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Einstein's General Theory of Relativity: With Modern Applications in Cosmology

Øyvind Grøn; Sigbjørn Hervik;

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No detectada 2007 SpringerLink

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Relativity Principles and Gravitation

Øyvind Grøn; Sigbjørn Hervik

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.

1 - Introduction: Newtonian Physics And Special Relativity | Pp. 3-19

The Special Theory of Relativity

Øyvind Grøn; Sigbjørn Hervik

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.

1 - Introduction: Newtonian Physics And Special Relativity | Pp. 21-47

Vectors, Tensors, and Forms

Øyvind Grøn; Sigbjørn Hervik

We shall present the theory of differential forms in a way so that the structure of the theory appears as clearly as possibly. In later chapters this formalism will be used to give a mathematical formulation of the fundamental principles of the general theory of relativity. It will also be employed to give an invariant formulation of Maxwell’s equations so that the equations can be applied with reference to an arbitrary basis in curved spacetime.

2 - The Mathematics Of The General Theory Of Relativity | Pp. 51-62

Basis Vector Fields and the Metric Tensor

Øyvind Grøn; Sigbjørn Hervik

In this chapter we are going to introduce the basic concepts necessary to grasp the geometrical significance of the metric tensor.

2 - The Mathematics Of The General Theory Of Relativity | Pp. 63-88

Non-inertial Reference Frames

Øyvind Grøn; Sigbjørn Hervik

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.

2 - The Mathematics Of The General Theory Of Relativity | Pp. 89-108

Differentiation, Connections, and Integration

Øyvind Grøn; Sigbjørn Hervik

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 explore differentiation on curved manifolds and reveal several interesting properties.

2 - The Mathematics Of The General Theory Of Relativity | Pp. 109-149

Curvature

Øyvind Grøn; Sigbjørn Hervik

We have seen, for example in rotating reference frames, that the geometry in a space with non-vanishing acceleration of gravity, may be non-Euclidean. It is easy to visualize curves and surfaces in three-dimensional space but it is difficult to grasp visually what curvature means in three-dimensional space, or worse still, in four-dimensional space-time. However the curvature of such spacesmay be discussed using the lower dimensional analogues of curves and surfaces. It is therefore important to have a good knowledge of the differential geometry of surfaces. Also the formalism used in describing surfaces may be taken over with minor modifications, when we are going to describe the geometric properties of curved space-time.

2 - The Mathematics Of The General Theory Of Relativity | Pp. 151-176

Einstein's Field Equations

Øyvind Grøn; Sigbjørn Hervik

Einstein’s field equations are the relativistic generalization of Newton’s law of gravitation. Einstein’s vision, based on the equality of inertial and gravitational masses, was that there is no gravitational force at all. What is said to be “particle motion under the influence of the gravitational force” in Newtonian theory, is according to the general theory of relativity, free motion along geodesic curves in a curved space-time.

3 - Einstein's Field Equations | Pp. 179-194

The Linear Field Approximation

Øyvind Grøn; Sigbjørn Hervik

Einstein’s theory of general relativity leads to Newtonian gravity in the limit when the gravitational field is weak and static and the particles in the gravitational field moves slowly compared to the velocity of light. In the case of mass distributions of limited extension the field is weak at distances much larger than the Schwarzschild radius of the mass (see Chapter 10). At such distances the absolute value of the gravitational potential is much less than 1, and there is approximately Minkowski spacetime.

3 - Einstein's Field Equations | Pp. 195-214

The Schwarzschild Solution and Black Holes

Øyvind Grøn; Sigbjørn Hervik

We have now established the Einstein field equations and explained their contents. In this chapter we will explore the first known non-trivial solution to these equations. The solution is due to the astronomer Karl Schwarzschild, and in his honour the solution is referred to as the . This solution represents a spacetime outside a non-rotating black hole. The Kerr solution representing spacetime outside a rotating black hole will also be deduced. Finally, interior solutions will be investigated.

3 - Einstein's Field Equations | Pp. 215-263

Homogeneous and Isotropic Universe Models

Øyvind Grøn; Sigbjørn Hervik

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 physical theory. But after the year 1917, things were different. Within two years after the birth of the General Theory of Relativity, Einstein realized that this theory actually could say something about the universe and constructed a static universemodel as a solution of the relativistic field equations. The era of modern cosmology had begun, which would revolutionize our view of the universe.

4 - Cosmology | Pp. 267-303

Universe Models with Vacuum Energy

Øyvind Grøn; Sigbjørn Hervik

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 gave cosmology a whole new view upon the first split second of our universe. A key ingredient in the inflationary model is the behaviour of models that have a cosmological constant-like behaviour. Secondly, recent observations may indicate that we live in an accelerated universe. The inclusion of a cosmological constant can give rise to such behaviour as we will show in this chapter. We will first start with the static solution that Einstein found and was the reason that Einstein introduced the cosmological constant in the first place.

4 - Cosmology | Pp. 305-365

Anisotropic and Inhomogeneous Universe Models

Øyvind Grøn; Sigbjørn Hervik

In this chapter we will investigate anisotropic and inhomogeneous universe models. If we relax the cosmological principles a bit we can get new and interesting models of our universe. Actually, one of the main goals of cosmology today is to the isotropy and homogeneity the universe has and in order to explain a certain property of the universe one has to consider sufficiently generalmodels that need not have this property. In this chapter we will investigate the Bianchi type I universe model and the inhomogeneous Lemaître-Tolman-Bondi (LTB) universe models. The Bianchi type I model is the simplest of the spatially homogeneous models which allows for anisotropy and the LTB-models are inhomogeneous universe models with spherically symmetric three-space.

4 - Cosmology | Pp. 367-385

Covariant Decomposition, Singularities, and Canonical Cosmology

Øyvind Grøn; Sigbjørn Hervik

In this chapter we will perform a 3+1 decomposition of the spacetime. This decomposition is very useful for various applications, in particular, we will use the 3+1 decomposition to derive a Lagrangian and Hamiltonian formalismof general relativity. We will also see howthe singularity theoremcan be described in this framework.

5 - Advanced Topics | Pp. 389-412

Spatially Homogeneous Universe Models

Øyvind Grøn; Sigbjørn Hervik

In this section we will explore the concept of symmetries even further. We introduced some of the basics in chapter 6, and we will pursue the ideas further here. In doing so, we will generalise the FRW models to the Bianchi models which are in general spatially homogeneous but not necessarily isotropic.

5 - Advanced Topics | Pp. 413-437

Israel's Formalism: The Metric Junction Method

Øyvind Grøn; Sigbjørn Hervik

A question that often arises in gravitational theory is what happens to the geometry of spacewhen there is a jump discontinuity in the energy-momentum tensor along a surface. For example, what is the connection between the curvature properties for the interior Schwarzschild solution and the exterior Schwarzschild solution? Here, along the boundary of some surface, the energy density experiences a jump discontinuity. Another analogous scenario is for example a shock wave propagating outwards from an exploding star. In models of such shock waves the density can be infinite.

5 - Advanced Topics | Pp. 439-451

Brane-worlds

Øyvind Grøn; Sigbjørn Hervik

In 1999, Lisa Randall and Raman Sundrumpresented a five-dimensional model for our universe [RS99b, RS99a]. They imagined our four-dimensional world as a or a surface layer in a five-dimensional bulk. This bulk may be infinite in size, but due to the special properties of the bulk the gravitational fields are effectively localised to the brane. The other standard model fields are confined to the brane; only gravity is allowed to propagate in the fifth dimension.

5 - Advanced Topics | Pp. 453-477

Kaluza-Klein Theory

Øyvind Grøn; Sigbjørn Hervik

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, this work was neglected for several decades.

5 - Advanced Topics | Pp. 479-499

Información

Tipo: libros

ISBN impreso

978-0-387-69199-2

ISBN electrónico

978-0-387-69200-5

Editor responsable

Springer Nature

País de edición

Reino Unido

Fecha de publicación