61. Letter by Einstein to the son and sister of Michele Besso, March 1955, in Albert Einstein and Michele Besso, Correspondence, 1903–1955 (Paris: Hermann, 1972).
62. 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 (1967): 240–47). Putnam uses Einstein’s definition of simultaneity. As we have seen in note 27, 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 nonrelativistic one through an approximation. But nonrelativistic simultaneity is a notion that 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.
63. That the discovery by physics of the impossibility of presentism implies that time is illusory is an argument put forward by Gödel in “A Remark about the Relationship between Relativity Theory and Idealistic Philosophy,” in Albert Einstein: Philosopher-Scientist, ed. P. A. Schlipp (Evanston, IL: Library of Living Philosophers, 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 Dorato, Che cos’è il tempo?, p. 77.
64. See, for instance, W. V. O. Quine, “On What There Is,” Review of Metaphysics 2 (1948): 21–38, and the fine discussion of the meaning of reality in J. L. Austin, Sense and Sensibilia (Oxford: Clarendon Press, 1962).
65. De Hebdomadibus of Boethius II.24, cited in C. H. Kahn, Anaximander and the Origins of Greek Cosmology (New York: Columbia University Press, 1960), pp. 84–85.
66. Some examples of important arguments where Einstein has strongly supported a thesis that 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.) . . .
8. DYNAMICS AS RELATION
67. The general form of a mechanical theory that describes the development of a system in time is given by a phase space and a Hamiltonian H. Evolution is described by the orbits generated by H, parametrized by the time t. The general form of a mechanical theory that describes the evolutions of variable with respect to each other is instead given by a phase space and a constraint C. The relations between the variables are given by the orbits generated by C in the subspace C=0. The parametrization of these orbits has no physical meaning. A detailed technical discussion can be found in chapter 3 of Carlo Rovelli, Quantum Gravity (Cambridge, UK: Cambridge University Press, 2004). For a concise technical account, see Carlo Rovelli, “Forget Time,” Foundations of Physics 41 (2011): 1475–90, https://arxiv.org/abs/0903.3832.
68. An accessible account of loop quantum gravity can be found in Rovelli, Reality Is Not What It Seems (op. cit.).
69. Bryce S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory,” Physical Review 160 (1967): 1112–48.
70. J. A. Wheeler, “Hermann Weyl and the Unity of Knowledge,” American Scientist 74 (1986): 366–75.
71. J. Butterfield and C. J. Isham, “On the Emergence of Time in Quantum Gravity,” in The Arguments of Time, ed. J. Butterfield (Oxford: Oxford University Press, 1999), pp. 111–68 (http://philsci-archive.pitt.edu/1914/1/EmergTimeQG=9901024.pdf); Zeh, Die Physik der Zeitrichtung; Craig Callender and Nick Huggett, eds., Physics Meets Philosophy at the Planck Scale (Cambridge, UK: Cambridge University Press, 2001); Sean Carroll, From Eternity to Here: The Quest for the Ultimate Theory of Time (New York: Dutton, 2010).
72. The general form of a quantum theory that describes the evolution of a system in time is given by a Hilbert space and the Hamiltonian operator H. The evolution is described by Schrödinger’s equation iħ∂tΨ = HΨ. The probability of measuring a pure state Ψ a time t after having measured a state Ψ′ is given by the transition amplitude 〈Ψ | exp[–iHt/ħ] | Ψ′〉. The general form of a quantum theory that describes the evolution of the variables with respect to one another is given by a Hilbert space and Wheeler-DeWitt equation CΨ = 0. The probability of measuring the state Ψ after having measured the state Ψ′ is determined by the amplitude 〈Ψ | ∫ dt exp[iCt/ħ]|Ψ′〉. A detailed technical discussion can be found in chapter 5 of Rovelli, Quantum Gravity (op. cit.). For a concise technical version, see Rovelli, “Forget Time” (op. cit.)
73. Bryce S. DeWitt, Sopra un raggio di luce (Rome: Di Renzo, 2005).
74. There are three: they define the Hilbert space of the theory where the elementary operators are defined, whose eigenstates describe the quanta of space and the probability of transitions between these.
75. Spin is the quantity that enumerates the representations of the group SO(3), the group of spatial symmetry. The mathematics that describes the spin networks has this feature in common with the mathematics of ordinary physical space.
76. These arguments are covered in detail in Rovelli, Reality Is Not What It Seems. (op. cit.).
9. TIME IS IGNORANCE
77. Ecclesiastes 3:2–4.
78. More precisely, the Hamiltonian H, that is, the energy as a function of position and speed.
79. dA/dt = {A, H}, where { , } are the Poisson brackets and A is any variable.
80. Ergodic.
81. The equations are more readable in the canonical formations of Boltzmann than in the microcanonical form to which I make reference in the text: the state ρ = exp[‒H/kT] is determined by the Hamiltonian H that generates evolution of time.
82. H = ‒klog[ρ] determines a Hamiltonian (up to a multiplicative constant), and thus a “thermal” time, starting from the state ρ.