Sigbjørn Hervik

University of Stavanger · Department of Mathematics and Natural Science

We consider Kundt solutions to vacuum Conformal Killing Gravity (CKG) proposed by Harada and find numerous solutions in four dimensions and in higher dimensions. In CKG theory the cosmological constant appears as an integration constant, and hence, is naturally embedded in the theory. However, by considering Kundt solutions to CKG we seemingly open...

We study left-invariant pseudo-Riemannian metrics on Lie groups using the bracket flow of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the G= O(p,q)-action; i.e., Lie algebras where zero is in the closure of the orbits. We provide examples of such Lie groups in various signatures and give some gene...

We introduce an approach to determine new pseudo-Riemannian Einstein spaces by deforming symmetric pseudo-Riemannian Einstein spaces. The metrics of the spaces we will deform are associated with complex hyperbolic spaces and are (para-)Kähler manifolds. That is, they admit a parallel field of skew-symmetric endomorphisms, called a (para-)complex st...

We explore how far one can go in constructing d -dimensional static black holes coupled to p -form and scalar fields before actually specifying the gravity and electrodynamics theory one wants to solve. At the same time, we study to what extent one can enlarge the space of black hole solutions by allowing for horizon geometries more general than sp...

In this paper, which is of programmatic rather than quantitative nature, we aim to further delineate and sharpen the future potential of the LISA mission in the area of fundamental physics. Given the very broad range of topics that might be relevant to LISA, we present here a sample of what we view as particularly promising directions, based in par...

What is the asymptotic future of a scalar-field model if the assumption of isotropy is relaxed in generic, homogeneous Universe models? This paper is a continuation of our previous work on Bianchi cosmologies with a p-form field (where p ∈ {1, 3})—or equivalently: an inhomogeneous, mass-less scalar gauge field with a homogeneous gradient. In this w...

We explore how far one can go in constructing $d$-dimensional static black holes coupled to $p$-form and scalar fields before actually specifying the gravity and electrodynamics theory one wants to solve. At the same time, we study to what extent one can enlarge the space of black hole solutions by allowing for horizon geometries more general than...

Recent results of arXiv:1907.08788 on universal black holes in $d$ dimensions are summarized. These are static metrics with an isotropy-irreducible homogeneous base space which can be consistently employed to construct solutions to virtually any metric theory of gravity in vacuum.

Continuing previous work, we show the existence of stable, anisotropic future attractors in Bianchi invariant sets with a p -form field () and a perfect fluid. In particular, we consider the not previously investigated Bianchi invariant sets (II), (IV), (VII0) and (VIIh) and determine their asymptotic behaviour. We find that the isolated equilibriu...

A bstract We prove that a generalized Schwarzschild-like ansatz can be consistently employed to construct d -dimensional static vacuum black hole solutions in any metric theory of gravity for which the Lagrangian is a scalar invariant constructed from the Riemann tensor and its covariant derivatives of arbitrary order. Namely, we show that, apart f...

We provide an updated assessment of the fundamental physics potential of LISA. Given the very broad range of topics that might be relevant to LISA, we present here a sample of what we view as particularly promising directions, based in part on the current research interests of the LISA scientific community in the area of fundamental physics. We org...

This paper is a continuation of our previous work on Bianchi cosmologies with a $p$-form field (where $p\,\in\,\{1,3\}$) -- or equivalently: an inhomogeneous, massless scalar gauge field with a homogeneous gradient. In this work we investigate such matter sector in General Relativity, and restrict to space-times of the particular Bianchi types VI$_...

Why is the Universe so homogeneous and isotropic? We summarize a general study of a $\gamma$-law perfect fluid alongside an inhomogeneous, massless scalar gauge field (with homogeneous gradient) in anisotropic spaces with General Relativity. The anisotropic matter sector is implemented as a $j$-form (field-strength level), where $j\,\in\,\{1,3\}$,...

Continuing previous work, we show the existence of stable, anisotropic future attractors in Bianchi invariant sets with a $p$-form field ($p\,\in\,\{1,3\}$) and a perfect fluid. In particular, we consider the not previously investigated Bianchi invariant sets $\mathcal{B}$(II), $\mathcal{B}$(IV), $\mathcal{B}$(VII$_0$) and $\mathcal{B}$(VII$_{h})$...

A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their application to gravity theories. It is conjectured that a CSI spacetime is either locally homogeneous (...

We consider the class of locally boost isotropic spacetimes in arbitrary dimension. For any spacetime with boost isotropy, the corresponding curvature tensor and all of its covariant derivatives must be simultaneously of alignment type ${\bf D}$ relative to some common null frame. Such spacetimes are known as type ${\bf D}^k$ spacetimes and are con...

We prove that a generalized Schwarzschild-like ansatz can be consistently employed to construct $d$-dimensional static vacuum black hole solutions in any metric theory of gravity for which the Lagrangian is a scalar invariant constructed from the Riemann tensor and its covariant derivatives of arbitrary order. Namely, the base space can be taken to...

We prove that a generalized Schwarzschild-like ansatz can be consistently employed to construct d-dimensional static vacuum black hole solutions in any metric theory of gravity for which the Lagrangian is a scalar invariant constructed from the Riemann tensor and its covariant derivatives of arbitrary order. Namely, the base space can be taken to b...

Motivated by Wick-rotations of pseudo-Riemannian manifolds, we study real geometric invariant theory (GIT)and compatible representations. We extend some of the results from earlier works (Helleland and Hervik, 2018), in particular, we give some sufficient as well as necessary conditions for when pseudo-Riemannian manifolds are Wick-rotatable to oth...

Any manifold equipped with a metric is called CSI if all polynomial scalar invariants constructed from the curvature tensor and its covariant derivatives are constant. All locally homogeneous spaces are CSI but for indefinite signature there are CSI spaces which are not locally homogeneous. In the Lorentzian case, the CSI spacetimes have been studi...

We study universal electromagnetic (test) fields, i.e. p-forms fields F that solve simultaneously (virtually) any generalized electrodynamics (containing arbitrary powers and derivatives of F in the field equations) in n spacetime dimensions. One of the main results is a sufficient condition: any null F that solves Maxwell's equations in a Kundt sp...

We consider four dimensional spaces of neutral signature and give examples of universal spaces of Walker type. These spaces have no analogue in other signatures in four dimensions and provide with a new class of spaces being universal.

Motivated by Wick-rotations of pseudo-Riemannian manifolds, we study real geometric invariant theory (GIT) and compatible representations. We extend some of the results from earlier works \cite{W2,W1}, in particular, we give sufficient and necessary conditions for when pseudo-Riemannian manifolds are Wick-rotatable to other signatures. For arbitrar...

We study universal electromagnetic (test) fields, i.e., p-forms fields F that solve simultaneously (virtually) any generalized electrodynamics (containing arbitrary powers and derivatives of F in the field equations) in n spacetime dimensions. One of the main results is a sufficient condition: any null F that solves Maxwell's equations in a Kundt s...

Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through exponentiation. The motivation behind studying these infinitesimal group actions is the investigation of $\mathcal{I}$...

Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through exponentiation. The motivation behind studying these infinitesimal group actions is the investigation of I-degenerate...

In this paper the dynamics of free gauge fields in Bianchi type I-VII$_{h}$ space-times is investigated. The general equations for a matter sector consisting of a $p$-form field strength ($p\,\in\,\{1,3\}$), a cosmological constant ($4$-form) and perfect fluid in Bianchi type I-VII$_{h}$ space-times are computed using the orthonormal frame method....

Universal spacetimes are exact solutions to all higher-order theories of gravity. We study these spacetimes in four dimensions and we show that all universal spacetimes in four dimensions are algebraically special and Kundt. It is also shown that Petrov type D universal spacetimes are necessarily direct products of two 2-spaces of constant and equa...

Universal spacetimes are exact solutions to all higher-order theories of gravity. We study these spacetimes in four dimensions and provide necessary and sufficient conditions for universality for all Petrov types except of type II. We show that all universal spacetimes in four dimensions are algebraically special and Kundt. Petrov type D universal...

We define Wick-rotations by considering pseudo-Riemannian manifolds as real slices of a holomorphic Riemannian manifold. From a frame bundle viewpoint Wick-rotations between different pseudo-Riemannian spaces can then be studied through their structure groups which are real forms of the corresponding complexified Lie group (different real forms $O(...

We define Wick-rotations by considering pseudo-Riemannian manifolds as real slices of a holomorphic Riemannian manifold. From a frame bundle viewpoint Wick-rotations between different pseudo-Riemannian spaces can then be studied through their structure groups which are real forms of the corresponding complexified Lie group (different real forms $O(...

We study type II universal metrics of the Lorentzian signature. These metrics solve vacuum field equations of all theories of gravitation with the Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order. We provide examples of type II universal metrics for al...

We apply the causal Israel-Stewart theory of irreversible thermodynamics to model the matter content of the universe as a dissipative fluid possessing bulk and shear viscosity. Along with the full transport equations we consider their widely used truncated version. By implementing a dynamical systems approach to Bianchi type IV and V cosmological m...

Universal spacetimes are vacuum solutions to all theories of gravity with the Lagrangian L = L(gab, Rabcd, ∇a1Rbcde,..., ∇a1...apRbcde). Well known examples of universal spacetimes are plane waves which are of the Weyl type N. Here, we discuss recent results on necessary and sufficient conditions for all Weyl type N spacetimes in arbitrary dimensio...

We show that a metric of arbitrary dimension and signature which allows for a Wick rotation to a Riemannian metric necessarily has a purely electric Riemann and Weyl tensor.

We investigate the three types of class B Bianchi cosmologies filled with a tilted perfect fluid undergoing velocity diffusion in a scalar field background. We consider the two most importantcases: dust and radiation. A complete numerical integration of the Einstein field equations coupled with the diffusion equations is done to demonstrate how the...

Universal spacetimes are spacetimes for which all conserved symmetric rank-2 tensors, constructed as contractions of polynomials from the metric, the Riemann tensor and its covariant derivatives of arbitrary order, are multiples of the metric. Consequently, metrics of universal spacetimes solve vacuum equations of all gravitational theories, with t...

In this paper we study pseudo-Riemannian spaces with a degenerate curvature structure i.e. there exists a continuous family of metrics having identical polynomial curvature invariants. We approach this problem by utilising an idea coming from invariant theory. This involves the existence of a boost, the existence of this boost is assumed to extend...

We apply the dynamical systems approach to ever-expanding Bianchi type VIII cosmologies filled with a tilted $\gamma$-fluid undergoing velocity diffusion on a scalar field. We determine the future attractors and investigate the late-time behaviour of the models. We find that at late times the normalized energy density $\Omega$ tends to zero, while...

We will construct explicit examples of four-dimensional neutral signature Walker (but not necessarily degenerate Kundt) spaces for which all of the polynomial scalar curvature invariants vanish. We then investigate the properties of some particular subclasses of Ricci flat spaces. We also briefly describe some four-dimensional neutral signature Ein...

It is well known that certain pp-wave metrics, belonging to a more general class of Ricci-flat type N, $\tau_i =0$, Kundt spacetimes, are universal and thus they solve vacuum equations of all gravitational theories with Lagrangian constructed from the metric, the Riemann tensor and its derivatives of arbitrary order. In this paper, we show (in an a...

We investigate a simple inhomogeneous anisotropic cosmology (plane symmetric $G_2$ model) filled with a tilted perfect fluid undergoing velocity diffusion on a scalar field. Considered are two types of fluid: dust and radiation. We solve the system of Einstein field equations and diffusion equations numerically and demonstrate how the universe evol...

We show that the recently found anti–de Sitter (AdS)-plane and AdS-spherical wave solutions of quadratic curvature gravity also solve the most general higher derivative theory in D dimensions. More generally, we show that the field equations of such theories reduce to an equation linear in the Ricci tensor for Kerr-Schild spacetimes having type-N W...

We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (i.e., a unit timelike vector field u), in any n. We study the cases where on...

Recent results on purely electric (PE) or magnetic (PM) spacetimes in n dimensions are summarized. These include: Weyl types; diagonalizability; conditions under which direct (or warped) products are PE/PM.

We refine the null alignment classification of the Weyl tensor of a five-dimensional spacetime. The paper focusses on the algebraically special alignment types {\bf {N}}, {\bf {III}}, {\bf {II}} and {\bf {D}}, while types {\bf {I}} and {\bf {G}} are briefly discussed. A first refinement is provided by the notion of spin type of the components of hi...

Motivated by the couplings of the dilaton in four-dimensional effective actions, we investigate the cosmological consequences of a scalar field coupled both to matter and a Maxwell-type vector field. The vector field has a background isotropy-violating component. New anisotropic scaling solutions which can be responsible for the matter and dark ene...

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). Using an algebraic classification of pseudo-Riemannian spaces in terms of the boost-weight decomposition, we first show more generally that a space which is not characterized by its invariants must poss...

By using invariant theory we show that a (higher-dimensional) Lorentzian metric that is not characterised by its invariants must be of aligned type II; i.e., there exists a frame such that all the curvature tensors are simultaneously of type II. This implies, using the boost-weight decomposition, that for such a metric there exists a frame such tha...

Recently an inflationary model with a vector field coupled to the inflaton was proposed and the phenomenology studied for the Bianchi type I spacetime. It was found that the model demonstrates a counter-example to the cosmic no-hair theorem since there exists a stable anisotropically inflationary fix-point. One of the great triumphs of inflation, h...

We consider higher dimensional Lorentzian spacetimes which are currently of interest in theoretical physics. It is possible to algebraically classify any tensor in a Lorentzian spacetime of arbitrary dimensions using alignment theory. In the case of the Weyl tensor, and using bivector theory, the associated Weyl curvature operator will have a restr...

We prove a generalization of the -property, namely that for any dimension and signature, a metric which is not characterized by its polynomial scalar curvature invariants; there is a frame such that the components of the curvature tensors can be arbitrary close to a certain 'background'. This 'background' is defined by its curvature tensors: it is...

There are a number of algebraic classifications of spacetimes in higher dimensions utilizing alignment theory, bivectors and discriminants. Previous work gave a set of necessary conditions in terms of discriminants for a spacetime to be of a particular algebraic type. We demonstrate the discriminant approach by applying the techniques to the Sorkin...

In this paper we present a number of four-dimensional neutral signature exact solutions for which all of the polynomial scalar curvature invariants vanish (VSI spaces) or are all constant (CSI spaces), which are of relevence in current theoretical physics.

A classical solution is called universal if the quantum correction is a multiple of the metric. Universal solutions consequently play an important role in the quantum theory. We show that in a spacetime which is universal all of the scalar curvature invariants are constant (i.e., the spacetime is CSI).

The Weyl and Ricci tensors can be algebraically classified in a Lorentzian spacetime of arbitrary dimensions using alignment theory. Used in tandem with the boost weight decomposition and curvature operators, the algebraic classification of the Weyl tensor and the Ricci tensor in higher dimensions can then be refined utilizing their eigenbivector a...

We consider arbitrary-dimensional pseudo-Riemannian spaces of signature $(k,k+m)$. We introduce a boost-weight decomposition and define a number of algebraic properties (e.g., the ${\bf S}_i$- and ${\bf N}$-properties) and present a boost-weight decomposition to classify the Weyl tensors of arbitrary signature and discuss degenerate algebraic types...

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of their polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the ${\bf S}_i$- and ${\bf N}$-properties, and show that if the curvature tenso...

We discuss (arbitrary-dimensional) Lorentzian manifolds and the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. Recently, we have shown that in four dimensions a Lorentzian spacetime metric is either \mathcal {I}-non-degenerate, and hence locally characterized by its scalar polynomial curvat...

In this paper we study the future asymptotics of spatially homogeneous Bianchi type II cosmologies with a tilted perfect fluid with a linear equation of state. By means of Hamiltonian methods we first find a monotone function for a special tilted case, which subsequently allows us to construct a new set of monotone functions for the general tilted...

We discuss negatively curved homogeneous spaces admitting a simply transitive group of isometries, or equivalently, negatively curved left-invariant metrics on Lie groups. Negatively curved spaces have a remarkably rich and diverse structure and are interesting from both a mathematical and a physical perspective. As well as giving general criteria...

We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterised by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of alignment theory and the bivector form of the Weyl operator in higher dimensions, and introduce the importan...

We illustrate the fact that the class of vacuum type D spacetimes which are $\mathcal{I}$-\emph{non-degenerate} are invariantly classified by their scalar polynomial curvature invariants.

We illustrate the fact that the class of vacuum type D spacetimes which are $\mathcal{I}$-\emph{non-degenerate} are invariantly classified by their scalar polynomial curvature invariants.

We display some simple cosmological solutions of gravity theories with quadratic Ricci curvature terms added to the Einstein-Hilbert lagrangian which exhibit anisotropic inflation. The Hubble expansion rates are constant and unequal in three orthogonal directions. We describe the evolution of the simplest of these homogeneous and anisotropic cosmol...

We develop the bivector formalism in higher dimensional Lorentzian spacetimes. We define the Weyl bivector operator in a manner consistent with its boost-weight decomposition. We then algebraically classify the Weyl tensor, which gives rise to a refinement in dimensions higher than four of the usual alignment (boost-weight) classification, in terms...

The locally rotationally symmetric tilted perfect fluid Bianchi type V cosmological model provides examples of future geodesically complete spacetimes that admit a `kinematic singularity' at which the fluid congruence is inextendible but all frame components of the Weyl and Ricci tensors remain bounded. We show that for any positive integer n there...

We discuss the geometrical properties of spacetimes in the context of higher dimensional theories of gravity. If the spacetime admits a covariantly constant time-like vector, the spacetime is static and (1+10)-decomposable, where the 10-dimensional transverse space is Riemannian. The second class of solutions consists of spacetimes that admit a cov...

In this paper we investigate four dimensional Lorentzian spacetimes with constant curvature invariants ($CSI$ spacetimes). We prove that if a four dimensional spacetime is $CSI$, then either the spacetime is locally homogeneous or the spacetime is a Kundt spacetime for which there exists a frame such that the positive boost weight components of all...

We study a class of constant scalar invariant (CSI) space–times which belong to the higher-dimensional Kundt class and which are solutions of supergravity. We review the known CSI supergravity solutions in this class and we explicitly present a number of new exact CSI supergravity solutions, some of which are Einstein.

In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an $\mathcal{I}$-non-degenerate spacetime metric, which implies that the spacetime metric is lo...

Kundt spacetimes are of great importance in general relativity in 4 dimensions and have a number of topical applications in higher dimensions in the context of string theory. The degenerate Kundt spacetimes have many special and unique mathematical properties, including their invariant curvature structure and their holonomy structure. We provide a...

Cosmological data seem to imply dynamical behaviour different to that implied by standard cosmological models. This dynamical behaviour is often modeled by an exotic form of matter with a non‐standard effective equation of state parameter called dark energy. We show that in general relativistic tilting perfect fluid cosmological models with an ultr...

In this note we complete the analysis of Hervik, van den Hoogen, Lim and Coley (2007 Class. Quantum Grav. 24 3859) of the late-time behaviour of tilted perfect fluid Bianchi type III models. We consider models with dust, and perfect fluids stiffer than dust, and eludicate the late-time behaviour by studying the centre manifold which dominates the b...

We follow a constructive approach and find higher-dimensional black holes with Ricci nilsoliton horizons. The spacetimes are solutions to the Einstein equation with a negative cosmological constant and generalise therefore, Anti-de Sitter black hole spacetimes. The approach combines a work by Lauret–which relates the so-called Ricci nilsolitons and...

In this talk we will discuss spacetimes with constant scalar invariants (CSI spacetimes). There are many examples of such spacetimes, among them spacetimes with vanishing curvature invariants and homogeneous spaces. Special emphasis will be put on a certain class of spacetimes to which all known inhomogeneous CSI spacetimes belong. The role of this...

The increasing prominence of general relativity in astrophysics and cosmology is reflected in the growing number of texts, particularly at the undergraduate level. A natural attitude before opening a new one is to ask i) what makes this different from those already published? And ii) does it follow the 'physics-first approach' as for instance the b...

Supersymmetric solutions of supergravity theories, and consequently metrics with special holonomy, have played an important role in the development of string theory. We describe how a Lorentzian manifold is either completely reducible, and thus essentially known, or not completely reducible so that there exists a degenerate holonomy invariant light...

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor $T_{\mu \nu}$ constructed from sums of terms the involving contractions of the metric and powers of arbitrary co...

The universe today, containing stars, galaxies and black holes, seems to have evolved from a very homogeneous initial state. From this it appears as if the entropy of the universe is decreasing, in violation of the second law of thermodynamics. It has been suggested by Roger Penrose that this inconsistency can be solved if one assigns an entropy to...

We consider the perihelion precession and bending of light in a class of Kaluza-Klein models and show that the "electric redshift" model, proposed in Zhang (2006) to explain the redshift of Quasars, does not agree with observations. As Zhang's model only considers the Jordan frame, we also compute the perihelion precession as seen in the Einstein f...

In this paper we study Lorentzian spacetimes for which all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes) in three dimensions. We determine all such CSI metrics explicitly, and show that for every CSI with particular constant invariants there is a locally homogeneous spac...

We study tilted perfect fluid cosmological models with a constant equation of state parameter in spatially homogeneous models of Bianchi type VI$_{-1/9}$ using dynamical systems methods and numerical simulations. We study models with and without vorticity, with an emphasis on their future asymptotic evolution. We show that for models with vorticity...

We study tilted perfect fluid cosmological models with a constant equation of state parameter in spatially homogeneous models of Bianchi type VI_h using dynamical systems methods and numerical experimentation, with an emphasis on their future asymptotic evolution. We determine all of the equilibrium points of the type VI_h state space (which corres...

We show that the higher-dimensional vanishing scalar invariant (VSI) spacetimes with fluxes and dilaton are solutions of type IIB supergravity, and we argue that they are exact solutions in string theory. We also discuss the supersymmetry properties of VSI spacetimes.

Many of us have experienced the same; fallen and broken something. Yet supposedly, gravity is the weakest of the fundamental forces; it is claimed to be 10-15 times weaker than electromagnetism. Still, every one of us has more or less had a personal relationship with gravity. Einstein's General Theory of Relativity: With Modern Applications in Cosm...

Already in 1914 – before Einstein had fulfilled the construction of the general theory of relativity – Gunnar Nordström1 had published a five-dimensional scalar-tensor theory of gravitation in an effort to unify gravitation and electromagnetism. Since it was based upon his own theory of gravitation which was soon surpassed by Einstein’s theory, thi...

Soon after Einstein had introduced the cosmological constant he withdrew it and called it “the biggest blunder” of his life. However, there has been developments in the last decades that have given new life to the cosmological constant. Firstly, the idea of inflation gave cosmology a whole new view upon the first split second of our universe. A key...

Forms prove to be a powerful tool in differential geometry and in physics. They have many wonderful properties that we shall explore further in this chapter. We know that in physics and mathematics, integration and differentiation are important, if not essential, operations that appear in almost all physical theories. In this chapter we will explor...

In this chapter we shall consider some consequences of the formalism developed so far, by studying the relativistic kinematics in two types of non-inertial reference frames: the rotating reference frame and the uniformly accelerating reference frame.

In this chapter we shall give a short introduction to the fundamental principles of the special theory of relativity, and deduce some of the consequences of the theory.

One of the most successful and useful applications of Einstein’s General Theory of Relativity is within the field of cosmology. Newton’s theory of gravitation, involves attraction between celestial bodies. However, very little is said of the evolution of the universe itself. The universe was believed to be static, and its evolution was beyond any p...

To obtain a mathematical description of physical phenomena, it is advantageous to introduce a reference frame in order to keep track of the position of events in space and time. The choice of reference frame has historically depended upon the view of human beings and their position in the Universe.