Pdf quasilocal energymomentum and angular momentum in. The quasilocal energy is also the value of the hamiltonian that generates unit magnitude proper time translations on in the direction orthogonal to. In einsteins theory of general relativity, the physical consequences of gravitational fields are stated in the following way. Spacetime is a fourdimensional noneuclidean continuum, and the curvature of the riemannian geometry of spacetime is produced by or related to the distribution of matter in the world. Quasilocal quantities for general relativity and other.
A quasilocal mass has been a long sought after quantity in general relativity. This is called the adm energy momentum and the bondi energy momentum when the system is viewed from spatial in. Also, the hamiltonian describing general relativity on a manifold. Pdf quasilocal definitions of energy in general relativity.
On the notion of quasilocal mass in general relativity. In general relativity, it is the object that is called the killing vector that generates the in. Many approaches to variational principles, conserved quantities and. Quasilocal mass in general relativity ucsb physics. For the latter, we find that this quantity interpolates between zero and the value for the. Their method employs a hamiltonjacobi analysis of an action functional for a spatially bounded spacetime m, and this analysis yields expressions for the quasilocal energy and momentum surface densities associated with the twoboundary b of a spacelike slice of such a spacetime. An investigation of quasilocal systems in general relativity nezihe uzun submitted in partial fulfilment of the requirements for the degree of doctor of philosophy, department of physics and astronomy, university of canterbury february 2016. Abstract using the firstprinciples technique of hamilton and jacobi, broandaad york have recently determined what geometric entities play the role of quasilocal energy in einsteins theory of general relativity. Next we apply it to study the behavior of quasilocal energy of the kerrsen family, which represents rotating, charged black hole solutions in heterotic string theory 5. Qle for charged kerr black holes the charged kerr solution in general relativity represents the spacetime of a rotating, charged black hole.
The result defies the role of the belrobinson superenergy as characterising the gravitational energy in higher dimensions, albeit the fact that it uniquely generalises. In this talk, i want to talk about quasilocal mass and momentum in general relativity. Surprisingly, the hawking energy and the brownyork energy exactly agree upon the small sphere limits across all dimensions. Spacetime mappings of the brownyork quasilocal energy. Many important statements in general relativity make sense only with the. To each optimal isometric embedding, a dual element of the lie algebra of the lorentz group is assigned. In general relativity theory the role of field energy is even more am biguous. My specialisation is geometric analysis, however most of my research is focused on problems coming from general relativity. The concept of mass in general relativity gr is more complex than the concept of mass in. General relativity gr, also known as the general theory of relativity gtr, is the geometric theory of gravitation published by albert einstein in 1915 and the current description of gravitation in modern physics. Canonical variables and quasilocal energy in general relativity. Angular momentum and an invariant quasilocal energy in general.
Quasilocal energy for rotating charged black hole solutions in general relativity and string theory. An investigation of quasilocal systems in general relativity. It is trivial for surfaces sitting in minkowski spacetime and future timelike for surfaces in spacetime which satis es the local energy condition. Recently brown and york1993 have devised a new method for defining quasilocal energy in general relativity. The role played by boundary conditions in the formalism is outlined and the quasilocal internal energy is. General relativity in an expanding universe does not have a conservation law for energy, the adm mass is conserved in an asymptotically flat spacetime, and this is the theoretical justification for saying, e. Maybe gravity is the curvature of spacetime caused by the mass energy of stuff within it plus the energy of space itself. Quasilocal mass in general relativity shingtung yau harvard university talk in munich, may 2011. Nowadays this theory is often used as a prototype for other, more intricate constructions to describe forces between elementary particles or other branches of fundamental physics. Quasilocal energy for rotating charged black hole solutions. Angular momentum and an invariant quasilocal energy in general relativity. Lecture notes on general relativity matthiasblau albert einstein center for fundamental physics institut fu. Quasilocal energy and conserved charges derived from the gravitational action j. General relativity generalizes special relativity and refines newtons law of universal gravitation, providing a unified description of gravity as a geometric property of space and.
In general relativity there is a greater choice of valid reference frames which define different quantities for the energy. We define quasilocal conserved quantities in general relativity by using the optimal isometric embedding in wang and yau commun math phys 2883. In four dimensions, the limits were shown be the stress tensor in nonvacuum and the belrobinson tensor in vacuum. Introduction a thermodynamic description of general relativity has been a longsought goal 1,2 which intensi. In special relativity, quasilocal massenergy is a wellde. It approaches neither the adm mass nor the adm energy. Richard epp has worked on many different areas of physics, ranging from quantum mechanics to black holes and general relativity. Energy, momentum, and center of mass in general relativity 3 in particular, m q e2. Despite these general formulations of energy conservation in general relativity there are some cosmologists who still take the view that energy conservation is only approximate or that it only works in special cases or that it reduces to a trivial identity. A new approach to defining energy momentum and angular momentum in general relativity is presented which avoids some of the difficulties of previous definitions and which can be applied quasilocally. Overall, quasilocal energy has been studied in the domain of formal relativity, but one ought to do better. Quasilocal mass should be nonnegative for surfaces in general spacetime and zero for surfaces in flat spacetime. In general relativity, the einstein equation is obtained by taking the.
First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasilocal quantities, are recalled. The notion of quasilocal energy is currently one of the most promising descriptions of energy in the context of general relativity, and can be characterized simply as follows. We show that the quasilocal mass defined by wang and yau is not welldefined at spatial infinity. Hamiltonian, energy and entropy in general relativity with. Potential energy in general relativity physics stack. Owing to its transformation property under local boosts, the brownyork quasilocal energy surface density is the analogue of e in the special relativity formula. This paper provides physical motivation for the general relativistic. Classical and quantum gravity, volume 16, number 4 article pdf.
In this paper, we prove the following theorem regarding the wangyau quasilocal energy of a spacelike twosurface in a spacetime. Gravity gravity gravitational fields and the theory of general relativity. Together with previous defined quasilocal energymomentum, this completes the definition of conserved quantities in general relativity at the quasilocal level. Angular momentum and an invariant quasilocal energy in. Boundary conditions and quasilocal energy in the canonical. Generalizing this definition to general relativity, however, is problematic.
Most studies in the literature focus on equilibrium thermodynamics of horizons, without stating the. Quasilocal hamiltonians in general relativity request pdf. The following is a list of papers published according to institution. Energy conservation in general relativity physics stack. This is called the adm energymomentum and the bondi energymomentum when the system is viewed from spatial in. Quasilocal energy and thermodynamic equilibrium conditions 2 1. Institute of field physics and theoretical astrophysics and relativity group department of physics and astronomy the university of north carolina chapel hill, nc 275993255 abstract.
Dirac versus reduced quantization of the poincare symmetry in scalar electrodynamics. The rigidity statement of the theorem asserts that m. General relativity is a beautiful scheme for describing the gravitational. Quasilocal energy and thermodynamic equilibrium conditions. The total energy in general relativity cannot be obtained by. Nevertheless, it has a reputation of being extremely di. This is why in an introduction to general relativity it is of. Based on the quasilocal energy definition of brown and york, we compute the integral of the trace of the extrinsic curvature over a codimension2 hypersurface. General relativity gr is the most beautiful physical theory ever invented.
Critical points of wangyau quasilocal energy springerlink. I am particularly interested in the problem of quasilocal mass hence the registration of the domain quasilocal. We will not delve into definitions of energy in general relativity such as the hamiltonian amusingly, the energy of a closed universe always works out to zero according to this definition, various kinds of energy one hopes to obtain by deparametrizing einsteins equations, or quasilocal energy. There are many reasons to search for such a concept. In general it depends on what you mean by energy, and what you mean by conserved. The total energy, including both matter and gravitational contributions. The misnersharphernandez mass defined in general relativity and in spherical symmetry has been recognized as having a.
Our approach uses the brown and york quasilocal stressenergymomentum tensor for matter and gravitational fields, plus the concept of a. This is not a problem either in practice or in principle. The rigidity statement of the theorem asserts that m 0 if and only if. In particular, we study the difference between the uncompactified minkowski spacetime and the toroidal kaluzaklein compactification. Then, the various specific constructions and their properties both successes and deficiencies are discussed. The theory of quasilocal observables in general relativity is far from being complete. In general relativity, the analog of a point particle is a black hole and the notion of binding energy must be replaced by quasilocal energy. First, the general ideas, concepts, and strategies, as well as the. Quasilocal energy and conserved charges derived from the. The liuyau mass as a good quasilocal energy in general. As such, the theory presented here is an ashtekarvariable reformulation of the metric theory of quasilocal stress energy momentum originally due to. The small sphere limit of quasilocal mass provides us the most local notion of gravitational energy. Only a few parts, including the treatment of the stress energy tensor are adapted in accordance with later reformulations of the theory, and contravariant coordinates are consistently labeled by superscripts. General relativity for di erential geometers with emphasis on world lines rather than space slices philadelphia, spring 2007 hermann karcher, bonn contents p.
Gravitational fields and the theory of general relativity. Pdf quasilocal angular momentum and center of mass in general. Pdf angular momentum and an invariant quasilocal energy. Isolating the three di erent e ects from newtonian gravitation, special relativity, and curved time. Quasilocal energymomentum and angular momentum in general. In flat spacetime the backdrop for special relativity you can phrase energy conservation in two ways. Abstract a key feature of the brownyork definition of quasilocal energy is that, under local boosts of the fleet of observers measuring the energy, the quasilocal energy surface density transforms as one would expect based on the equivalence principle, namely, like e in the special relativity formula. We suggest an alternative scheme which retains all the desirable characteristics of the wangyau mass and, in addition, asymptotes to the adm energy at infinity. In general relativity, the local gravitational energy is best characterised by the quasilocal mass. For example in the case of the cosmological solutions the choice of reference frame is usually taken to be comoving coordinates which leads to the formula above.
Nother charges, brownyork quasilocal energy and related topics. Pdf angular momentum and an invariant quasilocal energy in. In general relativity, objects following the motion of gravitation should simply be travelling by geodesics. Quasilocal energy, brownyork energy, lyndenbellkatz energy, black hole thermodynamics, gravitational energy density, perihelion precession. Quasilocal quantities for general relativity and other gravity theories. Conserved charges such as angular momentum are defined using the surface stress tensor and killing vector fields on. We will not delve into definitions of energy in general relativity such as the hamiltonian amusingly, the energy of a closed universe always works out to be zero according to this definition, various kinds of energy one hopes to obtain by deparametrizing einsteins equations, or quasilocal energy. The method used to derive the set of quasilocal densities is a hamiltonjacobi analysis of a suitable covariant action principle for the ashtekar variables. A generic spacetime does not admit any symmetry which, according. The cosmological constant is a quantity used in general relativity to describe some properties of spacetime. We explore the nonuniversality of martinezs conjecture, originally proposed for kerr black holes, within and beyond general relativity.
Canonical variables and quasilocal energy in general. Its spacetime metric and electromagnetic vector potential are given by 9. The conjecture states that the brownyork quasilocal energy at the outer horizon of such a black hole reduces to twice its irreducible mass, or equivalently, to kerr black hole. The present status of the quasilocal mass, energymomentum and. I often hear about how general relativity is very complicated because of all forms of energy are considered, including gravitations own gravitational binding energy. In this section we will rst describe this formalism for classical mechanics and then apply it to general relativity. Pdf quasilocal energy for rotating charged black hole. The integral of the energy density over such a twosurface b is the quasilocal energy associated.
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