arXiv subjects

Xiaokai He

Publications and source records attributed to Xiaokai He.

27 records · Page 2Linked to original sources

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↗

arXiv2020

Quasi-local energy and Oppenheimer-Snyder collapse

We consider a scenario of the gravitational collapse of the Oppenheimer-Snyder dust cloud with spatially constant matter density. Given a closed two-surface within the star, three versions of the quasi-local energy are investigated. We show that, during the collapse, (i) the Geroch energy of the surface is nonpositive and increases to zero; (ii) the Hawking energy of the surface is monotonically increasing and approaches to the Hayward energy of the initial surface; and (iii) the Hayward energy of the surface is conserved and remains positive. These results have no restrictions on the topology and the symmetry of the surface.

gr-qc↗

arXiv2020

Hayward Quasilocal Energy of Tori

This paper is dedicated to the investigation of the positivity of the Hayward quasilocal energy of tori. Marginally trapped tori have nonnegative Hayward energy. We consider a scenario of a spherically symmetric constant density star matched to an exterior Schwarzschild solution. We show that any generic tori within the star, distorted or not, trapped or not, have strictly positive Hayward energy. Surprisingly we find analytic examples of `thin' tori with negative Hayward energy in the outer neighborhood of the Schwarzschild horizon. These tori are swept out by rotating the standard round circles in the static coordinates but they are distorted in the isotropic coordinates. Numerical results also indicate that there exist horizontally dragged tori with strictly negative Hayward energy in the region between the boundary of the star and the Schwarzschild horizon.

gr-qc↗

arXiv2020

On limit behavior of quasi-local mass for ellipsoids at spatial infinity

We discuss the spatial limit of the quasi-local mass for certain ellipsoids in an asymptotically flat static spherically symmetric spacetime. These ellipsoids are not nearly round but they are of interest as an admissible parametrized foliation defining the Arnowitt-Deser-Misner (ADM) mass. The Hawking mass of this family of ellipsoids tends to $-\infty$. In contrast, we show that the Hayward mass converges to a finite value. Moreover, a positive mass type theorem is established. The limit of the mass has a uniform positive lower bound no matter how oblate these ellipsoids are. This result could be extended for asymptotically Schwarzschild manifolds. And numerical simulation in the Schwarzschild spacetime illustrates that the Hayward mass is monotonically increasing near infinity.

gr-qc↗

arXiv2018

Singular Values of Riemann Curvature Tensor

We introduce the concept of singular values for the Riemann curvature tensor, a central mathematical tool in Einstein's theory of general relativity. We study the properties related to the singular values, and investigate five typical cases to show its relationship to the Ricci scalar and other invariants.

math.DG↗

arXiv2018

Relationship between Bondi-Sachs quantities and source of gravitational radiation in asymptotically de Sitter spacetime

Gravitational radiation plays an important role in astrophysics. Based on the fact that our universe is expanding, the gravitational radiation when a positive cosmological constant is presented has been studied along with two different ways recently, one is the Bondi-Sachs (BS) framework in which the result is shown by BS quantities in the asymptotic null structure, the other is the perturbation approach in which the result is presented by the quadrupoles of source. Therefore, it is worth to interpret the quantities in asymptotic null structure in terms of the information of the source. In this paper, we investigate this problem and find the explicit expressions of BS quantities in terms of the quadrupoles of source in asymptotically de Sitter spacetime. We also estimate how far away the source is, the cosmological constant may affect the detection of the gravitational wave.

gr-qc↗

arXiv2018

Thermodynamical stability for perfect fluid

According to maximum entropy principle, it has been proved that the gravitational field equations could be derived by the extrema of total entropy for perfect fluid, which implies that thermodynamic relations contain information of gravity. In this manuscript, we obtain a criterion for thermodynamical stability of an adiabatic, self-gravitating perfect fluid system by the second variation of total entropy. We show, for Einstein's gravity with spherical symmetry spacetime, that the criterion is consistent with that for dynamical stability derived by Chandrasekhar and Wald. We also find that the criterion could be applied to cases without spherical symmetry, or under general perturbations. The result further establishes the connection between thermodynamics and gravity.

gr-qc↗

arXiv2017

Note on the gravitational and electromagnetic radiation for the Einstein-Maxwell equations with cosmological constant

In the middle of last century, Bondi and his coworkers proposed an out going boundary condition for the Einstein equations. Based on such boundary condition the authors theoretically solved the puzzle of the existence problem of gravitational wave. Currently many works on gravitational wave source modeling use Bondi's result to do the analysis including the gravitational wave calculation in numerical relativity. Recently more and more observations imply that the Einstein equations should be modified with an nonzero cosmological constant. In this work we use Bondi's original method to treat the Einstein equations with cosmological constant for theoretical curiosity. We find that the gravitational wave does not essentially exist if Bondi's original out going boundary condition is imposed. We find this unphysical result is due to the non-consistency between Bondi's original out going boundary condition and the Einstein equations with cosmological constant. Based on theoretical analysis, we propose an new Bondi-type out going boundary condition for the Einstein equations with cosmological constant. With this new boundary condition, the gravitational wave behavior for the Einstein equations with cosmological constant is similar to the Einstein equations without cosmological constant. In particular, when the cosmological constant goes to zero, the behavior of the Einstein equations without cosmological constant is recovered. Together with gravitational wave, electromagnetic wave is also analyzed. The boundary conditions for electromagnetic wave does not depend on the cosmological constant.

gr-qc↗

arXiv2017

Consistency between dynamical and thermodynamical stabilities for perfect fluid in $f(R)$ theories

We investigate the stability criterions for perfect fluid in $f(R)$ theories which is an important generalization of general relativity. Firstly, using Wald's general variation principle, we recast Seifert's work and obtain the dynamical stability criterion. Then using our generalized thermodynamical criterion, we obtain the concrete expressions of the criterion. We show that the dynamical stability criterion is exactly the same as the thermodynamical stability criterion. This result suggests that there is an inherent connection between the thermodynamics and gravity in $f(R)$ theories. It should be pointed out that using the thermodynamical method to determine the stability for perfect fluid is simpler and more directly than the dynamical method.

gr-qc↗