arXiv subjects

Xiaokai He

Publications and source records attributed to Xiaokai He.

At least 19 recordsLinked to original sources

arXiv2026

The total geodesic curvature and the $(2+1)$-dimensional hyperbolic mass

In this paper, we derive a novel geometric inequality involving the total geodesic curvature, serving as an analogue of the Brown-York mass in the setting of $(2+1)$-dimensional gravity. The asymptotic behaviour of this newly defined quasi-local mass is examined for large ellipses in the time-symmetric slices of the Bañados-Teitelboim-Zanelli black hole solutions. Moreover, we establish an improved upper bound for the $(2+1)$-dimensional hyperbolic Bartnik mass.

math.DG↗

arXiv2026

Effective metric for bound state in an effective-one-body theory based on the third-post-Minkowskian approximation

The effective-one-body (EOB) framework, originally formulated within the post-Newtonian (PN) expansion, is central to modeling and interpreting gravitational-wave signals. Recent developments have incorporated the post-Minkowskian (PM) expansion into EOB theory. In this work, we focus on the bound state dynamics in the PM approximation up to the third order and construct the core ingredients of EOB: the effective metric. We examine two correspondence strategies, one based on the radial action variable and the other on the precession angle. We demonstrate that the radial action variable provides a consistent correspondence, which is further verified by the correspondence based on the precession angle. Building on these results, we adopt an isotropic gauge with a Schwarzschild-like parametrization to fix the remaining freedom. Within this parametrization, the effective metric coefficients are determined at 3PM order. Our results provide a consistent effective metric for the bound state dynamics.

gr-qc↗

arXiv2026

Constructing the canonical harmonic coordinates of Kerr metric to the fourth post-Minkowskian order

In this paper we construct the canonical harmonic coordinates of the Kerr metric within the multipolar post-Minkowskian (MPM) formalism to the fourth post-Minkowskian (4PM) order. Based on the well known Geroch--Hansen moments of Kerr metric and Gürsel's theorem, we derive the exact canonical MPM moments $\mathrm{M}_L,\mathrm{S}_L$, which are free of any gauge moments. With these moments, we iteratively compute the gothic metric perturbation $h^{μν}_{\mathrm{can}}$ up to 4PM order and compute the 4PM canonical metric $g_{μν}^{\mathrm{can}}$. The resulting spatial and time components of these metrics are even functions of the spin parameter $a$ while the mixed components are odd. This parity property distinguishes the canonical coordinates from other harmonic coordinates. To contrast this minimal-gauge construction, we also extract the 1PM source moments of the Kerr metric in the Jiang--Lin coordinates. We find that the Jiang--Lin representation possesses non-vanishing gauge moments starting from the 1PM order, whereas in the canonical representation gauge moments vanish to all orders. This comparison highlights the canonical coordinates as the most gauge-pure representation of the Kerr metric in the MPM framework. The complete canonical metric for the Schwarzschild case is also computed to all PM orders. A recent independent construction by Damgaard et al. using momentum-space recursion yields 4PM equivalent results expressed as a power series in $a$, providing a cross-validation of our closed-form 4PM canonical metric. The coordinate transformation linking the canonical Kerr coordinates to previously known harmonic Kerr coordinates remains an open problem.

gr-qc↗

arXiv2026

The first law of black hole mechanics in conformal Einstein-Power-Yang-Mills theory

In the framework of the Iyer-Wald formalism, the first law of black hole mechanics is examined within the context of conformal Einstein-power-Yang-Mills (CEPYM) theory. By comparing two inffnitesimally neighbouring stationary black hole solutions, we obtain the explicit analytical expression for the first law of black hole thermodynamics in CEPYM theory.

gr-qc↗

arXiv2026

Spectral analysis of small amplitude periodic $b$-Novikov equation under transverse perturbations

This paper is devoted to the transverse stability problem for small-amplitude periodic traveling waves of the $b$-Novikov equation, which arises as a two-dimensional extension of the $b$-family hierarchy. The perturbations under consideration fall into two groups. One group matches the fundamental period of the underlying wave along the longitudinal direction, while the other consists of profiles that are bounded or localized in the transverse direction. We perform spectral analysis of the associated linearized operator and derive precise stability and instability thresholds. The resulting conditions exhibit intricate dependence on two key parameters including the cubic coupling strength $b$ and the wavenumber $k$.

math.AP↗

arXiv2026

Bondi Mass, Memory Effect and Balance Law of Polyhomogeneous Spacetime

Spacetimes with metrics admitting an expansion in terms of a combination of powers of 1/r and ln r are known as polyhomogeneous spacetimes. The asymptotic behaviour of the Newman-Penrose quantities for the vacuum polyhomogeneous spacetimes is presented under certain gauges. The Bondi mass is revisited via the Iyer-Wald formalism. The memory effect of the gravitational radiation in the vacuum polyhomogeneous spacetimes is also discussed. It is found that the appearance of the logarithmic terms does not affect the balance law and it remains identical to that of spacetimes with metrics admitting an expansion in terms of powers of 1/r.

gr-qc↗

arXiv2025

Monotonicity of the periodic waves for the perturbed generalized defocusing mKdV equation

In this paper, we study the existence of periodic waves for the perturbed generalized defocusing mKdV equation using the theory of geometric singular perturbation. By Abelian integral and involution operation, we prove that the limit wave speed c_0(h) is monotonic with respect to energy h,and the lower bound of the limit wave speed is found. These works extend the main result of Chen et al. (2018) to the generalized case. Some numerical simulations are conducted to verify the correctness of the theoretical analysis.

math.AP↗

arXiv2024

On the transverse-traceless gauge condition when matters are presented

The transverse-traceless gauge condition is an important concept in the theory of gravitational wave. It is well known that vacuum is one of the key conditions to guarantee the existence of the transverse-traceless gauge. Although it is thin, interstellar medium is ubiquitous in the universe. Therefore, it is important to understand the concept of gravitational wave when matter is presented. Bondi-Metzner-Sachs theory has solved the gauge problem related to gravitational wave. But it does not help with the cases when gravitational wave propagates in matters. This paper discusses possible extensions of the transverse-traceless gauge condition to Minkowski perturbation with matter presented.

gr-qc↗

arXiv2024

Self-consistent Effective-one-body theory for spinless binaries based on post-Minkowskian approximation I: Hamiltonian and decoupled equation for $ψ^B_{4}$

To build a self-consistent effective-one-body (EOB) theory, in which the Hamiltonian, radiation-reaction force and waveform for the "plus" and "cross" modes of the gravitational wave should be based on the same effective background spacetime, the key step is to look for the decoupled equation for $ψ^B_{4}=\ddot{h}_{+}-i\ddot{h}_{\times}$, which seems a very difficult task because there are non-vanishing tetrad components of the tracefree Ricci tensor for such spacetime. Fortunately, based on an effective spacetime obtained in this paper by using the post-Minkowskian (PM) approximation, we find the decoupled equation for $ψ^B_{4}$ by dividing the perturbation part of the metric into the odd and even parities. With the effective metric and decoupled equation at hand, we set up a frame of self-consistent EOB model for spinless binaries.

gr-qc↗

arXiv2023

Stability of smooth periodic travelling waves in the Dullin-Gottwald-Holm equation

The existence of smooth periodic traveling solutions in the Dullin-Gottwald-Holm (DGH) equation and the monotonicity of the period function are clarified. By introducing two suitable parameters, we show the existence of periodic travelling solutions of DGH equation in a concise way. The monotonicity of period function with respect to different variables are proved by using Chicone's criterion and the method developed by Geyer and Villadelprat. The problem of the spectral stability of smooth periodic waves in the DGH equation is discussed. Within the functional-analytic framework, we obtain a criterion for the spectral stability of smooth periodic traveling waves in DGH equation. In addition, we show the smooth periodic travelling solutions are orbitally stable under certain conditions.

math.DS↗

arXiv2023

The effect of the gravitational constant variation on the propagation of gravitational waves

Since the first detection of gravitational waves, they have been used to investigate various fundamental problems, including the variation of physical constants. Regarding the gravitational constant, previous works focused on the effect of the gravitational constant variation on the gravitational wave generation. In this paper, we investigate the effect of the gravitational constant variation on the gravitational wave propagation. The Maxwell-like equation that describes the propagation of gravitational waves is extended in this paper to account for situations where the gravitational constant varies. Based on this equation, we find that the amplitude of gravitational waves will be corrected. Consequently the estimated distance to the gravitational wave source without considering such a correction may be biased. Applying our correction result to the well known binary neutron star coalescence event GW170817, we get a constraint on the variation of the gravitational constant. Relating our result to the Yukawa deviation of gravity, we for the first time get the constraint of the Yukawa parameters in 10Mpc scale. This scale corresponds to a graviton mass $m_g\sim10^{-31}$eV.

gr-qc↗

arXiv2023

Thurston's sphere packings on 3-dimensional manifolds, I

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity that Thurston's Euclidean sphere packing can not be deformed (except by scaling) while keeping the combinatorial Ricci curvature fixed.

math.GT↗

arXiv2023

Lorentz transformation of three dimensional gravitational wave tensor

Recently there are more and more interest on the gravitational wave of moving sources. This introduces a Lorentz transformation problem of gravitational wave. Although Bondi-Metzner-Sachs (BMS) theory has in principle already included the Lorentz transformation of gravitational wave, the transformation of the three dimensional gravitational wave tensor has not been explicitly calculated before. Within four dimensional spacetime, gravitational wave have property of `boost weight zero' and `spin weight 2'. This fact makes the Lorentz transformation of gravitational wave difficult to understand. In the current paper we adopt the traditional three dimensional tensor description of gravitational wave. Such a transverse-traceless tensor describes the gravitational wave freedom directly. We derive the explicit Lorentz transformation of the gravitational wave tensor. The transformation is similar to the Lorentz transformation for electric field vector and magnetic field vector which are three dimensional vectors. Based on the deduced Lorentz transformation of the gravitational wave three dimensional tensor, we can construct the gravitational waveform of moving source with any speed if only the waveform of the corresponding rest waveform is given. As an example, we apply our method to the effect of kick velocity of binary black hole. The adjusted waveform by the kick velocity is presented.

gr-qc↗

arXiv2023

Accurate calculation of gravitational wave memory

Gravitational wave memory is an important prediction of general relativity. The detection of the gravitational wave memory can be used to test general relativity and to deduce the property of the gravitational wave source. Quantitative model is important for such detection and signal interpretation. Previous works on gravitational wave memory always use the energy flux of gravitational wave to calculate memory. Such relation between gravitational wave energy and memory has only been validated for post-Newtonian approximation. The result of numerical relativity about gravitational wave memory is not confident yet. Accurately calculating memory is highly demanded. Here we propose a new method to calculate the gravitational wave memory. This method is based on Bondi-Metzner-Sachs theory. Consequently our method does not need slow motion and weak field conditions for gravitational wave source. Our new method can accurately calculate memory if the non-memory waveform is known. As an example, we combine our method with matured numerical relativity result about non-memory waveform for binary black hole coalescence. We calculate the waveform for memory which can be used to aid memory detection and gravitational wave source understanding. Our calculation result confirms preliminary numerical relativity result about memory. We find out the dependence of the memory amplitude to the mass ratio and the spins of the two spin aligned black holes.

gr-qc↗

arXiv2022

Gravitational wave memory produced by cosmic background radiation

It is well known that energy fluxes will produce gravitational wave memory. The gravitational wave memory produced by background including cosmic microwave background (CMB), cosmic neutrino background (C$ν$B), and gravitational wave background is investigated in this work. We construct a theory relating the gravitational wave memory strength to the energy flux of a stochastic background. We find that the resulted gravitational wave memory behaves as a constantly varying metric tensor. Such a varying metric tensor will introduce a quadrupole structure to the universe expansion. The gravitational wave memory due to the CMB is too small to be detected. But the gravitational wave memory due to the C$ν$B and the gravitational wave background is marginally detectable. Interestingly, such detection can be used to estimate the neutrino masses and the properties of the gravitational wave background.

gr-qc↗

arXiv2022

On the angular momentum of compact binary coalescence

The supertranslation ambiguity issue of angular momentum is a long-standing problem in general relativity. Recently, there appeared the first definition of angular momentum at null infinity that is supertranslation invariant. However, in the compact binary coalescence community, supertranslation ambiguity is often ignored. This paper demonstrates that we have the happy circumstance that the newly defined angular momentum coincides with the classical definition at the quadrupole level.

gr-qc↗

arXiv2021

Gravitational wave memory of the binary black hole events in GWTC-2

Gravitational wave (GW) memory is an important prediction of general relativity. Existing works on the GW memory detection focus on the waveform analysis. It is hard for waveform analysis method to detect the GW memory due to its quasi-direct current behavior and weakness. We implement a completely different scheme in this work to estimate the GW memory. In this scheme, we firstly apply the Bondi-Metzner-Sachs method to calculate the GW memory of binary black hole based on numerical relativity simulation. Then we construct a surrogate model to relate binary black hole's parameters and the GW memory. Afterwards we apply this surrogate model together with Bayesian techniques to estimate the GW memory of the 48 binary black hole events recorded in GWTC-2. The GW memory corresponding to the all 48 events has been estimated. The most interesting results are for GW190814. The corresponding GW memory is about $-1\times10^{-23}$ and $1\times10^{-23}$ for Hanford detector and Livingston detector respectively. At the same time we find with 3$σ$ C.L. that the memory strain of GW190814 is negative on Hanford detector while positive on Livingston detector.

gr-qc↗

arXiv2021

From bending of light to positive mass: a non-PDE perspective

Penrose et al. investigated the physical incoherence of the spacetime with negative mass via the bending of light. Precise estimates of time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and make a reduction of this problem to a causality comparison in the compactified spacetimes regarding timelike connectedness near the conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues Penrose et al. met. It provides a new, substantially simple, and physically natural non-PDE viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions which are vacuum and stationary near infinity.

gr-qc↗