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

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 formalism is most effective when we introduce an orthonormal basis field. Using an orthonormal basis co-moving with an observer, i.e. where the time-like vector is equal to the 4-velocity of the observer, simplifies the physical interpretation of the calculations.

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

The linear field approximation of Einstein’s field equations is presented. The solutions of these equations inside and outside a rotating spherical shell are deduced. It is shown that inertial dragging is a consequence of the general theory of relativity. The equations of gravitomagnetism are deduced. Gravitational waves are described and related to their detection by the LIGO-detectors.

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 appears as a 2 order effect in v/c of electricity due to the Lorentz transformation.

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 in a rotating frame because this leads to a contradiction in a non-rotating frame. Gravitational time dilation and frequency shift, and also the Sagnac experiment are discussed. Finally we give an introduction to special relativistic kinematic in a uniformly accelerated reference frame in flat spacetime. It is pointed out that an observer experiences an acceleration of gravity in such a frame.

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 theory of relativity, and apply this first to a shell consisting of dust particles, and then to find a source of the conformally flat, spherically symmetric Levi-Civita—Bertotti—Robinson metric, and finally to the Kerr spacetime. Lastly we shall introduce a river model of space.

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 series in a pair is an increasingly stronger leading series, measured as LL-strength, to the other, CT ≈ - (minus) LL-strength. This may explain the frequent finding that many global warming time series, e.g., the Southern oscillation index and the Pacific decadal oscillation, show distinct cycle times (Power spectral analysis: 3-5, 7-8, 13-15, 22-24, and 29-30 years). An important implication of these findings is that processes that strengthen the impact of one ocean variable on another may cause more frequent adverse climate conditions.

Gravitational repulsion is defined to mean that a free particle instantaneously at rest is accelerated outwards from the mass center. It is shown that there is no gravitational repulsion in the Schwarzschild spacetime, and hence that there is no acceleration to high energy via gravitational repulsion in the Schwarzschild field.

We show that oceanic cycle lengths persist across oceanic cyclic time-series by comparing cycles in series that come from &ldquo, sister&rdquo, 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 is an extremely noisy series in its raw format, showed cycles at 7, 13, 20, 26 and 34 years that were common with, or overlapped, the other two series, and across increasing degrees of smoothing of the NAO series. At the 1960 midpoint of the hiatus period 1943&ndash, 1975, NAO was leading time-series to AMOC and AMO and AMO was a leading time-series to AMOC, but in 1975, at the end of the hiatus period, the leading relations were reversed.

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 (NAO), El Ni&ntilde, o, the Southern Oscillation Index (SOI), and the Pacific Decadal Oscillation (PDO). In the model, we let a stochastic movement in one oceanic oscillation cause a similar but lagging movement in another oceanic oscillation. The two interacting oscillations show distinct cycle lengths depending upon how strongly one oscillation creates lagging cycles in the other. The model and observations both show cycles around two to six, 13 to 16, 22 to 23, and 31 to 32 years. The ultimate cause for the distinct cycles is atmospheric and oceanic &ldquo, bridges&rdquo, that connect the ocean basins, but the distinct pattern in cycle lengths is determined by properties of statistical distributions. We found no differences in the leading or lagging strength between well separated basins (the North Atlantic and the Pacific) and overlapping ocean basins (both in the Pacific). The cyclic pattern before 1950 appears to be different from the cyclic pattern after 1950.

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 lagging (LL) relations between paired global warming variables have required that the time series show constant frequency (stationarity). Herein, we show that on orbital and multi-millennial time scales, the Vostok Antarctic ice core displays 9 periods of 8.7 kyr ± 5 kyr during which CO2 becomes a leading variable to temperature. Six of the 9 periods were associated with short-term pauses occurring during 4 major glaciation-deglaciation periods. We find that CO2 also leads temperature during short pauses in the major cyclic pattern of the Greenland time series. In the latter series, there are also two contrasting cycle developments. In the first contrasting cycle developments, lasting from 103.5 to 79 ka, there is an in-phase relation between CO2 and temperature, with a slope of 0.75. In the second contrasting cycle developments, lasting from 61.5 to 43.5 ka, there is an out-of-phase relation with a slope of −0.67. In addition, the latter shows a see-saw pattern between Arctic and Antarctic temperatures.

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 effect. Then, we calculate the mass inside the particle horizon of the flat $\Lambda$CDM-model integrated along the past light cone. The calculation shows that the Schwarzschild radius of this mass is around three times the radius of the particle horizon. This indicates that there is close to perfect inertial dragging in our universe. Hence, the calculation provides an explanation for the observation that the swinging plane of a Foucault pendulum follows the stars., Comment: 17 pages, 3 figures

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) universe models. We find the general, exact solution of the Nash field equations for empty space in the static case. The line element turns out to represent the Schwarzschild-de Sitter spacetime. Also we find the simplest non-trivial solution of the field equations in the cosmological case, which gives the scale factor corresponding to the de Sitter spacetime. Hence empty space in the Nash theory corresponds to a space with Lorentz Invariant Vacuum Energy (LIVE) in the Einstein theory. This suggests that dark energy may be superfluous according to the Nash theory. We also consider a radiation filled universe model in an effort to find out how energy and matter may be incorporated into the Nash theory. A tentative interpretation of the Nash theory as a unified theory of gravity and electromagnetism leads to a very simple form of the field equations in the presence of matter. It should be noted, however, that the Nash theory is still unfinished. A satifying way of including energy-momentum into the theory has yet to be found., Comment: 7 pages

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 finite maximum value. The effect of viscosity upon the entropy is also studied and we have found that its role is to decrease the entropy. The bigger the viscosity coefficient is the less the entropy will be. Furthermore, the radiation entropy for the ΛCDM Universe model with and without viscosity is investigated, and together with the cosmological event horizon entropy are used to examine the validity of the generalized second law of thermodynamics, which states that the total rate of change of entropy of the Universe is never negative, in this Universe model.

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 p R ,where R is the radius of the wheel, but 2 p R = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R 2 X 2 = c 2 q ,where X is the angular velocity of the wheels. In one solution, the wheel radius is constant as the velocity of the trolley increases, and in the other the wheels contract in the radial direction. We also explain two surprising facts. First that the shape of a rolling wheel is elliptical in spite of the fact that the upper part of the wheel moves faster than the lower part, and thus is more Lorentz contracted, and second that a Lorentz contracted wheel with relativistic velocity rolls out a larger distance between two successive touches of a point of the wheel on the rails than the length of a circle with the same radius as the wheels.

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 $ \Omega_{\Lambda 0}=0.737$ . The physical mechanism connecting the relativity of rotational motion with the energy density of dark energy is the inertial dragging effect. The predicted value is necessary in order to have perfect inertial dragging, which is required for rotational motion to be relative. If one accepts that due to the impossibility of defining motion for a single particle in an otherwise empty universe, the universe must be constructed so that all types of motion are relative, then this solves the so-called cosmological constant problem.

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 study observational viability of Natural Inflation with a tachyon field as inflaton. By obtaining the main perturbation parameters in this model, we perform a numerical analysis on the parameter space of the model and in confrontation with $68\%$ and $95\%$ CL regions of Planck2015 data. By adopting a warped background geometry, we find some new constraints on the width of the potential in terms of its height and the warp factor. We show that the Tachyon Natural Inflation in the large width limit recovers the tachyon model with a $\phi^{2}$ potential which is consistent with Planck2015 observational data. Then we focus on the reheating era after inflation by treating the number of e-folds, temperature and the effective equation of state parameter in this era. Since it is likely that the value of the effective equation of state parameter during the reheating era to be in the range $0\leq \omega_{eff}\leq \frac{1}{3}$, we obtain some new constraints on the tensor to scalar ratio as well as the e-folds number and reheating temperature in this Tachyon Natural Inflation model., Comment: 19 pages, 6 figures, Accepted for publication in JCAP

The main points in the history of the cosmological constant are briefly discussed. As a conceptual background, useful for teaching of physics at an elementary college and university level, Newton's theory formulated locally in terms of the Poisson equation is presented, and it is shown how it is modified by the introduction of the cosmological constant. The different physical interpretations of the cosmological constant, as introduced by Einstein in 1917 and interpreted by Lemaitre in 1934, are presented. Energy conservation in an expanding universe dominated by vacuum energy is discussed. The connection between the cosmological constant and the quantum mechanical vacuum energy is mentioned, together with the problem that a quantum mechanical calculation of the density of the vacuum energy gives a vastly too large value of the cosmological constant. The article is concluded by reviewing a solution of this problem that was presented on May 11, 2017.

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 meant as a detailed model, which is expected to be rather complicated, but rather as a 'pedagogic model' pointing out some important elements of a more complete model. Furthermore, the present model is sufficiently simple that it may be suitable as an 'astrophysical example' in the teaching of the special theory of relativity. In this model a particle is accelerated by ultrarelativistic shocks in a source of gamma ray bursts. No assumption as to the details of the accelerating mechanism is made except that the force acting on a charged particle depends only upon the charge of the particle and not upon its mass, and the product of the force and the thickness of the shock waves must be sufficiently great. It is important for the success of the model that the energy radiated by the particles is taken mainly from the Schott energy and not from the kinetic energy of the particles. It is shown how this model of the accelerating process can explain why electrons are accelerated to less energy than protons and heavier nuclei. The mechanism also explains how particles may be accelerated to energies greater than 1020 eV.

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 unstable. Interaction between the fluids may stabilize the universe. The condition for stability by perturbation of the scale factor from its static value is deduced for a class of universe models containing those investigated by Carneiro and Tavakol. Stability of the static state requires that energy is transformed to matter during such a perturbation.

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 inertial dragging been invoked, and no calculation have been offered by either side to substantiate their claims. Here I discuss the possible validity of the principle of relativity for rotating motion within the context of the general theory of relativity, and point out the significance of inertial dragging in this connection. Although my main points are of a qualitative nature, I also provide the necessary calculations to demonstrate how these points come out as consequences of the general theory of relativity, 17 pages

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 field does not, in general, come from the work performed upon the charge by an external force. The radiation energy also receives a contribution from a conversion of the velocity dependent nearfield to the acceleration dependent radiation field. We also exhibit the role of momentum conservation in connection with a radiating electric charge and its electromagnetic field.

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 the 6 to 8 years cyclic components of the variables are that during the period 1929 to 1936, CO2 significantly leads GTA. However, during the period 1960–2003, GTA apparently leads CO2, that is, the peaks (and troughs) in GTA are in front of, and close to, the peaks (and troughs) in CO2. For time windows outside these periods, we did not find significant before or after-relations. An alternative interpretation is that there is a shift between short (≈1.5 year) and long (≈5 years) durations between cause and effect. Relationships between GTA and TSI suggest that “inertia” of the global sea, land, and atmosphere system leads to delays longer than half their common cycle length of about 10 years. Based on the interaction patterns between the variables GTA, CO2, and TSI, we suggest the possibility that a new regime for how the variables interact started around 1960. From trend forms, and not considering physical mechanisms, we found that the trend in CO2 contributes ≈ 90 %, and the trend in TSI ≈ 10 %, to the trend in GTA during the last 130 years.

It is pointed out that a complete resolution of the twin paradox demands that the travelling twin takes into account the gravitational effect upon the rate of time when he predicts the ageing of his brother. Two ways of making this prediction are presented. The first one is formally very simple. However, it involves an assumption that is not obvious when the travelling twin stipulates the distance of his brother. The second method makes use of Lagrangian dynamics in a uniformly accelerated reference frame. Then no such assumption is necessary.

How can a universe coming from a point-like big bang event have infinite spatial extension? It is shown that the relativity of simultaneity is essential in answering this question. Space is finite as defined by the simultaneity of one observer, but it may be infinite as defined by the simultaneity of all the clocks participating in the Hubble flow.

In the present article I give a five-dimensional interpretation of the Coulomb field in the context of Kaluza-Klein theory. It is shown that the projection into our four-dimensional world of the inertial dragging field due to a neutral particle moving around a closed fifth dimension, is interpreted as the electrical field of a charged particle from our four-dimensional point of view.

We describe the electromagnetic field of a uniformly accelerated charge in its co-moving Rindler frame. It is shown that the electrical field lines coincide with the trajectories of photons. The self force of a charged particle at rest in Rindler space, and the increase of its weight due to its charge, is calculated. The general case of an accelerated charge in Rindler space is also considered. It is shown that the electrical field inside a uniformly charged spherical shell can be used as a measure of it 4-acceleration. A result that has earlier been deduced in a different way by Fugmann and Kretzschmar is confirmed, namely that the intensity of radiation from a point charge instantaneously at rest in an accelerated frame is proportional to the square of the relative acceleration of the charge and the observer. In particular it is shown that a freely falling charge in Rindler space radiates in accordance with Larmor’s formula. In this case the radiation energy is taken from the Schott energy. The energy of the electromagnetic field is analysed from the point of view of the Hirayama-separation, which generalizes the Teitelboim-separation to non-inertial frames, of the field in a bound part and an unbound part. A detailed account, with reference to the Rindler frame, of the field energy and particle energy is given for the case of a charge entering and leaving a region with hyperbolic motion. We also consider the electromagnetic field of a uniformly accelerated charge with reference to the Milne frame, which covers a different part of spacetime than the Rindler frame. The radiating part of the electromagnetic field is found in the Milne sector of spacetime.

We give a detailed description of the energy–momentum relationships of the Bondi–Gold field. The Schott field is separated into a near field and a radiation field. The physical significance of a plane acting as a barrier for transport of energy in the field of a charge with permanent hyperbolic motion, but which lets energy through in the case of hyperbolic motion for a finite time interval, is discussed. We deduce general expressions for energy and momentum of the near field and radiation field of charges with arbitrary motion. The field reaction on the particle producing the field is calculated. The formulae are applied to the Schott field and the Bondi–Gold field.

Analytical properties of a flat universe with cold matter and vacuum energy is presented., 10 pages, 11 figures