Space-time Signature Research Articles

Towards an Ashtekar formalism in 12 dimensions

We discuss the Ashtekar formalism from the point of view of twelve dimensions. We first focus on the 2+10 spacetime signature and then we consider the transition $2+10\to (2+2)+(0+8)$. We argue that both sectors 2+2 and 0+8, which are exceptional signatures, can be analyzed from the point of view of a self-dual action associated with the Ashtekar formalism.

Asymptotic flatness, little string theory, and holography

We argue that any non-gravitational holographic dual to asymptotically flat string theory in $d$-dimensions naturally resides at spacelike infinity. Since spacelike infinity can be resovled as a $(d-1)$-dimensional timelike hyperboloid (i.e., as a copy of de Sitter space in $(d-1)$ dimensions), the dual theory is defined on a Lorentz signature spacetime. Conceptual issues regarding such a duality are clarified by comparison with linear dilaton boundary conditions, such as those dual to little string theory. We compute both time-ordered and Wightman boundary 2-point functions of operators dual to massive scalar fields in the asymptotically flat bulk.

Blind detection for multi-rate DS-CDMA system with antenna array at the base station

In this paper, we propose the blind space-time high rate multi-user detector for synchronous uplink multi-rate Direct Sequence Code Division Multiple Access (DS-CDMA) systems with antenna array at the base station. By employing antenna array at the base stations, the spatial dimension is used efficiently to suppress co-channel interference and increase the capacity for multi-rate CDMA system. After low rate physical users in the system are modeled as corresponding high rate virtual users, we construct the space-time signature vectors of virtual users. And subspace projection algorithm is employed to estimate space-time signature vectors blindly. Then a soft-decision high rate multiuser detector is proposed based on the estimated signature vectors, which avoids estimating the ambiguous complex factors which are necessary in traditional blind detector. Numerical simulation results evaluate the performance in terms of Bit Error Rate (BER) for the proposed scheme. Simultaneously, it demonstrates that the system capability increases two times when using two-element antenna array.

The membrane at the end of the (de Sitter) universe

The original membrane at the end of the universe corresponds to a probe M2-brane of signature ( 2 , 1 ) occupying the S 2 × S 1 boundary of the ( 10 , 1 ) spacetime AdS 4 × S 7 , and is described by an OSp ( 4 / 8 ) SCFT. However, it was subsequently generalized to other worldvolume signatures ( s , t ) and other spacetime signatures ( S , T ) . An interesting special case is provided by the ( 3 , 0 ) brane at the end of the de Sitter universe dS 4 which has recently featured in the dS/CFT correspondence. The resulting CFT contains the one recently proposed as the holographic dual of a four-dimensional de Sitter cosmology. Supersymmetry restricts S, T, s, t by requiring that the corresponding bosonic symmetry O ( s + 1 , t + 1 ) × O ( S − s , T − t ) be a subgroup of a superconformal group. The case of dS 4 × AdS 7 is ‘doubly holographic’ and may be regarded as the near horizon geometry of N 2 M2-branes or equivalently, under interchange of conformal and R-symmetry, of N 5 M5-branes, provided N 2 = 2 N 5 2 . The same correspondence holds in the pp-wave limit of conventional M-theory.

Kerr–de Sitter black holes with NUT charges

The four-dimensional Kerr–de Sitter and Kerr–AdS black hole metrics have cohomogeneity 2, and they admit a generalisation in which an additional parameter characterising a NUT charge is included. In this paper, we study the higher-dimensional Kerr–AdS metrics, specialised to cohomogeneity 2 by appropriate restrictions on their rotation parameters, and we show how they too admit a generalisation in which an additional NUT-type parameter is introduced. We discuss also the supersymmetric limits of the new metrics. If one performs a Wick rotation to Euclidean spacetime signature, these yield new Einstein–Sasaki metrics in odd dimensions, and Ricci-flat metrics in even dimensions. We also study the five-dimensional Kerr–AdS black holes in detail. Although in this particular case the NUT parameter is trivial, our investigation reveals the remarkable feature that a five-dimensional Kerr–AdS “over-rotating” metric is equivalent, after performing a coordinate transformation, to an under-rotating Kerr–AdS metric.

Detectors and Asymptotic Analysis for Bandwidth-Efficient Space–Time Multiple-Access Systems

In this paper, a narrowband multiple-channel transmission scheme with multiple transmit antennas is proposed and analyzed. The channelization is based on space-time signature matrices, which do not expand bandwidth, unlike conventional schemes such as code-division or time-division multiplexing. The channels can be used by multiple independent users in an uplink or downlink scenario (multiple access or broadcast channels, respectively), or by one user in a multiplexing scenario. The data transmitted on each channel is convolutionally encoded, interleaved, and then space-time block encoded before space-time channelization. Each channel has a unique interleaver and space-time signature, but the convolutional encoder and space-time block code encoder can be identical across channels. It is shown that asymptotic single-user-like performance can be achieved with optimal detection and decoding in a Rayleigh fading channel. Practical receiver algorithms are developed based on the iterative (turbo) detection technique. The simulation results demonstrate that these suboptimal receivers achieve single-user performance at moderate signal-to-noise ratios, and moderate user loads. In the single-user multiplexing case, a significant performance gain over single-channel transmission with the same data rate is obtained.

Noncritical [formula omitted]-theory matrix model with an arbitrary time-dependent cosmological constant

Dimensional reduction of the D = 2 minimal super-Yang–Mills to the D = 1 matrix quantum mechanics is shown to double the number of dynamical supersymmetries, from N = 1 to N = 2 . We analyze the most general supersymmetric deformations of the latter, in order to construct the noncritical 3D M -theory matrix model on generic supersymmetric backgrounds. It amounts to adding quadratic and linear potentials with arbitrary time-dependent coefficients, namely, a cosmological ‘constant’, Λ ( t ) , and an electric flux background, ρ ( t ) , respectively. The resulting matrix model enjoys, irrespective of Λ ( t ) and ρ ( t ) , two dynamical supersymmetries which further reveal three hidden so ( 1 , 2 ) symmetries. All together they form the supersymmetry algebra, osp ( 1 | 2 , R ) . Each so ( 1 , 2 ) multiplet in the Hilbert space visualizes a dynamics constrained on either Euclidean or Minkowskian dS 2 / AdS 2 space, depending on its Casimir. In particular, all the unitary multiplets have the Euclidean dS 2 / AdS 2 geometry. We conjecture that the matrix model provides holographic duals to the 2D superstring theories on various backgrounds having the spacetime signature Minkowskian if Λ ( t ) > 0 , or Euclidean if Λ ( t ) < 0 .

EMERGENCE OF TIME FROM DIMENSIONAL REDUCTION IN NONCOMMUTATIVE GEOMETRY

By considering a new form of dimensional reduction for noncommutative field theory, we show that the signature of spacetime may be changed. In particular, it is demonstrated that a temporal dimension can emerge from a purely Euclidean geometry. We suggest that this mechanism may hint at the origin of time in the fundamental theory of quantum gravity.

The Fundamental Solution of the Hyperbolic Dirac Operator on $\mathbb{R}^{1,m}$ : a new approach

In this paper, the fundamental solution of the Dirac equation on hyperbolic space will be calculated by means of the fundamental solution for the wave-operator in the $(m+1)$-dimensional Minkowski space-time of signature $(1,m)$. This leads to addition formulas for the fundamental solution in terms of the solution in a lower-dimensional Minkowski space-time. Certain identities between hypergeometric functions can then be used to obtain a closed form for the fundamental solution of the Dirac equation.

Towards an Ashtekar formalism in eight dimensions

We investigate the possibility of extending the Ashtekar theory to eight dimensions. Our approach relies on two notions: the octonionic structure and the MacDowell–Mansouri formalism generalized to a spacetime of signature 1 + 7. The key mathematical tool for our construction is the self-dual (antiself-dual) four-rank fully antisymmetric octonionic tensor. Our results may be of particular interest in connection with a possible formulation of M -theory via matroid theory.

Time-like T-duality algebra

When compactifying M- or type II string-theories on tori of indefinite space-time signature, their low energy theories involve sigma models on E_{n(n)}/H_n, where H_n is a not necessarily compact subgroup of E_{n(n)} whose complexification is identical to the complexification of the maximal compact subgroup of E_{n(n)}. We discuss how to compute the group H_n. For finite dimensional E_{n(n)}, a formula derived from the theory of real forms of E_n algebra's gives the possible groups immediately. A few groups that have not appeared in the literature are found. For n=9,10,11 we compute and describe the relevant real forms of E_n and H_n. A given H_n can correspond to multiple signatures for the compact torus. We compute the groups H_n for all compactifications of M-, M*-, and M'-theories, and type II-, II*- and II'-theories on tori of arbitrary signature, and collect them in tables that outline the dualities between them. In an appendix we list cosets G/H, with G split and H a subgroup of G, that are relevant to timelike toroidal compactifications and oxidation of theories with enhanced symmetries.

Hermitian versus holomorphic complex and quaternionic generalized supersymmetries of the M-theory. A classification

Relying upon the division-algebra classification of Clifford algebras and spinors, a classification of generalized supersymmetries (or, with a slight abuse of language,``generalized supertranslations") is provided. In each given space-time the maximal, saturated, generalized supersymmetry, compatible with the division-algebra constraint that can be consistently imposed on spinors and on superalgebra generators, is furnished. Constraining the superalgebra generators in both the complex and the quaternionic cases gives rise to the two classes of constrained hermitean and holomorphic generalized supersymmetries. In the complex case these two classes of generalized supersymmetries can be regarded as complementary. The quaternionic holomorphic supersymmetry only exists in certain space-time dimensions and can admit at most a single bosonic scalar central charge. The results here presented pave the way for a better understanding of the various M algebra-type of structures which can be introduced in different space-time signatures and in association with different division algebras, as well as their mutual relations. In a previous work, e.g., the introduction of a complex holomorphic generalized supersymmetry was shown to be necessary in order to perform the analytic continuation of the standard M-theory to the 11-dimensional euclidean space. As an application of the present results, it is shown that the above algebra also admits a 12-dimensional, euclidean, F-algebra presentation.

Formula omitted]: sign of the times

We discuss the signature of space–time in the context of the E 11 -conjecture. In this setting, the space–time signature depends on the choice of basis for the “gravitational subalgebra” A 10 , and Weyl transformations connect interpretations with different signatures of space–time. Also the sign of the 4-form gauge field term in the Lagrangian enters as an adjustable sign in a generalized signature. Within E 11 , the combination of space–time signature ( 1 , 10 ) with conventional sign for the 4-form term, appropriate to M-theory, can be transformed to the signatures ( 2 , 9 ) and ( 5 , 6 ) of Hull's M * - and M ′ -theories (as well as ( 6 , 5 ) , ( 9 , 2 ) and ( 10 , 1 ) ) . Theories with other signatures organize in orbits disconnected from these theories. We argue that when taking E 11 seriously as a symmetry algebra, one cannot discard theories with multiple time-directions as unphysical. We also briefly explore links with the SL ( 32 , R ) conjecture.

A Simple Model of Space-Time Metric Signature Change

The problem of describing the space-time metric signature change is discussed. A simple model with the space-time metric signature determined by a nonlinear scalar field is suggested. It is demonstrated that both classical and quantum descriptions of the metric signature change are possible within the framework of the examined model, and the most characteristic features and differences of the classical and quantum descriptions are briefly discussed.

Quaternionic and octonionic spinors. A classification

Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and octonionic spinors is presented. In the octonionic case we further provide a systematic list of results and tables expressing, e.g., the relations of the octonionic Clifford algebras with the $G_2$ cosets over the Lorentz algebras, the identities satisfied by the higher-rank antisymmetric octonionic tensors and so on. Applications of these results range from the classification of octonionic generalized supersymmetries, the construction of octonionic superstrings, as well as the investigations concerning the recently discovered octonionic $M$-superalgebra and its superconformal extension.

Canonical quantization of general relativity in discrete space-times.

It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. We analyze discrete lattice general relativity and develop a canonical formalism that allows one to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a "dynamical gauge" fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. The problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories. A simple cosmological application exhibits quantum elimination of the singularity at the big bang.

Spacetime symplectic extension

It is conjectured that in the origin of space-time there lies a symplectic rather than metric structure. The complex symplectic symmetry Sp(2l,C), l\ge1 instead of the pseudo-orthogonal one SO(1,d-1), d\ge4 is proposed as the space-time local structure group. A discrete sequence of the metric space-times of the fixed dimensionalities d=(2l)^2 and signatures, with l(2l-1) time-like and l(2l+1) space-like directions, defined over the set of the Hermitian second-rank spin-tensors is considered as an alternative to the pseudo-Euclidean extra dimensional space-times. The basic concepts of the symplectic framework are developed in general, and the ordinary and next-to-ordinary space-time cases with l=1,2, respectively, are elaborated in more detail. In particular, the scheme provides the rationale for the four-dimensionality and 1+3 signature of the ordinary space-time.

Why Nature has made a choice of one time and three space coordinates?

We propose a possible answer to one of the most exciting open questions in physics and cosmology, that is the question why we seem to experience four- dimensional space-time with three ordinary and one time dimensions. We have known for more than 70 years that (elementary) particles have spin degrees of freedom, we also know that besides spin they also have charge degrees of freedom, both degrees of freedom in addition to the position and momentum degrees of freedom. We may call these ''internal degrees of freedom '' the ''internal space'' and we can think of all the different particles, like quarks and leptons, as being different internal states of the same particle. The question then naturally arises: Is the choice of the Minkowski metric and the four-dimensional space-time influenced by the ''internal space''? Making assumptions (such as particles being in first approximation massless) about the equations of motion, we argue for restrictions on the number of space and time dimensions. (Actually the Standard model predicts and experiments confirm that elementary particles are massless until interactions switch on masses.) Accepting our explanation of the space-time signature and the number of dimensions would be a point supporting (further) the importance of the ''internal space''.

Massless BTZ black holes in minisuperspace

We study aspects of the propagation of strings on BTZ black holes. After performing a careful analysis of the global spacetime structure of generic BTZ black holes, and its relation to the geometry of the SL(2,R) group manifold, we focus on the simplest case of the massless BTZ black hole. We study the SL(2,R) Wess-Zumino-Witten model in the worldsheet minisuperspace limit, taking into account special features associated to the Lorentzian signature of spacetime. We analyse the two- and three-point functions in the pointparticle limit. To lay bare the underlying group structure of the correlation functions, we derive new results on Clebsch-Gordan coefficients for SL(2,R) in a parabolic basis. We comment on the application of our results to string theory in singular time-dependent orbifolds, and to a Lorentzian version of the AdS/CFT correspondence.

On Metric-Connection Compatibility and the Signature Change of Space-Time

The problem of existence of metric-compatible linear connections for a given space-time metric, which is, generally, assumed to be semi-pseudo-Riemannian, is discussed and investigated. It is proved that, under sufficiently general conditions, such connections exist iff the rank and signature of the metric are constant. Some speculations on the space-time signature are made on this background.