The redshift factor z is an invariant quantity of fundamental interest in post-Newtonian and selfforce descriptions of compact binaries. It connects different approximation schemes, and plays a central role in the first law of binary black hole mechanics, which links local quantities to asymptotic measures of energy and angular momentum in these systems. Through this law, the redshift factor is conjectured to have a close relation to the surface gravity of the event horizons of black holes in circular orbits. We propose and implement a novel method for extracting the redshift factor on apparent horizons in numerical simulations of quasicircular binary inspirals. Our results confirm the conjectured relationship between z and the surface gravity of the holes and that the first law holds to a remarkable degree for binary inspirals. The redshift factor enables tests of analytic predictions for z in spacetimes where the binary is only approximately circular, giving a new connection between analytic approximations and numerical simulations.Introduction.-The relativistic two body problem is of fundamental importance in both general relativity and the astrophysics of compact objects. Compact binaries emit gravitational radiation and inspiral, eventually merging in a dynamic, nonlinear process. These mergers are the most promising sources of gravitational waves and provide a window into untested regimes of physics. The landmark detection of binary black hole (BH) mergers through gravitational waves [1-3] highlights both the sophistication of waveform models and the need for further improvements to search for and interpret gravitational wave signals. Current methods include post-Newtonian (PN) expansions in the slow velocity regime [4], selfforce (SF) approximations [5] for systems with high mass ratios, and direct numerical solutions [6][7][8] of inspirals beginning tens of orbits before merger. Each method has its limitations, and they are combined into effective one body (EOB) [9,10] and phenomenological waveform models [11]. In addition, connections and comparisons between the different approaches yield new insights into each of them [12,13]. Such insights deepen our understanding of relativity and maximize the scientific benefits of future gravitational wave observations. Invariant quantities play a crucial role in these comparisons, since each method uses different gauges and various approximation schemes. The invariant redshift factor z has proven essential in comparisons between analytic approximations, as first discussed for circular binaries [14]. Such systems remain stationary in the corotating frame, having a helical symmetry embodied in a helical Killing vector (HKV) K µ . In this context, the redshift factor allows for comparison of results obtained in distinct coordinate gauges [15,16], and has played a central role in the development of PN and EOB theory using SF, e.g. Refs. [17][18][19].For isolated BHs, the laws of black hole mechanics are relations between the area, angular momentum, and charge of the h...

Abstract. Binary black holes which are both eccentric and undergo precession remain unexplored in numerical simulations. We present simulations of such systems which cover about 50 orbits at comparatively high mass ratios 5 and 7. The configurations correspond to the generic motion of a nonspinning body in a Kerr spacetime, and are chosen to study the transition from finite mass-ratio inspirals to point particle motion in Kerr. We develop techniques to extract analogs of the three fundamental frequencies of Kerr geodesics, compare our frequencies to those of Kerr, and show that the differences are consistent with self-force corrections entering at first order in mass ratio. This analysis also locates orbital resonances where the ratios of our frequencies take rational values. At the considered mass ratios, the binaries pass through resonances in one to two resonant cycles, and we find no discernible effects on the orbital evolution. We also compute the decay of eccentricity during the inspiral and find good agreement with the leading order post-Newtonian prediction.

We have repurposed Google tensor processing units (TPUs), application-specific chips developed for machine learning, into large-scale dense linear algebra supercomputers. The TPUs’ fast intercore interconnects (ICIs), physically two-dimensional network topology, and high-bandwidth memory (HBM) permit distributed matrix multiplication algorithms to rapidly become computationally bound. In this regime, the matrix-multiply units (MXUs) dominate the runtime, yielding impressive scaling, performance, and raw size: Operating in float32 precision, a full 2,048-core pod of third-generation TPUs can multiply two matrices with linear size N = 2 20 = 1 , 048 , 576 in about 2 min. Via curated algorithms emphasizing large, single-core matrix multiplications, other tasks in dense linear algebra can similarly scale. As examples, we present 1) QR decomposition; 2) resolution of linear systems; and 3) the computation of matrix functions by polynomial iteration, demonstrated by the matrix polar factorization.

We demonstrate the use of Googles cloud-based Tensor Processing Units (TPUs) to accelerate and scale up conventional (cubic-scaling) density functional theory (DFT) calculations. Utilizing 512 TPU cores, we accomplish the largest such DFT computation to date, with 247848 orbitals, corresponding to a cluster of 10327 water molecules with 103270 electrons, all treated explicitly. Our work thus paves the way toward accessible and systematic use of conventional DFT, free of any system-specific constraints, at unprecedented scales.