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.

Two scientists more than anyone else have contributed in defining our understanding of gravity: Newton in 1679 and Einstein in 1915. The mathematical frameworks the two have developed and proposed, however, are very different. Newton’s gravity is the one we learn at school and is normally taught at university. It provides a very natural interpretation of what we experience - the apple falls from the tree because the Earth attracts it! Einstein’s gravity is studied only in the most advanced courses at the university and provides a very counterintuitive explanation, requiring the concepts of spacetime and curvature. This chapter will provide a first description of the Einstein equations and, although it will not enter into the mathematical aspects of the equations, it will explain the basic concepts behind them. Acquiring a first qualitative understanding of Einstein equations will be useful to comprehend better the concept of spacetime curvature discussed in Chapter 4.

Gravity attracts – this is such an obvious phenomenon that writing this book was not necessary to stress it. Less obvious is that, even before it appears in the form of physical interaction, gravity attracts our attention and our imagination. As soon as we are born, before developing a conscious relationship with the physical universe, we already know gravity at an instinctive level. For the rest of our lives, it will represent the only one of the four fundamental interactions of which we will have conscious awareness. And from which we will often try to escape.

This chapter explains how the researchers of the Event Horizon Telescope Collaboration were able to obtain the first picture of a black hole through radio-astronomical observations. In particular, we first describe the technological strategies that have been exploited in order to obtain a record-high angular resolution. We will also discuss the theoretical aspects that have allowed the collaboration to model the dynamics of the plasma falling onto the black and to produce a large database of synthetic images potentially describing an accreting supermassive black hole. The chapter reviews how the comparison between the theoretical images and the observations has allowed us to deduce the presence of a supermassive black hole with a mass of 6 billion solar masses in the very heart of the giant galaxy M87. The chapter will also summarise the lessons that have been learnt from this epochal achievement and the questions that are still left unanswered about black holes and gravity in the strongest regimes.

Gravitational waves are simply the solutions of the linearized Einstein equations, that is, the solution of the Einstein equations in weak gravitational fields or, equivalently, in spacetimes that are 'almost flat'. In this respect, they are not very different from the sound waves we hear when we listen to music or from the electromagnetic waves we receive when looking at the screen of our cellphone. However, it is not pressure or electromagnetic fields that are propagated, but rather the very same curvature we have encountered already. Hence, gravitational waves represent the propagation at the speed of light of small ripples in the curvature in spacetime. This chapter will make use of simple mechanical analogues to explain how gravitational waves are produced every time a mass or energy is set in motion, and how the amplitude of gravitational waves produced on Earth can only be extremely small.

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.

Gravity … attracts! This was obvious to you before you started reading this book and is even more obvious now that you have reached the end of it. At the same time, however, I hope you now agree with me that gravity is also attractive, which is far less obvious.

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.

Neutron stars are truly marvelous objects. They represent the end result of the evolution of very massive stars and are the “left-overs” of the enormous explosion that accompanies the death of these stars – namely, a supernova explosion. In a radius of a dozen of kilometers only, these stars can accumulate as much mass as twice that of the Sun, reach temperatures of tens of millions of degrees and magnetic fields that millions of billions larger than those on Earth. More importantly, by being so compact, these stars produce enormous gravitational fields, the largest gravitational fields for an object with a hard surface. This chapter will explain how neutron stars have been discovered and how we have learnt about their incredible properties. It will also stress that, although we now know quite a lot about neutron stars, they still represent a significant mystery in physics, since we have only a rather vague idea of what is inside neutron stars and how they can be built in nature.