$n$-body problem in general relativity up to the second post-Newtonian order from perturbative field theory

Motivated by experimental probes of general relativity, we adopt methods from perturbative (quantum) field theory to compute, up to certain integrals, the effective Lagrangian for its $n$-body problem. Perturbation theory is performed about a background Minkowski space-time to $\mathcal{O}[(v/c{)}^4]$ beyond Newtonian gravity, where $v$ is the typical speed of these $n$ particles in their center of energy frame. For the specific case of the 2-body problem, the major efforts underway to measure gravitational waves produced by inspiraling compact astrophysical binaries require their gravitational interactions to be computed beyond the currently known $\mathcal{O}[(v/c{)}^7]$. We argue that such higher order post-Newtonian calculations must be automated for these field theoretic methods to be applied successfully to achieve this goal. In view of this, we outline an algorithm that would in principle generate the relevant Feynman diagrams to an arbitrary order in $v/c$ and take steps to develop the necessary software. The Feynman diagrams contributing to the $n$-body effective action at $\mathcal{O}[(v/c{)}^6]$ beyond Newton are derived.

Figure 1

Newtonian gravity.

Figure 2

1 PN 2-body diagrams.

Figure 3

1 PN 3-body diagrams.

Figure 4

2 PN 2-body diagrams with no graviton vertices: 1 of 2.

Figure 5

2 PN 2-body diagrams with no graviton vertices: 2 of 2.

Figure 6

2 PN 2-body diagrams with graviton vertices: 1 of 4.

Figure 7

2 PN 2-body diagrams with graviton vertices: 2 of 4.

Figure 8

2 PN 2-body diagrams with graviton vertices: 3 of 4.

Figure 9

2 PN 2-body diagrams with graviton vertices: 4 of 4.

Figure 10

2 PN 3-body diagrams with no graviton vertices.

Figure 11

2 PN 3-body diagrams with graviton vertices: 1 of 2.

Figure 12

2 PN 3-body diagrams with graviton vertices: 2 of 2.

Figure 13

2 PN 4-body diagrams with no graviton vertices.

Figure 14

2 PN 4-body diagrams with graviton vertices.

Figure 15

3 PN 2-body diagrams.

Figure 16

3 PN 3-body diagrams containing 3 graviton vertices, 1 of 2.

Figure 17

3 PN 3-body diagrams containing 3 graviton vertices, 2 of 2.

Figure 18

3 PN 3-body diagrams with no graviton vertices.

Figure 19

3 PN 4-body diagrams with graviton vertices, 1 of 3.

Figure 20

3 PN 4-body diagrams with graviton vertices, 2 of 3.

Figure 21

3 PN 4-body diagrams with graviton vertices, 3 of 3.

Figure 22

3 PN 4-body diagrams with no graviton vertices, 1 of 2.

Figure 23

3 PN 4-body diagrams with no graviton vertices, 2 of 2.

Figure 24

3 PN 5-body diagrams with graviton vertices.

Figure 25

3 PN 5-body diagrams with no graviton vertices.