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Showing papers in "Classical and Quantum Gravity in 1995"


Journal ArticleDOI
TL;DR: Liouville theory is shown to describe the asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant as mentioned in this paper, and the anti-de Sitter boundary conditions implement the constraints that reduce the WZW model to the Liouville model.
Abstract: Liouville theory is shown to describe the asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant. This is because (i) Chern - Simons theory with a gauge group on a spacetime with a cylindrical boundary is equivalent to the non-chiral SL(2,R) WZW model; and (ii) the anti-de Sitter boundary conditions implement the constraints that reduce the WZW model to the Liouville theory.

508 citations


Journal ArticleDOI
TL;DR: Banados, Teitelboim and Zanelli as discussed by the authors reviewed the classical and quantum properties of the (2 + 1)-dimensional (3 + 1) black hole and showed that it shares many of the characteristics of the Kerr black hole: it has an event horizon, an inner horizon, and an ergosphere; it occurs as an endpoint of gravitational collapse.
Abstract: I review the classical and quantum properties of the (2 + 1)-dimensional black hole of Banados, Teitelboim and Zanelli. This solution of the Einstein field equations in three spacetime dimensions shares many of the characteristics of the Kerr black hole: it has an event horizon, an inner horizon, and an ergosphere; it occurs as an endpoint of gravitational collapse; it exhibits mass inflation; and it has a non-vanishing Hawking temperature and interesting thermodynamic properties. At the same time, its structure is simple enough to allow a number of exact computations, particularly in the quantum realm, that are impractical in 3 + 1 dimensions.

432 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the four-dimensional extreme dilaton black hole with dilaton coupling constant can be interpreted as a completely non-singular, non-dilatonic, black p-brane in (4+p) dimensions provided that p is odd.
Abstract: We show that the four-dimensional extreme dilaton black hole with dilaton coupling constant can be interpreted as a completely non-singular, non-dilatonic, black p-brane in (4+p) dimensions provided that p is odd. Similar results are obtained for multi-black-holes and dilatonic extended objects in higher spacetime dimensions. The non-singular black p-brane solutions include the self-dual 3-brane of ten-dimensional N=2B supergravity and a multi-5-brane solution of eleven-dimensional supergravity. In the case of a supersymmetric non-dilatonic p-brane solution of a supergravity theory, we show that it saturates a bound on the energy per unit p-volume of all field configurations of appropriate asymptotic behaviour which are non-singular on some initial hypersurface.

360 citations


Journal ArticleDOI
TL;DR: In this article, the authors prove a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold.
Abstract: We prove in detail a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold. This theorem determines which sets of CMC conformal data allow the constraint equations to be solved, and which sets of such data do not. The tools we describe and use here to prove these results have also been found to be useful for the study of non-constant mean curvature solutions of the Einstein constraints.

222 citations


Journal ArticleDOI
TL;DR: A survey of superspaces associated with four-dimensional Minkowski spacetime is given from a superconformal perspective in this article, which is closely related to twistor theory; in particular, the twistorial notion of a double fibration which links together sets of three such homogeneous supermanifolds is exploited to give a geometrical interpretation of super-formal transformations and is also applied to supersymmetric Yang-Mills theories.
Abstract: A survey of superspaces associated with four-dimensional Minkowski spacetime is given from a superconformal perspective. The discussion begins in the complex setting and focuses on a class of supermanifolds---flag supermanifolds---which includes not only conventional superspaces and chiral superspaces, but also supertwistor spaces and harmonic superspaces. All of them are homogeneous spaces of superconformal groups. The approach is closely related to twistor theory; in particular, the twistorial notion of a double fibration which links together sets of three such homogeneous supermanifolds is exploited to give a geometrical interpretation of superconformal transformations and is also applied to supersymmetric Yang--Mills theories. In the real setting the complex twistor space associated to Euclidean space is replaced in the supersymmetric case by supermanifolds which have a CR-structure. Representations of superconformal groups on fields are also studied; in the case of harmonic superspace this involves representations which differ slightly from standard induced representations.

185 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the two-dimensional theory is equivalent to planar symmetry in general relativity, and that the 2D theory admits black holes and free dilatons, and has a structure similar to 2D string theories.
Abstract: The Einstein--Hilbert action with a cosmological term is used to derive an action in 1+1 spacetime dimensions. It is shown that the two-dimensional theory is equivalent to planar symmetry in general relativity. The two-dimensional theory admits black holes and free dilatons, and has a structure similar to two-dimensional string theories. Since by construction these solutions also solve Einstein's equations, such a theory can bring two-dimensional results into the four-dimensional world. In particular the two-dimensional black hole is a black membrane in general relativity.

167 citations


Journal ArticleDOI
TL;DR: In this article, a simple quantization scheme is proposed to construct observables for a large class of finite-dimensional time reparametrization invariant systems by integration over the manifold of time labels.
Abstract: Within a simple quantization scheme, observables for a large class of finite-dimensional time reparametrization invariant systems may be constructed by integration over the manifold of time labels. This procedure is shown to produce a complete set of densely defined operators on a physical Hilbert space, for which an inner product is identified, and to provide reasonable results for simple test cases. Furthermore, many of these observables have a clear interpretation in the classical limit and we use this to demonstrate that, for a class of minisuperspace models including LRS Bianchi IX and the Kantowski--Sachs model, this quantization agrees with classical physics in predicting that such spacetimes recollapse.

156 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the opening angle at the horizon and the horizon area are canonical conjugates, and the dependence of the wavefunction on this new degree of freedom is governed by an extended Wheeler-DeWitt equation.
Abstract: The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black-hole topology, the opening angle at the horizon and the horizon area are canonical conjugates. It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity. The dependence of the wavefunction on this new degree of freedom is governed by an extended Wheeler--DeWitt equation. The trace of the Euclidean transition amplitude, which requires a sum over all horizon areas, yields a partition function whose logarithm gives the black-hole entropy.

156 citations


Journal ArticleDOI
TL;DR: In this paper, a generalization of the Leibniz rules of commutative geometry is proposed for linear connections on non-commutative algebras, and the covariant derivative obtained admits an extension to the tensor product of several copies of.
Abstract: A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalization of the Leibniz rules of commutative geometry and uses the bimodule structure of . A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of . The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois-Violette and then a general of the Dirac operator based differential calculus of Connes and other differential calculuses is given. The covariant derivative obtained admits an extension to the tensor product of several copies of . These constructions are illustrated with the example of the algebra of matrices.

149 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that there are no non-trivial black holes with a regular horizon in EH theory with any number of scalar fields and an arbitrary potential.
Abstract: We give a new simple proof of the fact that there are no non-trivial black holes with a regular horizon in Einstein-Higgs theory with any number of scalar fields and an arbitrary potential. We also give a brief discussion on the contrast between this theory and the Einstein-Yang-Mills theory that is responsible for the difference in the set of solutions allowed by each one.

147 citations


Journal ArticleDOI
TL;DR: In this paper, a review of all known exact D = 4 solutions with Minkowski signature in closed bosonic string theory with spacetime interpretation is presented. But the authors do not specify the corresponding conformal field theory explicitly.
Abstract: We review explicitly known exact D = 4 solutions with Minkowski signature in closed bosonic string theory. Classical string solutions with spacetime interpretation are represented by conformal sigma models. Two large (intersecting) classes of solutions are described by gauged WZW models and `chiral null models' (models with conserved chiral null current). The latter class includes plane-wave-type backgrounds (admitting a covariantly constant null Killing vector) and backgrounds with two null Killing vectors (e.g. fundamental string solution). D>4 chiral null models describe some exact D = 4 solutions with electromagnetic fields, for example, extreme electric black holes, charged fundamental strings and their generalizations. In addition, there exists a class of conformal models representing axially symmetric stationary magnetic flux tube backgrounds (including, in particular, the dilatonic Melvin solution). In contrast to spherically symmetric chiral null models for which the corresponding conformal field theory is not known explicitly, the magnetic flux tube models (together with some non-semisimple WZW models) are among the first examples of solvable unitary conformal string models with non-trivial D = 4 curved spacetime interpretation. For these models one is able to express the quantum Hamiltonian in terms of free fields and to find explicitly the physical spectrum and string partition function.

Journal ArticleDOI
TL;DR: In this article, the Einstein-Maxwell field equations for orthogonal Bianchi VI cosmologies with a -law perfect fluid and a pure, homogeneous source-free magnetic field are written as an autonomous differential equation in terms of expansion-normalized variables.
Abstract: The Einstein--Maxwell field equations for orthogonal Bianchi VI cosmologies with a -law perfect fluid and a pure, homogeneous source-free magnetic field are written as an autonomous differential equation in terms of expansion-normalized variables. The associated dynamical system is studied in order to determine the past, intermediate and future evolution of these models. All asymptotic states of the models, and the likelihood that they will occur, are described. In addition, it is shown that there is a finite probability that an arbitrarily selected model will be close to isotropy during some time interval in its evolution.

Journal ArticleDOI
TL;DR: In this article, it was shown that supercovariantly constant spinor fields admit super-consistency in SU(4) supergravity with axion and dilaton subject to an extra restriction.
Abstract: We find metrics admitting supercovariantly constant spinor fields in a sequence of supersymmetric theories of increasing complexity. With a dilaton with varying coupling, or with axion and dilaton in an -invariant theory we find all such metrics; in SU(4) supergravity with axion and dilaton, we find all such metrics subject to an extra restriction.

Journal ArticleDOI
TL;DR: In this paper, the authors generalize Wesson's procedure, whereby vacuum (4 + 1) dimensional field equations give rise to (3 + 1)-dimensional field equations with sources, to arbitrary dimensions.
Abstract: We generalize Wesson's procedure, whereby vacuum (4 + 1)-dimensional field equations give rise to (3 + 1)-dimensional equations with sources, to arbitrary dimensions. We then employ this generalization to relate the usual (3 + 1)-dimensional vacuum field equations to (2 + 1)-dimensional field equations with sources and derive the analogues of the classes of solutions obtained by Ponce de Leon. This way of viewing lower dimensional gravity theories can be of importance in establishing a relationship between such theories and the usual four-dimensional general relativity, as well as giving a way of producing exact solutions in (2 + 1) dimensions that are naturally related to the vacuum (3 + 1)-dimensional solutions. An outcome of this correspondence, regarding the nature of lower dimensional gravity, is that the intuitions obtained in (3 + 1) dimensions may not be automatically transportable to lower dimensions. We also extend a number of physically motivated solutions studied by Wesson and Ponce de Leon to (D + 1) dimensions and employ the equivalence between the (D + 1) Kaluza - Klein theories with empty D-dimensional Brans - Dicke theories (with ) to throw some light on the solutions derived by these authors.

Journal ArticleDOI
TL;DR: In this paper, the second post-Newtonian motion of compact binary-star systems with spin is solved by a generalized quasi-Keplerian parametrization, where first-order spin-orbit terms cross with Newtonian terms only since spin--orbit terms in compact binary star systems are numerically of second order.
Abstract: In this paper a full analytic solution for the second post-Newtonian motion of compact binaries with spin is presented. Advantage is taken of the following facts: (i) the second post-Newtonian motion of compact binaries without spin is already solved by a generalized quasi-Keplerian parametrization, (ii) first-order spin--orbit terms cross with Newtonian terms only since spin--orbit terms in compact binary-star systems are numerically of second post-Newtonian order, (iii) in the case of compact binary-star systems spin--spin interaction and quadrupole-deformation contributions are negligible at the second post-Newtonian approximation level. The analytic solution for the quasi-elliptic motion is given in a generalized quasi-Keplerian parametrization. Coordinate time and proper time of one of the bodies are used to parametrize the motion. As a by-product the first post-Newtonian motion of a satellite in the field of a rotating spherical mass is obtained.

Journal ArticleDOI
TL;DR: In this article, a class of (2+1)-dimensional spacetimes admitting Killing spinors appropriate to (2,0) adS-supergravity was found, subject to a condition on the matter currents and a conjecture concerning global obstructions to the existence of certain types of spinor fields.
Abstract: We find a class of (2+1)-dimensional spacetimes admitting Killing spinors appropriate to (2,0) adS-supergravity. The vacuum spacetimes include anti-de Sitter (adS) space and charged extreme black holes, but there are many others, including spacetimes of arbitrarily large negative energy that have only conical singularities, and the spacetimes of fractionally charged point particles. The non-vacuum spacetimes are those of self-gravitating solitons obtained by coupling (2,0) adS supergravity to sigma-model matter. We show, subject to a condition on the matter currents (satisfied by the sigma model), and a conjecture concerning global obstructions to the existence of certain types of spinor fields, that the mass of each supersymmetric spacetime saturates a classical bound, in terms of the angular momentum and charge, on the total energy of arbitrary field configurations with the same boundary conditions, although these bounds may be violated quantum mechanically.

Journal ArticleDOI
TL;DR: In this article, the complete and explicit general solution of the conformal Killing equation in static spherically symmetric spacetimes was found, thus unifying and generalizing previous special cases.
Abstract: We find the complete and explicit general solution of the conformal Killing equation in static spherically symmetric spacetimes, thus unifying and generalizing previous special cases. For non-conformally-flat spacetimes, there are at most two proper conformal motions. There are three classes of such spacetimes, and one or both of the conformal Killing vectors is non-inheriting. One of the classes includes self-similar spacetimes (i.e. with a homothetic motion). The conformally flat spacetimes (including the Schwarzschild interior metric) fall into three classes, and their eleven proper conformal Killing vectors are given in full. The only spacetimes with conformal motion that are regular at the centre are conformally flat. An addendum for this article has been published in 1996 Class. Quantum Grav. 13 317

Journal ArticleDOI
TL;DR: In this article, a consistent description of spinning strings with dislocations (chiral strings) as a missing part of the spacetime manifold (spacetime defect) is achieved by introducing a line of torsion along the string.
Abstract: A consistent description of spinning strings with dislocations (chiral strings) as a missing part of the spacetime manifold (spacetime defect) is achieved by introducing a line of torsion along the string. Also, the definition of a spin-dislocation tensor associated with the chiral string allows its interpretation in the context of the Einstein--Cartan theory of gravitation. Some generalizations, as well as the Dirac equations, are briefly discussed. In particular, we argue that the behaviour of the fermions near the chiral string can be used to test the torsion hypothesis.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the multidimensional cosmological model describing the evolution of n Einstein spaces in the presence of a multicomponent perfect fluid and the dynamics of the model near the singularity were reduced to a billiard on the (n-1)-dimensional Lobachevsky space.
Abstract: The multidimensional cosmological model describing the evolution of n Einstein spaces is considered in the presence of a multicomponent perfect fluid. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity are reduced to a billiard on the (n-1)-dimensional Lobachevsky space . The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of an (n-2)-dimensional sphere by point-like sources. Some generalizations of the considered scheme (including scalar field and quantum generalizations) are considered.

Journal ArticleDOI
TL;DR: In this article, it was shown that the topology of the event horizon of a four-dimensional asymptotically flat black-hole spacetime must be a 2-sphere.
Abstract: We prove that, under certain conditions, the topology of the event horizon of a four-dimensional asymptotically flat black-hole spacetime must be a 2-sphere. No stationarity assumption is made. However, in order for the theorem to apply, the horizon topology must be unchanging for long enough to admit a certain kind of cross section. We expect this condition is generically satisfied if the topology is unchanging for much longer than the light-crossing time of the black hole. More precisely, let M be a four-dimensional asymptotically flat spacetime satisfying the averaged null energy condition, and suppose that the domain of outer communication to the future of a cut K of is globally hyperbolic. Suppose further that a Cauchy surface for is a topological 3-manifold with compact boundary in M, and is a compact submanifold of with spherical boundary in (and possibly other boundary components in ). Then we prove that the homology group must be finite. This implies that either consists of a disjoint union of 2-spheres, or is non-orientable and contains a projective plane. Furthermore, , and will be a cross section of the horizon as long as no generator of becomes a generator of . In this case, if is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.

Journal ArticleDOI
TL;DR: In this article, it was shown that the topological censorship theorem of Friedman, Schleich and Witt implies, in the general setting of their result, that the domain of outer communication is simply connected.
Abstract: It is shown that the topological censorship theorem of Friedman, Schleich and Witt implies, in the general setting of their result, that the domain of outer communication is simply connected. This improves recent related results of Chrusciel and Wald and of Jacobson and Venkataramani.

Journal ArticleDOI
TL;DR: In this article, a tensor-variational approach yields field equations that possess a covariance similar to the gauge covariance of electromagnetism, and these equations are shown to possess solutions analogous to those found in the Einstein-Maxwell system.
Abstract: A simple modification to Einstein's theory of gravity in terms of a non-Riemannian connection is examined. A new tensor-variational approach yields field equations that possess a covariance similar to the gauge covariance of electromagnetism. These equations are shown to possess solutions analogous to those found in the Einstein-Maxwell system. In particular one finds gravi-electric and gravi-magnetic charges contributing to a spherically symmetric static Reissner - Nordstrom metric. Such Weyl `charges' provide a source for the non-Riemannian torsion and metric gradient fields instead of the electromagnetic field. The theory suggests that matter may be endowed with gravitational charges that couple to gravity in a manner analogous to electromagnetic couplings in an electromagnetic field. The nature of gravitational coupling to spinor matter in this theory is also investigated and a solution exhibiting a plane-symmetric gravitational metric wave coupled via non-Riemannian waves to a propagating spinor field is presented.

Journal ArticleDOI
TL;DR: In this paper, a formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an almost ideal clock, and the explicit relation between the physical Hilbert space and the usual Hilbert space for a system with an external time is derived.
Abstract: A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock'. Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution', the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, we derive the complete formalism of external time quantum mechanics. The explicit relation between the physical Hilbert space and the usual Hilbert space for a system with an external time is also derived and the limit of an ideal clock is found to be closely associated with the Klein - Gordon inner product and the Newton - Wigner formalism.

Journal ArticleDOI
TL;DR: In this article, a suitable modification of Utiyama's method for gauging the projective realization of the Galilei group associated with the free mass-point, together with connections between the consequent 11 external gauge fields and known facts about Galilean and Newtonian geometrical structures, are discussed from a unified point of view.
Abstract: Newton's standard theory of gravitation is reformulated in terms of a generally Galilei-covariant action principle as a gauge theory of the extended Galilei group. A suitable modification of Utiyama's method for gauging the projective realization of the Galilei group associated with the free mass-point, together with the connections between the consequent 11 external gauge fields and known facts about Galilean and Newtonian geometrical structures, are discussed from a unified point of view. Then the problem of the existence of an action principle for the dynamical evolution of the gauge fields is analysed. Since it is not known how to extend Utiyama's method from the case of `invariance' to the case of `quasi-invariance, modulo the equations of motion', which turns out to be the key factor in the Galilean case, an action principle is derived, starting from a suitable power expansion of the 4-metric tensor, as a contraction, for , of the ADM-De-Witt action of general relativity. The Galilean action depends on 27 fields (i.e. it contains 16 auxiliary fields besides the 11 gauge fields) and is indeed quasi-invariant, modulo the equations of motion under general Galilean coordinate transformations. The physical equivalence of this theory and Newton's theory of gravity is shown explicitly, by analysing its first- and second-class constraints. Finally, we discuss the feasibility of a symplectic reduction of the 27-fields theory to a minimal theory depending on only the 11 gauge fields, in the sense of Utiyama's method.

Journal ArticleDOI
TL;DR: In this paper, the ultrarelativistic limit of the Schwarzschild and Kerr geometries, together with their respective energy-momentum tensors, is derived based on tensor distributions making use of the underlying Kerr-Schild structure.
Abstract: The ultrarelativistic limit of the Schwarzschild and Kerr geometries, together with their respective energy-momentum tensors, is derived. Our approach is based on tensor distributions making use of the underlying Kerr-Schild structure, which remains stable under the ultrarelativistic boost.

Journal ArticleDOI
TL;DR: In this article, it was shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a symmetric compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation or the wave equation.
Abstract: It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a symmetric compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation (massless scalar field). In the spherically symmetric case it is further shown that if the spacetime admits a maximal slice then there are crushing singularities both in the past and in the future. The essential properties of the matter models chosen are that their energy--momentum tensors satisfy certain inequalities and that they do not develop singularities in a given regular background spacetime.

Journal ArticleDOI
TL;DR: In this paper, an upper mass limit is found, analogous to Buchdahl's theorem in 3+1 dimensions, and the possibility of collapse is discussed. And the case of uniform matter density is solved exactly and a new interior solution is presented.
Abstract: The hydrostatic equilibrium of a (2+1)-dimensional perfect fluid star in asymptotically anti-de Sitter space is discussed. The interior geometry matches the exterior 2+1 black-hole solution. An upper mass limit is found, analogous to Buchdahl's theorem in 3+1 dimensions, and the possibility of collapse is discussed. The case of a uniform matter density is solved exactly and a new interior solution is presented.

Journal ArticleDOI
TL;DR: In this paper, a quantization of the Bianchi IX cosmological model based on taking the constraint to be a self-adjoint operator in an auxiliary Hilbert space is considered.
Abstract: We consider a quantization of the Bianchi IX cosmological model based on taking the constraint to be a self-adjoint operator in an auxiliary Hilbert space. Using a WKB-style self-consistent approximation, the constraint chosen is shown to only have a continuous spectrum at zero. Nevertheless, the auxiliary space induces an inner product on the zero-eigenvalue generalized eigenstates such that the resulting physical Hilbert space has a countably infinite dimension. In addition, a complete set of gauge-invariant operators on the physical space is constructed by integrating differential forms over the spacetime. The behaviour of these operators indicates that this quantization preserves Wald's classical result that the Bianchi IX spacetimes expand to a maximum volume and then recollapse.

Journal ArticleDOI
K. P. Tod1
TL;DR: In this paper, an ansatz is presented which reduces the Toda field equation to special cases of the Painleve-III differential equation, which leads to new four-dimensional, scalar-flat, Kahler and hyper-Kahler metrics based on painleve transcendents.
Abstract: Motivated by the Bianchi-type-IX scalar-flat Kahler metric of Pedersen and Poon, an ansatz is presented which reduces the -Toda field equation to special cases of the Painleve-III differential equation. This leads to new four-dimensional, scalar-flat, Kahler and hyper-Kahler metrics based on Painleve transcendents. By considering the relation between Einstein--Weyl spaces and the -Toda field equation, we classify separable solutions of the latter equation and then characterize those Einstein--Weyl spaces which arise from it.

Journal ArticleDOI
TL;DR: In this article, a unique twistor-like Lorentz harmonic formulation for all N = 1 supersymmetric extended objects (super-p-branes) moving in spacetime of arbitrary dimension D (admissible for given p) is suggested.
Abstract: A unique twistor-like Lorentz harmonic formulation for all N=1 supersymmetric extended objects (super-p-branes) moving in spacetime of arbitrary dimension D (admissible for given p) is suggested. The equations of motion are derived, the explicit form of the -symmetry transformations is presented and the classical equivalence to the standard formulation is proved. The cases with minimal worldsheet dimensions p=1,2, namely the D=10 heterotic string and the D=11 supermembrane, are considered in detail. In particular, the explicit form of irreducible -symmetry transformations for the D=11 supermembrane is derived.