Effective field theory of gravity for extended objects

Abstract

Using Effective Field Theory (EFT) methods we present a Lagrangian formalism which describes the dynamics of non-relativistic 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: rs, 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 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 non-minimal 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 at arising at v 6 have no physically observable effect. Because in the EFT finite size features are encoded in the coefficients of non-minimal 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, do 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.

Towers of gravitational theories

“…For low orders in v, such divergences scale like powers of the cutoff, and are therefore not physical (they can be absorbed into mass renormalization). At order v 6 , however, one begins to encounter logarithmic divergences, which lead to non-trivial renormalization group flows in the Wilson coefficients of the EFT [1]. For objects that are perfectly spherical, such divergences are renormalized by worldline operators of the form dτ R, dτ R µν v µ v ν , which can be removed by field redefinitions of the metric, provided the black hole binary is in a background vacuum spacetime.…”

“…That is, we develop a tower of effective field theories in such a way that we may cleanly separate out the physics at disparate scales [1]. The shortest scales in the problem are those which dictate the "internal dynamics of the body".…”

Sources of Gravitational Waves: Theory and Observations

“…An important turning point occurred in 2006, when Goldberger and Rothstein proposed a more systematic use of Feynman diagrams within effective field theory (EFT) to describe non-relativistic extended objects coupled to gravity [130]. As discussed above, there are three relevant scales in the two-body problem: r s , the internal structure scale or the size of the compact object; r, the orbital separation, and λ GW ∼ r(c/v), the radiation wavelength, where v is the typical velocity of the body in the binary.…”

On-shell techniques and universal results in quantum gravity

“…In the gravitational case, there is also a low-energy effective Lagrangian for a massive system, described by its energy-momentum and spin [50,51]. This yields the leading couplings of two gravitons to the heavy particle, which is equivalent to the lowenergy limit of our gravitational Compton amplitudes.…”