Timeless Variation. A Contribution to the Metaphysics of Quantum Gravity (1/2)
1 In this article, we would like to show that the concept of time and that of variation do not encompass two variants of the same reality, which we could call “change” or “becoming,” but rather two fundamental aspects of reality, indeed its two most fundamental aspects. As a result, we will not only need to conceptualize the difference between the two, but more importantly the tension or opposition between them, by understanding that tension or opposition as constitutive of empirical reality. From the perspective of common sense, i.e. of that empirical reality, the hierarchy of these two terms seems to go without saying: time would encompass variation because all empirical variation takes place in time and stating the opposite would be meaningless. [1] And yet a reversal of this kind is indeed what we wish to establish by showing that all temporality in fact unfolds within a variation that encompasses it and founds it as such – on two conditions: first, on no longer considering that variation as empirical, purifying it of all empiricity; and second, on no longer thinking of time as fundamental but on the contrary as always already empirical. Between a variation conceived as pure and an irremediably empirical time, the possibility of a relationship of principle to consequence will then take shape, from explicans to explicandum. The purpose of our journey is consequently to reveal time as what it intrinsically is: an initial form of invariance, and therefore an “antivariation,” a partial suspension of “pure variation.”
2 This philosophical analysis echoes the work of recasting the most fundamental concepts, notably those of space and time, a project that the question of quantum gravity forces physicists to grapple with. If the smooth spacetime of general relativity must necessarily be the object of a Planckscale genesis, on what basis, of what nonspatiotemporal reality, can and must it be generated? Aren’t we induced to consider not only a geometrogenesis but also a chronogenesis? But what would a genesis of time itself be, which would therefore not take place within a given time? What fundamental properties of time need to be generated as a result? In particular, which ones encode the variable t that is present in every theory of physics? We would like to help shed light on this vast field of inquiry – without of course claiming to exhaust it – by showing that the abstract idea of time as well as the practical use in physics of the variable t obey a number of conditions. These conditions are minimal, to be sure, yet already restrictive (subsistence/invariance, continuity and succession), and neither the abstract idea of variation nor some of its concrete physical manifestations necessarily obey them. It will then be sufficient to remove the implicit limits within which we usually place the idea of variation (where any variation would be a variation in or of something) in order to understand that this “variational” aspect of reality is at a deeper level than its temporal aspect, making it possible to generate time through a simple addition of restrictive conditions. The idea of “pure variation” must then ultimately enable the specification of what could and should be a reality that would be the condition of possibility of all spacetime, without being spatiotemporal itself.
3 Let us take an arbitrary case of variation, expressed in an abstract form: A ® A’ ® A”. Neither the letters nor the arrows have any particular mathematical significance. Here, they just mean that “something” “goes” from one “state” to another, twice. What must we conclude from this elementary example? That an abstract variation of this nature from A to A’ and from A’ to A” can only become a concrete reality, i.e. receive a physical translation, through the involvement of three a priori necessary conditions: a) the subsistence of an invariant, i.e. of a “something” that varies or in relation to which variation is measured, which is therefore the subject, the substratum or the invariant measurement of the variation; b) the continuity of that subsistence, i.e. of the transition from one state to another; c) the succession of that continuity, i.e. of the distinct states of variation. We will see that each of these conditions manifests itself precisely as a constituent property of time such as it appears (minimally) in physics: the variable t which is an evolution parameter in every theory in the field. The question then arises whether such a variation can continue not only to be conceived in a logically coherent fashion, but also and especially to come into existence in a way that permits physics experiments, when each of these conditions has been eliminated one after the other. In fact, we are going to establish that the most fundamental physical theories that experiments have confirmed up to this point, general relativity and quantum mechanics, may not require us to eliminate the conditions, but they at least allow us to do so. What then is left of a variation once any invariance, any continuity, and any succession, in other words when each of the properties of the variable t, and ultimately all of them, have been removed from it? If something is left, it will allow us precisely to establish that the idea of variation is not only distinct but also more fundamental than that of time.
The First Condition of Any Physical Variation Is that Something Subsists in that Variation Without Varying Itself, Thereby Constituting A Priori the Substratum or the Invariant Measurement (Condition of Subsistence or Invariance)
4 In this case, what subsists from A to A’ and A” is obviously the letter A. But something at a deeper level subsists as well, which fundamentally concerns the intervals between the three terms of this variation. While the “speed” at which the intervals between A and A’, and between A’ and A”, are crossed, i.e. at which they succeed each other, may not be invariant, something else is: the possibility of measuring it, in other words the possibility of relating this traversal, this succession, or this flow to a supposedly standard or even flow. In relation to that flow, any other flow would have a certain “variation speed” (if that variation can be reduced to a continuous and differentiable function) or “frequency” (if it can be reduced to an iterative, periodic process). This first condition is neither trivial nor contrary to intuition: how can one envision or recognize the fact that something varies without having identified what varies and in relation to what it varies? Various a priori invariants – the subsistence of a subject (or substratum) of variation and/or the subsistence of a measurement (or framework) of variation – have followed each other over the history of philosophical and scientific thought. Recounting that history amounts to striding through the passageway that leads from Aristotelian (meta)physics to the GalileoNewtonian scientific revolution.
5 In order to take into account the fact of change or variation, Aristotelian theory creates a plethora of conceptual tools whose fundamental aim is to distinguish and combine two levels. One level consists of that which does not change, and on the sole basis of which change may take place (the level of the subject or substance as a figure of Parmenidean being). The other level consists of that which changes as a result of being variably attributed to that which does not change (the level of the attributes or properties as a figure of Heraclitean becoming). In fact, beyond the details that inhere to the Aristotelian theory of change (in particular the fact that it always has a direction), it is clear that every time we envision change on the basis of something that doesn’t change (material bodies, particles, fields, etc.) – which possesses properties, attributes, and qualities, in short variables that are supposedly secondary and located at an ontologically lower level (speed, position, mass, charge, etc.) – we continue to be beholden to the Aristotelian framework.
6 The modern scientific revolution that Galileo initiated and that Newton brought to completion only superficially appears, then, as the abandonment of final causes or the abstract recognition that the universe is “written in mathematical language.” On a structural level, it first consists in minimizing this Aristotelian invariant (transforming “substantial forms” into mere extended, material bodies). Then, to take into account the change or variation that these bodies undergo (movement, first of all), it consists in plunging them into another, more fundamental invariant: time – which is no longer a mere quality that is subordinated to movement. This act of elevating time to a fundamental factor, in relation to which anything may vary only insofar as time itself does not, presupposes several logical steps. First, the properties of bodies have to become constant quantities (for example the inertial mass) that characterize them but do not intrinsically belong to them, in such a way that no state of variation or value of any variable can have privilege over the others anymore (A in relation to A’ or A” in our example; rest in relation to movement; a given position in relation to another; etc.). On the contrary, all of these states become equivalent and appear as mere initial conditions in relation to the arbitrary bounding of the system under study. Second, the laws of variation within these variable quantities have to be highlighted by finding equivalences, i.e. the equations that link them to each other, the way some quantities depend on others. In this way, in Galileo’s law of falling bodies or the fundamental principle of dynamics as set out by Newton, time acts as an evolution parameter to the other variables, while not depending on any itself. Third, this variable or function needs to be usable as a measurement of the other variables by measuring it itself, in other words by checking on its monotonic progression, i.e. the regularity of the intervals or of their a priori invariance. This is where the difficulties begin, for how can one measure this regularity without correlating it to a movement that is itself periodic (which is the principle of any astronomical clock, whether mechanical or atomic)? In order to dispose of this difficulty, Newton succeeds in autonomizing the temporal function by positing the existence of an absolute time that, as such, possesses a twofold characteristic. It flows uniformly or regularly, thereby constituting an invariant framework within which any dynamic variable may vary. In addition, it defines a universal present or plane of global simultaneity, thereby making it possible to date the set of phenomena and order them in relation to each other. Consequently, within this framework, the “speed” with which A becomes A’ then A”, and the exact date at which it does so, can be determined unambiguously, independently from the chosen frame of reference.
7 But Einstein’s twopart revolution obviously shattered this framework. Time can no longer be asserted as a universal dating system and an invariant factor of the development of dynamic phenomena, insofar as it is not an independent variable at all anymore. On the one hand, with special relativity, it becomes dependent on the speed of the body serving as the frame of reference, in such a way that what is invariant in every frame of reference is no longer the intervals of time but just the interval of spacetime (pseudoEuclidean manifold or Lorentzian manifold of zero curvature). On the other hand – and at a deeper level – with general relativity, time becomes dependent on the matterenergy content in such a way that the metric itself becomes variable (pseudoRiemannian manifold or Lorentzian manifold of nonzero curvature), which means that the intervals of spacetime are no longer established once and for all, and vary locally.
8 The consequences are fundamental and twofold. On the one hand, in special and general relativity, unlike the GalileoNewtonian model, there is no unique temporal function anymore. There is either an infinite number of them, defining just as many planes of simultaneity and therefore just as many possible contradictory dating systems, or none at all in some “exotic” solutions of general relativity. When it is possible to parametrize space as a whole according to a given temporal function, this parametrization will always remain arbitrary, obeying conventional procedures, to meet certain needs. This is how cosmic time is created, making it possible to give approximate dates for the great cosmological events in relation to us since the Planck epoch. However, none of these functions can have an ontological privilege, as a result of the fundamental covariance of general relativity.
9 On the other hand, in general relativity, there is no temporal function that is external to what takes place within it anymore. Any function that is arbitrarily chosen to lay out a local set or even the global set of events will also, by necessity, be a variable that is internal to the theory, dependent upon equations describing the matterenergy content, which may itself be variable. General relativity therefore compels us to stop considering any dynamic change as something that takes place in time, and to see it instead as a mere correlation between variables where one may serve as an arbitrary clock for the others, and where none could, by definition, be perfect. [2]
10 Physicists and philosophers have not always adequately assessed what this abandonment of time as an a priori invariant exactly meant. Due to the fact, first, that nothing varies according to an invariant time any longer, and second, that no “present of variation” has a universal value any longer, it was thought possible to conclude that nothing varies any longer except in an illusory fashion and that no present exists differently from the past and the future – meaning everything is given for all eternity (eternalist metaphysics of the block universe). [3] By outlining the third condition of any physical variation – the condition of succession – we take a new look at the “nowness” effect produced by the idea of a variation in itself, the very effect of states following each other in succession with no supplementary dimension. But as for the fact that things vary, which general relativity does not refute at all, the only radical metaphysical conclusion that can be drawn from its equations is not the illusory existence of an invariant block universe but the reversal of the relationship that was traditionally established between time and variation: no variation ever moves from one state to another in a given underlying time since no invariant of this kind is established a priori. This means, on the one hand, that variations in metrics (for example a gravitational wave) and material variations (for example the formation of a black hole) are strictly correlated according to Einstein’s equations; and on the other, that all of the material variations that are or can be causally connected (time or light intervals, but not space intervals) are likely to produce persistences of one kind or another defining “world lines” that are endowed with “proper time.” This proper time, which is invariant in every frame of reference but strictly local, can itself serve as an invariant frame of reference to measure other times referred to then as “improper” (with Lorentz transformations allowing for the passage from one to the other and vice versa) and as an arbitrary basis for defining a global temporal function (which will have no a priori privilege over those defined by other proper times). In fact, no fourdimensional continuum precedes these local variations or establishes them a priori – it can only emerge from them or be a result of them a posteriori. [4] Therefore, spacetime, as well as all it contains, is never given but always “plotted out” by what occurs, by what varies and by the objects it acts upon as well as those objects that act upon it. The most astute heirs of Einstein have thus given the name “background independence” to the covariance of the theory, i.e. the absence of an invariant background on the basis of which variables vary, and they make it a fundamental constraint of general relativity with which any more fundamental theory, one that would have to deduce general relativity as one of its limit cases, must be equipped. [5]
11 [To be continued – the bibliography will appear at the end of the second part of this article, which will be published in the next issue]
Endnotes

[1]
See Sydney Shoemaker, “Time Without Change,” Journal of Philosophy 66.12 (1969), 363381: “The claim that time involves change must of course be distinguished from the truism that change involves time” (363).

[2]
See Carlo Rovelli, “Forget Time,” Cornell University / arxiv, arxiv.org, 2009, Web, 18 October 2022.

[3]
See Rovelli, The Order of Time, trans. Simon Carnell and Erica Segre (New York: Riverhead Books, 2018). Rovelli emphasizes that the abandonment of a unique time, a “single orderly succession,” does not at all mean that “nothing changes,” and that therefore the block universe and universalism are “the only way left” (chapter 7, “The Inadequacy of Grammar”). See also Henri Bergson’s analyses, which are still relevant today, in Durée et simultanéité [1922] (Paris, PUF, 2009), 5864 and 152166 [Duration and Simultaneity, trans. Leon Jacobson (Indianapolis / New York / Kansas City, BobbsMerrill, 1965), 5863 and 136149]. They demonstrate that the theory of the block universe, far from transcending the illusion of time passing, is a new vessel into which the oldest eternalist metaphysics is poured, one characterized by a twofold illusion consisting in simultaneously giving oneself more (since it presupposes an extra dimension) and less (since it presupposes, with no explanation, the existence of an illusory displacement throughout that dimension) than that which varies in itself, i.e. without a dimension.

[4]
Conceiving “the emergence of time” thus amounts to accounting for a threefold genesis. First, there is the genesis of time’s dimensionality, i.e. the consistently local nature of the trace or subsistence of a variation that is itself dimensionless (we shall therefore return to this point in the exposition of the third condition). Second, there is the genesis of time’s invariance and orientation, which are at first sight local as well. (The hypothesis of “thermal time” may account for this genesis: see Rovelli and Alain Connes, “Von Neumann Algebra Automorphisms and TimeThermodynamics Relation in General Covariant Quantum Theories,” Cornell University / arxiv, arxiv.org, 1994, Web, 19 October 2022.) Third, there is the genesis of time’s progressive globalization, which is a result of the two previous aspects understood as the causal intersection or shared past of all these local variations. This threefold genesis can be associated with the “theory of the growing block universe” (aka “nonfuturism”), a theory notably defended by Michael Tooley [Time, Tense and Causation (Oxford: Clarendon Press, 1997)], who explicitly built on the work of C. D. Broad [Scientific Thought (New York, Harcourt, Brace & Co, 1923)] while implicitly building on that of Bergson – Matière et mémoire (Paris: F. Alcan, 1896) [Matter and Memory, trans. Nancy Margaret Paul and W. Scott Palmer (New York: Macmillan, 1911)]. The whole interest of this theory lies in its ability to precisely account for this twofold character of the fourdimensional continuum, in contrast to the eternalism of the block universe. The fourth dimension, insofar as it would be a result of objectively shared durations, would be produced (and not given for all eternity) while manifesting a posteriori (not Translated from French by Allyn Hardycka priori) the asymmetry between past and future.

[5]
See Lee Smolin, The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next (Boston: Houghton Mifflin, 2006); Smolin, Time Reborn: From the Crisis in Physics to the Future of the Universe (Boston: Houghton Mifflin Harcourt, 2013).