Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem in General...

Yi-Zen Chu

BCCS 2008 @ Case Western Reserve University

9 December 2008

The n-body problem The n-body problem in in

General RelativityGeneral Relativityfrom from

perturbative field theoryperturbative field theory

arXiv: 0812.0012 [gr-qc]arXiv: 0812.0012 [gr-qc]

• System of n ≥ 2 System of n ≥ 2 gravitationally bound gravitationally bound compact objects:compact objects:

• Planets, neutron stars, Planets, neutron stars, black holes, etc.black holes, etc.

• What is their effective What is their effective gravitational interaction?gravitational interaction?

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• Compact objects ≈ point Compact objects ≈ point particlesparticles• n-body problem: Dynamics for n-body problem: Dynamics for the coordinates of the point the coordinates of the point particlesparticles• Assume non-relativistic Assume non-relativistic motionmotion

• GR corrections to GR corrections to Newtonian gravity: an Newtonian gravity: an expansion in (v/c)expansion in (v/c)22Nomenclature: O[(v/c)Nomenclature: O[(v/c)2Q2Q] = Q PN] = Q PN

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• Note that General Relativity Note that General Relativity is non-linear.is non-linear.

• Superposition does not Superposition does not holdhold• 2 body lagrangian is not 2 body lagrangian is not sufficient to obtain n-body sufficient to obtain n-body lagrangianlagrangian

Nomenclature: O[(v/c)Nomenclature: O[(v/c)2Q2Q] = Q PN] = Q PN

• n-body problem known up to n-body problem known up to O[(v/c)O[(v/c)22]: ]:

• Einstein-Infeld-Hoffman Einstein-Infeld-Hoffman lagrangianlagrangian• Eqns of motion used regularly to Eqns of motion used regularly to calculate solar system dynamics, calculate solar system dynamics, etc.etc.

• Precession of Mercury begins Precession of Mercury begins at this orderat this order

• O[(v/c)O[(v/c)44] only known partially.] only known partially.• Damour, Schafer (1985, 1987)Damour, Schafer (1985, 1987)• Compute using field theory? Compute using field theory? (Goldberger, Rothstein, 2004)(Goldberger, Rothstein, 2004)

• Solar system probes of GR Solar system probes of GR beginning to go beyond O[(v/c)beginning to go beyond O[(v/c)22]: ]:

• New lunar laser ranging observatory New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser APOLLO; Mars and/or Mercury laser ranging missions?ranging missions?• LATOR, GTDM, BEACON, ASTROD, LATOR, GTDM, BEACON, ASTROD, etc.etc.• See e.g. Turyshev (2008)See e.g. Turyshev (2008)

• n-body L gives not only dynamics but n-body L gives not only dynamics but also geometry.also geometry.

• Add a test particle, M->0: it moves Add a test particle, M->0: it moves along geodesic in the spacetime metric along geodesic in the spacetime metric generated by the rest of the n massesgenerated by the rest of the n masses• Metric can be read off its actionMetric can be read off its action

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• Gravitational wave observatories Gravitational wave observatories need the 2 body L beyond O[(v/c)need the 2 body L beyond O[(v/c)77]:]:

• LIGO, VIRGO, etc. can track LIGO, VIRGO, etc. can track gravitational waves (GWs) from gravitational waves (GWs) from compact binaries over O[10compact binaries over O[1044] orbital ] orbital cycles.cycles.• GW detection: Raw data integrated GW detection: Raw data integrated against theoretical templates to against theoretical templates to search for correlations.search for correlations.• Construction of accurate templates Construction of accurate templates requires 2 body dynamics.requires 2 body dynamics.• Currently, 2 body L known up to Currently, 2 body L known up to O[(v/c)O[(v/c)77]]• See e.g. Blanchet (2006)See e.g. Blanchet (2006)

• Starting at 3 PN, O[(v/c)Starting at 3 PN, O[(v/c)66], GR ], GR computations of 2 body L start to computations of 2 body L start to give divergences that were give divergences that were eventually handled by dimensional eventually handled by dimensional regularization.regularization.• Perturbation theory beyond Perturbation theory beyond O[(v/c)O[(v/c)77] requires systematic, ] requires systematic, efficient methods.efficient methods.

• Perturbation theory beyond Perturbation theory beyond O[(v/c)O[(v/c)77] requires systematic, ] requires systematic, efficient methods.efficient methods.• QFT gives systematic framework QFT gives systematic framework for for

• Renormalization & Renormalization & regularizationregularization• Computational algorithm – Computational algorithm – Feynman diagrams with Feynman diagrams with appropriate dimensional analysis.appropriate dimensional analysis.

• GR: Einstein-HilbertGR: Einstein-Hilbert• n point particles: any scalar functional of n point particles: any scalar functional of geometric tensors & geometric tensors & dd-velocities integrated -velocities integrated along world linealong world line

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• Non-minimal terms Non-minimal terms encode information on the encode information on the non-trivial structure of non-trivial structure of individual objects.individual objects.


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• Coefficients {cCoefficients {cxx} have to } have to

be tuned to match physical be tuned to match physical observables from full observables from full macroscopic description of macroscopic description of objects.objects.

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• For compact objects, For compact objects, up to O[(v/c)up to O[(v/c)88], only ], only minimal terms -Mminimal terms -Maa∫ds∫dsaa

neededneeded


• Expand GR and point particle Expand GR and point particle action in powers of graviton fields action in powers of graviton fields hhμνμν … …

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• … … but some dimensional analysis but some dimensional analysis before computation makes before computation makes perturbation theory much more perturbation theory much more systematicsystematic• The scales in the n-body problemThe scales in the n-body problem

• rr – typical separation between n – typical separation between n bodies.bodies.• vv – typical speed of point particles – typical speed of point particles• r/vr/v – typical time scale of n-body – typical time scale of n-body systemsystem 1

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• Schematically, conservative part Schematically, conservative part of effective action is a series:of effective action is a series:

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• Look at Re[Graviton propagator], Look at Re[Graviton propagator], non-relativistic limit:non-relativistic limit:

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• Look at Re[Graviton propagator], Look at Re[Graviton propagator], non-relativistic limit:non-relativistic limit:

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• With nWith n(w)(w) world line terms -M world line terms -Maa∫ds∫dsaa,,• With nWith n(v)(v) Einstein-Hilbert action Einstein-Hilbert action terms,terms,• With N total gravitons,With N total gravitons,• Every Feynman diagram scales asEvery Feynman diagram scales as

• n-graviton piece of -Mn-graviton piece of -Maa∫ds∫dsaa with with χχ powers of velocities scales aspowers of velocities scales as• n-graviton piece of Einstein-n-graviton piece of Einstein-Hilbert action with Hilbert action with ψψ time time derivatives scales asderivatives scales as

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)()()( wvw nN

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=1=1 Q PNQ PN

• With nWith n(w)(w) world line terms -M world line terms -Maa∫ds∫dsaa,,• With nWith n(v)(v) Einstein-Hilbert action Einstein-Hilbert action terms,terms,• With N total gravitons,With N total gravitons,• Every Feynman diagram scales asEvery Feynman diagram scales as

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)()()( wvw nN

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=1=1 Q PNQ PN• Limited form of superposition Limited form of superposition holdsholds

• At Q PN, i.e. O[(v/c)At Q PN, i.e. O[(v/c)2Q2Q], max ], max number of distinct terms from -number of distinct terms from -MMaa∫ds∫dsaa is is Q+2Q+2• 1 PN, O[(v/c)1 PN, O[(v/c)22]: 3 body problem]: 3 body problem• 2 PN, O[(v/c)2 PN, O[(v/c)44]: 4 body problem]: 4 body problem• … …

0

• Every Feynman diagram scales asEvery Feynman diagram scales as

2 body2 bodydiagramsdiagrams

3 body3 bodydiagramsdiagrams

Einstein-Infeld-HoffmanEinstein-Infeld-Hoffmandd-spacetime dimensions-spacetime dimensions

No gravitonNo gravitonverticesvertices

GravitonGravitonverticesvertices

No gravitonNo gravitonverticesvertices

GravitonGravitonverticesvertices

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• Perturbation theory beyond Perturbation theory beyond O[(v/c)7] for 2 body L requires O[(v/c)7] for 2 body L requires systematic, efficient methods.systematic, efficient methods.• QFT gives systematic framework QFT gives systematic framework for for

• Renormalization & Renormalization & regularizationregularization• Computational algorithm – Computational algorithm – Feynman diagrams with Feynman diagrams with appropriate dimensional analysis.appropriate dimensional analysis.• But the computation is still hard and But the computation is still hard and long – need for new technology.long – need for new technology.

• Perturbation theory beyond Perturbation theory beyond O[(v/c)O[(v/c)77] for 2 body L requires ] for 2 body L requires systematic, efficient methods.systematic, efficient methods.• QFT gives systematic framework QFT gives systematic framework for for

• Renormalization & Renormalization & regularizationregularization• Computational algorithm – Computational algorithm – Feynman diagrams with Feynman diagrams with appropriate dimensional analysis.appropriate dimensional analysis.• But the computation is still hard and But the computation is still hard and long – need for new technology.long – need for new technology.• http://www.stargazing.net/yizen/PN.hthttp://www.stargazing.net/yizen/PN.htmlml

Yi-Zen Chu BCCS 2008 @ Case Western Reserve University 9 December 2008 The n-body problem in General...

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