Narrow Results

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.

We set to weigh the black holes at their event horizons in various spacetimes and obtain masses which are substantially higher than their asymptotic values. In each case, the horizon mass of a Schwarzschild, Reissner–Nordström, or Kerr black hole is found to be twice the irreducible mass observed at infinity. The irreducible mass does not contain electrostatic or rotational energy, leading to the inescapable conclusion that particles with electric charges and spins cannot exist inside a black hole. This is proposed as the External Energy Paradigm. A higher mass at the event horizon and its neighborhood is obligatory for the release of gravitational waves in binary black hole merging. We describe how these horizon mass values are obtained in the quasi-local energy approach and applied to the black holes of the first gravitational waves GW150914.

Our topic concerns a long standing puzzle: The energy of gravitating systems. More precisely we want to consider, for gravitating systems, how to best describe energy–momentum and angular momentum/center-of-mass momentum (CoMM). It is known that these quantities cannot be given by a local density. The modern understanding is that (i) they are quasi-local (associated with a closed 2-surface), (ii) they have no unique formula, (iii) they have no reference frame independent description. In the first part of this work, we review some early history, much of it not so well known, on the subject of gravitational energy in Einstein's general relativity (GR), noting especially Noether's contribution. In the second part, we review (including some new results) much of our covariant Hamiltonian formalism and apply it to Poincaré gauge theories of gravity (PG), with GR as a special case. The key point is that the Hamiltonian boundary term has two roles, it determines the quasi-local quantities, and furthermore, it determines the boundary conditions for the dynamical variables. Energy–momentum and angular momentum/CoMM are associated with the geometric symmetries under Poincaré transformations. They are best described in a local Poincaré gauge theory. The type of spacetime that naturally has this symmetry is Riemann–Cartan spacetime, with a metric compatible connection having, in general, both curvature and torsion. Thus our expression for the energy–momentum of physical systems is obtained via our covariant Hamiltonian formulation applied to the PG.

We discuss our recent work [P.-N. Chen, M.-T. Wang and S.-T. Yau, Quasi-local mass in the gravitational perturbations of black holes, to appear.] 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. [S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences (Oxford University Press, New York, 1998).]

The evolution of a generally covariant theory is under-determined. One hundred years ago such dynamics had never before been considered; its ramifications were perplexing, its future important role for all the fundamental interactions under the name gauge principle could not be foreseen. We recount some history regarding Einstein, Hilbert, Klein and Noether and the novel features of gravitational energy that led to Noether’s two theorems. Under-determined evolution is best revealed in the Hamiltonian formulation. We developed a covariant Hamiltonian formulation. The Hamiltonian boundary term gives covariant expressions for the quasi-local energy, momentum and angular momentum. Gravity can be considered as a gauge theory of the local Poincaré group. The dynamical potentials of the Poincaré gauge theory of gravity are the frame and the connection. The spacetime geometry has in general both curvature and torsion. Torsion naturally couples to spin; it could have a significant magnitude and yet not be noticed, except on a cosmological scale where it could have significant effects.

We evaluate the mass of the black holes of GW150914 at their event horizons via quasi-local energy approach and obtain the values of 71 and 57 solar masses, compared to their asymptotic values of 36 and 29 units, respectively, as reported by LIGO. A higher mass at the event horizon is compulsory in order to overcome the huge negative gravitational potential energy surrounding the black holes and allow for the emission of gravitational waves during merging. We estimate the initial mass of the stars which collapsed to form the black holes from the horizon mass and obtain the impressive values of 95 and 76 solar masses for these progenitor stars.

We reveal three new discoveries in black hole physics previously unexplored in the Hawking era. These results are based on the remarkable 1971 discovery of the irreducible mass of the black hole by Christodoulou and Ruffini, and subsequently confirmed by Hawking.

1. The Horizon Mass Theorem shows that the mass at the event horizon of any black hole: neutral, charged, or rotating, depends only on twice its irreducible mass observed at infinity.

2. The External Energy Conjecture proposes that the electrostatic and rotational energy of a general black hole exist completely outside the horizon due to the nature of the irreducible mass.

3. The Moment of Inertia Property shows that every Kerr black hole has a moment of inertia. When the rotation stops, there is an irreducible moment of inertia as a result of the irreducible mass.

Thus after 50 years, the irreducible mass has gained a new and profound significance. No longer is it just a limiting value in energy extraction, it can also determine black hole dynamics and structure. What is believed to be a black hole is a physical body with an extended structure. Astrophysical black holes are likely to be massive compact objects from which light cannot escape.

We consider an unusual time slicing for the static spherically symmetric metrics. For the vacuum case this is the Schwarzschild metric in the Painlevé-Gullstrand form. For this slicing the spatial metric is fiat, and the lapse is just unity; all the dynamic geometry is encoded in what is supposed to be a gauge parameter: the shift vector. One consequence is that the standard ADM energy expression vanishes (contrary to the idea that vanishing energy should be Minkowski space). On the other hand, for an appropriate choice of reference and time displacement vector, our preferred quasilocal Hamiltonian boundary term expression gives a finite energy, namely 2M.

Our covariant Hamiltonian formalism gives a certain preferred Hamiltonian boundary term for quasi-local quantities which depends on the boundary conditions, plus a reference and displacement vector choice. With appropriate choices we found the quasi-local energy for the cosmological models. Homogeneous choices give vanishing energy for all regions of Bianchi class A models and positive energy for class B. Isotropic choices give energies proportional to the curvature parameter k: ie, vanishing for the flat case, positive for the closed model and negative(!) for the open model. Our values are consistent with the requirement that the energy vanishes for closed models. We have some conclusions regarding the best reference choice and two quasi-local desiderata: positivity and zero energy iff Minkowski space.

We discuss our recent work [P.-N. Chen, M.-T. Wang and S.-T. Yau, Quasi-local mass in the gravitational perturbations of black holes, to appear.] 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. [S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences (Oxford University Press, New York, 1998).]

The evolution of a generally covariant theory is under-determined. One hundred years ago such dynamics had never before been considered; its ramifications were perplexing, its future important role for all the fundamental interactions under the name gauge principle could not be foreseen. We recount some history regarding Einstein, Hilbert, Klein and Noether and the novel features of gravitational energy that led to Noether’s two theorems. Under-determined evolution is best revealed in the Hamiltonian formulation. We developed a covariant Hamiltonian formulation. The Hamiltonian boundary term gives covariant expressions for the quasi-local energy, momentum and angular momentum. Gravity can be considered as a gauge theory of the local Poincaré group. The dynamical potentials of the Poincaré gauge theory of gravity are the frame and the connection. The spacetime geometry has in general both curvature and torsion. Torsion naturally couples to spin; it could have a significant magnitude and yet not be noticed, except on a cosmological scale where it could have significant effects.

We investigate the energy distribution of a black hole in various spacetimes as reckoned by a distant observer using the quasi-local energy approach. In each case the horizon mass of a black hole: neutral, charged or rotating, is found to be twice the irreducible mass observed at infinity. This is known as the Horizon Mass Theorem. As a consequence, the electrostatic energy and the rotational energy of a general black hole are all external quantities. Matter carrying charges and spins could only lie outside the horizon. This result could resolve several long-standing paradoxes related to known black hole properties; such as why entropy is proportional to area and not to volume, the information loss problem, the firewall problem, the internal structure and the thin shell model of a black hole.

We set to weigh the black holes at their event horizons in various spacetimes and obtain masses which are substantially higher than their asymptotic values. In each case, the horizon mass of a Schwarzschild, Reissner–Nordström, or Kerr black hole is found to be twice the irreducible mass observed at infinity. The irreducible mass does not contain electrostatic or rotational energy, leading to the inescapable conclusion that particles with electric charges and spins cannot exist inside a black hole. This is proposed as the External Energy Paradigm. A higher mass at the event horizon and its neighborhood is obligatory for the release of gravitational waves in binary black hole merging. We describe how these horizon mass values are obtained in the quasi-local energy approach and applied to the black holes of the first gravitational waves GW150914.

Our covariant 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 using 4D isometric matching and extremizing the energy to determine the “best matched” reference metric and connection values.

Our topic concerns a long standing puzzle: The energy of gravitating systems. More precisely we want to consider, for gravitating systems, how to best describe energy–momentum and angular momentum/center-of-mass momentum (CoMM). It is known that these quantities cannot be given by a local density. The modern understanding is that (i) they are quasi-local (associated with a closed 2-surface), (ii) they have no unique formula, (iii) they have no reference frame independent description. In the first part of this work, we review some early history, much of it not so well known, on the subject of gravitational energy in Einstein’s general relativity (GR), noting especially Noether’s contribution. In the second part, we review (including some new results) much of our covariant Hamiltonian formalism and apply it to Poincaré gauge theories of gravity (PG), with GR as a special case. The key point is that the Hamiltonian boundary term has two roles, it determines the quasi-local quantities, and furthermore, it determines the boundary conditions for the dynamical variables. Energy–momentum and angular momentum/CoMM are associated with the geometric symmetries under Poincaré transformations. They are best described in a local Poincaré gauge theory. The type of spacetime that naturally has this symmetry is Riemann–Cartan spacetime, with a metric compatible connection having, in general, both curvature and torsion. Thus our expression for the energy–momentum of physical systems is obtained via our covariant Hamiltonian formulation applied to the PG.