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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.

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.

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.

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

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.

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.

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.

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.

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.

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.