Yi-Zen Chu BCCS Case Western Reserve University 9 December 2008 The n-body problem in General Relativity from perturbative field theory arXiv: [gr-qc]

System of n ≥ 2 gravitationally bound compact objects: System of n ≥ 2 gravitationally bound compact objects: Planets, neutron stars, black holes, etc. Planets, neutron stars, black holes, etc. What is their effective gravitational interaction? What is their effective gravitational interaction?

Compact objects ≈ point particles Compact objects ≈ point particles n-body problem: Dynamics for the coordinates of the point particles n-body problem: Dynamics for the coordinates of the point particles Assume non-relativistic motion Assume non-relativistic motion GR corrections to Newtonian gravity: an expansion in (v/c) 2 GR corrections to Newtonian gravity: an expansion in (v/c) 2 Nomenclature: O[(v/c) 2Q ] = Q PN

Note that General Relativity is non-linear. Note that General Relativity is non-linear. Superposition does not hold Superposition does not hold 2 body lagrangian is not sufficient to obtain n-body lagrangian 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 ]: n-body problem known up to O[(v/c) 2 ]: Einstein-Infeld-Hoffman lagrangian Einstein-Infeld-Hoffman lagrangian Eqns of motion used regularly to calculate solar system dynamics, etc. Eqns of motion used regularly to calculate solar system dynamics, etc. Precession of Mercury begins at this order Precession of Mercury begins at this order O[(v/c) 4 ] only known partially. O[(v/c) 4 ] only known partially. Damour, Schafer (1985, 1987) Damour, Schafer (1985, 1987) Compute using field theory? (Goldberger, Rothstein, 2004) Compute using field theory? (Goldberger, Rothstein, 2004)

Solar system probes of GR beginning to go beyond O[(v/c) 2 ]: 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? New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser ranging missions? LATOR, GTDM, BEACON, ASTROD, etc. LATOR, GTDM, BEACON, ASTROD, etc. See e.g. Turyshev (2008) See e.g. Turyshev (2008)

n-body L gives not only dynamics but also geometry. 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 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 Metric can be read off its action

Gravitational wave observatories need the 2 body L beyond O[(v/c) 7 ]: 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[10 4 ] orbital cycles. LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[10 4 ] orbital cycles. GW detection: Raw data integrated against theoretical templates to search for correlations. GW detection: Raw data integrated against theoretical templates to search for correlations. Construction of accurate templates requires 2 body dynamics. Construction of accurate templates requires 2 body dynamics. Currently, 2 body L known up to O[(v/c) 7 ] Currently, 2 body L known up to O[(v/c) 7 ] See e.g. Blanchet (2006) 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. 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 QFT gives systematic framework for Renormalization & regularization Renormalization & regularization Computational algorithm – Feynman diagrams with appropriate dimensional analysis. Computational algorithm – Feynman diagrams with appropriate dimensional analysis.

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

–M∫ds describes structureless point particle –M∫ds describes structureless point particle GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line

Non-minimal terms encode information on the non-trivial structure of individual objects. Non-minimal terms encode information on the non-trivial structure of individual objects. GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line

GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line Coefficients {c x } have to be tuned to match physical observables from full macroscopic description of objects. Coefficients {c x } have to be tuned to match physical observables from full macroscopic description of objects.

For compact objects, up to O[(v/c) 8 ], only minimal terms -M a ∫ds a needed For compact objects, up to O[(v/c) 8 ], only minimal terms -M a ∫ds a needed GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors & d-velocities integrated along world line 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 h μν … Expand GR and point particle action in powers of graviton fields h μν …

… but some dimensional analysis before computation makes perturbation theory much more systematic … but some dimensional analysis before computation makes perturbation theory much more systematic The scales in the n-body problem The scales in the n-body problem r – typical separation between n bodies. r – typical separation between n bodies. v – typical speed of point particles v – typical speed of point particles r/v – typical time scale of n-body system r/v – typical time scale of n-body system

Lowest order effective action Lowest order effective action Schematically, conservative part of effective action is a series: Schematically, conservative part of effective action is a series:

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

n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as With n (w) world line terms -M a ∫ds a, With n (w) world line terms -M a ∫ds a, With n (v) Einstein-Hilbert action terms, With n (v) Einstein-Hilbert action terms, With N total gravitons, With N total gravitons, Every Feynman diagram scales as Every Feynman diagram scales as

n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as =1 Q PN With n (w) world line terms -M a ∫ds a, With n (w) world line terms -M a ∫ds a, With n (v) Einstein-Hilbert action terms, With n (v) Einstein-Hilbert action terms, With N total gravitons, With N total gravitons, Every Feynman diagram scales as Every Feynman diagram scales as

=1 Q PN Limited form of superposition holds Limited form of superposition holds At Q PN, i.e. O[(v/c) 2Q ], max number of distinct terms from -M a ∫ds a is Q+2 At Q PN, i.e. O[(v/c) 2Q ], max number of distinct terms from -M a ∫ds a is Q+2 1 PN, O[(v/c) 2 ]: 3 body problem 1 PN, O[(v/c) 2 ]: 3 body problem 2 PN, O[(v/c) 4 ]: 4 body problem 2 PN, O[(v/c) 4 ]: 4 body problem … … Every Feynman diagram scales as Every Feynman diagram scales as

2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions

No graviton vertices Gravitonvertices

vertices Gravitonvertices

vertices Gravitonvertices

vertices Gravitonvertices

vertices Gravitonvertices

Perturbation theory beyond O[(v/c)7] for 2 body L requires systematic, efficient methods. Perturbation theory beyond O[(v/c)7] for 2 body L requires systematic, efficient methods. QFT gives systematic framework for QFT gives systematic framework for Renormalization & regularization Renormalization & regularization Computational algorithm – Feynman diagrams with appropriate dimensional analysis. Computational algorithm – Feynman diagrams with appropriate dimensional analysis. But the computation is still hard and long – need for new technology. 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. Perturbation theory beyond O[(v/c) 7 ] for 2 body L requires systematic, efficient methods. QFT gives systematic framework for QFT gives systematic framework for Renormalization & regularization Renormalization & regularization Computational algorithm – Feynman diagrams with appropriate dimensional analysis. Computational algorithm – Feynman diagrams with appropriate dimensional analysis. But the computation is still hard and long – need for new technology. But the computation is still hard and long – need for new technology.