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]
• System of n ≥ 2 gravitationally bound compact objects:
• Planets, neutron stars, black holes, etc.
• What is their effective gravitational interaction?
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• 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
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• 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
• 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)
• 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)
• 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
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• 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)
• 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.
• 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.
• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line
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• –M∫ds describes
structureless point
particle
• GR: Einstein-Hilbert• n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line
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• 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
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,
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3
2
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• 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.
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• 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
• Expand GR and point particle action in powers of graviton fields hμν …
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1
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,...}],,[{
exp
expexp
effeff
classical
gfeff
diagrams connectedFully
• … 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
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• Lowest order effective actionn
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• Schematically, conservative part of effective action is a series:
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20
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~~~
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vr
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eff
• Look at Re[Graviton propagator], non-relativistic limit:
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8
0],[],[0Re
;
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;
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8
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;
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• Look at Re[Graviton propagator], non-relativistic limit:
2/32/
0
2/12/
2/12/1
~
~
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vrh
vrh
vrh
d
d
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• 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
• 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
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
2 bodydiagrams
3 bodydiagrams
Einstein-Infeld-Hoffmand-spacetime dimensions
No gravitonvertices
Gravitonvertices
No gravitonvertices
Gravitonvertices
No gravitonvertices
Gravitonvertices
No gravitonvertices
Gravitonvertices
No gravitonvertices
Gravitonvertices
2121212121
2323
|||||||||| zxzxzyyxyx
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mnij
• 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.
• 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