Four-dimensional Space-time Manifold Research Articles

Primordial Black Holes from Spatially Varying Cosmological Constant Induced by Field Fluctuations in Extra Dimensions

The origin and evolution of supermassive black holes (SMBHs) in our universe have sparked controversy. In this study, we explore the hypothesis that some of these black holes may have seeded from the direct collapse of dark energy domains with density significantly higher than the surrounding regions. The mechanism of the origin of such domains relies on the inflationary evolution of a scalar field acting in D dimensions, which is associated with the cosmological constant in our four-dimensional spacetime manifold. Inner space quantum fluctuations of the field during inflation are responsible for the spatial variations of the dark energy density in our space. This finding holds particular significance, especially considering recent evidence from pulsar timing array observations, which supports the existence of a stochastic gravitational wave background consisting of SMBH mergers.

Postoji li promjena prema B-teoriji vremena?

The purpose of this paper is to explore the connection between change and the B-theory of time, sometimes also called the Scientific view of time, according to which reality is a four-dimensional spacetime manifold, where past, present and future things equally exist, and the present time and non-present times are metaphysically the same. I argue in favour of a novel response to the much-vexed question of whether there is change on the B-theory or not. In fact, B-theorists are often said to hold a ‘static’ view of time. But this far from being innocent label: if the B-theory of time presents a model of temporal reality that is static, then there is no change on the B-theory. From this, one can reasonably think as follows: of course, there is change, so the B-theory must be false. What I plan to do in this paper is to argue that in some sense there is change on the B-theory, but in some other sense, there is no change on the B-theory. To do so, I present three instances of change: Existential Change, namely the view that things change with respect to their existence over time; Qualitative Change, the view that things change with respect to how they are over time; Propositional Change, namely the view that things (i.e. propositions) change with respect to truth value over time. I argue that while there is a reading of these three instances of change that is true on the B-theory, and so there is change on the B-theory in this sense, there is a B-theoretical reading of each of them that is not true on the B-theory, and therefore there is no change on the B-theory in this other sense.

Global properties of warped solutions in general relativity with an electromagnetic field and a cosmological constant. II.

We consider general relativity with a cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional spacetime manifold is a warped product of two surfaces with Lorentzian and Euclidean signature metrics. Field equations imply that at least one of the surfaces must be of constant curvature leading to the symmetry of the metric (``spontaneous symmetry emergence''). We classify all global solutions in the case when the Lorentzian surface is of constant curvature (case C). These solutions are invariant with respect to the Lorentz $\mathbb{S}\mathbb{O}(1,2)$ or Poincare $\mathbb{I}\mathbb{O}(1,1)$ groups acting on the Lorentzian surface.

Yang–Mills Theory of Gravity

The canonical formulation of general relativity (GR) is based on decomposition space–time manifold M into R × Σ , where R represents the time, and Ksi is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general coordinates, and local Lorentz transformations. These symmetries are associated with conserved currents that are coupled to gravity. These symmetries are studied on a three dimensional space-like hypersurface Σ embedded in a four-dimensional space–time manifold. This implies continuous symmetries and conserved currents by Noether’s theorem on that surface. We construct a three-form E i ∧ D A i (D represents covariant exterior derivative) in the phase space ( E i a , A a i ) on the surface Σ , and derive an equation of continuity on that surface, and search for canonical relations and a Lagrangian that correspond to the same equation of continuity according to the canonical field theory. We find that Σ i 0 a is a conjugate momentum of A a i and Σ i a b F a b i is its energy density. We show that there is conserved spin current that couples to A i , and show that we have to include the term F μ ν i F μ ν i in GR. Lagrangian, where F i = D A i , and A i is complex S O ( 3 ) connection. The term F μ ν i F μ ν i includes one variable, A i , similar to Yang–Mills gauge theory. Finally we couple the connection A i to a left-handed spinor field ψ , and find the corresponding beta function.

Global properties of warped solutions in general relativity with an electromagnetic field and a cosmological constant

We consider general relativity with cosmological constant minimally coupled to electromagnetic field and assume that four-dimensional space-time manifold is the warped product of two surfaces with Lorentzian and Euclidean signature metrics. Einstein's equations imply that at least one of the surfaces must be of constant curvature. It means that the symmetry of the metric arises as the consequence of equations of motion (`spontaneous symmetry emergence'). We give classification of global solutions in two cases: (i) both surfaces are of constant curvature and (ii) the Riemannian surface is of constant curvature. The latter case includes spherically symmetric solutions (sphere S^2 with SO(3)-symmetry group), planar solutions (two-dimensional Euclidean space R^2 with IO(2)-symmetry group), and hyperbolic solutions (two-sheeted hyperboloid H^2 with SO(1,2)-symmetry). Totally, we get 37 topologically different solutions. There is a new one among them, which describes changing topology of space in time already at the classical level.

Space-Time Properties as Quantum Effects. Restrictions Imposed by Grothendieck’s Scheme Theory

In this paper we consider properties of the four-dimensional space-time manifold M caused by the proposition that, according to the scheme theory, the manifold M is locally isomorphic to the spectrum of the algebra A, M ≅ Spec (A), where A is the commutative algebra of distributions of quantum-field densities. Points of the manifold M are defined as maximal ideals of density distributions. In order to determine the algebra A, it is necessary to define multiplication on densities and to eliminate those densities, which cannot be multiplied. This leads to essential restrictions imposed on densities and on space-time properties. It is found that the only possible case, when the commutative algebra A exists, is the case, when the quantum fields are in the space-time manifold M with the structure group SO (3, 1) (Lorentz group). The algebra A consists of distributions of densities with singularities in the closed future light cone subset. On account of the local isomorphism M ≅ Spec (A) , the quantum fields exist only in the space-time manifold with the one-dimensional arrow of time. In the fermion sector the restrictions caused by the possibility to define the multiplication on the densities of spinor fields can explain the chirality violation. It is found that for bosons in the Higgs sector the charge conjugation symmetry violation on the densities of states can be observed. This symmetry violation can explain the matter-antimatter imbalance. It is found that in theoretical models with non-abelian gauge fields instanton distributions are impossible and tunneling effects between different topological vacua | n> do not occur. Diagram expansion with respect to the -algebra variables is considered.

Breaking of parallelograms in presence of torsion: An equivalent alternative approach to detect gravitational waves

The equations for gravitational plane waves produced by a typical binary system as a solution of linear approximation of Einstein equations are derived. The dynamics of the corresponding gravitational field is analyzed in a four-dimensional space-time manifold, endowed with a metric and taking into account torsion. In this context, the geometrical reason of the existence of torsion due to the presence of gravitational waves (GW) is highlighted and the geodesic deviation is obtained taking into account both curvature and torsion. In a laser interferometer gravitational detector, the delay time between the arrivals of the two laser beams traveling back and forth along the two arms in presence of gravitational waves is interpreted from this point of view. This delay is calculated for the NS–NS binary pulsar (1913 + 16) in two specific orientations with respect to the experimental device, corresponding to different polarizations of gravitational waves. In the specific case of this example, it is shown that the results obtained in the context of the standard general relativity (GR) and in the framework of teleparallel gravity are equivalent.

Some cosmological models coming from gravitational theories having torsional degrees of freedom

In this work we consider gravitational theories in which the effect of coupling characteristic classes, appropriately introduced as operators in the Einstein-Hilbert action, has been taken into account. As it is well known, this approach strays from the framework of general relativity since it results in theories in which torsion can be present. We consider here all the characteristic classes that are consistent with a four-dimensional space-time manifold. Then, we present explicit expressions for the contortion $1$-form and torsion $2$-form that can be obtained under general conditions in various cases of interest. Additionally, we use the same framework to study cosmological scenarios that are obtained mainly by selecting the flat FLRW metric and an ideal fluid.

On the Existence of Rigid Spheres in Four-Dimensional Spacetime Manifolds

This paper deals with the generalization of usual round spheres in the flat Minkowski spacetime to the case of a generic four-dimensional spacetime manifold $M$. We consider geometric properties of sphere-like submanifolds in $M$ and introduce conditions on external curvature and torsion, which lead to a definition of a {\em rigid sphere}. The main result is a local existence theorem concernig such spheres. For this purpose we apply the surjective implicit function theorem. The proof is based on a detailed analysis of the linearized problem and leads to an eight-parameter family of solutions in case when the metric tensor $g$ of $M$ is from a certain neighbourhood of the flat Minkowski metric. This contribution continues the study of rigid spheres in (Class. Quantum Grav. \textbf{30} (2013), 175010, doi:10.1088/0264-9381/30/17/175010, 18 pp.).

On Euler’s stretching formula in continuum mechanics

The Lie time-derivative of the material metric tensor field along the motion is the proper mathematical definition of the physical notion of strain rate or stretching. Its expression, as symmetric part of the velocity gradient in Euclid space, is provided by a celebrated formula conceived by the genius of Leonhard Euler around the middle of the eighteenth century and since then reproduced in articles, books and treatises on continuum mechanics. We present here a formulation, in the proper geometric context of the four-dimensional space-time manifold endowed with an arbitrary linear connection and referring to a material body of arbitrary dimensionality. The expression involves the material time-derivative of the metric field and torsion-form and gradient of the velocity field, according to the connection induced on the trajectory. As an application, the expressions of the Gram matrix of the stretching in natural and in normalized (or engineering) reference systems induced by orthogonal polar coordinates are provided.

A CPT anomaly

We consider chiral gauge theories defined over a four-dimensional spacetime manifold with a Cartesian product structure for at least one compact spatial dimension. For a simple setup, we calculate the effective gauge field action by integrating out the chiral fermions, while maintaining gauge invariance. Due to a combination of infrared and ultraviolet effects, there appears a CPT-odd term in the effective gauge field action. This CPT anomaly could occur in chiral gauge theories relevant to elementary particle physics, provided the spacetime manifold has the appropriate topology. Two possible applications for cosmology are discussed.

A topological extension of general relativity

A set of algebraic equations for the topological properties of space-time is derived, and used to extend general relativity into the Planck domain. A unique basis set of three-dimensional prime manifolds is constructed which consists of S3, S1 × S2, and T3. The action of a loop algebra on these prime manifolds yields topological invariants which constrain the dynamics of the four-dimensional space-time manifold. An extended formulation of Mach's principle and Einstein's equivalence of inertial and gravitational mass is proposed which leads to the correct classical limit of the theory. It is found that the vacuum possess four topological degrees of freedom corresponding to a lattice of three-tori. This structure for the quantum foam naturally leads to gauge groups O(n) and SU(n) for the fields, a boundary condition for the universe, and an initial state characterized by local thermal equilibrium. The current observational estimate of the cosmological constant is reproduced without fine-tuning and found to be proportional to the number of macroscopic black holes. The black hole entropy follows immediately from the theory and the quantum corrections to its Schwarzschild horizon are computed.

A topological extension of general relativity

A set of algebraic equations for the topological properties of space-time is derived, and used to extend general relativity into the Planck domain. A unique basis set of three-dimensional prime manifolds is constructed which consists of S 3, S 1 × S 2, and T 3. The action of a loop algebra on these prime manifolds yields topological invariants which constrain the dynamics of the four-dimensional space-time manifold. An extended formulation of Mach's principle and Einstein's equivalence of inertial and gravitational mass is proposed which leads to the correct classical limit of the theory. It is found that the vacuum possess four topological degrees of freedom corresponding to a lattice of three-tori. This structure for the quantum foam naturally leads to gauge groups O( n) and SU( n) for the fields, a boundary condition for the universe, and an initial state characterized by local thermal equilibrium. The current observational estimate of the cosmological constant is reproduced without fine-tuning and found to be proportional to the number of macroscopic black holes. The black hole entropy follows immediately from the theory and the quantum corrections to its Schwarzschild horizon are computed.

Philosophy of gravity: Intuitions of four-dimensional curved spacetime

General Relativity describes the universe as a four-dimensional spacetime manifold which is somehow ‘curved’. Accounts of curved spacetime often claim that most of us find it impossible to imagine four-dimensional spacetime, and so should not even try to do so. This paper examines the claim that the best description of the universe is counter-intuitive and suggests ways to re-educate our intuitions to make them compatible with what seems to be the best science around. The paper examines two world views, or pictures of the universe — the Aristotelian and the Newtonian — and argues that we have forgotten (or perhaps never realized) how much of an idealization the Newtonian world view actually is. For this reason we have come to believe that we can only ‘see’ the world in three spatial dimensions. The paper ends by describing an alternative idealization, or set of pictures, that would lead us to find the claim that the universe is a four-dimensional spacetime manifold ‘perfectly natural’.

Euclidean hyperspace and its physical significance

Contemporary approaches to quantum field theory and gravitation often use a four-dimensional space-time manifold of Euclidean signature (which we call “hyperspace”) as a continuation of the Lorentzian metric. To investigate what physical sense this might have we review the history of Euclidean techniques in classical mechanics and quantum theory. Schwinger’s Euclidean postulate gives a fundamental significance to such techniques and leads to a clearer understanding of the TCP theorem in terms of space-time uniformity. This is closely related to the principle of identicality that characterizes non-relativistic quantum theory. In quantum gravity we suggest that the Hartle-Hawking treatment of the wave function of the universe rests on a notion of space-time uniformity which can be related to the Euclidean postulate as a kind of “perfect cosmological principle” on the unobservable wave function of the universe which eliminates anya priori asymmetry between space and time. Euclidean hyperspace may mediate between the infinite-dimensional Hubert space of quantum theory (whose metric is Euclidean) and the four-dimensional Lorentzian space-time of physical observations.

Sufficient conditions for a curvature tensor to be Riemannian and for determining its metric

A simple necessary and sufficient condition that a curvature tensor be Riemannian (i.e., that its symmetric connection be metric) is found, in very general conditions, for a four-dimensional space-time manifold. This enables existing uniqueness results for the metric tensor and an associated algebraic procedure for obtaining the metric tensor components from the curvature tensor components (when the curvature tensor is known to be a Riemannian tensor), to be extended to the situation where it is not known a priori whether the curvature tensor has come from a Lorentz metric.

Spinor under space-time gauge and Dirac equation in curved space-time

We discussed space-time gauge transformations of a four-dimensional space-time manifold by using the method of tangent-space approach, and derived naturally the Dirac equation in curved space-time and the general form of covariant differentiations of spinor tensors.

The geometrical structure of a complexified theory of gravitation

Outlines the procedure for the complexification of the tangent bundle over a four-dimensional space-time manifold. By introducing a connection and metric compatible with the complex structure, the authors form the geometrical basis for a new (complexified) theory of gravitation whose fundamental gauge group is U(3,1). They further prove that the Lagrangian for the theory is necessarily real when the connection is compatible with the metric.

Extension of a space-time formalism in relativistic quantum mechanics

Recen t ly COLLINS and FA~C~I (~) in t roduced a new formula t ion in re la t iv is t ic quan tum mechanics for spin-zero particles. They modified the convent iona l t r ea tment , g iven by the Kle in-Gordon formal ism, in the sense tha t s ta te funct ions and operators were defined on a four-dimensional spacet ime manifold. The s tar t ing po in t was the es tabl ishment of a con t inu i ty equa t ion in Minkowski spacet ime

A redefinition of quantum-mechanical exchange and Coulomb integrals by a reinterpretation of atomic and molecular orbitals as Taylor expansions of a potential and a similar treatment of the electromagnetic field.

In previous papers a deterministic account of the dynamics underlying i) the Schrodinger equation, ii) the commutation relations of quantum theory and Planck's constant and iii) theS-matrix and the Klauder phenomenon, was presented. In the present paper, that topological analysis is extended to a deterministic interpretation of atomic orbitals, valence bonds, molecular orbitals and the electromagnetic field. It is demonstrated that atomic orbitals may be considered ask-jets, or the polynomial consisting of all terms of order less than or equal tok in the Taylor expansion of a potential function. The electromagnetic field is described in terms of a mapping of two sets of four Clebsch potentials defined on a four-dimensional Riemannian space-time manifold, onto a three-dimensional Euclidean space (Rund). The Clebsch potentials are related to the four parameters used in the quantum-mechanical orbital description.