arXiv subjects

Jorma Louko

Publications and source records attributed to Jorma Louko.

At least 109 records · Page 6Linked to original sources

arXiv2005

Refined algebraic quantisation with the triangular subgroup of SL(2,R)

We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL(2,R). The unreduced phase space is T^*R^{p+q} with p>0 and q>0, and the system has a distinguished classical o(p,q) observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O(p,q). The representation is nontrivial iff (p,q) is not (1,1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantisation that imposes the constraints in the sense H_a Psi = 0 and postulates self-adjointness of the o(p,q) observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantisation gives no quantum theory when (p,q) = (1,2) or (2,1), or when p>1, q>1 and p+q is odd.

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arXiv2005

Geons with spin and charge

We construct new geon-type black holes in D>3 dimensions for Einstein's theory coupled to gauge fields. A static nondegenerate vacuum black hole has a geon quotient provided the spatial section admits a suitable discrete isometry, and an antisymmetric tensor field of rank 2 or D-2 with a pure F^2 action can be included by an appropriate (and in most cases nontrivial) choice of the field strength bundle. We find rotating geons as quotients of the Myers-Perry(-AdS) solution when D is odd and not equal to 7. For other D we show that such rotating geons, if they exist at all, cannot be continuously deformed to zero angular momentum. With a negative cosmological constant, we construct geons with angular momenta on a torus at the infinity. As an example of a nonabelian gauge field, we show that the D=4 spherically symmetric SU(2) black hole admits a geon version with a trivial gauge bundle. Various generalisations, including both black-brane geons and Yang-Mills theories with Chern-Simons terms, are briefly discussed.

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arXiv2004

Einstein black holes, free scalars and AdS/CFT correspondence

We investigate AdS/CFT correspondence for two families of Einstein black holes in d > 3 dimensions, modelling the boundary CFT by a free conformal scalar field and evaluating the boundary two-point function in the bulk geodesic approximation. For the d > 3 counterpart of the nonrotating BTZ hole and for its Z_2 quotient, the boundary state is thermal in the expected sense, and its stress-energy reflects the properties of the bulk geometry and suggests a novel definition for the mass of the hole. For the generalised Schwarzschild-AdS hole with a flat horizon of topology R^{d-2}, the boundary stress-energy has a thermal form with energy density proportional to the hole ADM mass, but stress-energy corrections from compactified horizon dimensions cannot be consistently included at least for d=5.

hep-th↗

arXiv2004

Group averaging in the (p,q) oscillator representation of SL(2,R)

We investigate refined algebraic quantisation with group averaging in a finite-dimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group SL(2,R) and a distinguished o(p,q) observable algebra. The gauge group of the quantum theory is the double cover of SL(2,R), and its representation on the auxiliary Hilbert space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the o(p,q) quantum observable algebra. For p=q=1, the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form.

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arXiv2002

Brane worlds with bolts

We construct a family of (p+3)-dimensional brane worlds in which the brane has one compact extra dimension, the bulk has two extra dimensions, and the bulk closes regularly at codimension two submanifolds known as bolts. The (p+1)-dimensional low energy spacetime M_{low} may be any Einstein space with an arbitrary cosmological constant, the value of the bulk cosmological constant is arbitrary, and the only fields are the metric and a bulk Maxwell field. The brane can be chosen to have positive tension, and the closure of the bulk provides a singularity-free boundary condition for solutions that contain black holes or gravitational waves in M_{low}. The spacetimes admit a nonlinear gravitational wave whose properties suggest that the Newtonian gravitational potential on a flat M_{low} will behave essentially as the static potential of a massless minimally coupled scalar field with Neumann boundary conditions. When M_{low} is (p+1)-dimensional Minkowski with p\ge3 and the bulk cosmological constant vanishes, this static scalar potential is shown to have the long distance behaviour characteristic of p spatial dimensions.

hep-th↗

arXiv2001

The 2+1 Kepler Problem and Its Quantization

We study a system of two pointlike particles coupled to three dimensional Einstein gravity. The reduced phase space can be considered as a deformed version of the phase space of two special-relativistic point particles in the centre of mass frame. When the system is quantized, we find some possibly general effects of quantum gravity, such as a minimal distances and a foaminess of the spacetime at the order of the Planck length. We also obtain a quantization of geometry, which restricts the possible asymptotic geometries of the universe.

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arXiv2000

(2+1)-dimensional Einstein-Kepler problem in the centre-of-mass frame

We formulate and analyze the Hamiltonian dynamics of a pair of massive spinless point particles in (2+1)-dimensional Einstein gravity by anchoring the system to a conical infinity, isometric to the infinity generated by a single massive but possibly spinning particle. The reduced phase space Γ_{red} has dimension four and topology R^3 x S^1. Γ_{red} is analogous to the phase space of a Newtonian two-body system in the centre-of-mass frame, and we find on Γ_{red} a canonical chart that makes this analogue explicit and reduces to the Newtonian chart in the appropriate limit. Prospects for quantization are commented on.

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arXiv2000

On geodesic propagators and black hole holography

One of the most challenging technical aspects of the dualities between string theory on anti-de Sitter spaces and conformal field theories is understanding how location in the interior of spacetime is represented in the field theory. It has recently been argued that the interior of the spacetime can be directly probed by using intrinsically non-local quantities in the field theory. In addition, Balasubramanian and Ross [hep-th/9906226] argued that when the spacetime described the formation of an AdS_3 black hole, the propagator in the field theory probed the whole spacetime, including the region behind the horizon. We use the same approach to study the propagator for the BTZ black hole and a black hole solution with a single exterior region, and show that it reproduces the propagator associated with the natural vacuum states on these spacetimes. We compare our result with a toy model of the CFT for the single-exterior black hole, finding remarkable agreement. The spacetimes studied in this work are analytic, which makes them quite special. We also discuss the interpretation of this propagator in more general spacetimes, shedding light on certain issues involving causality, black hole horizons, and products of local operators on the boundary.

hep-th↗

arXiv2000

Refined Algebraic Quantization in the oscillator representation of SL(2,R)

We investigate Refined Algebraic Quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T^*R^4 and gauge group SL(2,R). The reduced phase space M is connected and contains four mutually disconnected `regular' sectors with topology R x S^1, but these sectors are connected to each other through an exceptional set where M is not a manifold and where M has non-Hausdorff topology. The RAQ physical Hilbert space H_{phys} decomposes as H_{phys} = (direct sum of) H_i, where the four subspaces H_i naturally correspond to the four regular sectors of M. The RAQ observable algebra A_{obs}, represented on H_{phys}, contains natural subalgebras represented on each H_i. The group averaging takes place in the oscillator representation of SL(2,R) on L^2(R^{2,2}), and ensuring convergence requires a subtle choice for the test state space: the classical analogue of this choice is to excise from M the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space H_{phys} and a finitely-generated observable subalgebra of A_{obs} is recovered through both Ashtekar's Algebraic Quantization and Isham's group theoretic quantization.

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arXiv1999

Single-exterior black holes

We discuss quantum properties of the single-exterior, geon-type black (and white) holes that are obtained from the Kruskal spacetime and the spinless Banados-Teitelboim-Zanelli hole via a quotient construction that identifies the two exterior regions. For the four-dimensional geon, the Hartle-Hawking type state of a massless scalar field is thermal in a limited sense, but there is a discrepancy between Lorentzian and Riemannian derivations of the geon entropy. For the three-dimensional geon, the state induced for a free conformal scalar field on the conformal boundary is similarly thermal in a limited sense, and the correlations in this state provide support for the holographic hypothesis in the context of asymptotically anti-de Sitter black holes in string theory.

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arXiv1999

The exponential law: Monopole detectors, Bogoliubov transformations, and the thermal nature of the Euclidean vacuum in RP^3 de Sitter spacetime

We consider scalar field theory on the RP^3 de Sitter spacetime (RP3dS), which is locally isometric to de Sitter space (dS) but has spatial topology RP^3. We compare the Euclidean vacua on RP3dS and dS in terms of three quantities that are relevant for an inertial observer: (i) the stress-energy tensor; (ii) the response of an inertial monopole particle detector; (iii) the expansion of the Euclidean vacuum in terms of many-particle states associated with static coordinates centered at an inertial world line. In all these quantities, the differences between RP3dS and dS turn out to fall off exponentially at early and late proper times along the inertial trajectory. In particular, (ii) and (iii) yield at early and late proper times in RP3dS the usual thermal result in the de Sitter Hawking temperature. This conforms to what one might call an exponential law: in expanding locally de Sitter spacetimes, differences due to global topology should fall off exponentially in the proper time.

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arXiv1999

Single-exterior black holes and the AdS-CFT conjecture

In the context of the conjectured AdS-CFT correspondence of string theory, we consider a class of asymptotically Anti-de Sitter black holes whose conformal boundary consists of a single connected component, identical to the conformal boundary of Anti-de Sitter space. In a simplified model of the boundary theory, we find that the boundary state to which the black hole corresponds is pure, but this state involves correlations that produce thermal expectation values at the usual Hawking temperature for suitably restricted classes of operators. The energy of the state is finite and agrees in the semiclassical limit with the black hole mass. We discuss the relationship between the black hole topology and the correlations in the boundary state, and speculate on generalizations of the results beyond the simplified model theory.

hep-th↗

arXiv1998

Instantons and unitarity in quantum cosmology with fixed four-volume

We find a number of complex solutions of the Einstein equations in the so-called unimodular version of general relativity, and we interpret them as saddle points yielding estimates of a gravitational path integral over a space of almost everywhere Lorentzian metrics on a spacetime manifold with topology of the "no-boundary" type. In this setting, the compatibility of the no-boundary initial condition with the definability of the quantum measure reduces reduces to the normalizability and unitary evolution of the no-boundary wave function ψ. We consider the spacetime topologies R^4 and RP^4 # R^4 within a Taub minisuperspace model with spatial topology S^3, and the spacetime topology R^2 x T^2 within a Bianchi type I minisuperspace model with spatial topology T^3. In each case there exists exactly one complex saddle point (or combination of saddle points) that yields a wave function compatible with normalizability and unitary evolution. The existence of such saddle points tends to bear out the suggestion that the unimodular theory is less divergent than traditional Einstein gravity. In the Bianchi type I case, the distinguished complex solution is approximately real and Lorentzian at late times, and appears to describe an explosive expansion from zero size at T=0. (In the Taub cases, in contrast, the only complex solution with nearly Lorentzian late-time behavior yields a wave function that is normalizable but evolves nonunitarily, with the total probability increasing exponentially in the unimodular "time" in a manner that suggests a continuous creation of new universes at zero volume.) The issue of the stability of these results upon the inclusion of more degrees of freedom is raised.

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arXiv1998

Hamiltonian evolution and quantization for extremal black holes

We present and contrast two distinct ways of including extremal black holes in a Lorentzian Hamiltonian quantization of spherically symmetric Einstein-Maxwell theory. First, we formulate the classical Hamiltonian dynamics with boundary conditions appropriate for extremal black holes only. The Hamiltonian contains no surface term at the internal infinity, for reasons related to the vanishing of the extremal hole surface gravity, and quantization yields a vanishing black hole entropy. Second, we give a Hamiltonian quantization that incorporates extremal black holes as a limiting case of nonextremal ones, and examine the classical limit in terms of wave packets. The spreading of the packets, even the ones centered about extremal black holes, is consistent with continuity of the entropy in the extremal limit, and thus with the Bekenstein-Hawking entropy even for the extremal holes. The discussion takes place throughout within Lorentz-signature spacetimes.

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arXiv1998

Inextendible Schwarzschild black hole with a single exterior: How thermal is the Hawking radiation?

Several approaches to Hawking radiation on Schwarzschild spacetime rely in some way or another on the fact that the Kruskal manifold has two causally disconnected exterior regions. We investigate the Hawking(-Unruh) effect for a real scalar field on the $\RPthree$ geon: an inextendible, globally hyperbolic, space and time orientable eternal black hole spacetime that is locally isometric to Kruskal but contains only one exterior region. The Hartle-Hawking-like vacuum~$\hhvacgeon$, which can be characterized alternatively by the positive frequency properties along the horizons or by the complex analytic properties of the Feynman propagator, turns out to contain exterior region Boulware modes in correlated pairs, and any operator in the exterior that only couples to one member of each correlated Boulware pair has thermal expectation values in the usual Hawking temperature. Generic operators in the exterior do not have this special form; however, we use a Bogoliubov transformation, a particle detector analysis, and a particle emission-absorption analysis that invokes the analytic properties of the Feynman propagator, to argue that $\hhvacgeon$ appears as a thermal bath with the standard Hawking temperature to any exterior observer at asymptotically early and late Schwarzschild times. A~(naive) saddle-point estimate for the path-integral-approach partition function yields for the geon only half of the Bekenstein-Hawking entropy of a Schwarzschild black hole with the same ADM mass: possible implications of this result for the validity of path-integral methods or for the statistical interpretation of black-hole entropy are discussed. Analogous results hold for a Rindler observer in a flat spacetime whose global properties mimic those of the geon.

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arXiv1998

Hamiltonian spacetime dynamics with a spherical null-dust shell

We consider the Hamiltonian dynamics of spherically symmetric Einstein gravity with a thin null-dust shell, under boundary conditions that fix the evolution of the spatial hypersurfaces at the two asymptotically flat infinities of a Kruskal-like manifold. The constraints are eliminated via a Kuchar-type canonical transformation and Hamiltonian reduction. The reduced phase space $\tildeΓ$ consists of two disconnected copies of $R^4$, each associated with one direction of the shell motion. The right-moving and left-moving test shell limits can be attached to the respective components of $\tildeΓ$ as smooth boundaries with topology $R^3$. Choosing the right-hand-side and left-hand-side masses as configuration variables provides a global canonical chart on each component of $\tildeΓ$, and renders the Hamiltonian simple, but encodes the shell dynamics in the momenta in a convoluted way. Choosing the shell curvature radius and the "interior" mass as configuration variables renders the shell dynamics transparent in an arbitrarily specifiable stationary gauge "exterior" to the shell, but the resulting local canonical charts do not cover the three-dimensional subset of $\tildeΓ$ that corresponds to a horizon-straddling shell. When the evolution at the infinities is freed by introducing parametrization clocks, we find on the unreduced phase space a global canonical chart that completely decouples the physical degrees of freedom from the pure gauge degrees of freedom. Replacing one infinity by a flat interior leads to analogous results, but with the reduced phase space $R^2 \cup R^2$. The utility of the results for quantization is discussed.

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arXiv1997

Reduced phase space formalism for spherically symmetric geometry with a massive dust shell

We perform a Hamiltonian reduction of spherically symmetric Einstein gravity with a thin dust shell of positive rest mass. Three spatial topologies are considered: Euclidean (R^3), Kruskal (S^2 x R), and the spatial topology of a diametrically identified Kruskal (RP^3 - {a point at infinity}). For the Kruskal and RP^3 topologies the reduced phase space is four-dimensional, with one canonical pair associated with the shell and the other with the geometry; the latter pair disappears if one prescribes the value of the Schwarzschild mass at an asymptopia or at a throat. For the Euclidean topology the reduced phase space is necessarily two-dimensional, with only the canonical pair associated with the shell surviving. A time-reparametrization on a two-dimensional phase space is introduced and used to bring the shell Hamiltonians to a simpler (and known) form associated with the proper time of the shell. An alternative reparametrization yields a square-root Hamiltonian that generalizes the Hamiltonian of a test shell in Minkowski space with respect to Minkowski time. Quantization is briefly discussed. The discrete mass spectrum that characterizes natural minisuperspace quantizations of vacuum wormholes and RP^3-geons appears to persist as the geometrical part of the mass spectrum when the additional matter degree of freedom is added.

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arXiv1997

Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons

We examine counterparts of the Reissner-Nordstrom-anti-de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface Sigma of constant negative or zero curvature. When horizons exist, the spacetimes are black holes with an asymptotically locally anti-de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not S^2. Maximal analytic extensions of the solutions are given. The local Hawking temperature is found. When Sigma is closed, we derive the first law of thermodynamics using a Brown-York type quasilocal energy at a finite boundary, and we identify the entropy as one quarter of the horizon area, independent of the horizon topology. The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution.

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