On A Quasi-local Mass
Xiao Zhang
我们修改了以前的准局域质量定义。新定义给出了准定域能量、准定域线动量和准定域质量的表达式。在空间无穷远处，它们与ADM表达式相等。此外，新的准局域能具有正性。
We modify previous quasi-local mass definition. The new definition provides expressions of the quasi-local energy, the quasi-local linear momentum and the quasi-local mass. And they are equal to the ADM expressions at spatial infinity. Moreover, the new quasi-local energy has the positivity property.
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The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the cosmic censorship conjecture in general relativity. In [14], Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy with reference in the Schwarzschild manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy with re…
We define an explicit quasi-local mass functional which is non-decreasing along all foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under fairly generic conditions.
This paper surveys recent progress on issues related to the Bartnik quasi-local mass $m_B$. In addition, we formulate a number of new problems and conjectures regarding foundational properties of the mass $m_B$. This work is dedicated with pleasure to Robert Bartnik in honor of his 60th birthday.
There are two important statements regarding the Trautman-Bondi mass [1,8,5] at null infinity: one is the positivity [7,6], and the other is the Bondi mass loss formula [1], which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect functi…
In [13], a new quasi-local energy is introduced for spacetimes with a non-zero cosmological constant. In this article, we study the small sphere limit of this newly defined quasi-local energy for spacetimes with a negative cosmological constant. For such spacetimes, the anti de-Sitter space is used as the reference for the quasi-local energy. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the quasi-local energy. The optimal embedding equation which identifies the critical points of the quasi-…
A spinorial quasilocal mass
本文利用一个流形上的旋量与另一个流形上拉回的对偶旋量，定义了带边紧流形相对于背景流形的拟局部能量，并在黎曼和洛伦兹情形下证明其正性。
Evaluating small sphere limit of the Wang-Yau quasi-local energy
本文研究了文[18，19]中Wang-Yau拟局域能的小球极限。给定时空$N$中的一个点$p$，我们考虑一个正则曲面族沿着它的未来零锥逼近$p$，并计算了Wang-Yau拟局域能量的极限.评估依赖于求解“最优嵌入方程”，其解表示准局部能量的临界点。对于具有物质场的时空，情形类似于在[7]中发现的大球极限。也就是说，存在一个局部极小值的自然解，其准局部能量的极限恢复了$p$处的应力-能量张量。在真空时空中，准局域能量向高阶消失，最优嵌入方程的求解更加复杂。然而，我们能够证明，存在一个解决方案，这是一个局部极小值，其准局部能量的限制是有关的贝尔-罗宾逊张量。与以前的工作[7]一起，这完成了Wang-Yau准局域能量与所有经典极限的一致性验证。
The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with non-negative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian Penrose inequality has also been established for the Einstein-Maxwell equations, where the lower bound on the mass also depends on the electric charge. In the context of quasi-local mass, one is interested in determining if, and for which quasi-local mass definitions, a quasi-local version of these inequalities also holds. It is known that the Brown-York quasi-…
For an admissible class of smooth compact initial data sets with boundary, we prove a comparison theorem between the Wang/Liu-Yau quasi-local mass of the boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs of the domain. Using this, we prove a quasi-local Penrose inequality that involves these quasi-local masses of the boundary and the area of an outermost marginally outer trapped surface (MOTS) in the domain or the area of minimizing minimal surface within the Jang graphs. Moreover, we obtain sufficient conditions for the (non)existence of a MOTS within a domain, in …
This article considers the quasi-local energy in reference to a general static spacetime. We follow the approach developed by the authors in [19, 20, 7, 9] and define the quasi-local energy as a difference of surface Hamiltonians, which are derived from the Einstein-Hilbert action. The new quasi-local energy provides an effective gauge independent measurement of how far a spacetime deviates away from the reference static spacetime on a finitely extended region.
Stability of Positive Mass Theorem for Static Quasi-Local Energy of Compact (Locally) Hyperbolic Graphical Manifolds
该研究针对（局部）双曲静态空间上的紧致带边界图流形，证明了其外边界静态拟局部Brown-York能量的正质量定理在Federer–Fleming平坦距离下的稳定性。
Minimizing properties of critical points of quasi-local energy
在相对论中，运动粒子的能量取决于观察者，静止质量是所有观察者中看到的最小能量。在[7]和[8]中引入的时空曲面的Wang-Yau准局部质量是通过极小化与曲面到Minkowski空间中的容许等距嵌入相关联的准局部能量来定义的。准局域能的临界点是满足欧拉-拉格朗日方程的等距嵌入。本文证明了Wang-Yau拟局部能量的临界点的局部和全局极小性质。特别地，在中曲率向量的条件下，我们证明了一个临界点局部最小化准局部能量。同样的条件也意味着，临界点是全局最小化的所有轴对称嵌入提供的图像相关联的等距嵌入位于一个全测地欧几里德3-空间。
Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface $Σ$, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasi-local mass at $Σ$. Combining the evolution formula for the quasi-local energy with a localized Penrose inequality proved in \cite{Lu-Miao}, we prove a rigidity theorem for compact $3…
We discuss our recent work [4] in which gravitational radiation was studied by evaluating the Wang-Yau quasi-local mass of surfaces of fixed size at the infinity of both axial and polar perturbations of the Schwarzschild spacetime, à la Chandrasekhar [1].
Positivity of Quasilocal Mass
受Brown和York关于拟局部能的重要工作的启发，本文给出了时空中具有正内禀曲率的类空2-曲面的拟局部能和动量表面能的定义.我们证明了满足局部能量条件的紧致类空超曲面的边界的拟局部能量是严格正的，除非时空沿着类空超曲面是平坦的。
The Bartnik quasi-local mass conjectures
本文提出了一个分析Bartnik静态真空延拓猜想的框架，证明该猜想一般不成立，但理解其有效范围仍是基本的开放问题。
Near horizon limit of the Wang--Yau quasi-local mass
本文计算了王-姚准局域质量的近视界极限，利用平均曲率向量模消失的特殊性质证明了最优嵌入的存在唯一性，还证明了该极限下准局域质量的连续性，补充了相关研究的理论结果。
A quasi-local mass for 2-spheres with negative Gauss curvature
本文将我们以前的准局部质量定义推广到高斯曲率为负的2-球面，并证明了它的正性。
In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang-Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a vacuum spacetime, the quasi-local mass decays faster and the leading order term is related to the Bel-Robinson tensor. Several new techniques of evaluating quasilocal mass are developed in this note.
Quasi-local energy for cosmological models
首先，我们简要回顾我们的协变哈密顿方法准本地能量，注意到哈密顿边界项准本地能量表达式取决于所选择的边界条件和参考配置。然后，我们提出了准本地的能量值所产生的形式主义适用于均匀的Bianchi宇宙学。最后，我们考虑准局域能量的FRW宇宙学。我们的结果不同意某些广泛接受的准本地标准。
Turnaround radius in an accelerated universe with quasi-local mass
本文应用Hawking-Hayward准定域能量结构严格地得到了广义相对论中宇宙结构的转向半径。这种准定域质量分裂成定域和宇宙部分，描述了定域引力和宇宙膨胀之间的相互作用。
Attractive gravity probe surface, positivity of quasi-local mass and Arnowitt-Deser-Misner mass expression
本文证明了在特定条件下，Geroch/Hawking拟局部质量的正性对高于三维的吸引性引力探针面成立，并对Arnowitt-Deser-Misner质量进行了评论。
On a Quasi-Local Energy Outside the Cosmological Horizon
本文研究了由儿玉公式自然引入的具有正宇宙学常数$Λ$的球对称时空的准局域能量。我们发现这种准局域能量在宇宙视界内表现良好。然而，当存在一个具有足够长康普顿波长的标量场时，准局域能量在宇宙视界外的时间演化过程中发散。这意味着准定域能量只有在宇宙视界内才有意义。
Massive quasi-normal mode
本文研究了大质量标量场的准简正模。我们，特别是调查的依赖QNM频率上的字段质量。通过研究，我们发现当场质量具有特定值时，存在寿命任意长的准简正模。当场质量超过这些值时，量子NM会消失。
Evaluation of Bartnik's quasilocal mass function
分析了Bartnik关于引力准局域能的定义。对于广泛的一类系统，Bartnik的功能是由一些真空扩展的ADM质量。作为一个例子，我们计算了史瓦西几何中非中心球的质量。在小体积的极限下，质量与体积之比变得奇异。
Quasi-Local Conservation Equations in General Relativity
利用广义相对论的一阶Kaluza-Klein形式，从四维时空（2，2）分裂的Einstein方程出发，导出了一组精确的准定域守恒方程.这些方程被解释为准本地能量，动量和角动量守恒方程。在渐近平坦时空的渐近区域中，准局域能量和能流积分分别约化为Bondi能量和能流。在球对称时空中，准局域能量变成了米斯纳-夏普能量。此外，在一般动态黑洞事件视界上，准定域能量守恒方程与Thorne { \it et al}研究的守恒方程一致.我们简要地讨论其余的准局部守恒方程。
Three Quasi-Local Masses
基于Penrose不等式提出了准局部质量的定义。通过测量指数映射的畸变给出了两个进一步的定义。
Positivity and Hoop Properties for Chen-Nester-Tung Quasi-local Energy with Analytic Reference in Spherical Symmetry
在球对称条件下，利用解析参照，给出了Chen-Nester-Tung准定域能为正的直接证明。还建立了这种能量的一个箍型定理。最后讨论了Chen-Nester-Tung准定域能与Brown-York准定域能之间的关系。
Quasi-local mass in the covariant Newtonian space-time
在广义相对论中，准定域能量-动量表达式已经由各种公式构造而成。然而，牛顿引力理论给出了一个众所周知的和唯一的准局部质量表达式（表面积分）。由于在协变的牛顿时空中建立了牛顿引力的几何表述，它提供了从相对论到牛顿理论的协变近似。利用这种近似，我们计算了协变牛顿时空中的Komar积分、Brown-York准局域能量和Dougan-Mason准局域质量。结果表明，Komar积分自然地给出了牛顿准局域质量表达式，然而，Brown-York和Dougan-Mason表达式需要进一步的条件（球对称）。
Quasi-Local Energy Conservation Law Derived From The Einstein's Equations
从广义相对论的正则（2，2）形式中类时边界面上的真空爱因斯坦方程出发，导出了准局域能量守恒定律。在渐近平坦极限和球对称时空中，准局域能量和能流积分与标准结果一致。
There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spacetime in the Bondi-Sachs coordinates. The quasi-local mass leads to a local description of the radiation that is purely gravitational at null infinity. In particular, the quasi-local mass is evaluated in terms of the news function of the Bondi-Sachs coordinates.
Relativistic tidal heating of Hamiltonian quasi-local boundary expressions
Purdue和Favata使用某些经典的伪张量计算潮汐加热。Booth和Creighton采用了Brown和York的准局部质量形式主义来证明同一主题。它们都给出了与牛顿理论相符合的结果。这里我们给出了另一个哈密顿准局部边界表达式，并且都给出了相同的期望值。这表明，潮汐加热是唯一的索恩预测。此外，我们还发现伪张量方法和拟局域方法有着本质的区别。
Positivity of quasi-local mass II
我们证明了gr-qc/0303019中关于拟局域质量正性的一个更强的证明：满足局域能量条件的紧致类空超曲面的边界的每个连通分支的拟局域能量（质量）是严格正的，除非时空沿类空超曲面是平坦的，并且类空超曲面的边界是连通的。
The mass formula for quasi-black holes
一个准黑洞，无论是非极端的还是极端的，都可以广义地定义为当物体的边界接近物体的准视界时的极限构型。我们考虑静态准黑洞的质量贡献和质量公式。当达到极限构形时，分析包括对表面应力的仔细检查.结果表明，准黑洞和纯黑洞的质量公式之间存在严格的对应关系。尽管在这两种情况下公式的推导和意义不同，但这种完美的平行性仍然存在。对于极值准黑洞，有限表面应力对总质量的贡献为零。这导致了广义相对论中的一种非常特殊的亚伯拉罕-洛伦兹电子，其中总质量具有纯电磁起源，尽管存在裸应力。
We give a brief review of the definition of the Wang-Yau quasilocal mass and discuss the evaluation of which on surfaces of unit size at null infinity of an axi-symmetric spacetime in Bondi-van der Burg-Metzner coordinates.
A general prescription for constructing quasi-local conserved quantities in General Relativity is proposed. The construction is applied to BMS symmetry generators in Newman-Unti gauge, so as to define quasi-local BMS charges. It is argued that the zero mode of this BMS charge is a promising definition of quasi-local energy.
Hawking-Hayward quasi-local energy under conformal transformations
我们推导出一个公式，描述的Hawking-Hayward准局域能量下的时空度规的共形重标度的变换。一个已知的公式转换的Misner-Sharp-Hernandez质量恢复作为一个特殊的情况。
Horizon news function and quasi-local energy-momentum flux near black hole
从视界的“准定域”定义出发，如孤立视界和动力学视界，利用标架对齐的思想，可以观察到视界附近的准定域能量动量的结果。特别地，我们从视界附近的渐近展开中找到了视界新闻函数，并用它来描述黑洞的引力通量和质量变化。
A comment on Liu and Yau's positive quasi-local mass
借助于一个简单的例子族，我们证明了由Kijowski和Liu和Yau定义的准局部质量，并证明了Liu和Yau是正的，对于平坦时空中的类空拓扑球面2-曲面，准局部质量可以是严格正的。
The Brown-York quasilocal energy is applied to three cosmological problems which have previously been studied with the Hawking-Hayward quasilocal energy (Newtonian simulations of large scale structure formation, turnaround radius in the present accelerating universe, and lensing by the cosmological constant). It is found that, in an appropriate gauge and to first order in the amplitude of the cosmological perturbations describing local structures, the Hawking-Hayward and the Brown-York quasilocal masses predict the same results.
Quasi-local mass on rigid or round spheres in Kerr spacetime
本文基于Hamilton方法引入并分析拓扑球面多极展开的准局部质量，并在Kerr时空中对刚性球与圆球两种曲面进行了测试。
Quasi-local masses in General relativity and their positivity: Spinor approach
本文用旋量方法证明广义相对论中拟局部质量的正性，给出Wang-Yau质量的纯拟局部证明，不依赖渐近延拓，并利用Dirac方程与特定边界条件实现。
Quasi-local variables, non-linear perturbations and back-reaction in spherically symmetric spacetimes
我们引入一个准局部积分泛函和标量准局部变量来研究一类球对称非齐次时空，推广了Lemaitre-Tolman-Bondi（LTB）尘埃解（“LTB”时空）。通过使用这些变量，我们可以将流体流动演化方程转化为Friedman-Lemaitre-Robertson-Walker（FLRW）宇宙学的非线性、协变、规范不变扰动的演化方程。在线性极限下，我们得到了长波近似下同步规范中的球微扰。形式主义有一个显着的潜力，宇宙学的应用，因为它允许一个检查各种各样的来源与不同的“状态方程”，推广已知的FLRW解决方案，理想化，但非平凡和非线性非均匀条件。准局部泛函可以被重新表述为加权的真体积平均分布，其权重因子由与准局部质能函数相关的标量不变量给出。在Buchert的适当平均形式主义中出现的反反应项可以表示为平均和准局部能量密度波动之间的差异。通过比较这个平均值与加权准本地的，我们可以定义一个结合能泛函的空间梯度的平均和准本地变量中出现的反反应条款。
A definition of quasi-local energy in a gravitational field based upon its embedding into flat space is discussed. The outcome is not satisfactory from many points of view.
Four lectures on quasi-local mass
本文基于作者于2015年9月28日至30日在蒙彼利埃大学的四次系列讲座。他从广义相对论中的质量概念开始，简要回顾了一些已知的准定域质量结构，然后讨论了丘成桐和作者在2009年提出的新的准定域能量和准定域质量。最后，简述了拟局部能正性的证明，并讨论了陈柏宁、丘成桐和作者提出的拟局部能临界点的稳定性定理。
The angular momentum and mass formulas for rotating stationary quasi-black holes
本文考虑了当边界接近于自身引力半径时，静止物体的准黑洞极限，即，准地平线结果表明，尽管准黑洞和黑洞的质量公式在推导过程和意义上存在差异，但它们的质量贡献与质量公式之间存在着完美的对应关系.对于极值准黑洞，有限表面应力对总质量的贡献为零。对于角动量，得到了与质量性质类似的结论。
The mathematical theory of isometric embedding is applied to study the notion of quasilocal mass in general relativity. In particular, I shall report some recent progress of quasilocal mass with reference to a cosmological spacetime, such as the de Sitter or the Anti-de Sitter spacetime, or a blackhole spacetime, such as the Schwarzschild spacetime. This article is based on joint work with Po-Ning Chen, Ye-Kai Wang, and Shing-Tung Yau.
Quasi-local charges and asymptotic symmetry generators
通过离壳方法的守恒电荷的准局部制定的推广，以涵盖渐近对称发电机。通过引入适合于渐近Killing向量的恒守恒电流，我们证明了渐近对称生成元可以理解为准局域电荷。我们还表明，这种结构是完全一致的壳方法。
The Schwarzschild-de Sitter (SdS) black hole solution, which has two event horizons, is considered to examine the relation between the quasi-localized energy complexes on ${\cal M}$ and the heat flows passing through its boundary $\partial {\cal M}$. Here ${\cal M}$ is the patch between cosmological event horizon and black hole event horizon of the SdS black hole solution. Conclusively, the relation, like the Legendre transformation, between the quasi-localized Einstein and Møller energy complex and the heat flows passing through the boundary is obeyed, and these two quasi-localized energy com…
Generalized Misner-Sharp quasi-local mass in Einstein-Gauss-Bonnet gravity
本文研究了在存在宇宙学常数的情况下，在爱因斯坦-高斯-邦纳引力中，具有对应于$(n-2)$维最大对称空间的对称性的高维时空中准局域质量的性质.我们假设高斯-博内耦合常数是非负的。准定域质量最近被一位作者定义为广义相对论中Misner-Sharp准定域质量的对应物。准局域质量是一个准局域守恒电荷，与由广义Kodama矢量构造的局域守恒电流相关联，并表现出与能量平衡定律相对应的统一第一定律。在渐近平坦的情况下，它收敛到类空无穷远处的Arnowitt-Deser-Misner质量，而在渐近AdS的情况下，它收敛到无穷远处的Padilla-Tekin和Padilla质量。在主能量条件下，我们证明了任何$k$的准局部质量的单调性，而对于$k=1$和具有附加条件的$k=0$，则证明了具有正则中心的非俘获超曲面上的正性，其中$k=\pm1,0$是等势面的每个空间截面的常截面曲率.在耦合常数之间的一种特殊关系下，对任何$k$，在没有上述假设的情况下，准定域质量都是正的。我们还利用广义Kodama向量对所有真空解进行了分类。最后，对进一步推广Lovelock引力中的准局域质量提出了几点建议。
A recent generalization of the Hawking-Hayward quasilocal energy to scalar-tensor gravity is adapted to general spherically symmetric geometries. It is then applied to several black hole and other spherical solutions of scalar-tensor and $f({\cal R}) $ gravity. The relations of this quasilocal energy with the Abreu-Nielsen-Visser gauge and the Kodama vector are discussed.
Quasi-Single Field Inflation with Large Mass
研究了准单场暴胀模型中等曲率子质量M大于哈勃参数H阶数时等曲率子对密度扰动的影响。我们分析计算校正功率谱，领先的顺序耦合，但准确的所有质量值。这验证了以前在0<M<3 H/2范围内的数值结果，并表明，在大质量极限下，修正是H^2/M^2量级的。
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