The concept to the metric is introduced. Various geometries, both flat and curved, are described including Euclidean space; Minkowski space-time; spheres; hyperbolic planes and expanding space-times. Lorentz transformations and relativistic time dilation in flat space-time is discussed as well as gravitational red-shift and the Global Positioning System. Hubble expansion and the cosmological red-shift are also explained.

One of the problems with the concept of spacetime is that it is hard for us to actually appreciate the implications of living in a curved spacetime, and the origin of this difficulty is that our local spacetime is essentially flat! Hence, all of our understanding of physics – of 'how things work' – has been built on the basis of perceptions that take place in almost flat spacetime. This chapter will provide a pragmatic approach to the measurement of spacetime by illustrating how it is actually not too difficult to obtain an estimate of local curvature by using simple physical quantities, such as the mass and the size of the object. In this manner, we will be able to appreciate that the curvature on Earth is only a few parts in a billion, hence explaining why we perceive everything in the actual absence of curvature. we will learn how to actually bend spacetime reaching the extreme values that are encountered near a neutron star and a black hole, both of which will be discussed more in detail in the following chapters.

A black hole can be rightfully thought as the most extreme manifestation of gravity – and thus of curvature! Besides being a unique source of puzzles and paradoxes for scientists, they have also been the inspiration for endless and breathtaking adventures in science-fiction novels and movies. This chapter will, therefore, explain the concept of black hole by making use of two different mechanical equivalents that have many points in common with black holes. In this way, it will become clear what is an event horizon and why it represents a one-way membrane, which can be entered, but from within which nothing can exit, not even light. Similarly, we will introduce the concept of spacetime singularity and explain why this is a problem that worries us physicists most, and for which we have not found any satisfactory solution yet. We will see that black holes are beautiful manifestations of nature and are not more monstrous than an erupting volcano.

Introducing the Minkowski diagram and Minkowski space; how do we represent motion? And how can we represent the phenomena of length contraction and time dilation graphically?

We look at the immediate consequences of the two axioms, and discover, qualitatively and then quantitatively, the phenomena of length contraction and time dilation.

Chapter 1 contains the problem statements of the 150 problems in special relativity theory. The chapter is divided into nine sections with problems organized by different topics defined by the keywords in the section headings.

Einstein’s 1905 special theory of relativity requires a profound revision of the Newtonian ideas of space and time that were reviewed in the previous chapter. In special relativity, the Newtonian ideas of Euclidean space and a separate absolute time are subsumed into a single four-dimensional union of space and time, called spacetime. This chapter reviews the basic principles of special relativity, starting from the non-Euclidean geometry of its spacetime. Einstein’s 1905 successful modification of Newtonian mechanics, which we call special relativity, assumed that the velocity of light had the same value, c, in all inertial frames, which requires a reexamination, and ultimately the abandonment, of the Newtonian idea of absolute time. Instead, he found a new connection between inertial frames that is consistent with the same value of the velocity of light in all of them. The defining assumption of special relativity is a geometry for four-dimensional spacetime.

This chapter covers the Special Theory of Relativity, introduced by Einstein in a pair of papers in 1905, the same year in which he postulated the quantization of radiation energy and showed how to use observations of diffusion to measure constants of microscopic physics. Special relativity revolutionized our ideas of space, time, and mass, and it gave the physicists of the twentieth century a paradigm for the incorporation of conditions of invariance into the fundamental principles of physics.

After many years of effort, Einstein discovered the general theory of relativity, the extension of special relativity to include the force of gravity. To do this, he extended the four-vector structure of special relativity to more general Riemannian geometries in which space-time is bent under the influence of gravity. The theory has successfully passed all the high precision tests now available. Predictions include the existence of black holes and gravitational waves. Both have now been convincingly observed. In this chapter, the basic concepts behind the theory are described and then illustrated by analysis of the Schwarzschild metric.

The farther we look, the more slowly the light from exploding stars (supernovae) fades. This tells us that the expansion of space affects even light, which arrives at Earth more spread out.

The farther we look, the more slowly the light from exploding stars (supernovae) fades. This tells us that the expansion of space affects even light, which arrives at Earth more spread out.