Four-dimensional Gravity Research Articles

Gravity on an extended brane in six-dimensional warped flux compactifications

We study linearized gravity in a six-dimensional Einstein-Maxwell model of warped braneworlds, where the extra dimensions are compactified by a magnetic flux. It is difficult to construct a strict codimension two braneworld with matter sources other than pure tension. To overcome this problem we replace the codimension two defect by an extended brane, with one spatial dimension compactified on a Kaluza-Klein circle. Our background is composed of a warped, axisymmetric bulk and one or two branes. We find that weak gravity sourced by arbitrary matter on the brane(s) is described by a four-dimensional scalar-tensor theory. We show, however, that the scalar mode is suppressed at long distances and hence four-dimensional Einstein gravity is reproduced on the brane.

Three-geometry and reformulation of the Wheeler–DeWitt equation

A reformulation of the Wheeler–DeWitt equation which highlights the role of gauge-invariant three-geometry elements is presented. It is noted that the classical super-Hamiltonian of four-dimensional gravity as simplified by Ashtekar through the use of gauge potential and densitized triad variables can furthermore be succinctly expressed as a vanishing Poisson bracket involving three-geometry elements. This is discussed in the general setting of the Barbero extension of the theory with arbitrary non-vanishing value of the Immirzi parameter, and when a cosmological constant is also present. A proposed quantum constraint of density weight 2 which is polynomial in the basic conjugate variables is also demonstrated to correspond to a precise simple ordering of the operators, and may thus help to resolve the factor ordering ambiguity in the extrapolation from classical to quantum gravity. An alternative expression of a density weight 1 quantum constraint which may be more useful in the spin network context is also discussed, but this constraint is non-polynomial and is not motivated by factor ordering. The paper also highlights the fact that while the volume operator has become a pre-eminent object in the current manifestation of loop quantum gravity, the volume element and the Chern–Simons functional can be of equal significance, and need not be mutually exclusive. Both these fundamental objects appear explicitly in the reformulation of the Wheeler–DeWitt constraint.

Is N=8 supergravity ultraviolet finite?

Conventional wisdom holds that no four-dimensional gravity field theory can be ultraviolet finite. This understanding is based mainly on power counting. Recent studies confirm that one-loop N = 8 supergravity amplitudes satisfy the so-called 'no-triangle hypothesis', which states that triangle and bubble integrals cancel from these amplitudes. A consequence of this hypothesis is that for any number of external legs, at one loop N = 8 supergravity and N = 4 super-Yang-Mills have identical superficial degrees of ultraviolet behavior in D dimensions. We describe how the unitarity method allows us to promote these one-loop cancellations to higher loops, suggesting that previous power counts were too conservative. We discuss higher-loop evidence suggesting that N = 8 supergravity has the same degree of divergence as N = 4 super-Yang-Mills theory and is ultraviolet finite in four dimensions. We comment on calculations needed to reinforce this proposal, which are feasible using the unitarity method.

Ricci flow and black holes

Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to general relativity, where gradient flow is Ricci flow, and focus on the example of four-dimensional Euclidean gravity with boundary S1 × S2, representing the canonical ensemble for gravity in a box. At high temperature the action has three saddle points: hot flat space and a large and small black hole. Adding a time direction, these also give static five-dimensional Kaluza–Klein solutions, whose potential energy equals the four-dimensional action. The small black hole has a Gross–Perry–Yaffe-type negative mode, and is therefore unstable under Ricci flow. We numerically simulate the two flows seeded by this mode, finding that they lead to the large black hole and to hot flat space respectively, in the latter case via a topology-changing singularity. In the context of string theory these flows are world-sheet renormalization group trajectories. We also use them to construct a novel free energy diagram for the canonical ensemble.

Inflaton perturbations in brane-world cosmology with induced gravity

We study cosmological perturbations in the brane models with an induced Einstein–Hilbertterm on a brane. We consider an inflaton confined to a de Sitter brane in a five-dimensionalMinkowski spacetime. Inflaton fluctuations excite Kaluza–Klein modes of bulk metricperturbations with mass and where is an integer. There are two branches(± branches) of solutions for the background spacetime. In the+ branch, which includes the self-accelerating universe, a resonance appears for a mode withm2 = 2H2 due to a spin-0perturbation with m2 = 2H2. The self-accelerating universe has a distinct feature becausethere is also a helicity-0 mode of spin-2 perturbations withm2 = 2H2. Inthe − branch, which can be thought as the Randall–Sundrum type brane-world withthe high energy quantum corrections, there is no resonance. At high energies,we analytically confirm that four-dimensional Einstein gravity is recovered,which is related to the disappearance of the van Dam–Veltman–Zakharovdiscontinuity in de Sitter spacetime. On sufficiently small scales, we confirm that thelinearized gravity on the brane is well described by the Brans–Dicke theory withω = 3Hrc in the− branchand ω = −3Hrc in the + branch, respectively, which confirms the existence of the ghost in the+ branch. We also study large scale perturbations. In the+ branch, the resonance induces a non-trivial anisotropic stress on the brane via theprojection of the Weyl tensor in the bulk, but no instability is shown to exist on the brane.

EFFECTIVE COSMOLOGICAL CONSTANT IN BRANE COSMOLOGY

We consider a brane-world of co-dimension one without reflection symmetry. Through it, we give a possible explanation of the great discrepancy between the vacuum energy and the observed cosmological constant without contradiction to the knowledge we have about our Universe. We also show the gravity observed will be standard four-dimensional gravity as long as the discrepancy of the bulk cosmological constant at different sides of the brane is small enough.

Effective gauge group of pure loop quantum gravity is SO(3): New estimate of the Immirzi parameter

We argue that the effective gauge group for {\it pure} four-dimensional loop quantum gravity(LQG) is SO(3) (or $SO(3,C)$) instead of SU(2) (or $SL(2,C)$). As a result, links with half-integer spins in spin network states are not realized for {\it pure} LQG, implying a modification of the spectra of area and volume operators. Our observations imply a new value of $\gamma \approx 0.170$ for the Immirzi parameter which is obtained from matching the Bekenstein-Hawking entropy to the number of states from LQG calculations. Moreover, even if the dominant contribution to the entropy is not assumed to come from configurations with the minimum spins, the results of both pure LQG and the supersymmetric extension of LQG can be made compatible when only integer spins are realized for the former, while the latter also contains half-integer spins, together with an Immirzi parameter for the supersymmetric case which is twice the value of the SO(3) theory. We also verify that the $-{1/2}$ coefficient of logarithmic correction to the Bekenstein-Hawking entropy formula is robust, independent of whether only integer, or also half-integer spins, are realized.

Boundary inflation and the WMAP data

Inflation in a five-dimensional brane world model with two boundary branes is studied. We make use of the moduli space approximation whereby the low energy theory reduces to a four-dimensional biscalar-tensor gravity plus a minimally coupled scalar field. After a detailed analysis of the inflationary solutions, we derive the evolution equations of the linear perturbations separating the adiabatic mode from two entropy modes. We then examine the primordial scalar and tensor power spectra and show that their tilt depends on the scalar-tensor coupling constant. Finally, the induced cosmic microwave background (CMB) anisotropies are computed and we present a Monte Carlo Markov chains exploration of the parameter space using the first year WMAP data. We find a marginalized probability bound for the associated Eddington parameter at the end of inflation $1\ensuremath{-}\ensuremath{\gamma}<2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, at 95% confidence level. This suggests that future CMB data could provide crucial information helping to distinguish scalar-tensor and standard inflationary scenarios.

A c-function for non-supersymmetric attractors

We present a c-function for spherically symmetric, static and asymptotically flat solutions in theories of four-dimensional gravity coupled to gauge fields and moduli. The c-function is valid for both extremal and non-extremal black holes. It monotonically decreases from infinity and in the static region acquires its minimum value at the horizon, where it equals the entropy of the black hole. Higher dimensional cases, involving $p$-form gauge fields, and other generalisations are also discussed.

Infinite-dimensional spin-2 symmetries in Kaluza-Klein theories

We consider the couplings of an infinite number of spin-2 fields to gravity appearing in Kaluza-Klein theories. They are constructed as the broken phase of a massless theory possessing an infinite-dimensional spin-2 symmetry. Focusing on a circle compactification of four-dimensional gravity we show that the resulting gravity/spin-2 system in D=3 has in its unbroken phase an interpretation as a Chern-Simons theory of the Kac-Moody algebra associated to the Poincare group and also fits into the geometrical framework of algebra-valued differential geometry developed by Wald. Assigning all degrees of freedom to scalar fields, the matter couplings in the unbroken phase are determined, and it is shown that their global symmetry algebra contains the Virasoro algebra together with an enhancement of the Ehlers group to its affine extension. The broken phase is then constructed by gauging a subgroup of the global symmetries. It is shown that metric, spin-2 fields and Kaluza-Klein vectors combine into a Chern-Simons theory for an extended algebra, in which the affine Poincare subalgebra acquires a central extension.

Localizing gravity on the triple intersection of 7-branes in 10D

It was recently proposed that our universe could naturally come to be dominated by 3-branes and 7-branes if the universe is ten-dimensional. In this paper, we explicitly demonstrate that gravity can be localized on the intersection of three 7-branes in AdS10 to give four-dimensional gravity. We derive the exact relations among the tensions of the branes, and show that they apply independently of the precise distribution of energy within the necessarily thickened branes. We demonstrate this with several technical sections showing a simple formula for the curvature tensor of a diagonal metric with isometries as well as for the curvature at a gravitational singularity. We also demonstrate a subtlety in applying Stoke's Theorem to this set-up.

Accelerating black diholes and static black dirings

We show how a recently discovered black ring solution with a rotating 2-sphere can be turned into two new solutions of Einstein-Maxwell-dilaton theory. The first is a four-dimensional solution describing a pair of oppositely charged, extremal black holes--known as a black dihole--undergoing uniform acceleration. The second is a five-dimensional solution describing a pair of concentric, static extremal black rings carrying opposite dipole charges--a so-called black di-ring. The properties of both solutions, which turn out to be formally very similar, are analyzed in detail. We also present, in an appendix, an accelerating version of the Zipoy-Voorhees solution in four-dimensional Einstein gravity.

Brane Cosmology From Heterotic String Theory

We consider brane cosmologies within the context of five-dimensional actions with O(a') higher curvature corrections. The actions are compatible with bulk string amplitude calculations from heterotic string theory. We find wrapped solutions that satisfy the field equations in an approximate but acceptable manner given their complexity, where the internal four-dimensional scale factor is naturally inflating, having an exponential De-Sitter form. The temporal dependence of the metric components is non-trivial so that this metric cannot be factored as in a conformally flat case. The effective Planck mass is finite and the brane solutions localize four-dimensional gravity, while the four-dimensional gravitational constant varies with time. The Hubble constant can be freely specified through the initial value of the scalar field, to conform with recent data.

COSMOLOGICAL SOLUTION FROM D-BRANE MOTION IN NS5-BRANES BACKGROUND

We study dynamics of a D3-brane propagating in the vicinity of k coincident NS5 branes. We show that when gs is small, there exists a regime in which dynamics of the D-brane is governed by Dirac–Born–Infeld action while higher order derivative in the expansion cannot be neglected. This leads to a restriction on how fast scalar field may roll. We analyze the motion of a rolling scalar field in this regime, and extend the analysis to cosmological systems obtained by coupling this type of field theory to four-dimensional gravity. It also leads to some FRW cosmologies, some of which are related to those obtained with tachyon matter.

METASTABLE MASSIVE GRAVITONS FROM AN INFINITE EXTRA DIMENSION

Motivated by stringy considerations, Randall and Sundrum have proposed a model where all the fields and particles of physics, save gravity, are confined on a four-dimensional brane embedded in five-dimensional anti-de Sitter space. Their scenario features a stable bound state of bulk gravity waves and the brane that reproduces standard general relativity. We demonstrate that in addition to this zero-mode, there is also a discrete set of metastable bound states that behave like massive four-dimensional gravitons which decay by tunneling into the bulk. These are resonances of the bulk-brane system akin to black hole quasinormal modes — as such, they give rise to the dominant corrections to four-dimensional gravity. The phenomenology of braneworld perturbations is greatly simplified when these resonant modes are taken into account, which is illustrated by considering gravitational radiation emitted from nearby sources and early universe physics.

Holography and brane–bulk energy exchange

Holography relates the low energy regime of the five-dimensional description of generalizedRandall–Sundrum cosmology to a four-dimensional theory. The four-dimensional theorydescribes observable matter interacting with a hidden (non-conformal) gauge theory. Thisprovides a new arena for implementing the cosmological brane–bulk energy exchangeprocesses studied previously. Unlike the five-dimensional description, there is no big-bangsingularity in four dimensions. Some simple solutions are presented in four dimensions. Agauge theory description of observable four-dimensional gravity is also advocated.

Quadrupole formula for Kaluza–Klein modes in the braneworld

The quadrupole formula in four-dimensional Einstein gravity is a useful tool to describe gravitational wave radiation. We derive the quadrupole formula for the Kaluza–Klein (KK) modes in the Randall–Sundrum braneworld model. The quadrupole formula provides a transparent representation of the exterior weak gravitational field induced by localized sources. We find that a general isolated dynamical source gives rise to the 1/r2 correction to the leading 1/r gravitational field. We apply the formula to an evaluation of the effective energy carried by the KK modes from the viewpoint of an observer on the brane. Contrary to the ordinary gravitational waves (zero mode), the flux of the induced KK modes by the non-spherical part of the quadrupole moment vanishes at infinity and only the spherical part contributes to the flux. Since the effect of the KK modes appears in the linear order of the metric perturbations, the effective energy flux observed on the brane is not always positive, but can become negative depending on the motion of the localized sources.

Mass, angular momentum and thermodynamics in four-dimensional Kerr-AdS black holes

In this paper, the connection between the Lorentz-covariant counterterms that regularize the four-dimensional AdS gravity action and topological invariants is explored. It is shown that demanding the spacetime to have a negative constant curvature in the asymptotic region permits the explicit construction of such series of boundary terms. The orthonormal frame is adapted to appropriately describe the boundary geometry and, as a result, the boundary term can be expressed as a functional of the boundary metric, extrinsic curvature and intrinsic curvature. This choice also allows to write down the background-independent Noether charges associated to asymptotic symmetries in standard tensorial formalism. The absence of the Gibbons-Hawking term is a consequence of an action principle based on a boundary condition different than Dirichlet on the metric. This argument makes plausible the idea of regarding this approach as an alternative regularization scheme for AdS gravity in all even dimensions, different than the standard counterterms prescription. As an illustration of the finiteness of the charges and the Euclidean action in this framework, the conserved quantities and black hole entropy for four-dimensional Kerr-AdS are computed.

Vacuum decay on a brane world

The bubble nucleation rate for a first order phase transition occurring on a brane world is calculated. Both the Coleman-de Luccia thin wall instanton and the Hawking-Moss instanton are considered. The results are compared with the corresponding nucleation rates for standard four-dimensional gravity.

Exact solutions with noncommutative symmetries in Einstein and gauge gravity

We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five-dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated by anholonomic frame transforms and parametrized by generic off-diagonal metrics. For certain particular cases, the new classes of metrics have explicit limits with Killing symmetries but, in general, they may be characterized by certain anholonomic noncommutative matrix geometries. We argue that different classes of noncommutative symmetries can be induced by exact solutions of the field equations in commutative gravity modeled by a corresponding moving real and complex frame geometry. We analyze two classes of black ellipsoid solutions (in the vacuum case and with cosmological constant) in four-dimensional gravity and construct the analytic extensions of metrics for certain classes of associated frames with complex valued coefficients. The third class of solutions describes 5D wormholes which can be extended to complex metrics in complex gravity models defined by noncommutative geometric structures. The anholonomic noncommutative symmetries of such objects are analyzed. We also present a descriptive account how the Einstein gravity can be related to gauge models of gravity and their noncommutative extensions and discuss such constructions in relation to the Seiberg–Witten map for the gauge gravity. Finally, we consider a formalism of vielbeins deformations subjected to noncommutative symmetries in order to generate solutions for noncommutative gravity models with Moyal (star) product.