1. Loss of Unity
1. This is the essence of the theory of general relativity (A. Einstein, ‘Die Grundlage der algemeinen Relativitätstheorie’, Annalen der Physik, 49, 1916, pp. 769–822).
2. In the approximation of a weak field, the metrics can be written 
where 
is the potential of Newton. Newtonian gravity follows from the sole modification of the temporal component of the metrics goo, that is, from the local slowing down of time. The geodesics of these metrics describe the fall of bodies: they bend towards the lowest potentiality, where time slows. (These and similar notes are for those who have some familiarity with theoretical physics.)
3. Carlo Rovelli, Che cos’è la scienza. La rivoluzione di Anassimandro, Mondadori, Milan, 2011. English translation: The First Scientist: Anaximander and His Legacy, Westholme, Yardley, 2011.
4. For example: 
where c is the speed of light, 
is the acceleration of Galileo and h is the height of the table.
5. They can also be written with a single variable, t, the ‘temporal coordinate’, but this does not indicate the time measured by a clock (determined by ds, not by dt) and may be changed arbitrarily without changing the world described. This t does not represent a physical quantity. What clocks measure is the proper time along a line of the universe γ, given by 
. The physical relation between this quantity and ds is discussed further on.
2. Loss of Direction
1. Rainer Maria Rilke, Duineser Elegien, in Sämtliche Werke, Insel, Frankfurt, vol. I, 1955, I, vv. 83–5.
2. The French Revolution was an extraordinary moment of scientific vitality in which the bases of chemistry, biology, analytic mechanics and much else were founded. The social revolution went hand in hand with the scientific one. The first revolutionary mayor of Paris was an astronomer; Lazare Carnot was a mathematician; Marat considered himself to be, above all else, a physicist. Lavoisier was active in politics. Lagrange was honoured by the different governments that succeeded each other in that tormented and magnificent moment in the history of humanity. See S. Jones, Revolutionary Science: Transformation and Turmoil in the Age of the Guillotine, Pegasus, New York, 2017.
3. Changing what is opportune: for instance, the sign of the magnetic field in the equations of Maxwell, charge and parity of elementary particles, etc. It is the invariance under CPT (Charge, Parity and Time reversal symmetry) that is relevant.
4. The equations of Newton determine how things accelerate, and the acceleration does not change if I project a film backwards. The acceleration of a stone thrown upwards is the same as that of a falling stone. If I imagine years running backwards, the moon turns around the Earth in the opposite direction but appears equally attracted to the Earth.
5. The conclusion does not change by adding quantum gravity. On the efforts to find the origin of the direction of time, see, for example, H. D. Zeh, Die Physik der Zeitrichtung, Springer, Berlin, 1984.
6. R. Clausius, ‘Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie’, Annalen der Physik, 125, 1865, pp. 353–400; p. 390.
7. In particular as a quantity of heat that escapes from a body divided by temperature. When the heat escapes from a hot body and enters a cold one, the total entropy increases because the difference in temperature makes it so that the entropy due to heat that escapes is less than that owed to the heat that enters. When all the bodies reach the same temperature, the entropy has reached its maximum: equilibrium has been reached.
8. Arnold Sommerfeld.
9. Wilhelm Ostwald.
10. The definition of entropy requires a coarse graining, that is to say, the distinction between microstates and macrostates. The entropy of a macrostate is determined by the number of corresponding microstates. In classic thermodynamics, the coarse graining is defined the moment it is decided to treat some variables of the system as ‘manipulable’ or ‘measurable’ from outside (the volume or pressure of a gas, for instance). A macrostate is determined by fixing these macroscopic variables.
11. That is to say, in a deterministic manner if you overlook quantum mechanics, and in a probabilistic manner if you take account of quantum mechanics instead. In both cases, in the same way for the future as for the past.
12. S = k lnW. Here, S is the entropy, W is the number of microscopic states, or the corresponding volume of phase space, and k is just a constant, today called Boltzmann’s constant, that adjusts the (arbitrary) dimensions.
3. The End of the Present
1. General relativity (A. Einstein, ‘Die Grundlage der allgemeinen Relativitätstheorie’, op. cit.).
2. Special relativity (A. Einstein, ‘Zur Elektrodynamik bewegter Körper’, Annalen der Physik, 17, 1905, pp. 891–921).
3. J. C. Hafele and R. E. Keating, ‘Around-the-World Atomic Clocks: Observed Relativistic Time Gains’, Science, 177, 1972, pp. 166–8.
4. That depends as much on t as on your speed and position.
5. Poincaré. Lorentz had tried to give a physical interpretation to t, but in a quite convoluted way.
6. Einstein frequently maintained that the experiments of Michelson and Morley were of no importance in allowing him to arrive at special relativity. I believe this to be true, and that it illustrates an important factor in the philosophy of science. In order to make advances in our understanding of the world, it is not always necessary to have new data. Copernicus had no more observational data than Ptolemy: he was able to deduce heliocentrism from the data available to Ptolemy by interpreting it better – as Einstein did with regard to Maxwell.
7. If I see my sister through a telescope celebrating her twentieth birthday and send her a radio message that will arrive on her twenty-eighth birthday, I can say that now is her twenty-fourth birthday: halfway between when the light departed from there (20) and when it returned (28). It’s a nice idea (not mine: it’s Einstein’s definition of ‘simultaneity’). But this does not define a common time. If Proxima b is moving away, and my sister uses the same logic to calculate the moment simultaneous to her twenty-fourth birthday, she does not obtain the present moment here. In other words, in this way of defining simultaneity, if for me a moment A in her life is simultaneous with a moment B in mine, the contrary is not the case: for her, A and B are not simultaneous. Our different speeds define different surfaces of simultaneity. Not even in this way do we obtain a notion of a common ‘present’.
8. The combination of events that are at space-like distance from here.
9. Among the first to realize this was Kurt Gödel (‘An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation’, Reviews of Modern Physics, 21, 1949, pp. 447–50). In his own words: ‘The notion of “now” is nothing more than a certain relation between a certain observer and the rest of the universe.’
10. Transitive.
11. Even the existence of a relation of partial order might be too strong with regard to reality, if closed temporal curves exist. On this subject see, for example, M. Lachièze-Rey, Voyager dans le temps. La Physique moderne et la temporalité, Éditions du Seuil, Paris, 2013.
12. The fact that there is nothing logically impossible about travels to the past is demonstrated clearly in an engaging article by one of the great philosophers of the last century, David Lewis (‘The Paradoxes of Time Travel’, American Philosophical Quarterly, 13, 1976, pp. 145–52, reprinted in The Philosophy of Time, eds. R. Le Poidevin and M. MacBeath, Oxford University Press, Oxford, 1993).
13. This is the representation of the causal structure of a black hole metric in Finkelstein coordinates.
14. Among the dissenting voices, there are those of two great scientists for whom I have a particular friendship, affection and admiration: Lee Smolin (Time Reborn, Houghton Mifflin Harcourt, Boston, 2013) and George Ellis (‘On the Flow of Time’, FQXi Essay, 2008, https://arxiv.org/abs/0812.0240; ‘The Evolving Block Universe and the Meshing Together of Times’, Annals of the New York Academy of Sciences, 1326, 2014, pp. 26–41; How Can Physics Underlie the Mind?, Springer, Berlin, 2016). Both insist that a privileged time and a real present must exist, even if these are not captured by current physics. Science is like affection: those who are dearest to us are those with whom we have the liveliest disagreements. An articulate defence of the fundamental aspect of the reality of time can be found in R. M. Unger and Lee Smolin, The Singular Universe and the Reality of Time (Cambridge University Press, Cambridge, 2015). Another dear friend who defends the idea of the real flowing of a singular time is Samy Maroun; with him I have explored the possibility of rewriting the physics of relativity, distinguishing the time that guides the rhythm of processes (‘metabolic’ time) from a ‘real’ universal time (S. Maroun and C. Rovelli, ‘Universal Time and Spacetime “Metabolism” ’, 2015). This is possible, and hence the point of view of Smolin, Ellis and Maroun is defensible. But is it fruitful? The choice is between forcing the description of the world so that it adapts to our intuition, or learning instead to adapt our intuition to what we have discovered about the world. I have few doubts that the second strategy is the most fruitful one.
4. Loss of Independence
1. On the effects of drugs on time perception, see R. A. Sewell et al., ‘Acute Effects of THC on Time Perception in Frequent and Infrequent Cannabis Users’, Psychopharmacology, 226, 2013, pp. 401–13; the direct experience is astonishing.
2. V. Arstila, ‘Time Slows Down during Accidents’, Frontiers in Psychology, 3, 196, 2012.
3. In our cultures. There are others with a profoundly different notion of time: D. L. Everett, Don’t Sleep, There are Snakes, Pantheon, New York, 2008.
4. Matthew 20:1–16.
5. P. Galison, Einstein’s Clocks, Poincaré’s Maps, Norton, New York, 2003, p. 126.
6. An excellent panoramic history of the way in which technology has progressively modified our concept of time can be found in A. Frank, About Time, Free Press, New York, 2001.
7. D. A. Golombek, I. L. Bussi and P. V. Agostino, ‘Minutes, Days and Years: Molecular Interactions among Different Scales of Biological Timing’, Philosophical Transactions of the Royal Society. Series B: Biological Sciences, 369, 2014.
8. Time is: ‘number of change, with regard to before and after’ (Aristotle, Physics, IV, 219 b 2; see also 232 b 22–3).
9. Aristotle, Physics, trans. Robin Waterfield with an introduction and notes by David Bostock, Oxford University Press, Oxford, 1999, p. 105.
10. Isaac Newton, Philosophiae Naturalis Principia Mathematica, Book I, def. VIII, scholium.
11. Ibid.
12. An introduction to the philosophy of space and of time can be found in B. C. van Fraassen, An Introduction to the Philosophy of Time and Space, Random House, New York, 1970.
13. Newton’s fundamental equation is F = m d2x/dt2. (Note that time t is squared: this indicates that the equation does not distinguish t from -t, that is to say, it is the same backwards or forwards in time, as I explain in Chapter 2.
14. Curiously, many contemporary manuals of the history of science present the discussion between Leibniz and the Newtonians as if Leibniz were the heterodox figure with audacious and innovative relationist ideas. In reality, the opposite was the case: Leibniz defended (with a new wealth of arguments) the dominant traditional understanding of space, which from Aristotle to Descartes had always been relationist.
15. Aristotle’s definition is more precise: the place of a thing is the inner boundary of that which surrounds the thing. An elegant and rigorous definition.
5. Quanta of Time
1. I speak of this in more depth in Reality is Not What It Seems, trans. Simon Carnell and Erica Segre, Allen Lane, London, 2016.
2. It is not possible to locate a degree of liberty in a region of its phase space within a volume smaller than the Planck constant.
3. The speed of light, the Newton constant and the Planck constant.
4. Maimonides, The Guide for the Perplexed, I, 73, 106a.
5. We can try to infer the thought of Democritus from the discussions of Aristotle (for example, in Physics, IV, 213), but the evidence seems insufficient to me. See Democrito. Raccolta dei frammenti, interpretazione e commentario di Salomon Luria, Bompiani, Milan, 2007.
6. Unless the de Broglie–Bohm theory is true, in which case it has it – but hides it from us. Which is perhaps not so different in the end.
7. Carlo Rovelli, ‘Relational Quantum Mechanics’, International Journal of Theoretical Physics, 35, 1637 (1996), http://arxiv.org/abs/quant-ph/9609002. See also ‘The Sky is Blue and Birds Fly Through It’, http://arxiv.org/abs/1712.02894.
8. Grateful Dead, ‘Walk in the Sunshine’.
6. The World is Made of Events, not Things
1. Nelson Goodman, The Structure of Appearance, Harvard University Press, Cambridge, MA, 1951.
7. The Inadequacy of Grammar
1. For opposing views, see Chapter 3, note 14.
2. In the terminology of a celebrated article by John McTaggart (‘The Unreality of Time’, Mind, N.S., 17, 1908, pp. 457–74; reprinted in The Philosophy of Time, op. cit.), this is equivalent to denying the reality of the A-series (the organization of time into ‘past–present–future’). The meaning of temporal determinations would then be reduced to only the B-series (the organization of time into ‘before-it, after-it’). For McTaggart, this implies denying the reality of time. To my mind, McTaggart is too inflexible: the fact that my car works differently from how I’d imagined it and how I’d originally defined it in my head does not mean that my car is not real.
3. Letter by Einstein to the son and sister of Michele Besso, 21 March 1955, in Albert Einstein and Michele Besso, Correspondance, 1903–1955, Hermann, Paris, 1972.
4. The classic argument for the block universe is given by the philosopher Hilary Putnam in a famous article published in 1967 (‘Time and Physical Geometry’, Journal of Philosophy, 64, pp. 240–47). Putnam uses Einstein’s definition of simultaneity. As we have seen in Chapter 3, note 7, if the Earth and Proxima b move with respect to one another, say they are approaching each other, an event A on Earth is simultaneous (for an earthling) to an event B on Proxima b, which in turn is simultaneous (for those on Proxima b) to an event C on Earth, that is in the future of A. Putnam assumes that ‘being simultaneous’ implies ‘being real now’, and deduces that the event in the future (such as C) is real now. The error is to assume that Einstein’s definition of simultaneity has an ontological value, whereas it is only a definition of convenience. It serves to identify a relativistic notion that may be reduced to the non-relativistic one through an approximation. But non-relativistic simultaneity is a notion which is reflexive and transitive, whereas Einstein’s is not, hence it makes no sense to assume that the two have the same ontological meaning beyond the approximation.
5. That the discovery by physics of the impossibility of presentism implies that time is illusory is an argument put forward by Gödel (‘A Remark about the Relationship between Relativity Theory and Idealistic Philosophy’, in Albert Einstein: Philosopher-Scientist, ed. P. A. Schlipp, Library of Living Philosophers, Evanston, 1949). The error always lies in defining time as a single conceptual block that is either all there or not there at all. The point is discussed lucidly by Mauro Dorato (Che cos’è il tempo?, op. cit., p. 77).
6. See, for instance, W. V. O. Quine, ‘On What There Is’, Review of Metaphysics, 2, 1948, pp. 21–38, and the fine discussion of the meaning of reality in J. L. Austin, Sense and Sensibilia, Clarendon Press, Oxford, 1962.
7. De Hebd., II, 24, cited in C. H. Kahn, Anaximander and the Origins of Greek Cosmology, Columbia University Press, New York, 1960, pp. 84–5.
8. Some examples of important arguments where Einstein has strongly supported a thesis which he later changed his mind about: 1. The expansion of the universe (first ridiculed, then accepted); 2. the existence of gravitational waves (first taken as obvious, then rejected, then accepted again); 3. the equations of relativity do not admit solutions without matter (a long-defended thesis that was abandoned – rightly so); 4. nothing exists beyond the horizon of Schwarzschild (wrong, though perhaps he never came to realize this); 5. the equations of the gravitational field cannot be general covariant (asserted in the work with Grossmann in 1912; three years later, Einstein argued the opposite); 6. the importance of the cosmological constant (first affirmed, then denied – having been right the first time) …