Effective field theory of gravity for extended objects

Using effective field theory (EFT) methods we present a Lagrangian formalism which describes the dynamics of nonrelativistic extended objects coupled to gravity. The formalism is relevant to understanding the gravitational radiation power spectra emitted by binary star systems, an important class of candidate signals for gravitational wave observatories such as LIGO or VIRGO. The EFT allows for a clean separation of the three relevant scales: $r_s$, the size of the compact objects, $r$, the orbital radius, and $r/v$, the wavelength of the physical radiation (where the velocity $v$ is the expansion parameter). In the EFT, radiation is systematically included in the $v$ expansion without the need to separate integrals into near zones and radiation zones. Using the EFT, we show that the renormalization of ultraviolet divergences which arise at $v^6$ in post-Newtonian (PN) calculations requires the presence of two nonminimal worldline gravitational couplings linear in the Ricci curvature. However, these operators can be removed by a redefinition of the metric tensor, so that the divergences arising at $v^6$ have no physically observable effect. Because in the EFT finite size features are encoded in the coefficients of nonminimal couplings, this implies a simple proof of the decoupling of internal structure for spinless objects to at least order $v^6$. Neglecting absorptive effects, we find that the power counting rules of the EFT indicate that the next set of short distance operators, which are quadratic in the curvature and are associated with tidal deformations, does not play a role until order $v^{10}$. These operators, which encapsulate finite size properties of the sources, have coefficients that can be fixed by a matching calculation. By including the most general set of such operators, the EFT allows one to work within a point-particle theory to arbitrary orders in $v$.

Figure 1

The incoming potential graviton with momentum $\mathbf{p}$ absorbs a radiation graviton with momentum $k$.

Figure 2

Diagrams contributing to $L{v}^0$ potentials. The self-energy graphs in (b), (c) are pure counterterm and have no physical effect.

Figure 3

NRGR diagram whose imaginary part gives rise to the quadrupolar gravitational radiation power spectrum.

Figure 4

Diagrams contributing to $L{v}^2$ terms in the two-body potential. The $\otimes$ in (a) denotes an insertion of the potential graviton kinetic term. A diagram similar to (c) with $1\leftrightarrow 2$ is not shown.

Figure 5

Diagrams with two and three potential graviton lines that also contribute to the $L{v}^2$ potentials. Similar diagrams with $1\leftrightarrow 2$ are not shown.

Figure 6

Diagrams that contribute to the matching of radiation graviton couplings. The diagram (a) is the multipole expansion carried out to order $v^2$. Diagrams (b) and (c) are needed to ensure gauge invariance of the radiation graviton couplings. Similar diagrams with $1\leftrightarrow 2$ are not shown.

Figure 7

Some contributions to the background graviton effective action in the relativistic point-particle theory. Diagram (b) has a power divergent term. Diagrams (c) and (d) are both power and logarithmically divergent. Two other diagrams that arise as permutations of (c) are not shown.

Figure 8

Sample contribution to two-body potentials at $v^6\ln \alpha$. All particle vertices are the leading order mass $v^0$ mass insertion. The $\otimes$ denotes an insertion of the potential graviton kinetic term.

Figure 9

One potential graviton exchange diagrams contributing to the two-body Lagrangian at $v^6\ln \alpha$. The heavy dot indicates an insertion of the $\int d\tau R$ or $\int d\tau R_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }$ vertex with $c_{R,V}(\mu )$ evaluated at $\mu \sim 1/r$. Similar diagrams with $1\leftrightarrow 2$ are not shown.

Figure 10

Two- and three-potential graviton exchange diagrams contributing to the two-body Lagrangian at $v^6\ln \alpha$. Similar diagrams with $1\leftrightarrow 2$ are not shown.

Figure 11

Vertex associated with the Feynman rule in Eq. (b9).