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.

Gravity has an irresistible grip on our curiosity and is able to drive our imagination to completely different theoretical spaces. This very fact alone sets gravity aside from all other types of physical interactions we know. Indeed, gravity is the only physical interaction of which we have a conscious experience and this awareness is with us every second of our life. In this book we set out to try to address the question: '…what is gravity and how does gravity actually work?'. This book is meant as a guide in a journey that will take us from our basic understanding of gravity, the one that is deeply coded in our brains even at an instinctive level, to the more physically detailed and yet incorrect description provided by Newton’s theory of gravity. The journey will then lead us to the mathematically beautiful and physically profound description that Einstein has proposed with his 'general theory of relativity', and that is elegantly embodied in his field equations.

The end result of Einstein’s revolutionary vision is that gravity is simply the manifestation of the curvature of spacetime. This is a concept that has a deep significance and is at the heart of the Einstein field equations. This Chapter will explain why we need to introduce the idea of 'spacetime' and how we can define the concept of spacetime curvature in this description. Starting from the example of a spacetime empty of matter – that is, a flat spacetime – we will move to the example of a spacetime containing matter and energy – that is, a curved spacetime. This chapter will explain why we find the description of gravity proposed by Newton very reasonable and why we have trouble appreciating the new vision proposed by Einstein. We will contrast the two descriptions with a simple example and show how the very same physical phenomenon – the orbit of the Earth around the Sun – can be seen with very different explanations by Newton and Einstein.

In this chapter we give an outline of the Cauchy problem in general relativity and show that, given certain data on a space like three-surface there is a unique maximal future Cauchy development D+() and that the metric on a subset of D+() depends only on the initial data on J–() ∩. We also show that this dependence is continuous if has a compact closure in D+(). This discussion is included here because of its intrinsic interest, its use of some of the results of the previous chapter, and its demonstration that the Einstein field equations do indeed satisfy postulate (a) of §3.2 that signals can only be sent between points that can be joined by a non-spacelike curve.

In §7.1, we discuss the various difficulties and give a precise formulation of the problem. In §7.2 we introduce a global background metric ĝ to generalize the relation which holds between the Ricci tensor and the metric in each coordinate patch to a single relation which holds over the whole manifold. We impose four gauge conditions on the covariant derivatives of the physical metric g with respect to the background metric ĝ.

The relation between local spacetime curvature and matter energy density is given by the Einstein equation – it is the field equation of general relativity in the way that Maxwell’s equations are the field equations of electromagnetism. Maxwell’s equations relate the electromagnetic field to its sources – charges and currents. Einstein’s equation relates spacetime curvature to its source – the mass-energy of matter. This chapter gives a very brief introduction to the Einstein equation; we consider the equation in the absence of matter sources (the vacuum Einstein equation) and will include matter sources in Chapter 22. Even the vacuum Einstein equation has important implications. Just as the field of a static point charge and electromagnetic waves are solutions of the source-free Maxwell’s equations, the Schwarzschild geometry and gravitational waves are solutions of the vacuum Einstein equation.