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Naqing Xie

Publications and source records attributed to Naqing Xie.

29 records · Page 2Linked to original sources
arXiv2015

The positive energy theorem for asymptotically anti-de Sitter spacetimes

We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary $t$-slice in anti-de Sitter spacetime. In particular, when $t=0$, it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We also show that the determinant of energy-momentum endomorphism ${\bf Q}$ is the geometric invariant of asymptotically anti-de Sitter spacetimes.

gr-qc↗
arXiv2013

Localized Penrose inequality for the Liu-Yau mass in spherical symmetry

For an asymptotically flat initial data, the Penrose inequality gives a lower bound of the Arnowitt-Deser-Misner total mass of a spacetime in terms of the area of certain surfaces representing black holes. This is a deep and beautiful refinement of the famous positive mass theorem and it plays an important role in the study of gravitational collapse. Gravitational collapse can also happen if sufficient mass is concentrated into a finite region. This motivates us to seek a localized version of the Penrose inequality. In this Letter, we successfully make a precise statement of this form for the Liu-Yau quasi-local mass in spherical symmetry.

gr-qc↗
arXiv2011

Critical points of Wang-Yau quasi-local energy

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $Σ$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $Ω$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $Σ$ has positive Gaussian curvature and all boundary components of $Ω$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $Σ$ in $Ω$ and $H_0$ is the mean curvature of $Σ$ when isometrically embedded in $R^3$. If $Ω$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $Σ$ in $Ω$ is a strict local minimum of the Wang-Yau quasi-local energy of $Σ$, 2. on a small perturbation $\tildeΣ$ of $Σ$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tildeΣ$.

math.DG↗
arXiv2010

Quasilocal Energy in General Relativity

We show that the quasilocal mass defined by Wang and Yau is not well-defined at spatial infinity. It approaches neither the ADM mass nor the ADM energy. We suggest an alternative scheme which retains all the desirable characteristics of the Wang-Yau mass and, in addition, asymptotes to the ADM energy at infinity.

gr-qc↗
arXiv2009

The Brown-York mass and the Thorne hoop conjecture

The Thorne hoop conjecture is an attempt to make precise the notion that gravitational collapse occurs if enough energy is compressed into a small enough volume, with the `size' being defined by the circumference. We can make a precise statement of this form, in spherical symmetry, using the Brown-York mass as our measure of the energy. Consider a spherical 2-surface in a spherically symmetric spacetime. If the Brown-York mass, $M_{BY}$, and the circumference, $C$, satisfy $C < 2πM_{BY}$, then the system must either have emerged from a white hole or will collapse into a black hole. We show that no equivalent result can hold true using either the Liu-Yau mass, $M_{LY}$ or the Wang-Yau mass, $M_{WY}$. This forms a major obstacle to any attempt to establish a Thorne-type hoop theorem in the general case based on either the Liu-Yau or the Wang-Yau mass.

gr-qc↗
arXiv2009

Positive mass theorems for asymptotically de Sitter spacetimes

We use planar coordinates as well as hyperbolic coordinates to separate the de Sitter spacetime into two parts. These two ways of cutting the de Sitter give rise to two different spatial infinities. For spacetimes which are asymptotic to either half of the de Sitter spacetime, we are able to provide definitions of the total energy, the total linear momentum, the total angular momentum, respectively. And we prove two positive mass theorems, corresponding to these two sorts of spatial infinities, for spacelike hypersurfaces whose mean curvatures are bounded by certain constant from above.

math.DG↗
arXiv2009

Some estimates of Wang-Yau quasilocal energy

Given a spacelike 2-surface $Σ$ in a spacetime $N$ and a constant future timelike unit vector $T_0 $ in $\R^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(Σ, X, T_0)$ for a given isometric embedding $X$ of $Σ$ into a flat 3-slice in $\R^{3,1}$. The quantity $ E(Σ, X, T_0) $ itself depends on the choice of $X$, however the infimum of $ E(Σ, X, T_0)$ over $ T_0 $ does not. In particular, when $Σ$ lies in a time symmetric 3-slice in $N$ and has nonnegative Brown-York quasilocal mass $\mby(Σ)$, our estimates show that $\inf\limits_{T_0}E(Σ, X, T_0)$ equals $ \mby (Σ)$. We also study the spatial limit of $ \inf\limits_{T_0}E(S_r,X_r,T_0)$, where $S_r$ is a large coordinate sphere in a fixed end of an asymptotically flat initial data set $(M, g, p)$ and $X_r$ is an isometric embeddings of $S_r$ into $\mathbb{R}^3 \subset \mathbb{R}^{3,1}$. We show that if $(M, g, p)$ has future timelike ADM energy-momentum, then $\lim\limits_{r\to\infty}\inf\limits_{T_0}E(S_r,X_r,T_0)$ equals the ADM mass of $(M, g, p)$.

math.DG↗