Brandon Carter

The Four laws of black hole mechanics

Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area A of the event horizon and the “surface gravity” κ of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which correspond to and in some ways transcend the four laws of thermodynamics.

Hamilton-Jacobi and Schrödinger separable solutions of Einstein's equations

This paper contains an investigation of spaces with a two parameter Abelian isometry group in which the Hamilton-Jacobi equation for the geodesies is soluble by separation of variables in such a way that a certain natural canonical orthonormal tetrad is determined. The spaces satisfying the stronger condition that the corresponding Schrodinger equation is separable are isolated in a canonical form for which Einstein’s vacuum equations and the source-free Einstein-Maxwell equations (with or without a Λ term) can be solved explicitly. A fairly extensive family of new solutions is obtained including the previously known solutions of de Sitter, Kasner, Taub-NUT, and Kerr as special cases.

Large number coincidences and the anthropic principle in cosmology

Prof. Wheeler has asked me to say something for the record about some ideas that I once suggested (at the Clifford Memorial meeting in Princeton in 1970) and to which Hawking and Collins have referred (Astrophys. J. 180, 317, 1973). This concerns a line of thought which I believe to be potentially fertile, but which I did not write up at the time because I felt (as I still feel) that it needs further development. However, it is not inappropriate that this matter should have cropped up again on the present occasion, since it consists basically of a reaction against exaggerated subservience to the ‘Copernican principle’.

Killing Horizons and Orthogonally Transitive Groups in Space‐Time

Some concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null hypersurface whose null tangent vector can be normalized to coincide with a Killing vector field) and of the related concepts of invertibility and orthogonal transitivity of an isometry group in an n‐dimensional pseudo‐Riemannian manifold (a group is said to be orthogonally transitive if its surfaces of transitivity, being of dimension p, say, are orthogonal to a family of surfaces of conjugate dimension n ‐ p). The relationships between these concepts are described and it is shown (in Theorem 1) that, if an isometry group is orthogonally transitive then a local isometry horizon occurs wherever its surfaces of transitivity are null, and that it is a Killing horizon if the group is Abelian. In the case of (n ‐ 2)‐parameter Abelian groups it is shown (in Theorem 2) that, under suitable conditions (e.g., when a sym...

The commutation property of a stationary, axisymmetric system

It is shown that in studies of space-time systems which are both stationary and axisymmetric, no generality is lost by considering only cases where the stationary and axisymmetric actions (or equivalently the two corresponding Killing vector fields) commute.

Elastic perturbation theory in general relativity and a variation principle for a rotating solid star

Perturbation analysis is applied to the theory of a General Relativistic perfectly elastic medium as developed by Carter and Quintana (1972). Formulae are derived for the Eulerian variations of the principal fields (density, pressure tensor, etc.) on which the description of such a medium is based, where the perturbations are induced both by infinitesimal displacements of the medium and by infinitesimal variations of the metric tensor. These formulae will be essential for problems such as the study of torsional vibration modes in a neutron star.As examples of their application, the variation formulae are used in the derivation firstly of a simple (dynamic) action principle for a perfectly elastic medium (this principle being a generalisation of the one given by Taub (1954) for a perfect fluid) and secondly in the derivation of a rather more sophisticated mass variation principle for a stationary rotating solid star (this principle being a generalisation of the one given by Hartle and Sharp (1967) for a perfect fluid star).

Cosmic vortons and particle physics constraints.

We investigate the cosmological consequences of particle physics theories that admit stable loops of superconducting cosmic string - {\it vortons}. General symmetry breaking schemes are considered, in which strings are formed at one energy scale and subsequently become superconducting in a secondary phase transition at what may be a considerably lower energy scale. We estimate the abundances of the ensuing vortons, and thereby derive constraints on the relevant particle physics models from cosmological observations. These constraints significantly restrict the category of admissible Grand Unified theories, but are quite compatible with recently proposed effects whereby superconducting strings may have been formed close to the electroweak phase transition.

Duality relation between charged elastic strings and superconducting cosmic strings

Abstract The mechanical properties of macroscopic electromagnetically coupled string models in a flat or curved background are treated using a covariant formalism allowing the construction of a duality transformation that relates the category of uniform “electric” string models, constructed as the (nonconducting) charged generalisation of ordinary uncoupled (violin type) elastic strings, to a category of “magnetic” string models comprising recently discussed varieties of “superconducting cosmic strings”.

Differential rotation of relativistic superfluid in neutron stars

ABSTRA C T It is shown how to set up a mathematically elegant and fully relativistic superfluid model that can provide a realistic approximation (neglecting small anisotropies due to crust solidity, magnetic fields, etc., but allowing for the regions with vortex pinning) of the global structure of a rotating neutron star, in terms of just two independently moving constituents. One of these represents the differentially rotating neutron superfluid, while the other part represents the combination of all the other ingredients, including the degenerate electrons, the superfluid protons in the core, and the ions in the crust, the electromagnetic interactions of which will tend to keep them locked together in a state of approximately rigid rotation. Order of magnitude estimates are provided for relevant parameters such as the resistive drag coefficient.

Outer curvature and conformal geometry of an imbedding

The differential geometry of an imbedded (e.g. string or membrane world sheet) surface in a higher-dimensional background is shown to be conveniently describable (except in the null limit case) in terms of what are designated as its first, second, and third fundamental tensors, which will have the respective symmetry properties ημν = η(μν) as a trivial algebraic identity, Kμνρ = K(μν)ρ as the “generalised Weingarten identity”, which is the (Frobenius type) integrability condition for the imbedding, and Ξλμνρ = Ξ(λμν)ρ as a “generalised Codazzi equation”, which depends on the background geometry being flat or of constant curvature, needing replacement by a more complicated expression for a generic value of the background curvature Bκλμν. The “generalised Gauss equation” expressing the dependence on this background curvature of the internal curvature tensor Rκλμν of the imbedded surface is converted into terms of the first and second fundamental tensors, and it is thereby demonstrated that the vanishing of the (conformally invariant) “conformation tensor”, i.e. the trace free part Cμνρ of the second fundamental tensor Kμvρ, is a sufficient condition for conformal flatness of the imbedded surface (and thus in particular for the vanishing of its (Weyl type) conformal curvature tensor Cκλμν) provided the background is itself conformally flat. In a trio of which the first two members are the generalised Gauss and Codazzi equations, the “third” member is shown to give an expression in terms of Cμνρ for the (trace free, conformally invariant) “outer curvature” tensor Ωκλμν whose vanishing is the condition for feasibility of the natural generalisation of the Walker frame transportation ansatz. The vanishing of Cμνρ is shown to be sufficient in a conformally flat background for the vanishing also of Ωκλμν.