Four-dimensional Minkowski Spacetime Research Articles

Numerical investigation of highly excited magnetic monopoles inSU(2)Yang-Mills-Higgs theory

Preliminary results concerning the time evolution of strongly exited $SU(2)$ Bogomolny-Prasad-Sommerfield (BPS) magnetic monopoles have been published in G. Fodor and I. R\'acz, Phys. Rev. Lett. 92, 151801 (2004). The behavior of these dynamical magnetic monopoles was investigated by means of numerical simulations in the four-dimensional Minkowski spacetime. The developed code---although applied to spherically symmetric configurations---incorporates both the techniques of conformal compactification and that of the hyperboloidal initial value problem. Our primary aim here is to provide a detailed account on the methods and results of the investigations reported in G. Fodor and I. R\'acz, Phys. Rev. Lett. 92, 151801 (2004). In addition, some important new results, which go much beyond the scope of these early studies, are also presented. In particular, to be able to distinguish linear and nonlinear effects, evolutions of monopoles deformed by various excitations, including both very small and extra large energy excitation, are investigated. In addition, a detailed account is provided on the spacetime dependence of the basic variables, as well as that of the other physically significant dynamical qualities such as the energy and energy current densities, the radial and angular pressures, and the magnetic charge density. A careful comprehensive study of the associated energy transfers and energy balances is also included.

Exact solutions of Einstein-Yang-Mills theory with higher-derivative coupling

We construct an exact classical solution of an Einstein-Yang-Mills system in ten space-time dimensions whose action contains a term quartic in Yang-Mills field strength. The solution provides compactification proposed by Cremmer and Scherk; the ten-dimensional space-time with a cosmological constant is compactified to the direct product of four-dimensional Minkowski space-time and a six-dimensional sphere ${S}^{6}$ whose radius is determined by coupling constants. We also construct a solution which compactifies the ten-dimensional space-time to ${\mathrm{AdS}}_{4}\ifmmode\times\else\texttimes\fi{}{S}^{6}$ when the cosmological constant is absent.

Non-commutative field theory with twistor-like coordinates

We consider quantum field theory in four-dimensional Minkowski spacetime, with the position coordinates represented by twistors instead of the usual world-vectors. Upon imposing canonical commutation relations between twistors and dual twistors, quantum theory of fields described by non-holomorphic functions of twistor variables becomes manifestly non-commutative, with Lorentz symmetry broken by a time-like vector. We discuss the free field propagation and its impact on the short- and long-distance behavior of physical amplitudes in perturbation theory. In the ultraviolet limit, quantum field theories in twistor space are generically less divergent than their commutative counterparts. Furthermore, there is no infrared–ultraviolet mixing problem.

Casimir Effect for a Massless Spin-3/2 Field in MinkowskiSpacetime

The Casimir effect has been studied for various quantum fields inboth flat and curved spacetimes. As a further step along thisline, we provide an explicit derivation of Casimir effect formassless spin-3/2 field with periodic boundary condition imposedin four-dimensional Minkowski spacetime. The corresponding resultswith Dirichlet and Neumann boundary conditions are also discussed.

An explicit quantum weak energy inequality for Dirac fields in curved spacetimes

The quantized Dirac field is known, by a result of Fewster and Verch, to satisfy a quantum weak energy inequality (QWEI) on its averaged energy density along timelike curves in arbitrary four-dimensional globally hyperbolic spacetimes. However, this result does not provide an explicit form for the bound. By adapting ideas from the earlier work, we give a simplified derivation of a QWEI for the Dirac field leading to an explicit bound. The bound simplifies further in the case of static curves in static spacetimes, and, in particular, coincides with a result of Fewster and Mistry in four-dimensional Minkowski spacetime. We also show that our QWEI is compatible with local covariance and derive a simple consequence.

A rotating Kaluza–Klein black hole with squashed horizons

We find a rotating Kaluza–Klein black hole solution with squashed S 3 horizons in five dimensions. This is a Kerr counterpart of the charged one found by Ishihara and Matsuno [H. Ishihara, K. Matsuno, Kaluza–Klein black holes with squashed horizons, hep-th/0510094] recently. The space–time is geodesic complete and free of naked singularities. Its asymptotic structure is a twisted S 1 fiber bundle over a four-dimensional Minkowski space–time. We also study the mass and thermodynamics of this black hole.

AdS Geometry, Projective Embedded Coordinates and Associated Isometry Groups

This work is intended to investigate the geometry of anti-de Sitter spacetime (AdS), from the point of view of the Laplacian Comparison Theorem (LCT), and to give another description of the hyperbolical embedding standard formalism of the de Sitter and anti-de Sitter spacetimes in a pseudoeuclidean spacetime. After Witten proved that general relativity is a renormalizable quantum system in (1+2) dimensions, it is possible to point out few interesting motivations to investigate AdS spacetime. A lot of attempts were made to generalize the gauge theory of gravity in (1+2) dimensions to higher ones. The first one was to enlarge the Poincaré group of symmetries, supposing an AdS group symmetry, which contains the Poincaré group. Also, the AdS/CFT correspondence asserts that a maximal supersymmetric Yang–Mills theory in four-dimensional Minkowski spacetime is equivalent to a type IIB closed superstring theory. The 10-dimensional arena for the type IIB superstring theory is described by the product manifold S 5× AdS, an impressive consequence that motivates the investigations about the AdS spacetime in this paper, together with the de Sitter spacetime. Classical results in this mathematical formulation are reviewed in a more general setting together with the isometry group associated to the de Sitter spacetime. It is known that, out of the Friedmann models that describe our universe, the Minkowski, de Sitter, and anti-de Sitter spacetimes are the unique maximally isotropic ones, so they admit a maximal number of conservation laws and also a maximal number of Killing vectors. In this paper it is shown how to reproduce some geometrical properties of AdS, from the LCT in AdS, choosing suitable functions that satisfy basic properties of Riemannian geometry. We also introduce and discuss the well-known embedding of a four-sphere and a four-hyperboloid in a five-dimensional pseudoeuclidean spacetime, reviewing the usual formalism of spherical embedding and the way how it can retrieve the Robertson–Walker metric. With the choice of the de Sitter metric static frame, we write the so-called reduced model in suitable coordinates. We assume the existence of projective coordinates, since de Sitter spacetime is orientable. From these coordinates, obtained when stereographic projection of the de Sitter four-hemisphere is done, we consider the Beltrami geodesic representation, which gives a more general formulation of the seminal full model described by Schr“odinger, concerning the” geometry and the topology of de Sitter spacetime. Our formalism retrieves the classical one if we consider the metric terms over the de Sitter splitting on Minkowski spacetime. From the covariant derivatives we find the acceleration of moving particles, Killing vectors and the isometry group generators associated to de the Sitter spacetime.

Quantum inequalities for a massless spin-3/2field in Minkowski spacetime

Quantum inequalities have been established for various quantum fields in both flat and curved spacetimes. In particular, for spin-$3/2$ fields, Yu and Wu have explicitly derived quantum inequalities for the massive case. Employing a similar method developed by Fewster and colleagues, this paper provides an explicit formula of quantum inequalities for a massless spin-$3/2$ field in four-dimensional Minkowski spacetime.

Minkowski vacuum stress tensor fluctuations

We study the fluctuations of the stress tensor for a massless scalar field in two- and four-dimensional Minkowski spacetime in the vacuum state. Covariant expressions for the stress tensor correlation function are obtained as sums of derivatives of a scalar function. These expressions allow one to express spacetime averages of the correlation function as finite integrals. We also study the correlation between measurements of the energy density along a world line. We find that these measurements may be either positively correlated or anticorrelated. The anticorrelated measurements can be interpreted as telling us that, if one measurement yields one sign for the averaged energy density, a successive measurement with a suitable time delay is likely to yield a result with the opposite sign.

The negative energy density for a three-single-electron state in the Dirac field

We examine the energy density produced by a state vector which is the superposition of three single electron states in the Dirac field in the four-dimensional Minkowski spacetime. We derive the conditions on which the energy density can be negative. We then show that the energy density satisfies two quantum inequalities in the ultrarelativistic limit.

Hawking radiation for Dirac spinors on the RP3 geon

We analyze the Hawking(-Unruh) effect for a massive Dirac spinor on the ${\mathbb{Z}}_{2}$ quotient of Kruskal spacetime known as the ${\mathbb{R}\mathbb{P}}^{3}$ geon. There are two distinct Hartle-Hawking-like vacua, depending on the choice of the spin structure, and suitable measurements in the static region (which on its own has only one spin structure) distinguish these two vacua. However, both vacua appear thermal in the usual Hawking temperature to certain types of restricted operators, including operators with support in the asymptotic future (or past). Similar results hold in a family of topologically analogous flat spacetimes, where we show the two vacua to be distinguished also by the shear stresses in the zero-mass limit. As a by-product, we display the explicit Bogolubov transformation between the Rindler-basis and the Minkowski-basis for massive Dirac fermions in four-dimensional Minkowski spacetime.

N=1geometries for M theory and type IIA strings with fluxes

We derive a set of necessary and sufficient conditions for obtaining $\mathcal{N}=1$ backgrounds of M theory and type IIA strings in the presence of fluxes. Our metrics are warped products of four-dimensional Minkowski space-time with a curved internal manifold. We classify the different solutions for irreducible internal manifolds as well as for manifolds with ${S}^{1}$ isometries by employing the formalism of group structures and intrinsic torsion. We provide examples within these various classes along with general techniques for their construction. In particular, we generalize the Hitchin flow equations so that one can explicitly build irreducible 7-manifolds with a 4-form flux. We also show how several of the examples found in the literature fit in our framework and suggest possible generalizations.

On the embedding of branes in five-dimensional spaces

We investigate the embedding of n-dimensional branes in (n + 1)-dimensional spaces. We firstly consider the case when the embedding space is a vacuum bulk whose energy–momentum tensor consists of a Dirac delta function with support in the brane. We then consider the embedding problem in the context of Randall–Sundrum-type models, taking into account Z2 symmetry and a cosmological constant. We employ the Campbell–Magaard theorem to construct the embeddings and are led to the conclusion that the content of energy-matter of the brane does not necessarily determine its curvature. Finally, as an application to illustrate our results, we construct the embedding of four-dimensional Minkowski spacetime filled with dust.

Null energy conditions in quantum field theory

For the quantised, massless, minimally coupled real scalar field in four-dimensional Minkowski space, we show (by an explicit construction) that weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below on the class of Hadamard states. Thus there are no quantum inequalities along null geodesics in four-dimensional Minkowski spacetime. This is in contrast to the case for two-dimensional flat spacetime, where such inequalities do exist. We discuss in detail the properties of the quantum states used in our analysis, and also show that the renormalized expectation value of the stress energy tensor evaluated in these states satisfies the averaged null energy condition (as expected), despite the nonexistence of a null-averaged quantum inequality. However, we also show that in any globally hyperbolic spacetime the null-contracted stress energy averaged over a timelike worldline does satisfy a quantum inequality bound (for both massive and massless fields). We comment briefly on the implications of our results for singularity theorems.

Twistor representation of null two-surfaces

We present a twistor description for null two-surfaces (null strings) in four-dimensional Minkowski space–time. The Lagrangian density for a variational principle is taken as a surface-forming null bivector. The proposed formulation is reparametrization invariant and free of any algebraic and differential constraints. The spinor formalism of Cartan–Penrose allows us to derive a nonlinear evolution equation for the world-sheet coordinate xa(τ,σ). An example of null two-surface given by the two-dimensional self-intersection (caustic) of a null hypersurface is studied.

Black holes on cylinders are not algebraically special

We give a Petrov classification for five-dimensional metrics. We give an almost complete classification of Ricci-flat metrics that are static, have an SO(3) isometry group and have Petrov type 22. We use this classification to look for the metric of a black hole on a cylinder, i.e., a black hole with asymptotic geometry four-dimensional Minkowski space times a circle. Although a black string wrapped around the S1 and the five-dimensional black hole are both algebraically special, it turns out that the black hole on a cylinder is not.

Four-dimensional conformal field theory models with rational correlation functions

Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in four-dimensional Minkowski spacetime. We consider in detail a model of a neutral scalar field of dimension two. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of is shown to coincide with a simpler one, generated by a bilocal scalar field V(x1,x2) of dimension (1,1). The modes of V together with the unit operator span an infinite-dimensional Lie algebra V whose vacuum (i.e. zero-energy lowest-weight) representations only depend on the central charge c. Wightman positivity (i.e. unitarity of the representations of V) is proven to be equivalent to c.

Massless 3-branes in M-theory

We construct supersymmetric M3-brane solutions in D=11 supergravity. They can be viewed as deformations of backgrounds taking the form of a direct product of four-dimensional Minkowski spacetime and a non-compact Ricci-flat manifold of G 2 holonomy. Although the 4-form field strength is turned on it carries no charge, and the 3-branes are correspondingly massless. We also obtain 3-branes of a different type, arising as M5-branes wrapped over S 2.

Non-standard compactifications with mass gaps and Newton's law

The four-dimensional Minkowski space-time is considered as a three-brane embedded in five dimensions, using solutions of five-dimensional supergravity. These backgrounds have a string theoretical interpretation in terms of D3-brane distributions. By studying linear fluctuations of the graviton we find a zero-mode representing the massless graviton in four-dimensional space-time. The novelty of our models is that the graviton spectrum has a genuine mass gap (independent of the position of the world-brane) above the zero-mode or it is discrete. Hence, an effective four-dimensional theory on a brane that includes the massless graviton mode is well defined. The gravitational force between point particles deviates from the Newton law by Yukawa-type corrections, which we compute explicitly. We show that the parameters of our solutions can be chosen such that these corrections lie within experimental bounds.

Massive spinning particles and the geometry of null curves

We study the simplest geometrical particle model associated with null paths in four-dimensional Minkowski space-time. The action is given by the pseudo-arclength of the particle worldline. We show that the reduced classical phase space of this system coincides with that of a massive spinning particle of spin s= α 2/ M, where M is the particle mass, and α is the coupling constant in front of the action. Consistency of the associated quantum theory requires the spin s to be an integer or half integer number, thus implying a quantization condition on the physical mass M of the particle. Then, standard quantization techniques show that the corresponding Hilbert spaces are solution spaces of the standard relativistic massive wave equations. Therefore this geometrical particle model provides us with an unified description of Dirac fermions ( s=1/2) and massive higher spin fields.