Abstract The Xanab field is located off the coast of Tabasco state in the southeastern Gulf of Me... more Abstract The Xanab field is located off the coast of Tabasco state in the southeastern Gulf of Mexico. In this field, oil reserves accumulate in dolomitized carbonate rocks at the Upper Jurassic Kimmeridgian formations. While drilling the Xanab 11 well, a narrow margin of ...
The microcanonical functional integral for an eternal black hole system is considered. This requi... more The microcanonical functional integral for an eternal black hole system is considered. This requires computing the microcanonical action for a spatially bounded spacetime region when its two disconnected timelike boundary surfaces are located in different wedges of the Kruskal diagram. The path integral is a sum over Lorentzian geometries and is evaluated semiclassically when its boundary data are chosen such that the system is approximated by any Lorentzian, stationary eternal black hole. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals S = A H /4 − A H /4 = 0 in the semiclassical approximation, where A H is the area of the black hole horizon. The functional integral reflects the properties of a pure state. The description of the black hole density of states in terms of the eternal black hole functional integral is also discussed.
We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding ... more We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding matter. The grand canonical ensemble is defined by the geometry of the boundary of a region of spacetime and the chemical potential evaluated on this boundary. Additivity of the actions of gravity and matter and the principle of equivalence then jointly imply that the total
We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding ... more We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding matter. The grand canonical ensemble is defined by the geometry of the boundary of a region of spacetime and the chemical potential evaluated on this boundary. Additivity of the actions of gravity and matter and the principle of equivalence then jointly imply that the total
Using our approach, we give an explicit one parameter set of metrics that are asymptotically flat... more Using our approach, we give an explicit one parameter set of metrics that are asymptotically flat and describe the collapse of apriori unspecified but physical matter fields. The black hole mass formula obtained from this metric exhibits a mass gap - that is, at the onset of black hole formation, the mass is finite and non-zero.
The microcanonical functional integral for an eternal black hole system is considered. This requi... more The microcanonical functional integral for an eternal black hole system is considered. This requires computing the microcanonical action for a spatially bounded spacetime region when its two disconnected timelike boundary surfaces are located in different wedges of the Kruskal diagram. The path integral is a sum over Lorentzian geometries and is evaluated semiclassically when its boundary data are chosen such that the system is approximated by any Lorentzian, stationary eternal black hole. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals ${\cal S} = A_H/4 - A_H/4 = 0$ in the semiclassical approximation, where $A_H$ is the area of the black hole horizon. The functional integral reflects the properties of a pure state. The description of the black hole density of states in terms of the eternal black hole functional integral is also discussed.
In recent years the proposal' to relate the Euclidean action of black holes to approximations of ... more In recent years the proposal' to relate the Euclidean action of black holes to approximations of certain functional integrals that can be interpreted as thermodynamic partition functions has been developed extensively. In this paper we outline briefly the key points of these developments and extend them to the treatment of stationary geometries, in particular the geometries of rotating black holes. Stationary holes are ostensibly more difficult to handle than static ones, primarily because neither the extrinsic curvature of the stationary constant-time slices nor the corresponding shift vector vanishes, while both do vanish in the case of static geometries. Consequently, as we shall see, there is no real Euclidean metric that represents a rotating black hole. Nevertheless, the hole can be described by a complex geometry that is an extremum of an appropriate real action-we shall call it the thermodynamical action.
The general principles and logical structure of a thermodynamic formalism that incorporates stron... more The general principles and logical structure of a thermodynamic formalism that incorporates strongly self-gravitating systems are presented. This framework generalizes and simplifies the formulation of thermodynamics developed by Callen. The definition of extensive variables, the homogeneity properties of intensive parameters, and the fundamental problem of gravitational thermodynamics are discussed in detail. In particular, extensive parameters include quasilocal quantities and are naturally incorporated into a set of basic general postulates for thermodynamics. These include additivity of entropies (Massieu functions) and the generalized second law. Fundamental equations are no longer homogeneous first-order functions of their extensive variables. It is shown that the postulates lead to a formal resolution of the fundamental problem despite non-additivity of extensive parameters and thermodynamic potentials. Therefore, all the results of (gravitational) thermodynamics are an outgrowth of these postulates. The origin and nature of the differences with ordinary thermodynamics are analyzed. Consequences of the formalism include the (spatially) inhomogeneous character of thermodynamic equilibrium states, a reformulation of the Euler equation, and the absence of a Gibbs-Duhem relation.
Kerr spacetime is presented. The calculations are based on a recent proposal \cite{by} in which q... more Kerr spacetime is presented. The calculations are based on a recent proposal \cite{by} in which quasilocal energy is derived from the Hamiltonian of spatially bounded gravitational systems. Three different classes of boundary surfaces for the Kerr slice are considered (constant radius surfaces, round spheres, and the ergosurface). Their embeddings in both the Kerr slice and flat three-dimensional space (required as a normalization of the energy) are analyzed. The energy contained within each surface is explicitly calculated in the slow rotation regime and its properties discussed in detail. The energy is a positive, monotonically decreasing function of the boundary surface radius. It approaches the Arnowitt-Deser-Misner (ADM) mass at spatial infinity and reduces to (twice) the irreducible mass at the horizon of the Kerr black hole. The expressions possess the correct static limit and include negative contributions due to gravitational binding. The energy at the ergosurface is compared with the energies at other surfaces. Finally, the difficulties involved in an estimation of the energy in the fast rotation regime are discussed.
We give an exact spherically symmetric solution for the Einstein-scalar field system. The solutio... more We give an exact spherically symmetric solution for the Einstein-scalar field system. The solution may be interpreted as an inhomogeneous dynamical scalar field cosmology. The spacetime has a timelike conformal Killing vector field and is asymptotically conformally flat. It also has black or white holelike regions containing trapped surfaces. We describe the properties of the apparent horizon and comment on the relevance of the solution to the recently discovered critical behaviour in scalar field collapse.
The entropy of a quantum-statistical system which is classically approximated by a general statio... more The entropy of a quantum-statistical system which is classically approximated by a general stationary eternal black hole is studied by means of a microcanonical functional integral. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals calS = A H /4 − A H /4 = 0 in the semiclassical approximation, where A H is the area of the black hole horizon. The functional integral reflects the properties of a pure state.
... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-p... more ... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-phase approxi ... thank HW Braden and BF Whiting for discussions in the early phases of this work, and G ... Research support was received from National Science Foundation Grants ...
We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatia... more We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatially bounded by two timelike surfaces. The results are applied to the particular case of a spacetime representing an eternal black hole. It is shown that in the case when the boundaries are located in two different wedges of the Kruskal diagram, the Hamiltonian is of the form $H = H_+ - H_-$, where $H_+$ and $H_-$ are the Hamiltonian functions for the right and left wedges respectively. The application of the obtained results to the thermofield dynamics description of quantum effects in black holes is briefly discussed.
... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-p... more ... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-phase approxi ... thank HW Braden and BF Whiting for discussions in the early phases of this work, and G ... Research support was received from National Science Foundation Grants ...
The features of the fundamental thermodynamical relation (expressing entropy as function of state... more The features of the fundamental thermodynamical relation (expressing entropy as function of state variables) that arise from the self-gravitating character of a system are analyzed. The models studied include not only a spherically symmetric hot matter shell with constant particle number but also a black hole characterized by a general thermal equation of state. These examples illustrate the formal structure of thermodynamics developed by Callen as applied to a gravitational configuration as well as the phenomenological manner in which Einstein equations largely determine the thermodynamical equations of state. We consider in detail the thermodynamics and quasi-static collapse of a self-gravitating shell. This includes a discussion of intrinsic stability for a one-parameter family of thermal equations of state and the interpretation of the Bekenstein bound. The entropy growth associated with a collapsing sequence of equilibrium states of a shell is computed under different boundary conditions in the quasi-static approximation and compared with black hole entropy. Although explicit expressions involve empirical coefficients, these are constrained by physical conditions of thermodynamical origin. The absence of a Gibbs-Duhem relation and the associated scaling laws for self-gravitating matter systems are presented.
The general principles and logical structure of a thermodynamic formalism that incorporates stron... more The general principles and logical structure of a thermodynamic formalism that incorporates strongly self-gravitating systems are presented. This framework generalizes and simplifies the formulation of thermodynamics developed by Callen. The definition of extensive variables, the homogeneity properties of intensive parameters, and the fundamental problem of gravitational thermodynamics are discussed in detail. In particular, extensive parameters include quasilocal quantities and are naturally incorporated into a set of basic general postulates for thermodynamics. These include additivity of entropies (Massieu functions) and the generalized second law. Fundamental equations are no longer homogeneous first-order functions of their extensive variables. It is shown that the postulates lead to a formal resolution of the fundamental problem despite non-additivity of extensive parameters and thermodynamic potentials. Therefore, all the results of (gravitational) thermodynamics are an outgrowth of these postulates. The origin and nature of the differences with ordinary thermodynamics are analyzed. Consequences of the formalism include the (spatially) inhomogeneous character of thermodynamic equilibrium states, a reformulation of the Euler equation, and the absence of a Gibbs-Duhem relation.
Abstract The Xanab field is located off the coast of Tabasco state in the southeastern Gulf of Me... more Abstract The Xanab field is located off the coast of Tabasco state in the southeastern Gulf of Mexico. In this field, oil reserves accumulate in dolomitized carbonate rocks at the Upper Jurassic Kimmeridgian formations. While drilling the Xanab 11 well, a narrow margin of ...
The microcanonical functional integral for an eternal black hole system is considered. This requi... more The microcanonical functional integral for an eternal black hole system is considered. This requires computing the microcanonical action for a spatially bounded spacetime region when its two disconnected timelike boundary surfaces are located in different wedges of the Kruskal diagram. The path integral is a sum over Lorentzian geometries and is evaluated semiclassically when its boundary data are chosen such that the system is approximated by any Lorentzian, stationary eternal black hole. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals S = A H /4 − A H /4 = 0 in the semiclassical approximation, where A H is the area of the black hole horizon. The functional integral reflects the properties of a pure state. The description of the black hole density of states in terms of the eternal black hole functional integral is also discussed.
We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding ... more We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding matter. The grand canonical ensemble is defined by the geometry of the boundary of a region of spacetime and the chemical potential evaluated on this boundary. Additivity of the actions of gravity and matter and the principle of equivalence then jointly imply that the total
We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding ... more We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding matter. The grand canonical ensemble is defined by the geometry of the boundary of a region of spacetime and the chemical potential evaluated on this boundary. Additivity of the actions of gravity and matter and the principle of equivalence then jointly imply that the total
Using our approach, we give an explicit one parameter set of metrics that are asymptotically flat... more Using our approach, we give an explicit one parameter set of metrics that are asymptotically flat and describe the collapse of apriori unspecified but physical matter fields. The black hole mass formula obtained from this metric exhibits a mass gap - that is, at the onset of black hole formation, the mass is finite and non-zero.
The microcanonical functional integral for an eternal black hole system is considered. This requi... more The microcanonical functional integral for an eternal black hole system is considered. This requires computing the microcanonical action for a spatially bounded spacetime region when its two disconnected timelike boundary surfaces are located in different wedges of the Kruskal diagram. The path integral is a sum over Lorentzian geometries and is evaluated semiclassically when its boundary data are chosen such that the system is approximated by any Lorentzian, stationary eternal black hole. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals ${\cal S} = A_H/4 - A_H/4 = 0$ in the semiclassical approximation, where $A_H$ is the area of the black hole horizon. The functional integral reflects the properties of a pure state. The description of the black hole density of states in terms of the eternal black hole functional integral is also discussed.
In recent years the proposal' to relate the Euclidean action of black holes to approximations of ... more In recent years the proposal' to relate the Euclidean action of black holes to approximations of certain functional integrals that can be interpreted as thermodynamic partition functions has been developed extensively. In this paper we outline briefly the key points of these developments and extend them to the treatment of stationary geometries, in particular the geometries of rotating black holes. Stationary holes are ostensibly more difficult to handle than static ones, primarily because neither the extrinsic curvature of the stationary constant-time slices nor the corresponding shift vector vanishes, while both do vanish in the case of static geometries. Consequently, as we shall see, there is no real Euclidean metric that represents a rotating black hole. Nevertheless, the hole can be described by a complex geometry that is an extremum of an appropriate real action-we shall call it the thermodynamical action.
The general principles and logical structure of a thermodynamic formalism that incorporates stron... more The general principles and logical structure of a thermodynamic formalism that incorporates strongly self-gravitating systems are presented. This framework generalizes and simplifies the formulation of thermodynamics developed by Callen. The definition of extensive variables, the homogeneity properties of intensive parameters, and the fundamental problem of gravitational thermodynamics are discussed in detail. In particular, extensive parameters include quasilocal quantities and are naturally incorporated into a set of basic general postulates for thermodynamics. These include additivity of entropies (Massieu functions) and the generalized second law. Fundamental equations are no longer homogeneous first-order functions of their extensive variables. It is shown that the postulates lead to a formal resolution of the fundamental problem despite non-additivity of extensive parameters and thermodynamic potentials. Therefore, all the results of (gravitational) thermodynamics are an outgrowth of these postulates. The origin and nature of the differences with ordinary thermodynamics are analyzed. Consequences of the formalism include the (spatially) inhomogeneous character of thermodynamic equilibrium states, a reformulation of the Euler equation, and the absence of a Gibbs-Duhem relation.
Kerr spacetime is presented. The calculations are based on a recent proposal \cite{by} in which q... more Kerr spacetime is presented. The calculations are based on a recent proposal \cite{by} in which quasilocal energy is derived from the Hamiltonian of spatially bounded gravitational systems. Three different classes of boundary surfaces for the Kerr slice are considered (constant radius surfaces, round spheres, and the ergosurface). Their embeddings in both the Kerr slice and flat three-dimensional space (required as a normalization of the energy) are analyzed. The energy contained within each surface is explicitly calculated in the slow rotation regime and its properties discussed in detail. The energy is a positive, monotonically decreasing function of the boundary surface radius. It approaches the Arnowitt-Deser-Misner (ADM) mass at spatial infinity and reduces to (twice) the irreducible mass at the horizon of the Kerr black hole. The expressions possess the correct static limit and include negative contributions due to gravitational binding. The energy at the ergosurface is compared with the energies at other surfaces. Finally, the difficulties involved in an estimation of the energy in the fast rotation regime are discussed.
We give an exact spherically symmetric solution for the Einstein-scalar field system. The solutio... more We give an exact spherically symmetric solution for the Einstein-scalar field system. The solution may be interpreted as an inhomogeneous dynamical scalar field cosmology. The spacetime has a timelike conformal Killing vector field and is asymptotically conformally flat. It also has black or white holelike regions containing trapped surfaces. We describe the properties of the apparent horizon and comment on the relevance of the solution to the recently discovered critical behaviour in scalar field collapse.
The entropy of a quantum-statistical system which is classically approximated by a general statio... more The entropy of a quantum-statistical system which is classically approximated by a general stationary eternal black hole is studied by means of a microcanonical functional integral. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals calS = A H /4 − A H /4 = 0 in the semiclassical approximation, where A H is the area of the black hole horizon. The functional integral reflects the properties of a pure state.
... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-p... more ... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-phase approxi ... thank HW Braden and BF Whiting for discussions in the early phases of this work, and G ... Research support was received from National Science Foundation Grants ...
We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatia... more We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatially bounded by two timelike surfaces. The results are applied to the particular case of a spacetime representing an eternal black hole. It is shown that in the case when the boundaries are located in two different wedges of the Kruskal diagram, the Hamiltonian is of the form $H = H_+ - H_-$, where $H_+$ and $H_-$ are the Hamiltonian functions for the right and left wedges respectively. The application of the obtained results to the thermofield dynamics description of quantum effects in black holes is briefly discussed.
... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-p... more ... We conclude that all of standard black-hole thermo-dynamics can be obtained as a stationary-phase approxi ... thank HW Braden and BF Whiting for discussions in the early phases of this work, and G ... Research support was received from National Science Foundation Grants ...
The features of the fundamental thermodynamical relation (expressing entropy as function of state... more The features of the fundamental thermodynamical relation (expressing entropy as function of state variables) that arise from the self-gravitating character of a system are analyzed. The models studied include not only a spherically symmetric hot matter shell with constant particle number but also a black hole characterized by a general thermal equation of state. These examples illustrate the formal structure of thermodynamics developed by Callen as applied to a gravitational configuration as well as the phenomenological manner in which Einstein equations largely determine the thermodynamical equations of state. We consider in detail the thermodynamics and quasi-static collapse of a self-gravitating shell. This includes a discussion of intrinsic stability for a one-parameter family of thermal equations of state and the interpretation of the Bekenstein bound. The entropy growth associated with a collapsing sequence of equilibrium states of a shell is computed under different boundary conditions in the quasi-static approximation and compared with black hole entropy. Although explicit expressions involve empirical coefficients, these are constrained by physical conditions of thermodynamical origin. The absence of a Gibbs-Duhem relation and the associated scaling laws for self-gravitating matter systems are presented.
The general principles and logical structure of a thermodynamic formalism that incorporates stron... more The general principles and logical structure of a thermodynamic formalism that incorporates strongly self-gravitating systems are presented. This framework generalizes and simplifies the formulation of thermodynamics developed by Callen. The definition of extensive variables, the homogeneity properties of intensive parameters, and the fundamental problem of gravitational thermodynamics are discussed in detail. In particular, extensive parameters include quasilocal quantities and are naturally incorporated into a set of basic general postulates for thermodynamics. These include additivity of entropies (Massieu functions) and the generalized second law. Fundamental equations are no longer homogeneous first-order functions of their extensive variables. It is shown that the postulates lead to a formal resolution of the fundamental problem despite non-additivity of extensive parameters and thermodynamic potentials. Therefore, all the results of (gravitational) thermodynamics are an outgrowth of these postulates. The origin and nature of the differences with ordinary thermodynamics are analyzed. Consequences of the formalism include the (spatially) inhomogeneous character of thermodynamic equilibrium states, a reformulation of the Euler equation, and the absence of a Gibbs-Duhem relation.
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Papers by Erik Martinez