In this talk I describe recent joint works with R.Schoen and with G.Gibbons and R.Schoen which prove the non-existence of certain asymptotically flat, stationary solutions of the Einstein equations with more than one body. The basic restriction is for example satisfied when spacetime has an isometry reversing the sign of the timelike Killing vector and fixing a hypersurface in the space of Killing trajectories which is disjoint from the bodies. I also give a detailed treatment of the Newtonian situation.
The intrinsic derivative and centrifugal forces in general relativity: II. Applications to circular orbits in some familiar stationary axisymmetric spacetimes
Using the time evolution equations of (cosmological) General Relativity in the first order Fischer-Marsden form, we construct an integral that measures the amount of non-stationary energy on a given spacelike hypersurface in $D$ dimensions. The integral vanishes for stationary spacetimes; and with a further assumption, reduces to Dain's invariant on the boundary of the hypersurface which is defined with the Einstein constraints and a fourth order equation defining approximate Killing symmetries.
We consider the so-called semiclassical variant of general relativity, where gravitational field is not quantized but matter is quantized, for the simplest composite quantum body - a hydrogen atom. We create a stationary electron quantum state in the atom in the absence of gravitational field and study its time evolution in the presence of the field, using the local Lorentz invariance property of spacetime. It is shown that this state with a definite energy in the absence of gravitational field is not anymore a stationary state in the field. Therefore, quantum measurements of passive gravitati…
We show that the introduction of two worldline parameters defines a different approach to computations in the effective field theory approach to the two-body problem in General Relativity and present some preliminary evidence for a reduction in computational complexity. These parameters obey a polynomial equation whose perturbative expansion recovers an infinite series of diagrams. Futhermore, we show that our equations define an effective two-body horizon for interacting particles in General Relativity; in the circular orbit case, it corresponds to the smallest conceivable separation up to wh…
We consider the equations for the coefficients of stationary rotating axisymmetric metrics governed by the Einstein-Euler equations, that is, the Einstein equations together with the energy-momentum tensor of a barotropic perfect fluid. Although the derived equations are not already known except for the case of the constant angular velocity described in the corotating coordinate system, the main content of this article is not to derive the equations, but to prove the equivalence of the derived equations with the full set of the Einstein equations, and to prove the consistency of the derived eq…
We consider anisotropic fluids with directional pressures $p_i = w_i ρ$ ($ρ$ is the density, $w_i = $const, $i = 1,2,3$) as sources of gravity in stationary cylindrically symmetric space-times. We describe a general way of obtaining exact solutions with such sources, where the main features are splitting of the Ricci tensor into static and rotational parts and using the harmonic radial coordinate. Depending on the values of $w_i$, it appears possible to obtain general or special solutions to the Einstein equations, thus recovering some known solutions and finding new ones. Three particular exa…
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}…
A particle orbiting a misaligned eccentric orbit binary undergoes nodal precession either around the binary angular momentum vector (a circulating orbit) or around a stationary inclination (a librating orbit). In the absence of general relativity, the stationary inclination is inclined by 90 degrees to the binary angular momentum vector (aligned with the binary eccentricity vector) and does not depend on the particle semi-major axis. General relativity causes apsidal precession of the binary orbit. Close to the binary, the behaviour of the particle is not significantly affected, a librating pa…
One of the open problems in developing binary black hole (BBH) waveforms for gravitational wave astronomy is to model the intermediate mass ratio regime and connect it to the extreme mass ratio regime. A natural approach is to employ the effective one body (EOB) approach to the two-body dynamics that, by design, can cover the entire mass ratio range and naturally incorporates the extreme mass ratio limit. Here we use recently obtained numerical relativity (NR) data with mass ratios $m_1/m_2=(7,15,\,32,\,64,\,128)$ to test the accuracy of the state-of-the-art EOB model TEOBResumS in the interme…
The two-body problem in general relativity is reduced to the problem of an effective particle (with an energy-dependent relativistic reduced mass) in an external field. The effective potential is evaluated from the Born diagram of the linearized quantum theory of gravity. It reduces to a Schwarzschild-like potential with two different `Schwarzschild radii'. The results derived in a weak field approximation are expected to be relevant for relativistic velocities.
Energy Localization Invariance of Tidal Work in General Relativity
众所周知，当外部广义相对论（电型）潮汐场E（t）与孤立物体的演化四极矩I（t）相互作用时，潮汐场确实对物体做功（"潮汐功"）--即，它将能量传递给物体--以牛顿理论中相同的公式给出的速率：dW/dt =-1/2 E dI/dt。索恩提出了以下问题：鉴于潮汐场和天体之间的引力相互作用能在E（t）I（t）量级上是不明确的，那么潮汐功是否也在这个量级上是不明确的，因此公式dW/dt =-1/2 E dI/dt是否只有在与I（t）的时间尺度相比，在更长的时间尺度上积分时才明确有效？本文完成了一个证明，答案是否定的，dW/dt是不是这样的模棱两可。更具体地说，本文表明，dW/dt是明确地由-1/2 E dI/dt独立的选择如何本地化的引力能在广义相对论。这是证明明确计算dW/dt使用各种引力应力-能量pseudotensors（爱因斯坦，朗道-Lifshitz，穆勒）以及伯格曼的保守量，其中概括了许多pseudotensors，包括一个任意函数的位置。本文还讨论了在广义相对论中建立守恒律的问题以及各种赝张量所起的作用。
