conformal metrics on • intrinsic geometry of • extrinsic curvature of

• Reisenberger – symplectic structure on the constraint free data null foliation

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

• Goldberg,Robinson,Soteriou(1991) – canonical formulation in Inverno,Vickers(1991) the complex Ashtekar variables constraint algebra becomes a Lie algebra

We want to analyze the real first order formulation on the light front • constraint free data exact path integral? • issue of zero modes

• Speziale,Zhang(2013) – null twisted geometries Technical motivation In tetrad formalism the null condition can be imposed in the tangent space

3+1 decomposition of the tetrad shift lapse

spatial metric Used in various approaches to quantum gravity determines the nature of the foliation (covariant LQG, spin foams…) spacelike lightlike timelike

light front formulation Perform canonical analysis for the real first order formulation of general relativity on a lightlike foliation Massless scalar field in 2d

Solution:

Light front formulation Primary constraint Hamiltonian

is of second class

Stability condition: first Identification: class zero mode

Conclusions: • the phase space is one-dimensional • the lost dimension is encoded in the Lagrange multiplier Massive theories

One generates the same constraint but different Hamiltonian

Stability inhomogeneous condition: equation

The existence of the zero mode contradicts to the natural boundary conditions In massive theories the light front constraints do not have first class zero modes In higher behave like dimensions: massive 2d case

dim. of num. of deg. = On the light front : phase space of freedom First order gravity (spacelike case)

Canonical variables:

Fix – normal to the foliation Linear simplicity constraints

Hamiltonian is a linear combination of constraints

3 1st class 6

2d class

secondary constraints dim. of phase space = 2×18 – 2(3+3+1)-(3+9+6)=4 Hamiltonian analysis on the light front

Canonical variables:

Light front condition Linear simplicity constraints Hamiltonian is a linear combination of constraints

2 7 where

The Hamiltonian constraint secondary constraints becomes second class + equation fixing the lapse Tertiary constraints The crucial observation:

and induced metric on the foliation has 2 null eigenvectors Projector on the null eigenvectors

There are two tertiary constraints

Stabilization procedure stops due to Summary

List of constraints: Gauss Gauss spatial primary secondary Hamiltonian tertiary preserving rotating diffeos simplicity simplicity

First class Second class

4 3 2 1 92 62 2 1 4 Lie algebra 4 2

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

as it should be on the light front Zero modes

The zero modes of constraints are determined by equations fixing Lagrange multipliers Potential first class constraints:

not expected to generate zero modes homogeneous equations Zero modes

1. time diffemorphisms independent of

Gauge transformation Light front condition

The zero mode corresponds to the residual gauge freedom 2. of the null foliation would generates shifts of the spin-connection not changing the tetrad cannot be a gauge generator

There are no (local) zero modes providing additional data needed to fix a solution Some future directions

• Better understanding of the zero modes?

• What are the appropriate boundary conditions along ?

• 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?