Radial Gauge reduced phase space of General Relativity

Jędrzej Świeżewski

in collaboration with: Norbert Bodendorfer and Jerzy Lewandowski

Tux, 17.02.2015

Radial Gauge

Motivation

is a certain gauge for canonical General Relativity

useful for discussing quantum spherical symmetry

Jędrzej Świeżewski, University of Warsaw

Preparations

qab

spatial slice

metric

Jędrzej Świeżewski, University of Warsaw

Preparations

observer

qab

Jędrzej Świeżewski, University of Warsaw

Preparations

qab

observer adapted coordinates(r, ✓)

Jędrzej Świeżewski, University of Warsaw

Preparations

observer adapted coordinates(r, ✓)

observablesqab Qab(r, ✓)

Jędrzej Świeżewski, University of Warsaw

Preparations

observer adapted coordinates(r, ✓)

observablesqab Qab(r, ✓)

{QAB(r, ✓), PCD(r0, ✓0)} = ���2

41 0 000

3

5QAB

in adapted coordinatesmetric has the form

appeared in: Duch, Kamiński, Lewandowski, JŚ JHEP05(2014)077

canonical subalgebra

Jędrzej Świeżewski, University of Warsaw

Gauge fixing

we want to fix the gauge: qra = �ra

Jędrzej Świeżewski, University of Warsaw

Gauge fixing

we want to fix the gauge: qra = �ra

Dirac matrixn

qra, C[ ~N ]o

= !ra is invertible (in Diff )obs

Jędrzej Świeżewski, University of Warsaw

Gauge fixing

we want to fix the gauge: qra = �ra

Dirac matrixn

qra, C[ ~N ]o

= !ra is invertible (in Diff )obs

~N(r, ✓) =

1

2!KJ(0)h

JLrnK

�@L +

1

2

Z r

0dr0 !rr(r

0, ✓)

�@r+

+

"Z r

0dr0 qBA(r0, ✓)

!rA(r

0, ✓)� 1

2@A

Z r0

0dr00 !rr(r

00, ✓)

!!#@B

Jędrzej Świeżewski, University of Warsaw

Gauge fixed dynamics

�qra, + C[ ~NH ]

= 0H[N ]

we need to find such that preserves our gauge~NH H[N ] + C[ ~NH ]

Jędrzej Świeżewski, University of Warsaw

Gauge fixed dynamics

we need to find such that preserves our gauge~NH H[N ] + C[ ~NH ]

H[N ]�qra, C[ ~NH ]

= �

�qra,

Jędrzej Świeżewski, University of Warsaw

Gauge fixed dynamics

we need to find such that preserves our gauge~NH H[N ] + C[ ~NH ]

H[N ]�qra, C[ ~NH ]

= �

�qra,

H[N ]|gauge-fix

=

ZN

✓1pdet q

G�p

det q(3)R

◆

where

G =1

2(eprr)2 + 2qABeprAeprB � qABp

ABeprr + (qACqBD � 1

2qABqCD)pABpCD

(3)R = (2)R� qABqAB,rr �3

4qAB

,rqAB,r �1

4(qABqAB,r)

2

Jędrzej Świeżewski, University of Warsaw

Gauge fixed dynamics

we need to find such that preserves our gauge~NH H[N ] + C[ ~NH ]

H[N ]�qra, C[ ~NH ]

= �

�qra,

H[N ]|gauge-fix

=

ZN

✓1pdet q

G�p

det q(3)R

◆

where

eprA(r, ✓) =Z 1

rdr0DBp

BA(r

0, ✓)

eprr(r, ✓) = �1

2

Z 1

rdr0

�pABqAB,r

�(r0, ✓)+

+

Z 1

rdr0DA

✓qAB(r0, ✓)

Z 1

r0dr00

�DCp

CB(r

00, ✓)�◆

G =1

2(eprr)2 + 2qABeprAeprB � qABp

ABeprr + (qACqBD � 1

2qABqCD)pABpCD

(3)R = (2)R� qABqAB,rr �3

4qAB

,rqAB,r �1

4(qABqAB,r)

2

Jędrzej Świeżewski, University of Warsaw

Example 1: Spherical symmetry

take metric in the form

2

4⇤2(r) 0 000

3

5R2(r)⌘AB

H[N ] =

Z 1

0drN

✓⇤P 2

⇤

2R2� PRP⇤

R+

RR00

⇤� RR0⇤0

⇤2+

R02

2⇤� ⇤

2

◆C[ ~N ] =

Z 1

0drNr (PRR

0 � ⇤P 0⇤)

the constraints are

Jędrzej Świeżewski, University of Warsaw

Example 1: Spherical symmetry2

4⇤2(r) 0 000

3

5R2(r)⌘AB H[N ] =

Z 1

0drN

✓⇤P 2

⇤

2R2� PRP⇤

R+

RR00

⇤� RR0⇤0

⇤2+

R02

2⇤� ⇤

2

◆C[ ~N ] =

Z 1

0drNr (PRR

0 � ⇤P 0⇤)

impose gauge ⇤ = 1

H[N ]|gauge-fix

=

Z 1

0

drN

1

2R2

✓�Z 1

rdr0 (PRR

0) (r0)

◆2

+PR

R

Z 1

rdr0 (PRR

0) (r0) +RR00 +R02

2� 1

2

!the Hamiltonian preserving the gauge is

Jędrzej Świeżewski, University of Warsaw

Example 1: Spherical symmetry2

4⇤2(r) 0 000

3

5R2(r)⌘AB H[N ] =

Z 1

0drN

✓⇤P 2

⇤

2R2� PRP⇤

R+

RR00

⇤� RR0⇤0

⇤2+

R02

2⇤� ⇤

2

◆C[ ~N ] =

Z 1

0drNr (PRR

0 � ⇤P 0⇤)

impose gauge ⇤ = 1

H[N ]|gauge-fix

=

Z 1

0

drN

1

2R2

✓�Z 1

rdr0 (PRR

0) (r0)

◆2

+PR

R

Z 1

rdr0 (PRR

0) (r0) +RR00 +R02

2� 1

2

!the Hamiltonian preserving the gauge is

it gives the following equations of motion

1

NR(r) = �F (r)

R(r)+R0(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

1

NPR(r) = �R00(r) +

P 2R(r)

R(r)� 2

PR(r)F (r)

R2(r)+

F 2(r)

R3(r)+ P 0

R(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

Jędrzej Świeżewski, University of Warsaw

Example 1: Spherical symmetry2

4⇤2(r) 0 000

3

5R2(r)⌘AB H[N ] =

Z 1

0drN

✓⇤P 2

⇤

2R2� PRP⇤

R+

RR00

⇤� RR0⇤0

⇤2+

R02

2⇤� ⇤

2

◆C[ ~N ] =

Z 1

0drNr (PRR

0 � ⇤P 0⇤)

impose gauge ⇤ = 1

H[N ]|gauge-fix

=

Z 1

0

drN

1

2R2

✓�Z 1

rdr0 (PRR

0) (r0)

◆2

+PR

R

Z 1

rdr0 (PRR

0) (r0) +RR00 +R02

2� 1

2

!the Hamiltonian preserving the gauge is

it gives the following equations of motion

1

NR(r) = �F (r)

R(r)+R0(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

1

NPR(r) = �R00(r) +

P 2R(r)

R(r)� 2

PR(r)F (r)

R2(r)+

F 2(r)

R3(r)+ P 0

R(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

Jędrzej Świeżewski, University of Warsaw

Example 1a: Minkowski

1

NR(r) = �F (r)

R(r)+R0(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

1

NPR(r) = �R00(r) +

P 2R(r)

R(r)� 2

PR(r)F (r)

R2(r)+

F 2(r)

R3(r)+ P 0

R(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

setting constantly in time, we obtainPR(r) = 0

R(r) = 0

0 = R00(r)

R(r) = r

Jędrzej Świeżewski, University of Warsaw

( Nr = 0 )

Example 1b: Schwarzschild

ds2 = �✓1� 2M

r

◆dt2 + 2

r2M

rdtdr + dr2 + r2d⌦2

⇤2N R2Nr

Schwarzschild metric in free-fall-coordinates is

1

NR(r) = �F (r)

R(r)+R0(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

1

NPR(r) = �R00(r) +

P 2R(r)

R(r)� 2

PR(r)F (r)

R2(r)+

F 2(r)

R3(r)+ P 0

R(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

Jędrzej Świeżewski, University of Warsaw

Example 1b: Schwarzschild

ds2 = �✓1� 2M

r

◆dt2 + 2

r2M

rdtdr + dr2 + r2d⌦2

⇤2N R2Nr

Schwarzschild metric in free-fall-coordinates is

1

NR(r) = �F (r)

R(r)+R0(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

1

NPR(r) = �R00(r) +

P 2R(r)

R(r)� 2

PR(r)F (r)

R2(r)+

F 2(r)

R3(r)+ P 0

R(r)

Z r

0dr0

✓PR(r0)

R(r0)� F (r0)

R2(r0)

◆

It turns out

NrH = Nr H[N ] + C[ ~NH ] = 0

R = 0

PR = 0

) P⇤ & PR

Jędrzej Świeżewski, University of Warsaw

Quantisation

rqAB

Radial gauge gives a reduced phase space of GR

Jędrzej Świeżewski, University of Warsaw

Radial gauge gives a reduced phase space of GR

What can we do with it?

quantum picturer

Jędrzej Świeżewski, University of Warsaw

EAi , Ai

A

Quantisation

E�(S) =

Z

SE

Ai �

i✏ABdrdx

B

he(A) = P exp

✓Z

eAAi⌧

idx

A

◆

Radial gauge gives a reduced phase space of GR

What can we do with it?

r

Jędrzej Świeżewski, University of Warsaw

E�(S) =

Z

SE

Ai �

i✏ABdrdx

B

he(A) = P exp

✓Z

eAAi⌧

idx

A

◆

EAi , Ai

A

can be obtained imposing prA = 0

eprA(r, ✓) =Z 1

rdr0DBp

BA(r

0, ✓)

quantum picture

Quantisation

or by averaging w.r.t. rigid rotations around the centre

Spherical Symmetry

Thank you for your attention

Bodendorfer, Lewandowski, JŚ General Relativty in the Radial Gauge I & II (to appear soon)

Bodendorfer, Lewandowski, JŚ A quantum reduction to spherical symmetry in loop quantum gravity arXiv:1410.5609

Extra slide

condition on canonical data in spherical symmetryZ 1

0drPRR

0 = 0

conditions on canonical data in general caseZ 1

0drDBp

BA = 0

�1

2

Z 1

0drpABqAB,r +

Z 1

0drDA

✓qAB

Z 1

rdr0DCp

CB

◆= 0

limr!0

�pABqAB

�,rr

� 12

�pABqAB,r

�,r�p

det q�,rr

= 0

Jędrzej Świeżewski, University of Warsaw