The post-Minkowskian approach to gravitationally interacting binary systems ({\it i.e.}, perturbation theory in $G$, without assuming small velocities) is extended to the computation of the dynamical effects induced by the tidal deformations of two extended bodies, such as neutron stars. Our derivation applies general properties of perturbed actions to the effective field theory description of tidally interacting bodies. We compute several tidal invariants (notably the integrated quadrupolar and octupolar actions) at the first post-Minkowskian order. The corresponding contributions to the scattering angle are derived.
Using the spinning worldline quantum field theory formalism we calculate the quadratic-in-spin momentum impulse $Δp_i^μ$ and spin kick $Δa_i^μ$ from a scattering of two arbitrarily oriented spinning massive bodies (black holes or neutron stars) in a weak gravitational background up to third post-Minkowskian (PM) order ($G^3$). Two-loop Feynman integrals are performed in the potential region, yielding conservative results. For spins aligned to the orbital angular momentum we find a conservative scattering angle that is fully consistent with state-of-the-art post-Newtonian results. Using the 2PM…
The Hamiltonian for a system of relativistic bodies interacting by their gravitational field is found in the post-Minkowskian approximation, including all terms linear in the gravitational constant. It is given in a surprisingly simple closed form as a function of canonical variables describing the bodies only. The field is eliminated by solving inhomogeneous wave equations, applying transverse-traceless projections, and using the Routh functional. By including all special relativistic effects our Hamiltonian extends the results described in classical textbooks of theoretical physics. As an ap…
High precision astrometry, space missions and certain tests of General Relativity, require the knowledge of the metric tensor of the solar system, or more generally, of a gravitational system of N extended bodies. Presently, the metric of arbitrarily shaped, rotating, oscillating and arbitrarily moving N bodies of finite extension is only known for the case of slowly moving bodies in the post-Newtonian approximation, while the post-Minkowskian metric for arbitrarily moving celestial objects is known only for pointlike bodies with mass-monopoles and spin-dipoles. As one more step towards the ai…
We compute the mass and current quadrupole tidal corrections to the four-momentum and energy flux radiated during the scattering of two spinless bodies, at leading order in $G$ and at all orders in the velocities, using the effective field theory worldline approach. In particular, we derive the conserved stress-energy tensor linearly coupled to gravity generated by the two bodies, including tidal fields, and the waveform in direct space. The integral is solved using scattering amplitude techniques. We show that our expressions are consistent with existing results up to the next-to-next-to-lead…
We study the gravitational radiation emitted during the scattering of two spinless bodies in the post-Minkowskian Effective Field Theory approach. We derive the conserved stress-energy tensor linearly coupled to gravity and the classical probability amplitude of graviton emission at leading and next-to-leading order in the Newton's constant $G$. The amplitude can be expressed in compact form as one-dimensional integrals over a Feynman parameter involving Bessel functions. We use it to recover the leading-order radiated angular momentum expression. Upon expanding it in the relative velocity bet…
Tools from scattering amplitudes and effective field theory have recently been repurposed to derive state-of-the-art results for the black hole binary inspiral in the post-Minkowskian expansion. In the present work we extend this approach to include the tidal effects of mass and current quadrupoles on the conservative dynamics of non-spinning neutron star mergers. We compute the leading and, for the first time, next-to-leading order post-Minkowskian finite size corrections to the conservative Hamiltonian, together with their associated scattering amplitudes and scattering angles. Our expressio…
The Hamiltonian for a system of relativistic bodies interacting by their gravitational field is found in the post-Minkowskian approximation, including all terms linear in the gravitational constant. It is given in a surprisingly simple closed form as a function of canonical variables describing the bodies only. The field is eliminated by solving inhomogeneous wave equations, applying transverse-traceless projections, and using the Routh functional. By including all special relativistic effects our Hamiltonian extends the results described in classical textbooks of theoretical physics. As an ap…
Conservative binary dynamics to third post-Minkowskian order beyond General Relativity
School of Physics & Shing-Tung Yau Center, Southeast University（东南大学物理学院及施雅天中心）；Niels Bohr International Academy & Center of Gravity, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所尼尔斯·玻尔国际学院及引力中心）
We present a general framework for calculating post-Minskowskian, classical, conservative Hamiltonians for $N$ non-spinning bodies in general relativity from relativistic scattering amplitudes. Novel features for $N>2$ are described including the subtraction of tree-like iteration contributions and the calculation of non-trivial many-body Fourier transform integrals needed to construct position space potentials. A new approach to calculating these integrals as an expansion in the hierarchical limit is described based on the method of regions. As an explicit example, we present the $\mathcal{O}…
Building upon recent progress in applying on-shell amplitude techniques to classical observables in general relativity, we propose a closed-form formula for the conservative Hamiltonian of a spinning binary system at the 1st post-Minkowskian (1PM) order. It is applicable for general spinning bodies with arbitrary spin multipole moments. The formula is linear in gravitational constant by definition, but exact to all orders in momentum and spin expansions. At each spin order, our formula implies that the spin-dependence and momentum dependence factorize almost completely. We expand our formula i…
We compute the four-momentum radiated during the scattering of two spinless bodies, at leading order in the Newton's contant $G$ and at all orders in the velocities, using the Effective Field Theory worldline approach. Following \cite{Mougiakakos:2021ckm}, we derive the conserved stress-energy tensor linearly coupled to gravity generated by localized sources, at leading and next-to-leading order in $G$, and from that the classical probability amplitude of graviton emission. The total emitted momentum is obtained by phase-space integration of the graviton momentum weighted by the modulo squared…
We present a novel calculation of the four-momentum that is radiated into gravitational waves during the scattering of two arbitrarily spinning bodies. Our result, which is accurate to leading order in $G$, to quadratic order in the spins, and to all orders in the velocity, is derived by using a Routhian-based worldline effective field theory formalism in concert with a battery of analytic techniques for evaluating loop integrals. While nonspinning binaries radiate momentum only along the direction of their relative velocity, we show that the inclusion of spins generically allows for momentum …
Building upon the worldline effective field theory (EFT) formalism for spinning bodies developed for the Post-Newtonian regime, we generalize the EFT approach to Post-Minkowskian (PM) dynamics to include rotational degrees of freedom in a manifestly covariant framework. We introduce a systematic procedure to compute the total change in momentum and spin in the gravitational scattering of compact objects. For the special case of spins aligned with the orbital angular momentum, we show how to construct the radial action for elliptic-like orbits using the Boundary-to-Bound correspondence. As a pa…
We present the two-body Hamiltonian and associated eikonal phase, to leading post-Minkowskian order, for infinitely many tidal deformations described by operators with arbitrary powers of the curvature tensor. Scattering amplitudes in momentum and position space provide systematic complementary approaches. For the tidal operators quadratic in curvature, which describe the linear response to an external gravitational field, we work out the leading post-Minkowskian contributions using a basis of operators with arbitrary numbers of derivatives which are in one-to-one correspondence with the world…
Expanding on the recent derivation of tidal actions for scalar particles, we present here the action for a tidally deformed spin-$1/2$ particle. Focusing on operators containing two powers of the Weyl tensor, we combine the Hilbert series with an on-shell amplitude basis to construct the tidal action. With the tidal action in hand, we compute the leading-post-Minkowskian tidal contributions to the spin-1/2 -- spin-1/2 amplitude, arising at $\mathcal{O}(G^{2})$. Our amplitudes provide evidence that the observed long range spin-universality for the scattering of two point particles extends to th…
The tidal Love numbers of a black hole vanish, and this is often taken to imply that the hole's tidally induced multipole moments vanish also. An obstacle to establishing a link between these statements is that the multipole moments of individual bodies are not defined in general relativity, when the bodies are subjected to a mutual gravitational interaction. In a previous publication [Phys. Rev. D 103, 064023 (2021)] I promoted the view that individual multipole moments can be defined when the mutual interaction is sufficiently weak to be described by a post-Newtonian expansion. In this view,…
Picture yourself in the wave zone of a gravitational scattering event of two massive, spinning compact bodies (black holes, neutron stars or stars). We show that this system of genuine astrophysical interest enjoys a hidden $\mathcal{N}=2$ supersymmetry, at least to the order of spin-squared (quadrupole) interactions in arbitrary $D$ spacetime dimensions. Using the ${\mathcal N}=2$ supersymmetric worldline action, augmented by finite-size corrections for the non-Kerr black hole case, we build a quadratic-in-spin extension to the worldline quantum field theory (WQFT) formalism introduced in our…
Hořava-Lifshitz gravity is an alternative theory to general relativity which breaks Lorentz invariance in order to achieve an ultraviolet complete and power-counting renormalizable theory of gravity. In the low energy limit, Hořava-Lifshitz gravity coincides with a vector-tensor theory known as khronometric gravity. The deviation of khronometric gravity from general relativity can be parametrized by three coupling constants: $α$, $β$, and $λ$. Solar system experiments and gravitational wave observations impose stringent bounds on $α$ and $β$, while $λ$ is still relatively unconstrained ($λ\les…
This paper employs the post-Newtonian approximations of scalar-tensor theory of gravity along with the Cartesian STF tensors and the Blanchet-Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies with arbitrary distribution of mass and velocity. We assume that spacetime can be covered by a global coordinate chart which is Minkowskian at infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field of each body is parametrized by an infinite set of the bod…
Equations of motion for compact binary systems in general relativity: Do they depend on the bodies' internal structure at the third post-Newtonian order?
Gravitational waves emitted by coalescing binary systems containing neutron stars (or other compact objects) carry signatures of the stars' internal equation of state, notably, through the influence of tidal deformations during the binary's inspiral stage. While the leading-order tidal effects for post-Newtonian binaries of compact bodies in general relativity are due to the bodies' mass-quadrupole moments induced by gravitoelectric tidal fields, we consider here the leading effects due to current-quadrupole moments induced by gravitomagnetic tidal fields. We employ an effective action approac…
In the first part of the paper we construct the metric of a tidally deformed, nonrotating black hole. The metric is presented as an expansion in powers of r/b << 1, in which r is the distance to the black hole and b the characteristic length scale of the tidal field --- the typical distance to the remote bodies responsible for the tidal environment. The metric is expanded through order (r/b)^4 and written in terms of a number of tidal multipole moments, the gravitoelectric moments E_{ab}, E_{abc}, E_{abcd} and the gravitomagnetic moments B_{ab}, B_{abc}, B_{abcd}. It differs from the similar c…
The study of scattering encounters continues to provide new insights into the general relativistic two-body problem. The local-in-time conservative dynamics of an aligned-spin binary, for both unbound and bound orbits, is fully encoded in the gauge-invariant scattering-angle function, which is most naturally expressed in a post-Minkowskian (PM) expansion, and which exhibits a remarkably simple dependence on the masses of the two bodies (in terms of appropriate geometric variables). This dependence links the PM and small-mass-ratio approximations, allowing gravitational self-force results to de…
Spinning Black Hole Scattering at $\mathcal{O}(G^3 S^2)$: Casimir Terms, Radial Action and Hidden Symmetry
University of Edinburgh（爱丁堡大学）；Florida State University（佛罗里达州立大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）；University of California at Los Angeles（加州大学洛杉矶分校）
