Massless Field Equations Research Articles

Quantum radiation from a5-dimensional rotating black hole

We study a massless scalar field propagating in the background of a five-dimensional rotating black hole. We showed that in the Myers-Perry metric describing such a black hole the massless field equation allows the separation of variables. The obtained angular equation is a generalization of the equation for spheroidal functions. The radial equation is similar to the radial Teukolsky equation for the 4-dimensional Kerr metric. We use these results to quantize the massless scalar field in the space-time of the 5-dimensional rotating black hole and to derive expressions for energy and angular momentum fluxes from such a black hole.

On the structure of degenerate solutions of the Einstein conformally invariant scalar system

We address the question of the existence and construction of nontrivial, regular solutions of the Einstein–conformally invariant massless scalar field equations, i.e., solutions (g,Φ) satisfying (1−αΦ2)Rμν=α(4∇μΦ∇νΦ−2Φ∇μ∇νΦ−gμν∇σΦ∇σΦ), ∇μ∇μΦ=0, and additionally geometry and the scalar field are regular across the degeneracy region defined as the zeros of (1−αΦ2). Under the assumptions (1) the solution (g,Φ) is minimally of class C3 and admits a hypersurface orthogonal, timelike Killing vector field ξ, and (2) relative to the three spacelike hypersurfaces perpendicular to the Killing field, the degeneracy region constitute regular two-surfaces, and the induced positive definite three metric possesses a degenerate Ricci, we show that the conformal system admits nontrivial, regular across the degeneracy region solutions and we demonstrate that any such solution necessarily admits an additional local G(3) group of isometries possessing two-dimensional orbits of constant Gaussian curvature coinciding with the Φ=cons- equipotential two surfaces. Those solutions exhibit similar properties as the Levi–Civita–Ehlers–Kundt class of static solutions of Einstein’s vacuum equations. We investigate this coincidence and in particular we probe the origin of the additional local G(3) group of isometries exhibited by both classes of solutions. From the partial differential equations point of view, both systems, i.e., conformal system as well as the vacuum system, degenerate or become singular, the conformal system along solutions subject to αΦ2=1 within the static region, the vacuum along solutions subject to V=(−ξ⋅ξ)1/2→0+. We demonstrate that as a consequence of the singular nature of the dynamical equations, among all solutions possessing degenerate Ricci in the open vicinity of αΦ2=1, respectively, V→0+, the only regular across degeneracy region solutions are those characterized by a vanishing York–Cotton tensor and, furthermore, such solutions necessarily admit an additional local G(3) group of isometries possessing two-dimensional orbits of constant Gaussian curvature.

Star-product and massless free field dynamics in AdS4

Generic solution of free equations for massless fields of an arbitrary spin in AdS 4 is built in terms of the star-product algebra with spinor generating elements. A class of “plane wave” solutions is described explicitly.

Weyl Spinor and Solution of Massless Free Field Equations

The massless field equations for arbitrary spin in curved space-time arereconsidered. The general solution of the field equation in Robertson-Walkerspace-time that was previously determined is briefly discussed after explicitlyshowing that the Weyl spinor vanishes. The case of the Lemaitre-Tolman-Bondispace-time is studied in detail. The general expression of the corresponding Weylspinor is obtained and some particular situations exploited. The spin-3/2 andspin-2 massless field equations are solved explicitly. The solutions are simplifiedby the existence of nontrivial algebraic constraints. The angular part of theequations is separated by the usual separation method and integrated directly.The other equations that are not separated in the radial and time dependence arereduced to a simple form. The results obtained are extended, as a consequenceof previous results, to the case of arbitrary spin. The solution of the general caseessentially reduces to the treatment of spin 3/2 and spin 2.

Twistor integral representations of fundamental solutions of massless field equations

We consider the general dimensional (complex) Minkowski spaces and the extended twistor spaces. We show that the fundamental solutions of the complex wave or Laplace equations are explicitly represented by the integrals of some closed forms on the twistor spaces. The closed form is defined from labeled trees explained in graphs theory, and is written, as the cohomology class, by the linear combination of the logrithmic forms on some hyperplane configuration complement in some complex affine space.

Spherical black holes cannot support scalar hair

The static spherically symmetric ``black hole solution" of the Einstein - conformally invariant massless scalar field equations known as the BBMB ( Bocharova, , Bronikov, Melinkov, Bekenstein) black hole is critically examined. It is shown that the stress energy tensor is ill-defined at the horizon, and that its evaluation through suitable regularization yields ambiguous results. Consequently, the configuration fails to represent a genuine black hole solution. With the removal of this solution as a counterexample to the no hair conjecture, we argue that the following appears to be true: Spherical black holes cannot carry any kind of classical scalar hair.

Invariance of the Massless Field Equations under Changes of the Metric

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field Fijand the current density four-vector Ji, are invariant under the change of the metric gijby gij′= gij+ liljif liis a principal null direction of Fijand that an analogous result holds in the case of the massless Klein-Gordon equation if liis null and orthogonal to the gradient of the field and in the case of the null dust equations if liis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (gij, Fij) be an exact solution of the Einstein-Maxwell equations and let libe a principal null direction of Fij, then (gij+ lilj, Fij) is also an exact solution of the Einstein-Maxwell equations if and only if (lilj, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (gij, Fij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.

A Note on the Rarita-Schwinger Equations

Assuming that the background spacetime is a solution of the Einstein vacuum equations without cosmological constant, we analyze how the Rarita-Schwinger equations can be obtained via a particular generalization of the usual spin-3/2 massless free field equations. On the basis of this analysis we speculate on the possibility of finding other generalizations of the Rarita-Schwinger equations.

Problème de Cauchy global pour les équations linéaires de champs sans masse de spin 3/2 en métrique de Schwarzschild

We study the Dirac form of linear spin 3/2 massless field equations outside an uncharged spherical black-hole. We solve the Global Cauchy problem, first for minimum regularity (L 2 ) solutions, then in the successive domains of the Hamiltonian, and we prove the conservation of the constraints under evolution.

Electromagnetic interpretation of the massless spin-1 field equation in curved space-time

The source-free Maxwell equations associated to the massless spin-1 free field equation are considered in curved space-time. The gauge invariance of the theory is discussed by using as starting point the notion of the spinor potential. The structure of the electromagnetic field in the case of the Robertson-Walker space-time is discussed by using the solutions of the massless spin-1 equations previously determined. The flat-universe case of the standard cosmology is studied exactly and considered in some limiting physical situations.

On the W-geometrical origins of massless field equations and of gauge invariance

We show how to obtain all covariant field equations for massless particles of arbitrary integer, or half-integer, helicity in four dimensions from the quantization of the rigid particle, whose action is given by the integrated extrinsic curvature of its world-line, i.e. S = α ʃ ds κ. This geometrical particle system possesses one extra gauge invariance besides reparametrizations, and the full gauge algebra has been previously identified as classical W 3. The key observation is that the covariantly reduced phase space of this model can be naturally identified with the spinor and twistor descriptions of the covariant phase spaces associated with massless particles of helicity s = α. Then, standard quantization techniques require α to be quantized and show how the associated Hilbert spaces are solution spaces of the standard relativistic massless wave equations with s = α. Therefore it provides us with a simple particle model for Weyl fermions ( α = 1 2 ), Maxwell fields ( α = 1), and higher spin fields. Moreover, one can go a little further and in the Maxwell case show that, after a suitable redefinition of constraints, the standard Dirac quantization procedure for first-class constraints leads to a wave function which can be identified with the gauge potential A μ . Gauge symmetry appears in the formalism as a consequence of the invariance under W 3-morphisms, that is, exclusively in terms of the extrinsic geometry of paths in Minkowski space. When all gauge freedom is fixed one naturally obtains the standard Lorentz gauge condition on A μ , and Maxwell equations in that gauge. This construction has a direct generalization to arbitrary integer values of α, and we comment on the physically interesting case of linearized Einstein gravity ( α = 2).

Separation of the massless field equations for arbitrary spin in the Robertson–Walker space–time

The massless spin 2 free-field equation is studied in the Robertson–Walker space–time via the Newman–Penrose formalism and separated by using a Chandrasekhar–Teukolski method. The resulting temporal and angular equations are explicitly integrated. The radial equations are solved in the flat universe case. The closed universe case shows, in principle, the existence of a discrete spectrum of the energy of the massless particles. In the general case of spin greater than 2 the massless field equations are shown to be separable by induction. The separated equations admit the same recurrence structure and analog interpretation properties of the spin 2 and less than 2 cases.

Separation of the massless spin-1 equation in Robertson-Walker space-time

The massless spin-1 free field equation is studied via the Newman-Penrose formalism and separated by the Chandrasekhar-Teukolski method. The temporal and angular equations are explicitly integrated. The radial equations are solved in the flat-universe case. The closed-universe case shows, in principle, the existence of a discrete spectrum of the energy of the massless particles.

SPIN-$\frac{3}{2}$ POTENTIALS IN BACKGROUNDS WITH BOUNDARY

This paper studies the two-spinor form of the Rarita-Schwinger potentials subject to local boundary conditions compatible with local supersymmetry. The massless Rarita-Schwinger field equations are studied in four-real-dimensional Riemannian backgrounds with boundary. Gauge transformations on the potentials are shown to be compatible with the field equations providing the background is Ricci-flat, in agreement with previous results in the literature. However, the preservation of boundary conditions under such gauge transformations leads to a restriction of the gauge freedom. The recent construction by Penrose of secondary potentials which supplement the Rarita-Schwinger potentials is then applied. The equations for the secondary potentials, jointly with the boundary conditions, imply that the background four-geometry is further restricted to be totally flat. The analysis of other gauge transformations confirms that, in the massless case, the only admissible class of Riemannian backgrounds with boundary is totally flat.

Twistor spaces for real four-dimensional Lorentzian manifolds

It is R. Penrose who constructed the twistor theory which gives a correspondence between complex space-times and 3-dimensional complex manifolds called twistor spaces. He and his colleagues investigated conformally invariant equations (e.g. massless field equations, self-dual Yang-Mills equations) on the space-time by transforming them into objects in complex analytical geometry. See e.g. Penrose-Ward [P-W] or Ward-Wells [W-W]. After that, Atiyah-Hitchin-Singer ([A-H-S], cf. [Fr]) constructed the twistor spaces corresponding to real 4-dimensional Riemannian manifolds. Their construction as well as that of Penrose is mainly effective under the condition of the self-duality. In this paper we will construct twistor spaces more geometrically from real 4-dimensional Lorentzian manifolds under a suitable curvature condition.

Curvature invariants for Kantowski–Sachs metrics with source: A conformally invariant scalar field

Xanthopoulos and Zannias have solved the coupled Einstein conformally invariant massless scalar field equations under the assumption that the metric admits a four-parameter group of isometries with spacelike generators when the three-parameter subgroup of isometries acts on two-dimensional surfaces of positive, negative, and zero curvature. In this paper all known independent second order curvature invariants of these metrics are constructed in order to discuss the scalar curvature singularities. The plane metrics are, except for a subset of measure zero, singular free.

Kantowski–Sachs metrics with source: A conformally invariant scalar field

Solutions of the coupled Einstein conformally invariant massless scalar field equations are derived under the assumption that the metric admits a four-parameter group of isometries with spacelike generators, where further, the three-parameter subgroup of isometries acts multiply transitive on two-dimensional surfaces. The general solution of the system is obtained and some properties concerning the singularity structure are briefly discussed. Results for all three cases are presented, i.e., for the case where the three-dimensional group of isometries acts on two-dimensional orbits of positive, negative, or zero curvature. The first two classes belong to the Kantowski–Sachs class of metrics, while the third one is of Bianchi type I with additional rotational symmetry.

Kantowski–Sachs metrics with source: A massless scalar field

Solutions of the coupled Einstein massless scalar field equation are discussed under the assumption that the metric belongs to the ‘‘Kantowski–Sachs’’ class of metrics. More precisely, the general solution of the coupled Einstein massless scalar field equations is derived under the following assumptions: (a) The scalar field is massless minimally coupled to gravity and obeys the Klein–Gordon equation; and (b) the metric belongs to the spherical Kantowski–Sachs class of metrics. The main feature of the results is a tractable, very simple, compact form of the space-time metric. This is accomplished by expressing the solution in a special coordinate gauge. An analysis of the geometry shows the presence of curvature singularities which consist of a spacelike part and a null part. The spacelike singularity appears to ‘‘squeeze’’ the SO(3) orbits to zero proper area while simultaneously stretching their proper separation to infinity length. The null part appears to squeeze to zero values both SO(3) orbits and their proper separation as well. There also exists a subclass of metrics for which the singularity structure is different. The null singularity is replaced by a spacelike part. Both ‘‘initial’’ and ‘‘final’’ singularities exhibit the same qualitative features and they are of finite proper time away from each other.

Generalized Darboux maps and massless spin-s fields

We consider conformally invariant massless spin-s field equations on a spherically symmetrical space-time. Precisely when these equations are consistent appropriately defined field components are shown to satisfy wave equations related by a generalization of the classical Darboux map.

An L2-cohomology construction of negative spin mass zero equations for U( p, q)

In this paper a geometric construction is given of the ladder representations of G = U( p, q) which correspond to negative spin solutions of the massless field equations. The construction is based on the unitary model of the ladder representations in a Bargmann-Segal-Fock space of square-integrable (0, q)-forms. The representations are constructed as holomorphic sections of certain vector bundles over G K , and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics. Finally, the resulting differential equations are exhibited explicitly over G K .