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The specification of energy for gravitating systems has been an unsettled issue since Einstein proposed his pseudotensor. It is now understood that energy-momentum is \emph{quasi-local} (associated with a closed 2-surface). Here we consider quasi-local proposals (including pseudotensors) in the Lagrangian-Noether-Hamiltonian formulations. There are two ambiguities: (i) there are many possible expressions, (ii) they depend on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. The Hamiltonian perspective helped us to make a remarkable discovery: with an isometric Minkowski reference a large class of expressions---namely all those that agree with the Einstein pseudotensor's Freud superpotential to linear order---give a common quasi-local energy value. Moreover, with a best-matched reference on the boundary this is the Wang-Yau mass value., Comment: 14 pages, accepted by Gen Rel Grav, selected as an Editor's Choice

The energy of gravitating systems has been an issue since Einstein proposed general relativity: considered to be ill defined, having no proper local density. Energy-momentum is now regarded as \emph{quasi-local} (associated with a closed 2-surface). We consider the pseudotensor and quasi-local proposals in the Lagrangian-Noether-Hamiltonian formulations. There are two ambiguities: (i) many expressions, (ii) each depends on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. Our remarkable discovery is that with a 4D isometric Minkowski reference a large class of expressions---those that agree with the Einstein pseudotensor's Freud superpotential to linear order---give a common quasi-local energy value. With a best-matched reference on the boundary this value is the non-negative Wang-Yau mass., Comment: 7 pages, Honorable Mention in the 2018 Essay Competition of the Gravity Research Foundation

In this paper, we show the rigidity of isometric immersions for a Riemannian manifold of dimension $n-1$ into the light cone of $n+1$ dimensional Minkowski, de Sitter and anti-de Sitter spacetimes for $n\geq 3$., Comment: Fix a mistake. All comments are welcome

We study the quasi-local energy (QLE) and the surface geometry for Kerr spacetime in the Boyer-Lindquist coordinates without taking the slow rotation approximation. We also consider in the region $r\leq2m$, which is inside the ergosphere. For a certain region, $r>r_{k}(a)$, the Gaussian curvature of the surface with constant $t,r$ is positive, and for $r>\sqrt{3}a$ the critical value of the QLE is positive. We found that the three curves: the outer horizon $r=r_{+}(a)$, $r=r_{k}(a)$ and $r=\sqrt{3}a$ intersect at the point $a=\sqrt{3}m/2$, which is the limit for the horizon to be isometrically embedded into $\mathbb{R}^3$. The numerical result indicates that the Kerr QLE is monotonically decreasing to the ADM $m$ from the region inside the ergosphere to large $r$. Based on the second law of black hole dynamics, the QLE is increasing with respect to the irreducible mass $M_{\mathrm{ir}}$. From a results of Chen-Wang-Yau, we conclude that in a certain region, $r>r_{h}(a)$, the critical value of the Kerr QLE is a global minimum., Comment: 20 pages

The Hamiltonian for dynamic geometry generates the evolution of a spatial region along a vector field. It includes a boundary term which determines both the value of the Hamiltonian and the boundary conditions. The value gives the quasi-local quantities: energy-momentum, angular-momentum and center-of-mass. The boundary term depends not only on the dynamical variables but also on their reference values; the latter determine the ground state (having vanishing quasi-local quantities). For our preferred boundary term for Einstein's GR we propose 4D isometric matching and extremizing the energy to determine the reference metric and connection values., Comment: 6 pages, contribution to the Proceedings of the Second LeCosPA Symposium "Everything about Gravity", Taipei, 14-18 Dec., 2015

In this paper, we show that the Chen-Nester-Tung (CNT) quasi-local energy is closely related to the Wang-Yau (WY) quasi-local mass. As a particular example, we compute the second variation of the CNT quasi-local energy for axially symmetric Kerr-like spacetimes with axially symmetric embeddings at the obvious critical point (0,0) and find that it is a saddle critical point in most of the cases. Also, as a byproduct, we generalize a previous result about the coincidence of CNT quasi-local energy and Brown-York mass for Kerr-like spacetimes by Tam and the first author to general spacetimes., Comment: 18 pages

In this work we study the quasilocal energy as in [11] for a constant radius surface in Kerr spacetime in Boyer-Lindquist coordinates. We show that under suitable conditions for isometric embedding, for a stationary observer the quasilocal energy defined in [11] for constant radius in a Kerr like spacetime is exactly equal to the Brown-York quasilocal energy [2]. By some careful estimations, we show that for a constant radius surface in the Kerr spacetime which is outside the ergosphere the embedding conditions for the previous result are satisfied. We discuss extremal solutions as described in [14]. We prove a uniqueness result. We find all extremal solutions in the Minkowski spacetime. Finally, we show that near the horizon of the Kerr spacetime for the small rotation case the extremal solutions are trivial., Comment: There are some contents added in section 4 and also some refinement on abstract and introduction

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.

The Hamiltonian for physical systems and dynamic geometry generates the evolution of a spatial region along a vector field. It includes a boundary term which not only determines the value of the Hamiltonian, but also, via the boundary term in the variation of the Hamiltonian, the boundary conditions. The value of the Hamiltonian comes from its boundary term; it gives the quasi-local quantities: energy-momentum and angular-momentum/center-of-mass. This boundary term depends not only on the dynamical variables but also on their reference values; these reference values determine the ground state---the state having vanishing quasi-local quantities. Here our concern is how to select on the two-boundary the reference values. To determine the "best matched" reference metric and connection values for our preferred boundary term for Einstein's general relativity, we propose on the boundary two-surface (i) \emph{four dimensional} isometric matching, and (ii) extremizing the value of the energy., Comment: 4 pages

The boundary term of the gravitational Hamiltonian can be used to give the values of the quasi-local quantities as long as one can provide a suitable evolution vector field and an appropriate reference. On the two-surface boundary of a region we have proposed using {\em four dimensional isometric matching} between the dynamic spacetime and the reference geometry along with energy extremization to find both the optimal reference matching and the appropriate quasi-Killing vectors. Here we consider the axisymmetric spacetime case. For the Kerr metric in particular we can explicitly solve the equations to find the best matched reference and quasi-Killing vectors. This leads to the exact expression for the quasi-local boundary term and the values of our optimal quasi-local energy and angular momentum., Comment: 19 pages, 1 figure

The Hamiltonian for dynamic geometry generates the evolution of a spatial region along a vector field. It includes a boundary term which determines both the value of the Hamiltonian and the boundary conditions. The value gives the quasi-local quantities: energy-momentum, angular-momentum/center-of-mass. The boundary term depends not only on the dynamical variables but also on their reference values; the latter determine the ground state (having vanishing quasi-local quantities). For our preferred boundary term for Einstein's GR we propose 4D isometric matching and extremizing the energy to determine the reference metric and connection values., Comment: submitted to the proceedings of the conference on Relativity and Gravitation: 100 Years after Einstein in Prague

We present two complementary approaches for determining the reference for the covariant Hamiltonian boundary term quasi-local energy and test them on spherically symmetric spacetimes. On the one hand, we isometrically match the 2-surface and extremize the energy. This can be done in two ways, which we call programs I (without constraint) and II (with additional constraints). On the other hand, we match the orthonormal 4-frames of the dynamic and the reference spacetimes. Then, if we further specify the observer by requiring the reference displacement to be the timelike Killing vector of the reference, the result is the same as program I, and the energy can be positive, zero, or even negative. If, instead, we require that the Lie derivatives of the two-area along the displacement vector in both the dynamic and reference spacetimes to be the same, the result is the same as program II, and it satisfies the usual criteria: the energies are non-negative and vanish only for Minkowski (or anti-de Sitter) spacetime., Comment: 16 pages, no figures

For a given timelike displacement vector the covariant Hamiltonian quasi-local energy expression requires a proper choice of reference spacetime. We propose a program for determining the reference by embedding a neighborhood of the two-sphere boundary in the dynamic spacetime into a Minkowski reference, so that the two sphere is embedded isometrically, and then extremizing the energy to determine the embedding variables. Applying this idea to Schwarzschild spacetime, we found that for each given future timelike displacement vector our program gives a unique energy value. The static observer measures the maximal energy. Applied to the Friedmann-Lemaitre-Robertson-Walker spacetime, we find that the maximum energy value is nonnegative; the associated displacement vector is the unit dual mean curvature vector, and the expansion of the two-sphere boundary matches that of its reference image. For these spherically symmetric cases the reference determined by our program is equivalent to isometrically matching the geometry at the two-sphere boundary and taking the displacement vector to be orthogonal to the spacelike constant coordinate time hypersurface, like the timelike Killing vector of the Minkowski reference., Comment: 12 pages

A quasi-local energy for Einstein's general relativity is defined by the value of the preferred boundary term in the covariant Hamiltonian formalism. The boundary term depends upon a choice of reference and a time-like displacement vector field (which can be associated with an observer) on the boundary of the region. Here we analyze the spherical symmetric cases. For the obvious analytic choice of reference based on the metric components, we find that this technique gives the same quasi-local energy values using several standard coordinate systems and yet can give different values in some other coordinate systems. For the homogeneous-isotropic cosmologies, the energy can be non-positive, and one case which is actually flat space has a negative energy. As an alternative, we introduce a way to determine the choice of both the reference and displacement by extremizing the energy. This procedure gives the same value for the energy in different coordinate systems for the Schwarzschild space, and a non-negative value for the cosmological models, with zero energy for the dynamic cosmology which is actually Minkowski space. The timelike displacement vector comes out to be the dual mean curvature vector of the two-boundary., Comment: 21 pages; revised version to appear in CQG

We have proposed a program for determining the reference for the quasi-local energy defined in the covariant Hamiltonian formalism. Our program has been tested by applying it to the spherically symmetric spacetimes. With respect to different observers we found that the quasi-local energy can be positive, zero, or even negative. The observer measuring the maximum energy was identified; the associated energy is positive for both the Schwarzschild and the Friedmann-Lema\^{i}tre-Robertson-Walker spacetimes., Comment: 4 pages

First we briefly review our covariant Hamiltonian approach to quasi-local energy, noting that the Hamiltonian-boundary-term quasi-local energy expressions depend on the chosen boundary conditions and reference configuration. Then we present the quasi-local energy values resulting from the formalism applied to homogeneous Bianchi cosmologies. Finally we consider the quasi-local energies of the FRW cosmologies. Our results do not agree with certain widely accepted quasi-local criteria., Comment: Contributed to International Symposium on Cosmology and Particle Astrophysics (CosPA 2006), Taipei, Taiwan, 15-17 Nov 2006