We show that the Gauss-Bonnet term can have physical effects in four dimensions. Specifically, th... more We show that the Gauss-Bonnet term can have physical effects in four dimensions. Specifically, the entropy of a black hole acquires a correction term that is proportional to the Euler characteristic of the cross sections of the horizon. While this term is constant for a single black hole, it will be a non-trivial function for a system with dynamical topologies such as black-hole mergers: it is shown that for certain values of the Gauss-Bonnet parameter, the second law of black-hole mechanics can be violated.
An isolated horizon (IH) is a null hypersurface at which the geometry is held fixed. This general... more An isolated horizon (IH) is a null hypersurface at which the geometry is held fixed. This generalizes the notion of an event horizon so that the black hole is an object that is in local equilibrium with its (possibly) dynamic environment. The first law of IH mechanics that arises from the framework relates quantities that are all defined at the
We review the implications of modern higher-dimensional theories of gravity for astrophysics and ... more We review the implications of modern higher-dimensional theories of gravity for astrophysics and cosmology. In particular, we discuss the latest developments of space-time-matter theory in connection with dark matter, particle dynamics and the cosmological constant, as well as related aspects of quantum theory. There are also more immediate tests of extra dimensions, notably involving perturbations of the cosmic 3K microwave background and the precession of a supercooled gyroscope in Earth orbit. We also outline some general features of embeddings, and include pictures of the big bang as viewed from a higher dimension. Contents 1 Introduction 2 2 Solitons and Dark Matter 6 3 Particle Dynamics 9 4 The Cosmological Constant 12 5 Aspects of Quantum Theory 14 6 The Big Bang Revisited 19 7 Conclusion 20 A Conventions 22
The field equations for gravitation and electromagnetism with sources in four dimensions can be i... more The field equations for gravitation and electromagnetism with sources in four dimensions can be interpreted as arising from the vacuum Einstein equations in five dimensions. Gauge invariance of the electromagnetic potentials leads to a "generalized" electromagnetic field tensor. We use the action principle to derive the equations of motion for free electromagnetic fields in flat spacetime, and isolate an effective electromagnetic current with a source that is purely higher-dimensional in origin. This current provides, at least in principle, a means of detecting extra dimensions experimentally.
We discuss various physical aspects of nonextremal, extremal and supersymmetric black holes in as... more We discuss various physical aspects of nonextremal, extremal and supersymmetric black holes in asymptotically anti-de Sitter (ADS) spacetimes. Specifically, we discuss how the isolated horizon (IH) framework leads to an ambiguityfree description of rotating black holes in these spacetimes. We then apply this framework to investigate the properties of supersymmetric isolated horizons (SIHs) in four-dimensional N = 2 gauged supergravity. Among other results we find that they are necessarily extremal, that rotating SIHs must have nontrivial electromagnetic fields, and that non-rotating SIHs necessarily have constant curvature horizon cross sections and a magnetic (though not electric) charge.
We study the mechanics of D-dimensional isolated horizons (IHs) for Einstein gravity in the prese... more We study the mechanics of D-dimensional isolated horizons (IHs) for Einstein gravity in the presence of arbitrary p-form matter fields. This generalizes the analysis of [K. Copsey and G. T. Horowitz Phys. Rev. D 73 024015 (2006)] to non-stationary spacetimes and therefore the local first law to include nonmonopolar (dipole) charges. The only requirement for the local first law to hold is that the action has to be differentiable. The resulting conserved charges are all intrinsic to the horizon and are independent of the topology of the horizon cross sections. We explicitely calculate the local charges for five-dimensional black holes and black rings that are relevant within the context of superstring theory. We conclude with some comments on the black-hole/string correspondence principle and argue that IHs (or some other quasilocal variant) should play a fundamental role in superstring theory.
We present an exact solution to the Einstein field equations which is Ricci and Riemann flat in f... more We present an exact solution to the Einstein field equations which is Ricci and Riemann flat in five dimensions, but in four dimensions is a good model for the early vacuum-dominated universe.
We study a five-dimensional cosmological model, which suggests that the universe began as a disco... more We study a five-dimensional cosmological model, which suggests that the universe began as a discontinuity in a scalar (Higgs-type) field, or alternatively as a conventional four-dimensional phase transition.
We discuss the theory of knots, and describe how knot invariants arise naturally in gravitational... more We discuss the theory of knots, and describe how knot invariants arise naturally in gravitational physics. The focus of this review is to delineate the relationship between knot theory and the loop representation of non-perturbative canonical quantum general relativity (loop quantum gravity). This leads naturally to a discussion of the Kodama wavefunction, a state which is conjectured to be the ground state of the gravitational field with positive cosmological constant. This review can serve as a self-contained introduction to loop quantum gravity and related areas. Our intent is to make the paper accessible to a wider audience that may include topologists, knot-theorists, and other persons innocent of the physical background to this approach to quantum gravity.
We construct a covariant phase space for rotating weakly isolated horizons in Einstein-Maxwell-Ch... more We construct a covariant phase space for rotating weakly isolated horizons in Einstein-Maxwell-Chern-Simons theory in all (odd) D ≥ 5 dimensions. In particular, we show that horizons on the corresponding phase space satisfy the zeroth and first laws of black-hole mechanics. We show that the existence of a Killing spinor on an isolated horizon in four dimensions (when the Chern-Simons term is dropped) and in five dimensions requires that the induced (normal) connection on the horizon has to vanish, and this in turn implies that the surface gravity and rotation one-form are zero. This means that the gravitational component of the horizon angular momentum is zero, while the electromagnetic component (which is attributed to the bulk radiation field) is unconstrained. It follows that an isolated horizon is supersymmetric only if it is extremal and nonrotating. A remarkable property of these horizons is that the Killing spinor only has to exist on the horizon itself. It does not have to exist off the horizon. In addition, we find that the limit when the surface gravity of the horizon goes to zero provides a topological constraint. Specifically, the integral of the scalar curvature of the cross sections of the horizon has to be positive when the dominant energy condition is satisfied and the cosmological constant Λ is zero or positive, and in particular rules out the torus topology for supersymmetric isolated horizons (unless Λ < 0) if and only if the stress-energy tensor T ab is of the form such that T ab ℓ a n b = 0 for any two null vectors ℓ and n with normalization ℓ a n a = −1 on the horizon.
The isolated horizon framework was introduced in order to provide a local description of black ho... more The isolated horizon framework was introduced in order to provide a local description of black holes that are in equilibrium with their (possibly dynamic) environment. Over the past several years, the framework has been extended to include matter fields (dilaton, Yang-Mills etc) in D = 4 dimensions and cosmological constant in D ≥ 3 dimensions. In this article we present a further extension of the framework that includes black holes in higher-dimensional Einstein-Gauss-Bonnet (EGB) gravity. In particular, we construct a covariant phase space for EGB gravity in arbitrary dimensions which allows us to derive the first law. We find that the entropy of a weakly isolated and non-rotating horizon is given by
We show that the on-shell path integral for asymptotically flat Euclidean spacetimes can be given... more We show that the on-shell path integral for asymptotically flat Euclidean spacetimes can be given in the first-order formulation of general relativity, without assuming the boundary to be isometrically embedded in Euclidean space and without adding infinite counter-terms. For illustrative examples of our approach, we evaluate the first-order action for the four-dimensional Euclidean Schwarzschild and NUT-charged spacetimes to derive the corresponding onshell partition functions, and show that the correct thermodynamic quantities for the solutions are reproduced.
The Barbero-Immirzi parameter γ appears in the real connection formulation of gravity in terms of... more The Barbero-Immirzi parameter γ appears in the real connection formulation of gravity in terms of the Ashtekar variables, and gives rise to a oneparameter quantization ambiguity in Loop Quantum Gravity. In this paper we investigate the conditions under which γ will have physical effects in Euclidean Quantum Gravity. This is done by constructing a well-defined Euclidean path integral for the Holst action with non-zero cosmological constant on a manifold with boundary. We find that two general conditions must be satisfied by the spacetime manifold in order for the Holst action and its surface integral to be non-zero: (i) the metric has to be non-diagonalizable; (ii) the Pontryagin number of the manifold has to be non-zero. The latter is a strong topological condition, and rules out many of the known solutions to the Einstein field equations. This result leads us to evaluate the on-shell first-order Holst action and corresponding Euclidean partition function on the Taub-NUT-ADS solution. We find that γ shows up as a finite rotation of the on-shell partition function which corresponds to shifts in the energy and entropy of the NUT charge. In an appendix we also evaluate the Holst action on the Taub-NUT and Taub-bolt solutions in flat spacetime and find that in that case as well γ shows up in the energy and entropy of the NUT and bolt charges. We also present an example whereby the Euler characteristic of the manifold has a non-trivial effect on black-hole mergers.
The first-order Holst action with negative cosmological constant is rendered finite by requiring ... more The first-order Holst action with negative cosmological constant is rendered finite by requiring functional differentiability on the configuration space of tetrads and connections. The surface terms that arise in the action for anti de Sitter gravity are equivalent to the Euler and Pontryagin densities with fixed weight factors; these terms modify the Noether charges that arise from diffeomorphism invariance of the action.
We show that the Gauss-Bonnet term can have physical effects in four dimensions. Specifically, th... more We show that the Gauss-Bonnet term can have physical effects in four dimensions. Specifically, the entropy of a black hole acquires a correction term that is proportional to the Euler characteristic of the cross sections of the horizon. While this term is constant for a single black hole, it will be a non-trivial function for a system with dynamical topologies such as black-hole mergers: it is shown that for certain values of the Gauss-Bonnet parameter, the second law of black-hole mechanics can be violated.
An isolated horizon (IH) is a null hypersurface at which the geometry is held fixed. This general... more An isolated horizon (IH) is a null hypersurface at which the geometry is held fixed. This generalizes the notion of an event horizon so that the black hole is an object that is in local equilibrium with its (possibly) dynamic environment. The first law of IH mechanics that arises from the framework relates quantities that are all defined at the
We review the implications of modern higher-dimensional theories of gravity for astrophysics and ... more We review the implications of modern higher-dimensional theories of gravity for astrophysics and cosmology. In particular, we discuss the latest developments of space-time-matter theory in connection with dark matter, particle dynamics and the cosmological constant, as well as related aspects of quantum theory. There are also more immediate tests of extra dimensions, notably involving perturbations of the cosmic 3K microwave background and the precession of a supercooled gyroscope in Earth orbit. We also outline some general features of embeddings, and include pictures of the big bang as viewed from a higher dimension. Contents 1 Introduction 2 2 Solitons and Dark Matter 6 3 Particle Dynamics 9 4 The Cosmological Constant 12 5 Aspects of Quantum Theory 14 6 The Big Bang Revisited 19 7 Conclusion 20 A Conventions 22
The field equations for gravitation and electromagnetism with sources in four dimensions can be i... more The field equations for gravitation and electromagnetism with sources in four dimensions can be interpreted as arising from the vacuum Einstein equations in five dimensions. Gauge invariance of the electromagnetic potentials leads to a "generalized" electromagnetic field tensor. We use the action principle to derive the equations of motion for free electromagnetic fields in flat spacetime, and isolate an effective electromagnetic current with a source that is purely higher-dimensional in origin. This current provides, at least in principle, a means of detecting extra dimensions experimentally.
We discuss various physical aspects of nonextremal, extremal and supersymmetric black holes in as... more We discuss various physical aspects of nonextremal, extremal and supersymmetric black holes in asymptotically anti-de Sitter (ADS) spacetimes. Specifically, we discuss how the isolated horizon (IH) framework leads to an ambiguityfree description of rotating black holes in these spacetimes. We then apply this framework to investigate the properties of supersymmetric isolated horizons (SIHs) in four-dimensional N = 2 gauged supergravity. Among other results we find that they are necessarily extremal, that rotating SIHs must have nontrivial electromagnetic fields, and that non-rotating SIHs necessarily have constant curvature horizon cross sections and a magnetic (though not electric) charge.
We study the mechanics of D-dimensional isolated horizons (IHs) for Einstein gravity in the prese... more We study the mechanics of D-dimensional isolated horizons (IHs) for Einstein gravity in the presence of arbitrary p-form matter fields. This generalizes the analysis of [K. Copsey and G. T. Horowitz Phys. Rev. D 73 024015 (2006)] to non-stationary spacetimes and therefore the local first law to include nonmonopolar (dipole) charges. The only requirement for the local first law to hold is that the action has to be differentiable. The resulting conserved charges are all intrinsic to the horizon and are independent of the topology of the horizon cross sections. We explicitely calculate the local charges for five-dimensional black holes and black rings that are relevant within the context of superstring theory. We conclude with some comments on the black-hole/string correspondence principle and argue that IHs (or some other quasilocal variant) should play a fundamental role in superstring theory.
We present an exact solution to the Einstein field equations which is Ricci and Riemann flat in f... more We present an exact solution to the Einstein field equations which is Ricci and Riemann flat in five dimensions, but in four dimensions is a good model for the early vacuum-dominated universe.
We study a five-dimensional cosmological model, which suggests that the universe began as a disco... more We study a five-dimensional cosmological model, which suggests that the universe began as a discontinuity in a scalar (Higgs-type) field, or alternatively as a conventional four-dimensional phase transition.
We discuss the theory of knots, and describe how knot invariants arise naturally in gravitational... more We discuss the theory of knots, and describe how knot invariants arise naturally in gravitational physics. The focus of this review is to delineate the relationship between knot theory and the loop representation of non-perturbative canonical quantum general relativity (loop quantum gravity). This leads naturally to a discussion of the Kodama wavefunction, a state which is conjectured to be the ground state of the gravitational field with positive cosmological constant. This review can serve as a self-contained introduction to loop quantum gravity and related areas. Our intent is to make the paper accessible to a wider audience that may include topologists, knot-theorists, and other persons innocent of the physical background to this approach to quantum gravity.
We construct a covariant phase space for rotating weakly isolated horizons in Einstein-Maxwell-Ch... more We construct a covariant phase space for rotating weakly isolated horizons in Einstein-Maxwell-Chern-Simons theory in all (odd) D ≥ 5 dimensions. In particular, we show that horizons on the corresponding phase space satisfy the zeroth and first laws of black-hole mechanics. We show that the existence of a Killing spinor on an isolated horizon in four dimensions (when the Chern-Simons term is dropped) and in five dimensions requires that the induced (normal) connection on the horizon has to vanish, and this in turn implies that the surface gravity and rotation one-form are zero. This means that the gravitational component of the horizon angular momentum is zero, while the electromagnetic component (which is attributed to the bulk radiation field) is unconstrained. It follows that an isolated horizon is supersymmetric only if it is extremal and nonrotating. A remarkable property of these horizons is that the Killing spinor only has to exist on the horizon itself. It does not have to exist off the horizon. In addition, we find that the limit when the surface gravity of the horizon goes to zero provides a topological constraint. Specifically, the integral of the scalar curvature of the cross sections of the horizon has to be positive when the dominant energy condition is satisfied and the cosmological constant Λ is zero or positive, and in particular rules out the torus topology for supersymmetric isolated horizons (unless Λ < 0) if and only if the stress-energy tensor T ab is of the form such that T ab ℓ a n b = 0 for any two null vectors ℓ and n with normalization ℓ a n a = −1 on the horizon.
The isolated horizon framework was introduced in order to provide a local description of black ho... more The isolated horizon framework was introduced in order to provide a local description of black holes that are in equilibrium with their (possibly dynamic) environment. Over the past several years, the framework has been extended to include matter fields (dilaton, Yang-Mills etc) in D = 4 dimensions and cosmological constant in D ≥ 3 dimensions. In this article we present a further extension of the framework that includes black holes in higher-dimensional Einstein-Gauss-Bonnet (EGB) gravity. In particular, we construct a covariant phase space for EGB gravity in arbitrary dimensions which allows us to derive the first law. We find that the entropy of a weakly isolated and non-rotating horizon is given by
We show that the on-shell path integral for asymptotically flat Euclidean spacetimes can be given... more We show that the on-shell path integral for asymptotically flat Euclidean spacetimes can be given in the first-order formulation of general relativity, without assuming the boundary to be isometrically embedded in Euclidean space and without adding infinite counter-terms. For illustrative examples of our approach, we evaluate the first-order action for the four-dimensional Euclidean Schwarzschild and NUT-charged spacetimes to derive the corresponding onshell partition functions, and show that the correct thermodynamic quantities for the solutions are reproduced.
The Barbero-Immirzi parameter γ appears in the real connection formulation of gravity in terms of... more The Barbero-Immirzi parameter γ appears in the real connection formulation of gravity in terms of the Ashtekar variables, and gives rise to a oneparameter quantization ambiguity in Loop Quantum Gravity. In this paper we investigate the conditions under which γ will have physical effects in Euclidean Quantum Gravity. This is done by constructing a well-defined Euclidean path integral for the Holst action with non-zero cosmological constant on a manifold with boundary. We find that two general conditions must be satisfied by the spacetime manifold in order for the Holst action and its surface integral to be non-zero: (i) the metric has to be non-diagonalizable; (ii) the Pontryagin number of the manifold has to be non-zero. The latter is a strong topological condition, and rules out many of the known solutions to the Einstein field equations. This result leads us to evaluate the on-shell first-order Holst action and corresponding Euclidean partition function on the Taub-NUT-ADS solution. We find that γ shows up as a finite rotation of the on-shell partition function which corresponds to shifts in the energy and entropy of the NUT charge. In an appendix we also evaluate the Holst action on the Taub-NUT and Taub-bolt solutions in flat spacetime and find that in that case as well γ shows up in the energy and entropy of the NUT and bolt charges. We also present an example whereby the Euler characteristic of the manifold has a non-trivial effect on black-hole mergers.
The first-order Holst action with negative cosmological constant is rendered finite by requiring ... more The first-order Holst action with negative cosmological constant is rendered finite by requiring functional differentiability on the configuration space of tetrads and connections. The surface terms that arise in the action for anti de Sitter gravity are equivalent to the Euler and Pontryagin densities with fixed weight factors; these terms modify the Noether charges that arise from diffeomorphism invariance of the action.
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