arXiv subjects

C. N. Pope

Publications and source records attributed to C. N. Pope.

At least 109 records · Page 6Linked to original sources

arXiv2010

Homogeneous Einstein Metrics on SO(n)

It is well known that every compact simple Lie group G admits an Einstein metric that is invariant under the independent left and right actions of G. In addition to this bi-invariant metric, with G x G symmetry, it was shown by D'Atri and Ziller that every compact simple Lie group except SU(2) and SO(3) admits at least one further homogeneous Einstein metric, invariant under G x H, where H is some proper subgroup of G. In this paper we consider the Lie groups G=SO(n) for arbitrary n, and provide an explicit construction of (3k-4) inequivalent homogeneous Einstein metrics on SO(2k), and (3k-3) inequivalent homogeneous Einstein metrics on SO(2k+1).

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arXiv2009

Kac-Moody and Virasoro Symmetries of Principal Chiral Sigma Models

It is commonly asserted that there is a \hat G\times G centreless Kac-Moody extension of the manifest G\times G global symmetry of the two-dimensional principal chiral model (PCM) for the group manifold G. Here, we show that the symmetry is in fact larger, namely \hat G\times \hat G, the full centreless Kac-Moody extension of the entire manifest G\times G global symmetry. Extending previous results in the literature, we also obtain an explicit realisation of the Virasoro-like symmetry of the PCM, generated by K_n=L_{n+1} - L_{n-1} for both positive and negative n. We show that these generators obey Sugarawara-type commutation relations with the two commuting copies of the Kac-Moody algebra \hat G.

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arXiv2009

Non-Abelian Black Holes in D=5 Maximal Gauged Supergravity

We investigate static non-abelian black hole solutions of anti-de Sitter Einstein-Yang-Mills-Dilaton gravity, which is obtained as a consistent truncation of five-dimensional maximal gauged supergravity. If the dilaton is (consistently) set to zero, the remaining equations of motion, with a spherically-symmetric ansatz, may be derived from a superpotential. The associated first-order equations admit an explicit solution supported by a non-abelian SU(2) gauge potential, which has a logarithmically growing mass term. In an extremal limit the horizon geometry becomes AdS$_2\times S^3$. If the dilaton is also excited, the equations of motion cannot easily be solved explicitly, but we obtain the asymptotic form of the more general non-abelian black holes in this case. An alternative consistent truncation, in which the Yang-Mills fields are set to zero, also admits a description in terms of a superpotential. This allows us to construct explicit wormhole solutions (neutral spherically-symmetric domain walls). These solutions may be generalised to dimensions other than five.

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arXiv2009

Solutions to Horava Gravity

Recently Horava proposed a non-relativistic renormalisable theory of gravitation, which reduces to Einstein's general relativity at large distances, and that may provide a candidate for a UV completion of Einstein's theory. In this paper, we derive the full set of equations of motion, and then we obtain spherically symmetric solutions and discuss their properties. We also obtain solutions for the Friedman-Lemaitre-Robertson-Walker cosmological metric.

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arXiv2009

Einstein Metrics on Group Manifolds and Cosets

It is well known that every compact simple group manifold G admits a bi-invariant Einstein metric, invariant under G_L\times G_R. Less well known is that every compact simple group manifold except SO(3) and SU(2) admits at least one more homogeneous Einstein metric, invariant still under G_L but with some, or all, of the right-acting symmetry broken. (SO(3) and SU(2) are exceptional in admitting only the one, bi-invariant, Einstein metric.) In this paper, we look for Einstein metrics on three relatively low dimensional examples, namely G=SU(3), SO(5) and G_2. For G=SU(3), we find just the two already known inequivalent Einstein metrics. For G=SO(5), we find four inequivalent Einstein metrics, thus extending previous results where only two were known. For G=G_2 we find six inequivalent Einstein metrics, which extends the list beyond the previously-known two examples. We also study some cosets G/H for the above groups G. In particular, for SO(5)/U(1) we find, depending on the embedding of the U(1), generically two, with exceptionally one or three, Einstein metrics. We also find a pseudo-Riemannian Einstein metric of signature (2,6) on SU(3), an Einstein metric of signature (5,6) on G_2/SU(2)_{diag}, and an Einstein metric of signature (4,6) on G_2/U(2). Interestingly, there are no Lorentzian Einstein metrics among our examples.

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arXiv2009

Extremal Black Hole/CFT Correspondence in (Gauged) Supergravities

We extend the investigation of the recently proposed Kerr/CFT correspondence to large classes of rotating black hole solutions in gauged and ungauged supergravities. The correspondence, proposed originally for four-dimensional Kerr black holes, asserts that the quantum states in the near-horizon region of an extremal rotating black hole are holographically dual to a two-dimensional chiral theory whose Virasoro algebra arises as an asymptotic symmetry of the near-horizon geometry. In fact in dimension D there are [(D-1)/2] commuting Virasoro algebras. We consider a general canonical class of near-horizon geometries in arbitrary dimension D, and show that in any such metric, the [(D-1)/2] central charges each imply, via the Cardy formula, a microscopic entropy that agrees with the Bekenstein-Hawking entropy of the associated extremal black hole. In the remainder of the paper we show for most of the known rotating black hole solutions of gauged supergravity, and for the ungauged supergravity solutions with four charges in D=4 and three charges in D=5, that their extremal near-horizon geometries indeed lie within the canonical form. This establishes that in all these examples, the microscopic entropies of the dual CFTs agree with the Bekenstein-Hawking entropies of the extremal rotating black holes.

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arXiv2009

Naturalness of CP Violation in the Standard Model

We construct a natural measure on the space of CKM matrices in the standard model, assuming the fermion mass matrices are randomly selected from a distribution which fits the observed quark mass hierarchy. This measure allows us to assess the likelihood of Jarlskog's CP violation parameter J taking its observed value J $\approx$ 3 x 10^{-5}. While well below the mathematically allowed maximum, we find that this value is in fact typical, once the observed quark masses are assumed.

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arXiv2009

Extremal Static AdS Black Hole/CFT Correspondence in Gauged Supergravities

A recently proposed holographic duality allows the Bekenstein-Hawking entropy of extremal rotating black holes to be calculated microscopically, by applying the Cardy formula to the two-dimensional chiral CFTs associated with certain reparameterisations of azimuthal angular coordinates in the solutions. The central charges are proportional to the angular momenta of the black hole, and so the method degenerates in the case of static (non-rotating) black holes. We show that the method can be extended to encompass such charged static extremal AdS black holes by using consistent Kaluza-Klein sphere reduction ansatze to lift them to exact solutions in the low-energy limits of string theory or M-theory, where the electric charges become reinterpreted as angular momenta associated with internal rotations in the reduction sphere. We illustrate the procedure for the examples of extremal charged static AdS black holes in four, five, six and seven dimensions.

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arXiv2009

Measures on Mixing Angles

We address the problem of the apparently very small magnitude of CP violation in the standard model, measured by the Jarlskog invariant J. In order to make statements about probabilities for certain values of J, we seek to find a natural measure on the space of Kobayashi-Maskawa matrices, the double quotient U(1)^2\SU(3)/U(1)^2. We review several possible, geometrically motivated choices of the measure, and compute expectation values for powers of J for these measures. We find that different choices of the measure generically make the observed magnitude of CP violation appear finely tuned. Since the quark masses and the mixing angles are determined by the same set of Yukawa couplings, we then do a second calculation in which we take the known quark mass hierarchy into account. We construct the simplest measure on the space of 3 x 3 Hermitian matrices which reproduces this known hierarchy. Calculating expectation values for powers of J in this second approach, we find that values of J close to the observed value are now rather likely, and there does not seem to be any fine tuning. Our results suggest that the choice of Kobayashi-Maskawa angles is closely linked to the observed mass hierarchy. We close by discussing the corresponding case of neutrinos.

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arXiv2008

Kerr-AdS/CFT Correspondence in Diverse Dimensions

It was proposed recently that the near-horizon states of an extremal four-dimensional Kerr black hole could be identified with a certain chiral conformal field theory whose Virasoro algebra arises as an asymptotic symmetry algebra of the near-horizon Kerr geometry. Supportive evidence for the proposed duality came from the equality of the microscopic entropy of the CFT, calculated by means of the Cardy formula, and the Bekenstein-Hawking entropy of the extremal Kerr black hole. In this paper we examine the proposed Kerr/CFT correspondence in a broader context. In particular, we show that the microscopic entropy and the Bekenstein-Hawking entropy agree also for the extremal Kerr-AdS metric in four dimensions, and also for the extremal Kerr-AdS metrics in dimensions 5, 6 and 7. General formulae for all higher dimensions are also presented.

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arXiv2008

AdS Wormholes

We obtain a large class of smooth Lorentzian p-brane wormholes in supergravities in various dimensions. They connect two asymptotically flat spacetimes. In cases where there is no dilaton involved in the solution, the wormhole can connect an AdS_n\times S^m in one asymptotic region to a flat spacetime in the other. We obtain explicit examples for (n,m)=(4,7), (7,4), (5,5), (3,3), (3,2). These geometries correspond to field theories with UV conformal fixed points, and they undergo decompactification in the IR region. In the case of AdS_3, we compute the central charge of the corresponding conformal field theory.

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arXiv2008

New Charged Black Holes in Five Dimensions

We obtain new stationary charged solutions of five-dimensional minimal supergravity. We first obtain purely dipole charged solutions, by extending a technique that we developed for five-dimensional Ricci-flat metrics in a previous paper, which could be viewed as being analogous to a four-dimensional construction by Demianski and Plebanski. The further introduction of electric charge is achieved by means of a solution-generating technique, which exploits the global SL(2,R) symmetry of five-dimensional minimal supergravity reduced on a timelike direction to four dimensions. We present this procedure in detail, since it provides a particularly simple general way of adding charge to any stationary solution of five-dimensional minimal supergravity. The new charged solutions we obtain limit in special cases to black rings carrying electric and magnetic dipole charge, or to charged Myers-Perry rotating black holes. We analyse the general solutions in detail, showing that they can describe asymptotically locally flat black holes whose horizon is a lens space L(n;m)=S^3/Γ(n;m). At infinity they approach Minkowski_5/Γ(m;n).

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arXiv2008

New Black Holes in Five Dimensions

We construct new stationary Ricci-flat metrics of cohomogeneity 2 in five dimensions, which generalise the Myers-Perry rotating black hole metrics by adding a further non-trivial parameter. We obtain them via a construction that is analogous to the construction by Plebanski and Demianski in four dimensions of the most general type D metrics. Limiting cases of the new metrics contain not only the general Myers-Perry black hole with independent angular momenta, but also the single rotation black ring of Emparan and Reall. In another limit, we obtain new static metrics that describe black holes whose horizons are distorted lens spaces L(n;m)= S^3/Γ(n;m), where m\ge n+2\ge 3. They are asymptotic to Minkowski spacetime factored by Γ(m;n). In the general stationary case, by contrast, the new metrics describe spacetimes with an horizon and with a periodicity condition on the time coordinate; these examples can be thought of as five-dimensional analogues of the four-dimensional Taub-NUT metrics.

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arXiv2008

Metrics With Vanishing Quantum Corrections

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor $T_{μν}$ constructed from sums of terms the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called {\it universal} if, when evaluated on that Einstein metric, $T_{μν}$ is a multiple of the metric. A Ricci flat classical solution is called {\it strongly universal} if, when evaluated on that Ricci flat metric, $T_{μν}$ vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalisation; Einstein metrics with holonomy ${\rm Sim} (n-2)$ in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalised Ghanam-Thompson metric is weakly universal and that the Goldberg-Kerr metric is strongly universal; indeed, we show that universality extends to all 4-dimensional ${\rm Sim}(2)$ Einstein metrics. We also discuss generalizations to higher dimensions.

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arXiv2008

Infinite-Dimensional Symmetries of Two-Dimensional Coset Models

It has long been appreciated that the toroidal reduction of any gravity or supergravity to two dimensions gives rise to a scalar coset theory exhibiting an infinite-dimensional global symmetry. This symmetry is an extension of the finite-dimensional symmetry G in three dimensions, after performing a further circle reduction. There has not been universal agreement as to exactly what the extended symmetry algebra is, with different arguments seemingly concluding either that it is $\hat G$, the affine Kac-Moody extension of G, or else a subalgebra thereof. Exceptional in the literature for its explicit and transparent exposition is the extremely lucid discussion by Schwarz, which we take as our starting point for studying the simpler situation of two-dimensional flat-space sigma models, which nonetheless capture all the essential details. We arrive at the conclusion that the full symmetry is described by the Kac-Moody algebra G, although truncations to subalgebras, such as the one obtained by Schwarz, can be considered too. We then consider the explicit example of the SL(2,R)/O(2) coset, and relate Schwarz's approach to an earlier discussion that goes back to the work of Geroch.

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arXiv2008

Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)

The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, SIM(n-2). Ricci-flat metrics with SIM(2) holonomy were constructed by Kerr and Goldberg, and a single four-dimensional example with a non-zero cosmological constant was exhibited by Ghanam and Thompson. Here we reduce the problem of finding the general $n$-dimensional Einstein metric of SIM(n-2) holonomy, with and without a cosmological constant, to solving a set linear generalised Laplace and Poisson equations on an (n-2)-dimensional Einstein base manifold. Explicit examples may be constructed in terms of generalised harmonic functions. A dimensional reduction of these multi-centre solutions gives new time-dependent Kaluza-Klein black holes and monopoles, including time-dependent black holes in a cosmological background whose spatial sections have non-vanishing curvature.

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arXiv2007

New Rotating Non-Extremal Black Holes in D=5 Maximal Gauged Supergravity

We obtain new non-extremal rotating black hole solutions in maximal five-dimensional gauged supergravity. They are characterised by five parameters, associated with the mass, the two angular momenta, and two independently-specifiable charge parameters. Two of the three charges associated with the U(1)^3 Cartan subgroup of the SO(6) gauge group are equal, whilst the third can be independently specified. These new solutions generalise all the previously-known rotating solutions in five-dimensional gauged supergravity with independent angular momenta. They describe regular black holes, provided the parameters lie in appropriate ranges so that naked singularities and closed-timelike curves (CTCs) are avoided. We also construct the BPS limit, and show that regular supersymmetric black holes or topological solitons arise if the parameters are further restricted in an appropriate manner.

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arXiv2007

General Very Special Relativity is Finsler Geometry

We ask whether Cohen and Glashow's Very Special Relativity model for Lorentz violation might be modified, perhaps by quantum corrections, possibly producing a curved spacetime with a cosmological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous deformations, but none of these give rise to non-commutative translations analogous to those of the de Sitter deformation of the Poincaré group: spacetime remains flat. Only a 1-parameter family DISIM_b(2) of deformations of SIM(2) is physically acceptable. Since this could arise through quantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be taken into account. The Lorentz-violating point particle action invariant under DISIM_b(2) is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive DISIM_b(2)-invariant wave equations for particles of spins 0, 1/2 and 1. The experimental bound, $|b|<10^{-26}$, raises the question ``Why is the dimensionless constant $b$ so small in Very Special Relativity?''

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