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Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds ( Σ ≅ S 2 , g ) (\Sigma \cong \mathbb {S}^2,g) , with g g satisfying λ 1 ≔ λ 1 ( − Δ g + K ( g ) ) > 0 \lambda _1 ≔\lambda _1(-\Delta _g + K(g))>0 , where λ 1 \lambda _1 is the first eigenvalue of the operator − Δ g + K ( g ) -\Delta _g+K(g) and K ( g ) K(g) is the Gaussian curvature of g g , with control on the ADM mass of the extension. Remarkably, this procedure allowed them to compute the Bartnik mass in this so-called minimal case; the Bartnik mass is a notion of quasi-local mass in General Relativity which is very challenging to compute. In this survey, we describe the Mantoulidis–Schoen construction, its impact and influence in subsequent research related to Bartnik mass estimates when the minimality assumption is dropped, and its adaptation to other settings of interest in General Relativity.

The central (surface) energy-density, $E_0 (E_R)$, which appears in the expression of total static and spherical mass, $M$ (corresponding to the total radius $R$) is defined as the density measured only by one observer located at the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF). Since the mass, $M$, depends only on the central (surface) density for most of the equations of state (EOSs) and/or exact analytic solutions of Einstein's field equations available in the literature, the central (surface) density measured in the preferred frame (that is, in the MCRF) appears to be not in agreement with the coordinate invariant form of the field equations that result for the source mass, $M$. In order to overcome the use of any preferred coordinate system (the MCRF) defined for the central (surface) density in the literature, we argue for the first time that the said density may be defined in the coordinate invariant form, that is, in the form of the average density ($3M/4\pi R^3)$ of the the configuration which turns out to be independent of the radial coordinate $`r'$ and depends only on the central (surface) density of the configuration. In this connection, we further argue that the central (surface) density of the structure should be {\em independent} of the density measured on the other boundary (surface/central) because there exists no a priori relation between the radial coordinate $`r'$ and the proper distance from the centre of the sphere to its surface \cite{Ref1}. In the light of this reasoning, the various EOSs and analytic solutions of Einstein's field equations in which the central and the surface density are {\em interdependent} can not fulfill the definition of central (surface) density measured only by one observer located in the MCRF at the centre (surface) of the configuration., Comment: 6 pages

In this chapter we address the important but subtle concept of mass in general relativity. Following Wald (General relativity, University of Chicago Press, 1984), we start by defining the Komar mass for stationary spacetimes. We then discuss field theory and introduce the Einstein–Hilbert action as a means of motivating the definition of the ADM mass. Finally, we prove the (Riemannian) positive mass theorem and the (Riemannian) Penrose inequality for graphs, following Lam (The graphs cases of the Riemannian positive mass and Penrose inequalities in all dimensions, arXiv:1010.4256, 2010).

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., Comment: to appear in Advanced Studies in Pure Mathematics 85, 2020, The Role of Metrics in the Theory of Partial Differential Equations

The self-consistency of general relativity requires that the spacetime total energy must be positive for isolated gravitational systems with positive mass density. It is called the positive energy conjecture. It plays a fundamental role in general relativity, moreover, the total mass is well-defined if it holds true. In the case of zero cosmological constant, the positive energy conjecture was first proved by Schoen and Yau in the late 1970s, and soon later by Witten using different method. In this paper, we provide a review on its recent development for 4-dimensional physical spacetimes, which includes the positive energy theorem, Kerr constraint, the positive energy theorem near null infinity and the positivity of the Bondi energy, as well as the positive energy theorem for positive cosmological constant and for negative cosmological constant.

Geometrically the phase space of a mechanical system involves the co-tangent bundle of the configuration space. The phase space of a relativistic field theory is infinite dimensional and can be endowed with a symplectic structure defined in a perfectly co-variant manner that is very useful for discussing symmetries and conserved quantities of the system. In general relativity the symplectic structure takes Darboux form and it is shown in this work that the presence of a cosmological constant does not change this conclusion. For space-times that admit time-like Killing vectors the formalism can be used to define mass in general relativity and it is known that, for asymptotically flat black holes, this mass is identical the the usual Arnowitt-Desner-Misner mass while for asymptotically anti-de Sitter Kerr metrics it is the same as the Henneaux-Teitelboim mass. We show that the same formalism can also be used to derive the Brown-York mass and the Bondi mass for stationary space times, in particular the Brown-York mass has a natural interpretation in terms of differential forms on the space of solutions of the theory., Comment: 34 pages, 2 figures

In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects that the mass should vanish. We describe the Wang-Yau quasi-local mass whose definition is motivated by this rigidity property and by the Hamilton-Jacobi analysis of the Einstein-Hilbert action. In addition, we survey recent results on the minimizing property the Wang-Yau quasi-local mass., Comment: 11 pages

This chapter explains the notions of mass in general relativity. The positive mass theorem, when it exists, guarantees stability. We also introduce the idea of Wick rotation to imaginary time, and explain how to use this technique to study possible phase transitions between various gravitational configurations. This includes, specifically, the Hawking-Page phase transition between a Schwarzschild-AdS black hole and a “thermal AdS” spacetime; and a similar phase transition discovered by Horowitz and Myers, between toral AdS black holes and a type of soliton; which configurations are preferred is dictated by their free energy.

The de Sitter special relativity on the Beltrami-de Sitter-spacetime and Snyder's model in the momentumspace can be combined together with an IR-UV duality to get the complete Yang model at both classical and quantumlevels, which are related by the proposed Killing quantization.It is actually a ...

In this paper, an ADM mass formula for asymptotically de Sitter(dS) space-time is derived from the energy-momentum tensor. We take the vacuum dS space as the background and investigate the ADM mass of the (d + 3)-dimensional sphere-symmetric space with a positive cosmological constant, and find that the ADM mass of asymptotically dS space is based on the ADM mass of Schwarzschild field and the cosmological background brings some small mass contribution as well.

In this paper, we will show that the limit of the Brown–York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.

A problem in general relativity is how to extract physical information from solutions to the Einstein equations. Most often information is found from special conditions, e.g., special vector fields, symmetries or approximate symmetries. Our concern is with asymptotically flat space–times with approximate symmetry: the BMS group. For these spaces the Bondi four-momentum vector and its evolution, found at infinity, describes the total energy–momentum and the energy–momentum radiated. By generalizing the simple idea of the transformation of (electromagnetic) dipoles under a translation, we define (analogous to center of charge) the center of mass for asymptotically flat Einstein–Maxwell fields. This gives kinematical meaning to the Bondi four-momentum, i.e., the four-momentum and its evolution is described in terms of a center of mass position vector, its velocity and spin-vector. From dynamical arguments, a unique (for our approximation) total angular momentum and evolution equation in the form of a conservation law is found.

The two exterior solutions for a charged line-mass are examined. In both cases the mass per unit length is negative.

This chapter introduces the physics of horizons and reviews the basic stationary black holes of General Relativity (Schwarzschild, Reissner-Nordstrom, Kerr, and Kerr-Newman) in various coordinate systems. The various energy conditions of relativistic gravity are also discussed.

Boundary charges in gauge theories (like the ADM mass in general relativity) can be understood as integrals of linear conserved n-2 forms of the free theory obtained by linearization around the background. These forms are associated one-to-one to reducibility parameters of this background (like the time-like Killing vector of Minkowski space-time). In this paper, closed n-2 forms in the full interacting theory are constructed in terms of a one parameter family of solutions to the full equations of motion that admits a reducibility parameter. These forms thus allow one to apply Stokes theorem without bulk contributions and, provided appropriate fall-off conditions are satisfied, they reduce asymptotically near the boundary to the conserved n-2 forms of the linearized theory. As an application, the first law of black hole mechanics in asymptotically anti-de Sitter space-times is derived., 17 pages Latex file, improved presentation, main results unchanged, additional section on first law, additional references

This paper addresses the question of whether Witten's proof of positive ADM energy for classical general relativity can be extended to give a proof of positive energy for a non-perturbative quantization of general relativity. To address this question, a set of conditions is shown to be sufficient for showing the positivity of a Hamiltonian operator corresponding to the ADM energy. One of these conditions is a particular factor ordering for the constraints of general relativity, in a representation where the states are functionals of the Ashtekar connection, and the auxiliary, Witten spinor. These developments are partly based on results derived with Artem Starodubtsev (unpublished notes, 2004)., 14 pages, no figures

The Special Theory of Relativity: Preface 1. Relativity and Physical Principles 2. The Experimental Basis 3. Relativity and Length 4. Time 5. Velocity and Acceleration 6. Mass, Energy and Force 7. Electromagnetic Measurements 8. Transition to General Relativity

In this sequel paper we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient conditions of divergence type for the monotonicity of the Hawking mass. These flows of surfaces may have connections to the problem in general relativity of bounding the total mass of a spacetime from below by the quasi-local mass of spacelike 2-surfaces in the spacetime., 19 pages, 1 figure

We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy-momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig-\'O Murchadha-Regge-Teitelboim center of mass and study analogous examples in the Schwarzschild spacetime., Comment: References are updated, and typos and computational errors are corrected. Accepted by Ann. Henri Poincare

In relativity, mass depends on the observer, i.e. mass is an observer dependent quantity. Chapter 9 discusses three methods for defining relativistic mass formula. The experiment made by Guye and Lavanchy to verify relativistic mass formula is also explained. Lorentz Transformation of relativistic mass is explained.

The purpose of this work is to study the structure and nature of the singularities of wavefronts in flat space-time. We computed the behavior at the singularities of important objects that take place in the null surface formulation of general relativity. As a secondary result we show that the Minkowski space-time with non-trivial null surfaces is a solution of the null surface approach to general relativity.

Scalar fields bounded by their own gravitational field can form absolutely stable boson stars, resembling neutron stars. Within general relativity, we construct for the first time the corresponding differentially rotating, localized configurations via numerical integration. The ratio of conserved angular momentum and particle number turns out to be an integer a . The resulting axisymmetric metric, the energy density and the Tolman mass are completely regular.

An attempt is made here to integrate and consolidate foundational ideas of electrodynamics and special relativity theories into a consistent structure, from the point of view of the application-oriented applied scientist and engineer. A discussion of this kind is best served by minimizing the sophisticated mathematical tools to the bare minimum, and avoiding lengthy calculations. Some of the difficulties encountered in educating engineers and applied scientist are exactly of this kind, hence a topsy-turvy presentation of special relativity is used, in which fundamental postulates and conclusions exchange roles. With the recent interest in direct time-domain methods, the pre-sent discussion strives to elucidate the fundamental problems involved. Essentially it is argued that appending Maxwell's theory with constitutive relations is in general valid for systems homogeneous in time and space. Spatiotemporal constitutive relations are thus becoming differential operators. Strictly speaking, inhomogeneous systems in time and space are appropriate for non-dispersive systems, or as an approximation. Although Electrodynamics in moving media is often considered a purely academic subject, the understanding of its implementation as proposed by Minkwoski is a crucial cornerstone to our understanding of the physical models at hand. To that end, a generalized approach to the Minkowski method for electrodynamics in moving media is presented. This applies to linear as well as nonlinear media. Finally, a novel representation is proposed for nonlinear constitutive differential operators. This contributes to the spatiotemporal representation of Volterra type nonlinear constitutive parameters.

We obtain a general covariant conservation law of energy momentum in complex general relativity by general displacement transformation in terms of Ashtekar new variables. The energy is exactly the ADM Hamiltonian on the constraint surface on condition that an appropriate time function is chosen. The energy momentum is gauge-covariant and commutes with all the constraints whence they are physical observables. Furthermore, the Poisson brackets of the momentum and the internal charges form a three-Poincare algebra.

A post-Newtonian approximation scheme for general relativity is defined using the Arnowitt-Deser-Misner formalism. The scheme is applied to perfect fluids and point-mass systems. The two-body point-mass Hamiltonian is given explicitly up to the post2.5-Newtonian order.

We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $\gamma$ be a smooth curve in $\mathbb{R}^3$ that is the graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We give a new proof that if $\gamma$ has nonnegative (or nonpositive) torsion, then $\gamma$ has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian $\mathbb{R}^{2,1}$ which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity., Comment: 13 pages, 7 figures; references added in second version

Chapter 2 begins with the introduction of the metric matrix and the effect of a homogeneous field of gravitation on a mass particle. Then the motion on geodesic lines in a gravitational field is considered. The general transformation of coordinates leads to the Christoffel matrix and the Riemannian curvature matrix. With the help of the Ricci matrix the Theory of General Relativity of Einstein can then be formulated.

In order to avoid a speed-of-light catastrophe in General Relativity with an electromagnetic source, gauge invariance with respect to the electric charge is broken with the photon acquiring mass. The general equations for the Einstein-Maxwell system are derived for the case with massive photons. Nonminimal couplings which might compete with the small minimal photon mass term are included.

By means of pseudotensor calculations in general relativity, we calculate the electric energy in two particular cases for the Reissner-Nordstrom metric.

We consider the Denisov-Solov'ov example which shows that the inertial mass is not well defined in General Relativity. It is shown that the mathematical reason why this is true is a wrong application of the Stokes theorem. Then we discuss the role of the order of asymptotically flatness in the definition of the mass. In conclusion some comments on conservation laws in General Relativity are presented.

We systematically analyze the nonlinear partial differential equation that determines the behaviour of a bounded radiating spherical mass in general relativity. Four categories of solution are possible. These are identified in terms of restrictions on the gravitational potentials. One category of solution can be related to the horizon function transformation which was recently introduced. A Lie symmetry analysis of the resulting Riccati equation shows that several new classes of exact solutions are possible. The relationship between the horizon function, Euclidean star models and other earlier investigations is clarified., 17 pages, submitted for publication

The notion of center of mass is reviewed and three natural definitions for the relativistic regime are proposed. This construction can be explicitly calculated by means of an algorithm which is described below.

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