Radial Gauge · reduced phase space of General Relativity Jędrzej Świeżewski in collaboration...
Transcript of Radial Gauge · reduced phase space of General Relativity Jędrzej Świeżewski in collaboration...
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Radial Gauge reduced phase space of General Relativity
Jędrzej Świeżewski
in collaboration with: Norbert Bodendorfer and Jerzy Lewandowski
Tux, 17.02.2015
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Radial Gauge
Motivation
is a certain gauge for canonical General Relativity
useful for discussing quantum spherical symmetry
Jędrzej Świeżewski, University of Warsaw
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Preparations
qab
spatial slice
metric
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Preparations
observer
qab
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Preparations
qab
observer adapted coordinates(r, ✓)
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Preparations
observer adapted coordinates(r, ✓)
observablesqab Qab(r, ✓)
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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
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Gauge fixing
we want to fix the gauge: qra = �ra
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Gauge fixing
we want to fix the gauge: qra = �ra
Dirac matrixn
qra, C[ ~N ]o
= !ra is invertible (in Diff )obs
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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
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Gauge fixed dynamics
�qra, + C[ ~NH ]
= 0H[N ]
we need to find such that preserves our gauge~NH H[N ] + C[ ~NH ]
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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
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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
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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
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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
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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
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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
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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
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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 )
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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
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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
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Quantisation
rqAB
Radial gauge gives a reduced phase space of GR
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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
◆
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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
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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
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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