Oyvind Gron

Oslo Metropolitan University

We show that there are distinct periods when three ocean variability series in the Atlantic and the Pacific Oceans persistently lead or lag each other, as well as distinct periods when ocean variability series lead the rate of changes in global temperature anomaly (∆GTA) and in atmospheric CO2 concentration (1880–2019). The superimposed lead-lag (L...

The Schwarzschild solution describing spacetime outside a spherical mass distribution is deduced. In this deduction we give a detailed prescription of how one calculates the components of Einstein’s curvature tensor using differential forms as decomposed in an orthonormal basis. The predictions for the classical tests of Einstein’s theory—gravitati...

Spacetime outside black holes with and without rotation—i.e. the Kerr and Schwarzschild spacetimes—is studied. By considering the motion of free particles in the Kerr spacetime we find an exact expression for the angular velocity of the inertial dragging. Hawking radiation from a non-rotating black hole is also studied.

This chapter gives a concise and yet rather complete introduction to the special theory of relativity. Minkowski diagrams are used to illustrate several concepts such as the relativity of simultaneity. Special relativity is a theory of flat spacetime admitting accelerated and rotating reference frames. In this chapter we also show how magnetism app...

This chapter begins with an introduction to the formalism used to project four-dimensional spacetime into a 3-dimensional spatial 3-space. Then we apply this formalism to deduce the spatial geometry in a rotating reference frame and discuss Ehrenfest’s paradox. Also we show that it is impossible to Einstein synchronize clocks around a closed path i...

This chapter starts with a hydrodynamical description of energy–momentum conservation in a Newtonians context in order to give some intuition about the relativistic formulation of energy–momentum conservation as represented by a vanishing divergence of the energy–momentum tensor. Einstein demanded that energy–momentum conservation should follow fro...

The Lemaître–Friedmann–Robertson–Walker (LFRW) universe models are deduced as solutions of Einstein’s field equations, and the Hubble–Lemaître expansion law is found as a general property of these models. It is shown that the cosmic redshift due to the expansion of the space contains both the kinematic Doppler effect due to the velocity of the emit...

In this chapter we develop the main mathematical concepts used in this book. First vectors, not only as quantities with length and direction, but as differential operators. Then tensors of arbitrary rank are introduced. As a preparation for using Cartan’s formalism we introduce forms, i.e. antisymmetric covariant tensors. This antisymmetric tensor...

In this chapter we shall first find a general expression of the acceleration of gravity due to a mass distribution. Then we shall deduce the solution of Einstein’s field equations inside an incompressible star—the internal Schwarzschild solution. Furthermore we shall present Israel’s formalism for describing singular mass shells in the general theo...

TheCovariant Differentiation theory of formsForms is a theory of antisymmetric tensorsAntisymmetric tensor. In such a theory we need an antisymmetric version of the covariant derivativeCovariant derivative such that the derivative of a form is a form. Hence in this chapter we first introduce the covariant derivative and then the antisymmetric exter...

TheLinear field approximationlinearGravitational waves field approximation of Einstein’s field equationsEinstein’s field equations is presented. The solutions of these equations inside and outside a rotating spherical shell are deduced. It is shown that inertial draggingInertial dragging is a consequence of the general theory of relativityGeneral t...

In this chapter we first deduce Newton’s law of gravitationNewton’s law of gravitation in its local form as a preparation for comparing Newton’s and Einstein’s theories, including a discussion of tidal forcesTidal force. Then we give a presentation of the main conceptual foundation of the general theory of relativityGeneral theory of relativity, em...

TheCurvatureRiemann curvature tensorRiemann curvature tensor is introduced, and the expression of its components in terms of the derivatives of the metric and the structure coefficients is deduced. Tidal forcesTidal force are discussed in a relativistic context, and it is pointed out that the relativistic gravitational field has both a non-tidal co...

Cycle times found in many oceanic time series have been explained with references to external mechanisms that act on the systems. Here we show that when we extract cycle times from 100 sets of paired random series, we find six distinct clusters of common cycle times ranging from about 3 years to about 32 years. Cycle times, CT, get shorter when one...

What causes cycles in oceanic oscillations, and is there a change in the characteristics of oscillations in around 1950? Characteristics of oceanic cycles and their sources are important for climate predictability. We here compare cycles generated in a simple model with observed oceanic cycles in the great oceans: The North Atlantic Oscillation (NA...

Three large ocean currents are represented by proxy time series: the North Atlantic Oscillation (NAO), the Southern Oscillation Index (SOI), and the Pacific Decadal Oscillation (PDO). We here show how proxies for the currents interact with each other and with the global temperature anomaly (GTA). Our results are obtained by a novel method, which id...

We show that oceanic cycle lengths persist across oceanic cyclic time-series by comparing cycles in series that come from “sister” measurements in the North Atlantic Ocean. These are the North Atlantic oscillation (NAO), the Atlantic multidecadal oscillation (AMO) and the Atlantic meridional overturning circulation (AMOC). The raw NAO series, which...

Alexander Friedmann, Carl Wilhelm Wirtz, Vesto Slipher, Knut E. Lundmark, Willem de Sitter, Georges H. Lemaître, and Edwin Hubble all contributed to the discovery of the expansion of the universe. If only two persons are to be ranked as the most important ones for the general acceptance of the expansion of the universe, the historical evidence poin...

In Antarctica, ice-core temperature has traditionally been regarded as a leading variable to carbon dioxide, CO2 during the last 400,000 years before present (B.P.). This finding is in contrast to most reports on global mean surface temperature and atmospheric CO2 for the last 150 years. However, previous techniques for establishing leading or lagg...

I give a review of predictions of values of spectral parameters for a large number of inflationary models. The present review includes detailed deductions and information about the approximations that have been made, written in a style that is suitable for text book authors. The Planck data have the power of falsifying several models of inflation a...

From a hydrodynamicist's point of view the inclusion of viscosity concepts in the macroscopic theory of the cosmic fluid would appear most natural, as an ideal fluid is after all an abstraction (excluding special cases such as superconductivity). Making use of modern observational results for the Hubble parameter plus standard Friedmann formalism,...

In order to provide a better understanding of rotating universe models, and in particular the G\"{o}del universe, we discuss the relationship between cosmic rotation and perfect inertial dragging. In this connection, the concept of \emph{causal mass} is defined in a cosmological context, and discussed in relation to the cosmic inertial dragging eff...

John Nash has proposed a new theory of gravity. We define a Nash-tensor equal to the curvature tensor appearing in the Nash field equations for empty space, and calculate its components for two cases. 1. A static, spherically symmetric space, and 2. The expanding, homogeneous and isotropic space of the Friedmann-Lemaitre-Robertson-Walker (FLRW) uni...

We apply a novel method based upon “before” and “after” relationships to investigate and quantify interconnections between global temperature anomaly (GTA), as response variable, and greenhouse gases (CO2) and total solar irradiance (TSI) as candidate causal variables for the period 1880 to 2010. The most likely interpretations of our results for t...

The “Mitra paradox” refers to the fact that while the de Sitter spacetime appears non-static in a freely falling reference frame, it looks static with reference to a fixed reference frame. The coordinate-independent nature of the paradox may be gauged from the fact that the relevant expansion scalar, θ = 3 Λ , is finite if Λ > 0 . The trivial resol...

I show here that there are some interesting differences between the predictions of warm and cold inflation models focusing in particular upon the scalar spectral index n s and the tensor-to-scalar ratio r. The first thing to be noted is that the warm inflation models in general predict a vanishingly small value of r. Cold inflationary models with t...

This article is a celebration of the centenary of Schwarzschild's presentations of his external and internal solutions describing spacetime outside and inside an incompressible, spherically symmetric body. I give a review of these solutions and how they have been interpreted physically.

We present an apparent paradox within the special theory of relativity, involving a trolley with relativistic velocity and its rolling wheels. Two solutions are given, both making clear the physical reality of the Lorentz contraction, and that the distance on the rails between each time a specific point on the rim touches the rail is not equal to 2...

We show that Einstein’s general theory of relativity, together with the assumption that the principle of relativity encompasses rotational motion, predicts that in a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) universe model with dust and Lorentz Invariant Vacuum Energy (LIVE), the density parameter of vacuum energy must have the value (Formula...

We explore a new method for identifying leaders and followers, LF, in repeated games by analyzing an experimental, repeated (50 rounds) game where Row player shifts the payoff between small and large values-a type of "investor" and Column player determines who gets the payoff-a type of "manager". We found that i) the Investor (Row) most often is a...

Data and example calculation for Seip. Leader and followers.xlsx. (XLSX)

The Levi-Civita-Bertotti-Robinson (LBR) solution of Einstein's field equations may be interpreted to describe spacetime in a region without matter outside a charged spherical domain wall. In the present paper we investigate the physical properties of some solutions of the Einstein-Maxwell equations, and show that they are generalizations of the LBR...

The research on relativistic universe models with viscous fluids is reviewed. Viscosity may have been of significance during the early inflationary era, and may also be of importance for the late time evolution of the Universe. Bulk viscosity and shear viscosity cause exponential decay of anisotropy, while nonlinear viscosity causes power-law decay...

A new source of the C-metric is described using Israel's formalism. This source is a singular accelerated shell. By construction, perfect inertial dragging is realized inside the shell. The equation of state and energy conditions for the shell are discussed.

The cosmological event horizon entropy and the apparent horizon entropy of the ΛCDM and the Bianchi type I Universe model with viscosity has been calculated numerically, and analytically in the large time limit. It is shown that for these Universe models the cosmological event horizon entropy increases with time and for large times it approaches a...

The electrodynamics of a radiating charge and its electromagnetic field based upon the Lorentz–Abraham–Dirac (LAD ) equation are discussed both with reference to an inertial reference frame and a uniformly accelerated reference frame. It is demonstrated that energy and momentum are conserved during runaway motion of a radiating charge and during fr...

We give a review of viscous relativistic universe models that have been presented during the period from 1990 until the present time. In particular we discuss the properties of isotropic and homogeneous universe models, and of anisotropic and homogeneous Bianchi type I models. We consider these types of models both in the context of the non-causal...

The main topic of this paper is a description of the generation of entropy at the end of the inflationary era. As a generalization of the present standard model of the Universe dominated by pressureless dust and a Lorentz invariant vacuum energy (LIVE), we first present a flat Friedmann universe model, where the dust is replaced with an ideal gas....

We deduce a simple expression for the Kretschmann curvature scalar of a conformally flat spacetime with a perfect fluid. Conformally flat, static, spherically symmetric spacetimes are investigated in various coordinate systems. The equation for a vanishing Weyl tensor and Einstein's field equations are integrated in curvature coordinates. We find c...

The concept of negative temperatures has occasionally been used in connection with quantum systems. A recent example of this sort is reported in the paper of S. Braun et al. [Science 339,52 (2013)], where an attractively interacting ensemble of ultracold atoms is investigated experimentally and found to correspond to a negative-temperature system s...

In this paper we study the evolution of spatially homogeneous and anisotropic Bianchi type-I Universe models with the cosmological constant, \Lambda, and filled with nonlinear viscous fluid. The dynamical equations for these models are obtained and solved for some special cases. We calculate the statefinder parameters for the models and display the...

We explore flat {\Lambda}CDM models with bulk viscosity, and study the role of the bulk viscosity in the evolution of these universe models. The dynamical equations for these models are obtained and solved for some cases of bulk viscosity. We obtain differential equations for the Hubble parameter H and the energy density of dark matter {\rho}, for...

In the standard formulation of the twin paradox an accelerated twin considers himself as at rest and his brother as moving. Hence, when formulating the twin paradox, one uses the general principle of relativity, i.e. that accelerated and rotational motion is relative. The significance of perfect inertial dragging for the validity of the principle o...

The Levi-Civita Bertotti Robinson (LBR) spacetime is investigated in various coordinate systems. By means of a general formalism for constructing coordinates in conformally flat spacetimes, coordinate transformations between the different coordinate systems are deduced. We discuss the motion of the reference frames in which the different coordinate...

The effect of gravity upon changes of the entropy of a gravity-dominated system is discussed. In a universe dominated by vacuum energy, gravity is repulsive, and there is accelerated expansion. Furthermore, inhomogeneities are inflated and the universe approaches a state of thermal equilibrium. The difference between the evolution of the cosmic ent...

Some difficulties with the deduction of Goto et al (2010 Class. Quantum Grav. 27 025005) that the gravitational mass of a charged particle is greater than its inertial mass are pointed out. It is concluded that the gravitational and inertial masses of a charged particle are equal.

The theory of electrodynamics of radiating charges is reviewed with special emphasis on the role of the Schott energy for the conservation of energy for a charge and its electromagnetic field. It is made clear that the existence of radiation from a charge is not invariant against a transformation between two reference frames that has an accelerated...

Within the theory of general relativity gravitational phenomena are usually attributed to the curvature of four-dimensional spacetime. In this context we are often confronted with the question of how the concept of ordinary physical three-dimensional space fits into this picture. In this work we present a simple and intuitive model of space for bot...

The kinematical properties of the de Sitter space-time are reviewed and investigated. The properties of the static sections are clarified. A deduction of the analytic extension, analogous to that of Kruskal and Szekeres for the Schwarzschild space-time, of the static section to the region outside the horizon is given. The representation of the de S...

Cartesian coordinate systems have straight coordinate lines orthogonal to each other. The coordinate basis vector fields are constant, and all the vectors of each field are unit vectors with the same direction. This is the simplest coordinate system that one can imagine. It seems strange that it can be advantageous to introduce coordinate systems w...

In order to be able to understand Einstein’s field equations we should first consider some important concepts of Newtonian physics.

This chapter is written for those people who have the courage to approach the mathematics of general relativity without being familiar with differential calculus. The use of this fabulous creation by Newton and Leibniz is essential and omnipresent on our way to Einstein’s field equations.

We have now completed our intended introduction to the mathematics used in the general theory of relativity. It remains to explain the central physical contents of the theory. Let us first offer a brief summary of the fundamental concept of Newton’s theory of gravitation.

Mathematically the general theory of relativity is a theory of vectors and quantities that generalize vectors, namely tensors. If one wants to master, or at least obtain some familiarity, with the mathematical apparatus of this theory, one should manage to have intercourse with vectors as dear friends. It is wise from the very start to listen to th...

The metric tensor is perhaps the most important mathematical quantity in the theory of relativity. From a knowledge of the metric tensor one may compute the geometry of spacetime and for example the motion of the planets in the solar system. In this chapter we shall give a thorough introduction to the metric tensor and its physical significance in...

‘Spacetime is curved’. It is, of course, not easy to understand adequately what is meant by that sentence as it occurs in the general theory of relativity. There are two principal axes of ‘precisation’, one leads into pure mathematics, the other into physics and cosmology. We shall start with the mathematical.1

Curves of particular interest in physics are those representing paths of moving particles. Such curves may be described by giving the coordinates of the particle as functions of time. In the following we shall consider the path of a particle thrown horizontally, as an illustrating example (see Fig. 3.1).

‘Geodesy’ comes from Greek γη, Earth, and \(\delta \alpha \acute{\iota }\omega \), divide, i.e.‘Earth division’. ‘Geodesic’ will be used in a rather special geometric sense in the following, but it will be related to the old problem of measuring the shortest path on the curved surface of the Earth. From the Euclidean geometry of a plane surface, we...

Choreography is the art of composing dances and the recording of movements on paper by means of convenient signs and symbols. Consider for a moment a most disturbing and uncanny experience suffered by a well established choreographer. He was supposed to record on paper the movements of certain fairly simple dances, but in a faraway, strange place.

The first eleven chapters of our text were devoted to the development of the mathematical structure of Einstein’s theory of relativity. In chapter 12 we discussed the physical principles of the theory. But there is a third region of inquiry which the reader may want to enter: the multitude of applications of the general theory. Applied to the world...

Cosmology may be said to be that part of physical science that aims at giving a description of the universe at large. Such descriptions are called universe models, and are mathematical models interpreted physically. They are based upon observations and physical laws. These laws represent our deepest insights as to the behaviour of the material worl...

We explore flat ΛCDM models with bulk viscosity, and study the role of the bulk viscosity in the evolution of these universe models. The dynamical equations for these models are obtained and solved for some cases of bulk viscosity. We obtain differential equations for the Hubble parameter H and the energy density of dark matter ρ m , for which we g...

We derive and discuss the physical interpretation of a conformally flat, static solution of the Einstein-Maxwell equations. It is argued that it describes a conformally flat, static spacetime outside a charged spherically symmetric domain wall. The acceleration of gravity is directed away from the wall in spite of the positive gravitational mass of...

We deduce general expressions for the line element of universe models with positive spatial curvature described by conformally flat spacetime coordinates. Models with dust, radiation and vacuum energy are exhibited. Discussing the existence of particle horizons we show that there is continual annihilation of space, matter and energy in a dust and r...

The 3 -space of a universe model is defined at a certain simultaneity. Hence space depends on which time is used. We find a general formula generating all known and also some new transformations to conformally flat-spacetime coordinates. A general formula for the recession velocity is deduced.

We deduce general expressions for the line element of universe models with negative and vanishing spatial curvature described by conformally flat-spacetime coordinates. The empty Milne universe model and models with dust, radiation and vacuum energy are exhibited. Discussing the existence of particle horizons we show that there is continual creatio...

It is shown that energy and momentum are conserved during the runaway motion of a radiating charge and during free fall of a charge in a gravitational field. The runaway motion demonstrates the consistency of classical electrodynamics and the Lorentz-Abraham-Dirac equation. The important role of the Schott (acceleration) energy in this connection i...

The question whether rotational motion is relative according to the general theory of relativity is discussed. Einstein's ambivalence concerning this question is pointed out. In the present article I defend Einstein's way of thinking on this when he presented the theory in 1916. The significance of the phenomenon of perfect inertial dragging in con...

The investigation of the stability properties of certain variants of Einstein’s static universe performed by Carneiro and Tavakol (in Stability of the Einstein static universe in the presence of vacuum energy) is generalized. It is shown that all versions of Einstein’s static universe without interaction between the two fluids it contains are unsta...

In the present article we find a new class of solutions of Einstein's field equations. It describes stationary, cylindrically symmetric spacetimes with closed timelike geodesics everywhere outside the symmetry axis. These spacetimes contain a magnetic field parallel to the axis, a perfect fluid with constant density and pressure, and Lorentz invari...

It is demonstrated that energy is conserved during runaway motion of a radiating charge, and during free fall of a charge in a field of gravity. The decisive role of the Schott energy in this connection is made clear. Also it is pointed out that a proton and a neutron fall with the same acceleration in a uniform gravitational field although the pro...

Two important questions concerning cosmic rays are: Why are electrons in the cosmic rays less efficiently accelerated than nuclei? How are particles accelerated to great energies in ultra-high energy cosmic rays? In order to answer these questions we construct a simple model of the acceleration of a charged particle in the cosmic ray. It is not mea...

The 3-space of a universe model is defined at a certain simultaneity. Hence space depends on which time is used. We find a general formula generating all known transformations to conformally flat spacetime coordinates, and work out the physical interpretation of conformal coordinate systems in different universe models. We show that continual creat...

Recently Abramowicz and Bajtlik [ArXiv: 0905.2428 (2009)] have studied the twin paradox in Schwarzschild spacetime. Considering circular motion they showed that the twin with a non-vanishing 4-acceleration is older than his brother at the reunion and argued that in spaces that are asymptotically Minkowskian there exists an absolute standard of rest...

Energy–momentum conservation for a Newtonian fluid in terms of the divergence of the energy momentum tensor can be shown as follows.

This is a solution of the vacuum field equations Emn=0E_{\mu\nu}=0 for a static spherically symmetric spacetime. One can then choose the following form of the line element (employing units so that c=1):

For a spherical mass distribution, V([(r)\vec])=-mG\fracMrV(\vec r)=-mG\frac{M}{r} , with zero potential infinitely far from the centre of MM . Newton’s law of gravitation is valid for “small” velocities, i.e. velocities much smaller than the velocity of light and “weak” fields. Weak fields are fields in which the gravitational potential energy...

Let [(e)\vec][^0]\vec{e}_{\hat{0}} be the 4-velocity field ( x0=ct, c=1, x0=tx^0=ct, c=1, x^0=t ) of the reference particles in a reference frame R. We are going to find the metric tensor gij\gamma_{ij} in a tangent space orthogonal to [(e)\vec][^0]\vec{e}_{\hat{0}} , expressed by the metric tensor gmng_{\mu\nu} of spacetime.

We must have a method of differentiation that maintains the anti-symmetry, thus making sure that what we end up with after differentiation is still a form.

Gravitational acceleration: g=-nablaφ=-dφ/dre_r Finite g in r=0 demands K_1=0.. Assume that the mass distribution has a radius R. Demand continuous potential at r=R.

Surface gravity is denoted by kappa_1 and is defined by where r_+ is the horizon radius, r_+=R_S for the Schwarzschild spacetime, u^t is the time component of the 4-velocity. The 4-velocity of a free particle instantaneously at rest in the Schwarzschild spacetime: The only component of the 4-acceleration different from zero is a_r

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.

The covariant directional derivative of a vector field \(\vec{A}\) along a vector \(\vec{u}\) was defined and interpreted geometrically in Sect. 5.2, as follows.

In the present article we analyze, by means of the statefinder parameter formalism, some universe models introduced by Brevik and co-workers. We determine constants that earlier were left unspecified, in terms of observable quantities. It is verified that a Big Bang universe model with a fluid having a certain non-linear equation of state behaves i...

An observer at rest in a uniformly accelerated reference frame experiences a parallel gravitational field, where the rays of light are formed as circular arcs instead of being straight lines. We deduce how a sphere at rest in the Rindler frame, and a freely falling sphere, will appear to an observer at rest in the frame. Comment: 14 pages and 6 fig...

We find a class of solutions of Einstein's field equations representing spacetime outside a spinning cosmic string surrounded by a gas of non-spinning cosmic strings, and show that there exist closed timelike geodesics in this spacetime.

The significance of the Schott four-momentum in the energy-momentum conservation account of a charged particle and its electromagnetic field is analyzed. Periods with preacceleration and run away motion are discussed. The existence of pre-radiation is demonstrated. The Schott energy is identified as field energy localized just outside the particle....

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...

A charge moving freely in orbit around the Earth radiates according to Larmor's formula. If the path is closed, it would constitute a perpetuum mobile. The solution to this energy paradox is found in an article by C. M. DeWitt and B. DeWitt from 1964. The main point is that the equation of motion of a radiating charge is modified in curved spacetim...

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...

Machs principle and the principle of relativity have been discussed by H. I. Hartman and C. Nissim-Sabat in this journal. Several phenomena were said to violate the principle of relativity as applied to rotating motion. These claims have recently been contested. However, in neither of these articles have the general relativistic phenomenon of inert...

Heras (2006 Am. J. Phys. 74 1025–30) has recently claimed that preradiation does not exist according to classical electrodynamics. In the present paper, we show that the theory does indeed predict the existence of preradiation, and that the Schott energy plays an essential role in this connection.

We give a systematic development of the theory of the radiation field of an accelerated charged particle with reference to an inertial reference frame in flat spacetime. Special emphasis is given to the role of the Schott energy and momentum in the energy–momentum balance of the charge and its field. It is shown that the energy of the radiation fie...

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...