The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen...
Transcript of The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen...
![Page 1: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/1.jpg)
Yi-Zen Chu
BCCS 2008 @ Case Western Reserve University
9 December 2008
The n-body problem in
General Relativityfrom
perturbative field theory
arXiv: 0812.0012 [gr-qc]
![Page 2: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/2.jpg)
• System of n ≥ 2 gravitationally bound compact objects:
• Planets, neutron stars, black holes, etc.
• What is their effective gravitational interaction?
?...||2
1
2
1
1
2n
ba ba
baNn
a
aaxx
MMGvML
eff
![Page 3: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/3.jpg)
?...||2
1
2
1
1
2n
ba ba
baNn
a
aaxx
MMGvML
eff
• Compact objects ≈ point particles• n-body problem: Dynamics for the coordinates of the point particles• Assume non-relativistic motion
• GR corrections to Newtonian gravity: an expansion in (v/c)2
Nomenclature: O[(v/c)2Q] = Q PN
![Page 4: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/4.jpg)
?...||2
1
2
1
1
2n
ba ba
baNn
a
aaxx
MMGvML
eff
• Note that General Relativity is non-linear.
• Superposition does not hold• 2 body lagrangian is not sufficient to obtain n-body lagrangian
Nomenclature: O[(v/c)2Q] = Q PN
![Page 5: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/5.jpg)
• n-body problem known up to O[(v/c)2]: • Einstein-Infeld-Hoffman lagrangian• Eqns of motion used regularly to calculate solar system dynamics, etc.
• Precession of Mercury begins at this order
• O[(v/c)4] only known partially.• Damour, Schafer (1985, 1987)• Compute using field theory? (Goldberger, Rothstein, 2004)
![Page 6: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/6.jpg)
• Solar system probes of GR beginning to go beyond O[(v/c)2]:
• New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser ranging missions?• LATOR, GTDM, BEACON, ASTROD, etc.• See e.g. Turyshev (2008)
![Page 7: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/7.jpg)
• n-body L gives not only dynamics but also geometry.
• Add a test particle, M->0: it moves along geodesic in the spacetime metric generated by the rest of the n masses• Metric can be read off its action
...2
1
2
1
2
11
000
2
dt
dz
dt
dzg
dt
dzggM
dt
zdM
dt
dz
dt
dzgdtM
ji
ij
i
i
gg
![Page 8: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/8.jpg)
• Gravitational wave observatories need the 2 body L beyond O[(v/c)7]:
• LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[104] orbital cycles.• GW detection: Raw data integrated against theoretical templates to search for correlations.• Construction of accurate templates requires 2 body dynamics.• Currently, 2 body L known up to O[(v/c)7]• See e.g. Blanchet (2006)
![Page 9: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/9.jpg)
• Starting at 3 PN, O[(v/c)6], GR computations of 2 body L start to give divergences that were eventually handled by dimensional regularization.• Perturbation theory beyond O[(v/c)7] requires systematic, efficient methods.
![Page 10: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/10.jpg)
• Perturbation theory beyond O[(v/c)7] requires systematic, efficient methods.• QFT gives systematic framework for
• Renormalization & regularization• Computational algorithm – Feynman diagrams with appropriate dimensional analysis.
![Page 11: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/11.jpg)
• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line
a
a
a
a
a
aaa
a
n
a
app
dd
plGR
ppGR
Npld
pl
ds
dxu
dt
dx
dt
dxgdtds
uuuuRRc
RRcRRcdsMS
RgxdMS
SSS
GMM
hg
,
...
1
2
32,
3
2
21
1
2
2/1
1)2/(
![Page 12: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/12.jpg)
a
a
a
a
a
aaa
a
n
a
app
dd
plGR
ppGR
Npld
pl
ds
dxu
dt
dx
dt
dxgdtds
uuuuRRc
RRcRRcdsMS
RgxdMS
SSS
GMM
hg
,
...
1
2
32,
3
2
21
1
2
2/1
1)2/(
• –M∫ds describes
structureless point
particle
• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line
![Page 13: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/13.jpg)
a
a
a
a
a
aaa
a
n
a
app
dd
plGR
ppGR
Npld
pl
ds
dxu
dt
dx
dt
dxgdtds
uuuuRRc
RRcRRcdsMS
RgxdMS
SSS
GMM
hg
,
...
1
2
32,
3
2
21
1
2
2/1
1)2/(
• Non-minimal terms
encode information on the
non-trivial structure of
individual objects.
• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line
![Page 14: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/14.jpg)
a
a
a
a
a
aaa
a
n
a
app
dd
plGR
ppGR
Npld
pl
ds
dxu
dt
dx
dt
dxgdtds
uuuuRRc
RRcRRcdsMS
RgxdMS
SSS
GMM
hg
,
...
1
2
32,
3
2
21
1
2
2/1
1)2/(
• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line• Coefficients {cx} have to
be tuned to match physical
observables from full
macroscopic description of
objects.
![Page 15: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/15.jpg)
a
a
a
a
a
aaa
a
n
a
app
dd
plGR
ppGR
Npld
pl
ds
dxu
dt
dx
dt
dxgdtds
uuuuRRc
RRcRRcdsMS
RgxdMS
SSS
GMM
hg
,
...
1
2
32,
3
2
21
1
2
2/1
1)2/(
• For compact objects,
up to O[(v/c)8], only
minimal terms -Ma∫dsa
needed
• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line
![Page 16: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/16.jpg)
• Expand GR and point particle action in powers of graviton fields hμν …
j
a
i
aij
i
ai
d
plaa
n
a
a
dd
pl
d
pl
aaa
d
xxhxhhMxdtM
RgxdMS
M
hg
xxxLL
SSiDhLdti
000
2/12
1
2
1)2/(
1
0
21
2
,...}],,[{
exp
expexp
effeff
classical
gfeff
diagrams connectedFully
![Page 17: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/17.jpg)
• … but some dimensional analysis before computation makes perturbation theory much more systematic• The scales in the n-body problem
• r – typical separation between n bodies.• v – typical speed of point particles• r/v – typical time scale of n-body system
1
00
0
~
~][~
vrxd
r
vxx
dx
d
dd
![Page 18: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/18.jpg)
• Lowest order effective actionn
bad
ba
ba
d
Nn
a
aaxx
MMGxMdtS
3
1)2/(
1
2
||2
1
0
• Schematically, conservative part of effective action is a series:
2
3
1)2/(
321)2/(2
0
0
4
4
2
20
~
~~~
~...,~
vr
MG
MvrrMGdtMvdtS
SSvSvSSS
d
d
N
dd
N
n
eff
![Page 19: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/19.jpg)
• Look at Re[Graviton propagator], non-relativistic limit:
2
2
][||
]2/)3[(
8
0],[],[0Re
;
00
32/)1(
;
00
dP
xxxx
diP
xxhxxhT
dd
cv
![Page 20: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/20.jpg)
][||
]2/)3[(
8
0],[],[0Re
00
32/)1(
;
00
xxxx
diP
xxhxxhT
dd
cv
• Look at Re[Graviton propagator], non-relativistic limit:
2/32/
0
2/12/
2/12/1
~
~
~
vrh
vrh
vrh
d
d
i
d
![Page 21: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/21.jpg)
• n-graviton piece of -Ma∫dsa with χpowers of velocities scales as• n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as
222/1
0
nn vS
422/1
0
nn vS
2220
)()()(
wvw n
Nnn
vS0
• With n(w) world line terms -Ma∫dsa,• With n(v) Einstein-Hilbert action terms,• With N total gravitons,• Every Feynman diagram scales as
![Page 22: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/22.jpg)
• n-graviton piece of -Ma∫dsa with χpowers of velocities scales as• n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as
222/1
0
nn vS
422/1
0
nn vS
2220
)()()(
wvw n
Nnn
vS=1 Q PN
• With n(w) world line terms -Ma∫dsa,• With n(v) Einstein-Hilbert action terms,• With N total gravitons,• Every Feynman diagram scales as
![Page 23: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/23.jpg)
2220
)()()(
wvw n
Nnn
vS
=1 Q PN
• Limited form of superposition holds• At Q PN, i.e. O[(v/c)2Q], max number of distinct terms from -Ma∫dsa is Q+2• 1 PN, O[(v/c)2]: 3 body problem• 2 PN, O[(v/c)4]: 4 body problem• …
0
• Every Feynman diagram scales as
![Page 24: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/24.jpg)
![Page 25: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/25.jpg)
2 bodydiagrams
3 bodydiagrams
Einstein-Infeld-Hoffmand-spacetime dimensions
![Page 26: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/26.jpg)
No gravitonvertices
Gravitonvertices
![Page 27: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/27.jpg)
No gravitonvertices
Gravitonvertices
![Page 28: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/28.jpg)
No gravitonvertices
Gravitonvertices
![Page 29: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/29.jpg)
No gravitonvertices
Gravitonvertices
![Page 30: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/30.jpg)
No gravitonvertices
Gravitonvertices
2121212121
2323
|||||||||| zxzxzyyxyx
zdyd
encjbmai
mnij
![Page 31: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/31.jpg)
• Perturbation theory beyond O[(v/c)7] for 2 body L requires systematic, efficient methods.• QFT gives systematic framework for
• Renormalization & regularization• Computational algorithm – Feynman diagrams with appropriate dimensional analysis.• But the computation is still hard and long – need for new technology.
![Page 32: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/32.jpg)
![Page 33: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/33.jpg)
![Page 34: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/34.jpg)
![Page 35: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/35.jpg)
![Page 36: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/36.jpg)
![Page 37: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/37.jpg)
![Page 38: The n-body problem in General Relativity from perturbative field theory · 2008-12-09 · Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem](https://reader034.fdocuments.us/reader034/viewer/2022042116/5e94119f0e1c9f345c5bf05b/html5/thumbnails/38.jpg)
• Perturbation theory beyond O[(v/c)7] for 2 body L requires systematic, efficient methods.• QFT gives systematic framework for
• Renormalization & regularization• Computational algorithm – Feynman diagrams with appropriate dimensional analysis.• But the computation is still hard and long – need for new technology.• http://www.stargazing.net/yizen/PN.html