3. The philosopher Mauro Dorato has insisted on the necessity to render the elementary conceptual framework of physics coherent with our experience; see Mauro Dorato, Che cos’è il tempo? (Rome: Carocci, 2013).
1. LOSS OF UNITY
4. This is the essence of the theory of general relativity. Albert Einstein, “Die Grundlage der algemeinen Relativitätstheorie,” Annalen der Physik 49 (1916): 769–822.
5. In the approximation of a weak field, the metrics can be written ds2 = (1 + 2ϕ(x)) dt² – dx², where ϕ(x) 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 toward the lowest potentiality, where time slows. (These and similar notes are for those who have some familiarity with theoretical physics.)
6. Carlo Rovelli, Che cos’è la scienza: La rivoluzione di Anassimandro (Milan: Mondadori, 2011); English translation, The First Scientist: Anaximander and His Legacy (Chicago: Westholme, 2011).
7. For example: (ttable − tground) = gh/c2tground where c is the speed of light, g = 9.8m/s2 is the acceleration of Galileo, and h is the height of the table.
8. 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 tγ = ∫γ√ gab (x)dxadxb. The physical relation between this quantity and gab(x) is discussed further on.
2. LOSS OF DIRECTION
9. Rainer Maria Rilke, Duineser Elegien, in Sämtliche Werke, vol. 1, I, vv. 83–85 (Frankfurt: Insel, 1955).
10. 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. Antoine Lavoisier was active in politics. The mathematician Joseph-Louis Lagrange was honored by the different governments that succeeded each other in that tormented and magnificent moment in the history of humanity. See Steve Jones, Revolutionary Science: Transformation and Turmoil in the Age of the Guillotine (New York: Pegasus, 2017).
11. 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.
12. The equations of Newton determine how things accelerate, and the acceleration does not change if I project a film backward. The acceleration of a stone thrown upward is the same as that of a falling stone. If I imagine years running backward, the moon turns around the Earth in the opposite direction but appears equally attracted to the Earth.
13. The conclusion does not change by adding quantum gravity. On the efforts to fund the origin of the direction of time, see, for example, H. D. Zeh, Die Physik der Zeitrichtung (Berlin: Springer, 1984).
14. Rudolf Clausius, “Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie,” Annalen der Physik 125 (1865): 353–400; 390.
15. 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.
16. Arnold Sommerfeld.
17. Wilhelm Ostwald.
18. 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.
19. 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.
20. S = k log W. 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
21. General relativity (A. Einstein, “Die Grundlage der algemeinen Relativitätstheorie,” op. cit.).
22. Special relativity (Albert Einstein, “Zur Elektrodynamik bewegter Körper,” Annalen der Physik 17 [1905]: 891–921).
23. J. C. Hafele and Richard E. Keating, “Around-the-World Atomic Clocks: Observed Relativistic Time Gains,” Science 177 (1972): 168–70.
24. That depends as much on t as on your speed and position.
25. Poincaré. Lorentz had tried to give a physical interpretation to t, but in a quite convoluted way.
26. 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.
27. 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.”