Some Mathematical Examples of Emergent Intuitive Local Time Flow

Abstract

After reviewing important historical and present day ideas about the concept of time, we develop some instances of mathematical examples where, from the interaction of concepts that model interactions of things in the observable world, time flow emerges in an intuitive and local interpretation. We present several instances of emergence of time flow in mathematical contexts, to wit, by specific parametrisation of deterministic and stochastic curves or of geodesics in Riemann manifolds.

1. Introduction

“Again, time does not exist in itself, but a sense results from things themselves, what has passed in time, and what things remain, and what will follow next; nor should we confess that we should feel any time by ourselves aloof from the motion of things, and in a calm quietness”.

“tempus item per se non est, sed rebus ab ipsis consequitur sensus, transactum quid sit in aevo tum quae res instet, quid porro deinde sequatur; nec per se quemquam tempus sentire fatendumst semotum ab rerum motu placidaque quiete”.

Titus Lucretius Carus; c. 94 a.C.—c. 50 a.C; De rerum natura, vers. 460.

Time in itself is indefinable. Only its flow is sensible and can be thought of. These are two intuitive propositions—perhaps coming to mind after some reflection—that summarise what is both ancient and present day knowledge about time; for instance, in [1] (p. 100) we have this statement: “Let us look first at the question ‘What is time?’ Time is an inescapable—perhaps the most inescapable—act of experience, but we cannot define it”.

We can imagine what could be a primordial human representation of time. Let us imagine the life of a primitive hominid in an African savanna three million years ago. Acquiring consciousness of the regularity of repetition of the alternating periods day–night was surely essential for organising activities in order to maximise survival. A less primitive hominid would also notice the repetition of the most conspicuous phases of the moon, full moon–absent moon, and they would notice the variations of the natural phenomena in consonance with the moon phase. An even less primitive hominid, willing to anticipate some of the phenomena associated with the repetition of the moon phases, would count, for instance, the approximate number of days between two full moons and, with this, would create a measure of the time flow.

This imagination exercise shows us that the essence for the construction of a primitive individual representation of time is periodicity. Local time flow—materialised in language with the terms before, now, after—requires the identification of simultaneity; that is, a decision process on the possibility of events being simultaneous, or not. Moreover, the possibility of sharing the notion of time relies on the possibility of a common time measuring unit and, as a consequence, that a primitive unit measure of time is essentially the period of a pendulum, that is, the period of some noticeable periodic regularly repetitive event.

In this work, we deal mostly with concepts regarding time flow.

There are three caveats for the presentation in this work.

• The analysis is conditioned by our own conceptions and historical perspectives. This being so, we want to explore the notion of emergent local time flow, that is, the notion of a flow—represented either by an increasing sequence of numbers or by a real number interval $[0,T]$—that can be taken as a flow of time; in the sense of the points of the flow being tags for observable interactions of things in an observable universe composed of things. Local time, in the context of this work, may be thought of as time in an inertial time frame shared by observers.
• Our purpose is to achieve a description of examples of emergent time in mathematical models: relativistic, quantum mechanistic, thermodynamic, etc.; that is, instances of mathematical descriptions and results where a special time flow appears as a consequence of the nature of the mathematical description and associated results; an example of such implicit time flow, that we may consider as an emergent time, is the choice of the arc length parametrisation of a geodesic in a Riemannian manifold, that may be thought of, in relativity theory, as Minkowski’s proper time, that is, time measured by a clock along a time-like world line.
• It is possible to provide a mathematical theoretical formulation of emergent time flow as considered in this work but we opted for a more intuitive presentation of examples in order to illustrate the main characteristics of the emergence of time flow in well-known mathematical contexts. We stress that, for us, the most relevant rigorous formulation of emergent time can be found in the works of Alain Connes and Carlo Rovelli, briefly described in Section 2.5.

A summary of one of the contributions of this work of a substantial part of Section 3, for the more mathematically oriented reader, is that there are several examples in mathematics where if we have:

• a finite set of vectors of a finite-dimensional space whose components represent observations, i.e., measurements, which we consider to be simultaneous;
• that these vectors are in a certain total order, that is, we can say that given any two, one is “before” the other;

then it is possible to define a special canonical flow of time, that is, a function of an interval of $\mathbb{R}$ in vector space, that goes through the vectors (exactly or approximately), that may be thought as a flow of time and that is either unique or has minimisation properties that characterise it.

Other parts of Section 3 deal with the emergence of time from the existence of integral curves of Lipschitz vector fields or from the existence of geodesics in complete Riemannian manifolds.

Finally, Section 3 also deals with the emergence of time for stochastic processes in the case of processes that can be amenable either to Brownian motion or to a stochastic line integral with respect to Brownian motion.

Let us briefly describe the contents of this work.

• In Section 2, we present several ideas about the concepts of time by quoting some very important authors—such as Kant, Newton, Einstein, Wittgenstein, Rovelli, and Connes—that, one way or another, substantiate our approach to the idea of intuitive local time flow, an idea that we may consider implicit in some sets of mathematical concepts.
• In Section 3, we present some examples of mathematical situations where we can isolate intuitive local time flow that, in some instances, is somehow canonical by reason of its uniqueness or by reason of some invariance property.

2. Concepts of Time

2.1. Classical Roman and Greek Times

We just observe that the citation above of Titus Lucretius Carus predates the contemporary general ideas of time, as in Carlo Rovelli (see Section 2.5).

2.2. Kant’s Time

It is a strenuous task to completely analyse Kant’s ideas about time. Is it remarkable that in his epoch Kant was a philosopher in the classic meaning of the term, mastering the works of Newton and lengthy lecturing about the current state of cosmology at his time. We now quote some excerpts of a profound and exhaustive analysis in [2] that substantiates the importance of time in Kant’s natural philosophy. We have this first quote: “All Kantian reflection starts from a completely different assumption. To understand the present kinematic order of the different worlds we have to go back to previous states of Nature, dynamically relating the present situation with a configuration and a game of forces that have already disappeared. Time thus acquires a qualitatively central place”. (see [2] (pp. 33–35)). And we have this second quote: “The novelty of the Kantian perspective lies in the understanding of natural laws as a result of the action of the elementary forces of matter over time. The world would have changed its configuration, not because of variations in the basic forces that regulate natural laws, but because the mode of expression of the laws itself needs time to update itself”. (see [2] (pp. 35–36)). As so, it is perfectly established that time, in its capacity of allowing the description and analysis of change, was an essential a priori ingredient in Kant’s thoughts about the world. We now detail some more specific characteristics that Kant considers in his ideas about time.

In a very rough first approach, for non-specialists, we can say that Kant seems to consider at least two stages of intelligence. The first stage is the one of perception—with its counterpart at the cerebral construction of images and memories—created by the interactions of the senses of the observer with the world. The second stage is the build up of complex rational associations of images and memories in an internal representation of the world. When defining time, it is possible to detect these two stages, for instance in this quote: “Time is nothing other than the form of inner sense, i.e., of the intuition of our self and our inner state. For time cannot be a determination of outer appearances; it belongs neither to a shape or a position, etc., but on the contrary determines the relation of representations in our inner state. And just because this inner intuition yields no shape we also attempt to remedy this lack through analogies, and represent the temporal sequence through a line progressing to infinity, in which the manifold constitutes a series that is of only one dimension, and infer from the properties of this line to all the properties of time, with the sole difference that the parts of the former are simultaneous but those of the latter always exist successively. From this it is also apparent that the representation of time is itself an intuition (see [3] (pp. 163, B50))”. From this text it is clear to us that Kant is perfectly aware of the two identifiable aspects of time: the flow of time and the need for a definition of simultaneous events, that is also essential in Einstein’s views on time (see Section 2.3).

2.3. Isaac Newton’s and Albert Einstein’s Concepts of Time

A common assertion about the notion of time in Newton’s work is paraphrased in [4] as: “Absolute, true, and mathematical time, from its own nature, passes equably without relation to anything external, and thus, without reference to any change or way of measuring of time (e.g., the hour, day, month, or year)”. Also, in [5] we have the following appreciation: “Newtonian time is assumed to be “flowing uniformly”, even when nothing happens, with no influence from events, and to have a metric structure: we can say when two time intervals have equal duration”. So, according to Newton, time is defined by an immutable flow that cannot be altered. It follows that the quality of a measure of time only depends on the procedure.

Basically, Albert Einstein is interested in time as a measurable quantity useful in mechanics (see [6] (pp. 9–11)). It is possible to describe the ideas of time implicit in the relativity theories in a mathematically oriented form; for the moment we will mainly underline that Einstein considered that in order to measure time, in a way that could be communicated to others, an a priori definition of simultaneous events was necessary. In fact, in [7], the author says: “We therefore need such a definition of simultaneity as to give us a method by which we can decide, …, by experiments, whether the two lightning strikes were simultaneous or not”. And, in the first article on relativity theory (see [8] for the translation from the German in [9]), he writes: “We have to bear in mind that all our judgements involving time are always judgements about simultaneous events”. This statement corresponds to an idea—also found in [5]—that coincides with the following proposition: “In GR (general relativity), therefore, there are two distinct kinds of temporal notions. The first is the simple fact that all events are localised with respect to one another”.

In an apparent contradiction in Einstein’s relativity, since the goal is to describe gravity via space with a particular way of defining distances in that space—namely, a Lorentzian manifold equipped with a Riemannian metric—time does not appear explicitly in the equations defining the gravitational field (see [10] (p. 11)). A field that, in the presence of a mass distribution, allows the quantitative appreciation of gravity. A particular family of solutions to the Einstein field equations was found by the renowned logician Kurt Gödel (see [11]). Due to the very specific set of conditions on the type and distribution of mass in Gödel’s equations (see [12,13] for developments and visualisations) there is the possibility of closed geodesics, which in the absence of a preferable origin choice induces the idea of the absence of time. In the recent essay [14] on Godel’s work on universe models, Reinhard Kahle draws attention both to the possibility of the existence of solutions to Einstein’s equations in which “there can be no objective course of time” and also to the philosophical consequences, from a logician’s perspective, of such a premise. It is remarkable that Gödel himself, in the note [15], written with the purpose of casting light on the relationship between Einstein’s relativity and Kant’s philosophy, highlights the idea that “Kant, insofar as he attributes any reality to time, such as we perceive it, really means that temporal properties are certain relations of the things to the perceiving subject…”, a statement that agrees with our view on Kant’s ideas in Section 2.2. As we will show, the existence of regular geodesics allows the definition of a notion of a local canonical concept of quantitative time by means of the parametrisation by arc length of the geodesic.

In Einstein’s works, a fundamental idea, necessary to identify time as a measurable quantity, is the relation of events to one another, and so, the possibility of determining simultaneous happenings. We take this idea as one of the starting points of our approach to the mathematical examples of the emergence of time in Section 3.

2.4. Time in Ludwig Wittgenstein’s Philosophy

Ludwig Wittgenstein is a very important philosopher of the 20th century and we consider it essential to analyse the conceptions of time he proposed. Proceeding chronologically, we have, in a preparatory text of the Tractatus, an observation that seems to pertain to the later philosophy of Wittgenstein, since it addresses the interpretation of a particular formulation of a property of time, to wit: ““Time has only one direction” must be a piece of nonsense. Having only one direction is a logical property of time. For if one were to ask someone how he imagines having only one direction he would say: Time would not be confined to one direction if an event could be repeated. But the impossibility of an event’s being repeated, like that of a body’s being in two places at once, is involved in the logical nature of the event”. (see [16] (p. 84e)). This reflection seems to point to the idea that time flow is an inherent property of time, a property that constitutes the logic structure of time. This idea is somehow present in our approach. Later on, we have the following statement in the Tractatus (see [17] (p. 83)). “We cannot compare any process with the “passage of time”—there is no such thing—but only with another process (say, with the movement of the chronometer). Hence, the description of the temporal sequence of events is only possible if we support ourselves on another process”. Our interpretation is that, for Wittgenstein, the flow of time requires—for its detection and appreciation—the interaction of things in the world and, in a more or less implicit way, it also requires the consideration of simultaneity. In his Philosophical Grammar—a text in which Wittgenstein explores a method for the production of rules guiding the thoughts expressed in propositions—he says: “How does it happen that every fact of experience can be brought into a relationship with what is shown by a clock?” (see [18] (p. 216)). Again, in our interpretation, in the transformation of sensorial inputs to rational thoughts the mind always attaches a time tag to a processed sensorial input; this attachment requires the notions of simultaneity and of time flow given by a device such as a clock.

Another further analysis of a very particular idea about the concepts related to time in Wittgenstein’s thoughts is provided in [19].

2.5. Time According to Carlo Rovelli and Alain Connes

Carlo Rovelli, in his book The Order of Time, gives a historical account of ideas about time in order to defend a thesis that is perfectly summarised in the following paragraph: “What we need, if we want to do science, is a theory that tells us how the variables change with respect to each other. That is to say, how one changes when others change. The fundamental theory of the world must be constructed in this way; it does not need a time variable: it needs to tell us only how the things that we see in the world vary with respect to each other. That is to say, what the relations may be between these variables” (see [20] (p. 106)). It is a Herculean task to describe in a simple way the ideas of Alain Connes about time. One of these ideas that relates to the approach of Rovelli (see [21] (p. 523)) is to consider that a quantum statistical system is defined by a set of observables—with particular algebraic properties, namely, a $C^{★}$ algebra, that is, a set of invertible transformations that act on the states of the system—and the time evolution given by a family indexed by the real numbers of one to one maps on the observables—that respect the $C^{★}$ algebra and that constitute a group for the composition of maps. The existence of a canonical evolution such as that referred to—which is a prominent example of emergent time—is one of the major mathematical contributions of Connes to a mathematical understanding of the notion of time flow. The synthesis of the ideas of Connes and Rovelli can be found in the article [22], where the authors suggest that the flow of time has a thermodynamical origin. For the more mathematically minded reader, let us elaborate on the contribution of Connes and Rovelli based on the article text. The temporal progression within a system emerges from and relies on its thermodynamic configuration, thereby establishing a thermodynamic basis for the flow of time. Within the framework of a quantum system’s von Neumann algebra of observables, one can perceive the temporal evolution as delineated by the one-parameter group of modular automorphisms, as stipulated by the Tomita–Takesaki theorem, corresponding to a faithful normal state on the algebra. Despite the state dependence of the modular automorphisms group, its uniqueness, modulo the inner automorphisms of the algebra, is guaranteed by the cocycle Radon–Nikodým theorem. This conceptualisation offers a cohesive approach to various foundational issues in quantum gravity’s temporal concept, the statistical mechanics of general relativity, and thermodynamics in quantum field theory.

3. Mathematical Examples of Emergence of a Local Flow of Time

Since our construction of the intuitive concept of local time derives from sensorial perceptions that require periodicity to build a referential time flow, most of our mathematical concepts describing the interactions of things in the world, such as the phenomena of evolution or change, require an a priori time variable. In mathematics, the flow of time is usually described either by a sequence $t_0,t_1,\cdots ,t_n,\cdots$ of real numbers—a sequence that is often identified with the integers $\{0,1,\cdots ,n,\cdots \}$—or, by a real variable t belonging to some interval $I\subseteq \mathbb{R}$ of the real line; in the first case, we have an instance of the so-called discrete time and in the second case, an instance of the so-called continuous time. In order to describe evolution—or variation in time—of measured quantities we consider maps, say f, defined in some time flow—discrete or continuous—and taking values in the space that contains the measured values; if this space has two or more dimensions, say N, then a value $f\left(t\right)=(x_1,x_2,\cdots ,x_N)$ means that all the measures $x_1,x_2,\cdots ,x_N$ were taken simultaneously. Our approach here is the following. We start with some set of possible time-dependent phenomena and we represent, mathematically, this set just as interacting concepts depending on simultaneous observations but possibly with only an implicit notion of time flow, and we then show that there exists a privileged—or canonical—time flow derived or associated, in most instances, from an invariant of the interacting object or concepts.

3.1. Curves in Euclidean Real Spaces

There are simple and common situations in elementary mathematics that can be taken as examples for the emerging of intuitive local time phenomena. Let us refer to an example with practical relevance that illustrates two important ideas.

• The idea of a totally ordered sequence of simultaneous interactions of objects or concepts is inherent to the perception of a time flow.
• The method of finding invariants in a phenomenon may allow the definition of a time that is intrinsic to the phenomenon.

Consider Figure 1 where the left and right components were taken from [23].

On the left, there is the evolution, in time, with two measured quantities: viral charge, $z\left(t\right)$, and CT4 antibody count, $u\left(t\right)$. On the right, the curve in the plane is drawn by taking the successive simultaneous measures of the two quantities to define the coordinates of the point in the plane, that is, a specific point of the curve corresponds to a pair $(u\left(t_0\right),z\left(t_0\right))$ of simultaneous measurements performed at a certain time $t_0\in [0,30]$.

Now, suppose that you start to analyse the right-hand side of Figure 1 with no prior information on time. If you assume time is at play, that is, each point of the curve is defined by its coordinates $(u\left(t_0\right),z\left(t_0\right))$ of simultaneous measurements performed at a certain value of a variable time $t_0$, but without knowing the governing law of this variable, there are many possibilities to define time dependency. But one of these possibilities deserves attention since it is a canonical one: it is the parametrisation by arc length of regular curves (see [24] (p. 9), [25] (pp. 38, 68), and [26] (pp. 76–80) for the French “reparamétrage par l’abcisse curviligne”); note that a regular curve is a curve with a non-zero tangent vector (see [24] (p. 8)). Let us briefly present the idea. If a plane curve is regular it is possible to define the length of any portion of it. We will briefly consider the heuristic idea about the length of a curve; suppose, for a planar curve C defined with $f:I\subset \mathbb{R}⟼\mathbb{R}$, a real-valued function defined on an interval I, such that $y=f\left(x\right)$, that is, $C=\{(x,f\left(x\right))\in {\mathbb{R}}^2:x\in I\}$; consider that the parametrisation is given by $x=t$. Then, an infinitesimal segment of the curve has length equal to $\sqrt{d{x}^2+d{y}^2}$, and so, the total length $\mathcal{L}\left(C\right)$ of the curve is given by (see [27] (p. 108)):

$$\mathcal{L}\left(C\right)={\int }_I\sqrt{d{x}^2+d{y}^2}={\int }_I\sqrt{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2}dx={\int }_I\sqrt{1+f^{\prime }{\left(x\right)}^2}dx.$$

Now, consider a curve C defined by a function $f:I\subset \mathbb{R}⟼{\mathbb{R}}^d$ that corresponds to a parametrisation of the curve, that is, $C=\{f\left(t\right)\in {\mathbb{R}}^d:t\in I\}$. Then, for $0<\Delta t\ll 1$, we have that,

$${∥f(t+\Delta t)-f\left(t\right)∥}_d={∥{\displaystyle \frac{f(t+\Delta t)-f\left(t\right)}{\Delta t}}∥}_d\Delta t,$$

and so,

$$\mathcal{L}\left(C\right){\approxeq }_{\Delta t\approxeq 0}\sum _{k=1}^{k\approxeq +\infty }{∥{\displaystyle \frac{f(t_0+(k+1)\Delta t)-f(t_0+k\Delta t)}{\Delta t}}∥}_d\Delta t,$$

with $t_0$ the variable value for the origin of the curve C, and so we should have,

$$\mathcal{L}\left(C\right)={\int }_I{∥{\displaystyle \frac{df\left(t\right)}{dt}}∥}_{d}dt={\int }_I{∥f^{\prime }\left(t\right)∥}_{d}dt,$$

(1)

which is an expression that we can find, for instance, in the form of a definition in terms of speed of the curve in [25] (p. 68) and [24] (p. 9) and in a deductive way with proofs in [26] (p. 78). We observe that the rigorous proof of Formula (1) requires a set of adequate hypotheses.

In order to have an unambiguous definition of $\mathcal{L}\left(C\right)$ in Formula (1), we have to show that the length of the curve does not depend on the parametrisation; let us detail next the invariance of the length of the curve by the change in parametrisation. Consider two parametrisations for the curve C given by $C=\{f_1\left(t\right)\in {\mathbb{R}}^d:t\in I=[a_1,b_1]\}$ and $C=\{f_2\left(t\right)\in {\mathbb{R}}^d:t\in I=[a_2,b_2]\}$. Consider also a derivable function h such that $h:[a_1,b_1]⟼[a_2,b_2]$ and observe that since $a_1<b_1$ and $a_2<b_2$, if $a_1<a_2$ we have that $h^{\prime }>0$. Then, by Formula (1) we have, by change of variables, that

$$\mathcal{L}\left(C\right)={\int }_{a_2}^{b_2}{∥f_2^{\prime }\left(u\right)∥}_{d}du={\int }_{a_2}^{b_2}{∥f_2^{\prime }\left(h\left(t\right)\right)∥}_d{h}^{\prime }\left(t\right)dt={\int }_{a_1}^{b_1}{∥f_1^{\prime }\left(t\right)∥}_{d}dt=\mathcal{L}\left(C\right).$$

(2)

If we have a regular curve, upon choosing an origin—let us say, one of the “ends” of the curve—the length of the portion of the curve between the chosen origin and some arbitrary point may be taken as the time stamp associated with that point. Since, as shown above, we have that the length of a curve is invariant by a change in parametrisation of the curve, we have that for any regular curve there is an intrinsic time associated with the curve given by its arc length parametrisation (see, for instance, Proposition 1.1.5 in [24] (p. 9)). As a final step, we now can formally detail the definition of the intrinsic flow of time emergent from a curve. Consider a curve C defined by a function $f_1:I=[a,b]\subset \mathbb{R}⟼{\mathbb{R}}^d$, which corresponds to a parametrisation given by $C=\{f_1\left(t\right)\in {\mathbb{R}}^d:t\in I\}$. Using Formula (1), define, for all $s\in I=[a,b]$, ${\mathcal{A}}_l\left(C\right)$, the arc length function of the curve C given by

$${\mathcal{A}}_l\left(C\right)\left(s\right)={\int }_a^s{∥f_1^{\prime }\left(t\right)∥}_{d}dt,$$

(3)

that is, the length of the piece of the curve C from the origin defined to be $f_1\left(a\right)$ until the point of the curve $f_1\left(s\right)$. Under the condition that for all $t\in I$ we have ${∥f_1^{\prime }\left(t\right)∥}_d>0$, we have that the arc length function ${\mathcal{A}}_l\left(C\right)$ defined from $I=[a,b]$ onto the interval $[0,\mathcal{L}(C\left)\right]$—recall that ${\mathcal{A}}_l\left(C\right)\left(b\right)$ is equal to the total length $\mathcal{L}\left(C\right)]$—is a one-to-one differentiable map. Let us consider $h:[0,\mathcal{L}(C\left)\right]⟼I=[a,b]$ defined by the inverse of ${\mathcal{A}}_l\left(C\right)$, that is, $h(·):={\mathcal{A}}_l{\left(C\right)}^{-1}(·)$ and consider $f_2(·)=(f_1\circ h)(·)$. We now have, using the fact that

$$h^{\prime }\left(t\right)={\displaystyle \frac{1}{{\mathcal{A}}_l{\left(C\right)}^{\prime }\left({\mathcal{A}}_l{\left(C\right)}^{-1}\left(t\right)\right)}}={\displaystyle \frac{1}{{∥f_1^{\prime }\left({\mathcal{A}}_l{\left(C\right)}^{-1}\left(t\right)\right)∥}_d}},$$

and the invariance of the length of curve with respect to parametrisation shown in Formula (2), that for all time $u\in [0,\mathcal{L}(C\left)\right]$,

$${\mathcal{A}}_l\left(C\right)\left(u\right)={\int }_0^u{∥f_2^{\prime }\left(t\right)∥}_{d}dt={\int }_0^u{∥{(f_1\circ h)}^{\prime }\left(t\right)∥}_{d}dt={\int }_0^u{∥(f_1^{\prime }\circ h)\left(t\right)∥}_d{h}^{\prime }\left(t\right)dt={\int }_0^{u}dt=u,$$

that is, the length of the curve C, at time u, is u. From this deduction, two observations can be pointed out. Firstly, that the parametrisation by arc length given by $f_2$ is such that ${∥f_2^{\prime }(·)∥}_d\equiv 1$, and secondly, that this parametrisation using arc length in the form $f_2(·)=(f_1\circ h)(·)$, with $h(·):=\mathcal{L}{\left(C\right)}^{-1}(·)$, allows us to define an intrinsic time—given by the function $\mathcal{L}{\left(C\right)}^{-1}(·)$ obtained from the arc length function defined in Formula (3)—that perfectly discriminates the points of the curve C, possibly corresponding to simultaneous measures of d quantities.

Consider now the example of a geodesic in a modified torus presented in Figure 2 (a figure that was prepared using software in [28]).

Let us observe that the existence of a canonical curve joining two sets of simultaneous measures is assured in many relevant situations; for instance, we may consider an idealised sphere around Earth—or any other regular surface (a regular surface is a surface having an injective differential map (see [24] (p. 33))) such as a torus—at a given altitude, for which a set of measures is taken at two points on the sphere: temperature, atmospheric pressure, relative humidity, etc. It is possible to show that there exists a curve of minimal length joining the two points on the said sphere, that is, a geodesic on the sphere, starting at one point and ending at the other.

The technical details of this statement may be summarised by quoting the renowned Hopf–Rinow theorem (see [29] (p. 26)) that shows that under mild conditions on a Riemannian manifold there always exists a geodesic of minimal length joining two points. For completeness we recall that a Riemannian manifold is, simply put, a manifold where a local notion of distance is defined by means of a positive definite symmetric matrix with entries given by smooth functions of the points of the manifold (see [29] (p. 13)). As a consequence, a manifold is a set that can locally be described by a set of coordinates taking values in ${\mathbb{R}}^d$ for some integer d; so, essentially, a manifold is locally akin to a Euclidean space. Of course, the structure of a Riemannian manifold—such as a sphere or a torus—is much richer than the structure of a Euclidean space and allows us to describe some very important examples of sets where simultaneous sets of measures—of physical quantities—can be given.

By choosing the parametrisation by arc length, we have a canonical quantitative local time flow emerging from the ordered sets of simultaneous measures; note that local in this context may be interpreted with the concept of proper time such as it is defined in [10] (p. 4). The general existence of a result of the type needed for this particular situation is well known in Riemannian geometry (see the first chapter of [29]).

We may conclude this instance of emergent time by saying that, in many relevant situations, whenever we have a set of simultaneous quantitative measures for which there exists a total ordering, there is an intrinsic time measure in the form of a flow of time. This shows that the definition of quantitative measures of (intuitive local) time can be achieved by means of a totally ordered set of simultaneous measures of different quantities for a set of different objects. In order to detail this idea suppose that you have a sequence of measures of d quantities $\mathcal{X}={(x_1^n,\cdots ,x_d^n)}_{1\le n\le N}$ that is completely ordered, meaning that

$$\forall k\in \{1,\cdots ,N-1\},(x_1^k,\cdots ,x_d^k)\prec (x_1^{k+1},\cdots ,x_d^{k+1}),$$

with ≺ denoting the total order we use for the points of $\mathcal{X}\subset {\mathbb{R}}^d$, an ordering possibly specific for the sequence $\mathcal{X}$. Suppose that there is a smooth map, that is, having a continuous derivative, $f:I=[a,b]\subset \mathbb{R}⟼{\mathbb{R}}^d$ such that

$$\forall k\in \{1,\cdots ,N\},\exists t_1,t_2,\cdots ,t_N\in [a,b],t_1<t_2<\cdots <t_N,f\left(t_k\right)=(x_1^k,\cdots ,x_d^k).$$

In practice, f may be a numerical interpolation curve for the sequence $\mathcal{X}$. By the results previously presented, there is an intrinsic time associated with the sequence $\mathcal{X}$ which is given by the arc length parametrisation of the curve initially parametrised by f.

3.2. Curves in Special Spaces

Since the dawn of the discovery of the calculus, the techniques developed from differential equations have been a most important modelling tool. The following passage that we quote from [30] (p. 7) shows that one of the creators of calculus already anticipated the usefulness of the approach: “Newton’s main discovery, the one which he believed essential to keep secret and which he only published in the form of an anagram, lies in the following sentence: “Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa”. We would translate this into modern mathematical language as: “It is useful to solve differential equations”.

Our next example of emerging time comes from the theory of differential equations, namely, the existence of integral curves for a given vector field. We may define a vector field as a map from a subset of a vector space—having a topological structure and possibly with an infinite number of dimensions—into the vector space. A function, defined on some interval of the real numbers into the vector space, is an integral curve of the vector field if, for each vector obtained as the function image of a real number of the said interval, the tangent vector to the vector is given by the vector field; the formulation of the problem of the existence and unicity of integral curves for a given vector field is a problem of the existence and unicity of solutions of differential equations. The important result that instantiates the emergence of time is that, for vector fields satisfying regularity hypotheses, there exists integral curves—that is, there exists a curve on the vector space—and moreover, that these integral curves are unique. This unicity shows that the problem of determining integral curves for a vector field implies the existence of a time flow for a family of possibly infinite simultaneous measures. It is possible to detail some aspects of this idea following [31,32]. For that purpose, we summarise the excellent exposition in [31] (pp. 132–136) in what follows. We consider a Banach space E, that is, a complete vectorial normed vector space; readers may take E as ${\mathbb{R}}^d=\{(x_1^{,}\cdots ,x_k,\cdots ,x_d):x_k\in \mathbb{R}\}$, the usual Euclidian space, with the usual norm of a vector x given by $∥x∥=\sqrt{x_1^2+\cdots +x_d^2}$, in case they are not familiar with the notion of Banach space. Consider an open set U in E and a mapping $f:U⟼E$, that is, a map that to each vector of U associates a vector of E. We suppose that the mapping f is $p\ge 1$ differentiable and that the p differential is continuous. Choose a certain point $u_0\in U$. We say that there exists an integral curve for the vector field f with initial condition $u_0$ if there exists a real function $\alpha :J⟼U$, with J an open interval of the real numbers containing zero, such that $\alpha \left(0\right)=u_0$, and,

$$\forall t\in J,{\alpha }^{\prime }\left(t\right)={\displaystyle \frac{d\alpha }{dt}}\left(t\right)=f\left(\alpha \left(t\right)\right),$$

(4)

an expression that can be interpreted by saying that the tangent vector of the vector $\alpha \left(t\right)$, a vector of the curve in E parametrised by the map $\alpha$, is given by the value of the vector field f at the point $\alpha \left(t\right)$, that is, $f\left(\alpha \right(t\left)\right)$. Formula (4) is a differential equation. The Cauchy–Lipschitz theorem shows—see Theorem 3.1 in [31] (p. 133)—that if we have for some $K>0$ that

$$\forall u,v\in U,∥f\left(u\right)-f\left(v\right)∥\le K∥u-v∥,$$

a condition known as a Lipschitz condition, then there exists an integral curve for the vector field f with initial condition $u_0$, and that this integral curve is essentially unique. Being so, we have that if the vector field verifies a Lipschitz condition then there is a curve depending on a canonical time in the space such that each point of the curve at a given time has a tangent vector given by the vector field.

3.3. Stochastic Curve Processes

In this section, we detail some instances of emergent time for the interaction of two random mathematical objects. For that purpose, we should first identify a suitable invariant of the interaction of the objects. We want to consider stochastic processes. A stochastic process, with continuous trajectories, may be seen as a random function for which a realisation—the equivalent in a game of dice being the result of a throw—is a particular function, a trajectory of the stochastic process. If we want to consider important processes—such as the Brownian process—an approach similar to the approach in Section 3.1 is impossible since, in general, the trajectories of such stochastic processes have infinite length in any of their pieces. So the attack strategy must be to look for some adequate invariant with respect to the variable that initially takes the role of time—at least when modelling some practical problem—and then, trying to detect a canonical time induced by this invariant.

The most promising starting idea at this point has to do with the scale invariance of the Brownian process also known as the Wiener process. We know that if ${\left(W_t\right)}_{t\ge 0}$ is a Wiener process, then for every $\alpha >0$ we have that

$${\displaystyle \frac{1}{\alpha }}{W}_{{\alpha }^{2}t}\stackrel{\mathrm{Law}}{\equiv }{W}_t,$$

(5)

that is, the scale-changed process on the left has the same probability law—the law or probability distribution that describes the way the particular trajectory is chosen at random—that the original process on the right has. This example points out that we can only expect invariance in the sense of identical probability laws—or probability distributions—and that a canonical time, for a wide class of stochastic processes, should be somehow connected to the change in time and to the change in measure in the sense of [33].

Let us suppose that we have a sequence of ordered points in ${\mathbb{R}}^2$, corresponding to a sequence of values of an ordered sequence of positive real numbers, and that there is a two-dimensional Brownian process with trajectories that are conditioned to have at each instant the corresponding value in the plane. Such a process exists under mild regularity assumptions; an example of an application of this idea is given in Figure 3 taken from [34], where the technical details of the procedure are presented.

We now have that Formula (5) can be used to define a canonical time associated with the curves defined by the trajectories of the conditioned two-dimensional Wiener process. In fact, for an end-time instance corresponding to a point of the curve there exists, for each interpolating trajectory, a value of the constant c such that the variable on the left-hand side of Formula (5) has the same probability law as the variable on the right-hand side. Let us detail this idea; suppose that you have a sequence of measures of two quantities $\mathcal{X}={(x_1^n,x_2^n)}_{1\le n\le N}$ that is completely ordered, meaning that,

$$\forall k\in \{1,2,\cdots ,N-1\},(x_1^k,x_2^k)\prec (x_1^{k+1},x_2^{k+1}),$$

with ≺ denoting the total order used in ${\mathbb{R}}^2$, an ordering possibly specific for the sequence $\mathcal{X}$. Suppose that there is a conditioned two-dimensional Wiener process ${(W_t^1,W_t^2)}_{t\ge 0}$ such that, almost surely for all trajectory indices $\omega \in \Omega$,

$$\forall k\in \{1,\cdots ,N\},\exists t_1,t_2,\cdots ,t_N\in [a,b],t_1<t_2<\cdots <t_N,(W_{t_k\left(\omega \right)}^1\left(\omega \right),W_{t_k\left(\omega \right)}^2\left(\omega \right))=(x_1^k,x_2^k).$$

(6)

In practice (see [34]), the realisations of the random variables $(W_{t_k}^1,W_{t_k}^2)$ are obtained by simulation and constitute an approximation; that being so, it is possible, using the neighbourhood recurrence of the non-conditioned two-dimensional Wiener process, to replace the points $(x_1^k,x_2^k)$ by neighbourhoods of these points. Let us use Formula (5) to define an intrinsic time for the interpolating random curve defined by the two-dimensional Wiener process. Choose a trajectory indexed by, say, ${\omega }_0\in \Omega$, such that Formula (6) holds. Then, there exists $c=c\left({\omega }_0\right)$ such that $c{\left({\omega }_0\right)}^2=t_N\left({\omega }_0\right)$. Now, we have that

$$\left({\displaystyle \frac{1}{c\left({\omega }_0\right)}}{W}_{c{\left({\omega }_0\right)}^{2}t}^1,{\displaystyle \frac{1}{c\left({\omega }_0\right)}}{W}_{c{\left({\omega }_0\right)}^{2}t}^2\right)\stackrel{\mathrm{Law}}{\equiv }(W_t^1,W_t^2),$$

So that there is a new stochastic process,

$${({\tilde{W}}_t^1,{\tilde{W}}_t^2)}_{t\in [0,1]}:={\left({\displaystyle \frac{1}{c\left({\omega }_0\right)}}{W}_{c{\left({\omega }_0\right)}^{2}t}^1,{\displaystyle \frac{1}{c\left({\omega }_0\right)}}{W}_{c{\left({\omega }_0\right)}^{2}t}^2\right)}_{t\in [0,1]},$$

such that the trajectory indexed by ${\omega }_0\in \Omega$ interpolates the sequence $\mathcal{X}$. We observe that $c{\left(\omega \right)}^2=t_N\left(\omega \right)$, where $t_N\left(\omega \right)$ is the random time such that Formula (6), verified for the point $(x_1^k,x_2^k)$, is a well-defined random variable that we may take as the definition of the intrinsic time of the interpolation process. The association of the constant c to each interpolation trajectory may be taken as the intrinsic random time for the interpolating stochastic curve.

3.4. Ito Line Integrals

In this next example, we consider stochastic lines in the plane that may be used to model functions of coupled simultaneous measures. In the case considered in [35], the stochastic line is defined as a pair of integrated diffusion processes—integrated in order to have smooth lines, that is, with a continuous derivative—one coordinate of the line for the price of a stock and the other coordinate for the volume of the same stock.

Consider now Figure 4, in which we have an example of such a line in a plane denoted the risk plane, since we may say that the higher the volume and the price, the higher the risk and vice versa.

It is possible to define a stochastic line integral of a non-anticipative process having smooth trajectories with respect to the usual Brownian motion, and the interesting result is that this integral is invariant in probability law by a change in the line parametrisation. Let us detail the particular case mentioned. We consider the price ${\left(S_t\right)}_{t\ge 0}$ and volume ${\left(V_t\right)}_{t\ge 0}$ of a stock as diffusion processes, solutions of stochastic differential equations.

$$\left\{\begin{array}{cc}d{S}_t=\mu (t,S_t,V_t)dt+\sigma (t,S_t,V_t)d{B}_t^{\left(1\right)}\hfill & S_0=S\left(0\right)\hfill \\ d{V}_t=\nu (t,S_t,V_t)dt+\rho (t,S_t,V_t)d{B}_t^{\left(2\right)}\hfill & V_0=V\left(0\right),\hfill \end{array}\right.$$

With these processes, we can associate the $\Delta t$ (e.g., daily) return processes defined by $r_t=r_t^{\Delta t}=ln(S_{t+\Delta t}/S_t)$ and $w_t=w_t^{\Delta t}=ln(V_{t+\Delta t}/V_t)$. And with the daily returns we can associate a stochastic line in the plane given by,

$$\gamma \left(t\right)=\left({\int }_0^t{r}_{u}du,{\int }_0^t{w}_{u}du\right),$$

with $\gamma \left(t\right)$ defined for $t\in [a,b]$, initial point $A=\gamma \left(a\right)$, and final point $B=\gamma \left(b\right)$. Now, observe that for a $\gamma$ stochastic, continuous, piecewise differentiable line, having $z=z\left(t\right)$ as a generic point of the line and f being a smooth enough process, the usual line integral is,

$${\int }_{\gamma }f\left(z\right)dz={\int }_a^{b}f\left(\gamma \left(t\right)\right){\gamma }^{\prime }\left(t\right)dt.$$

Note that, for f being locally Lipschitz continuous the Ito stochastic line integral with respect to the usual Brownian motion ${\left(B_t\right)}_{t\ge 0}$ may be defined by,

$${\int }_{\gamma }f\left(z\right)d{B}_z:={\int }_a^{b}f\left(\gamma \left(t\right)\right)·\sqrt{\mid {\gamma }^{\prime }\left(t\right)\mid }·exp(i{\displaystyle \frac{Arg{\gamma }^{\prime }\left(t\right)}{2}})d{B}_t.$$

For f being locally Lipschitz continuous we have Ito’s isometry-like formula, given by,

$$\mathbb{E}\left[{\left|{\int }_{\gamma }f\left(z\right)d{B}_z\right|}^2\right]=\mathbb{E}\left[{\int }_a^b\mid f\left(\gamma \left(t\right)\right){\mid }^2\mid {\gamma }^{\prime }\left(t\right)\mid dt\right].$$

It is also proved in [35] that with $\psi$, almost surely, a $C^1$ one-to-one map from $[c,d]$ onto $[a,b]$, adapted to the Brownian filtration, such that $\psi \left(c\right)=a$ and $\psi \left(d\right)=b$, then ${\gamma }_1:=\gamma \circ \psi$ is another stochastic line, defined in $[c,d]$, with the same initial and final points as $\gamma$, and the following integrals represent random variables with the same distribution or probability law, that is,

$${\int }_{{\gamma }_1}f\left(z\right)d{B}_z={\int }_{\gamma }f\left(z\right)d{B}_z,\mathrm{with\; the\; equality\; in\; distribution\; or\; in\; law}.$$

We can then, for almost all $\omega \in \Omega$, parametrise the random curve $\gamma \left(t\right)$ by arc length and have for each observed trajectory a local intuitive time flow. This being so, we have, as in Section 3.1, the possibility to define an intrinsic intuitive time flow by the arc length parametrisation of the curve. The difference resides in the fact that the time flow so defined is random, it changes with the trajectory according to some probability distribution.

4. Conclusions

We present historical and philosophical arguments allowing to isolate important characteristics of a notion of intuitive local time flow; intuitive in the sense of being generated by the ways sensorial information is processed by humans and local in the sense that it may be obtained by an individual without tools allowing them to take measurements at large distances and speeds, and so, corresponding to a particular inertial referential frame. These characteristics are, firstly, time flow—which requires periodic repetition of observable interactions among things in the world of an individual—and secondly, a notion of simultaneity of interactions among things in the world.

We next present some examples of mathematical constructions where, from what we could classify as measures of simultaneous interactions—given by vectors in a real or complex spaces—it is possible to determine a flow of time in the form of a curve in the space of observable interactions. That is, this flow of time is in the form of a map defined on an interval of real numbers and takes values in the space of observable interactions. These examples are first described in natural language in order to be understandable by readers with possibly no training in the mathematical formalism; following this more intuitive presentation, we develop mathematical formalism to rigorously express the main ideas of the examples. The mathematical tools used in this formalism pertain to the usual content of a four-year university course study in Mathematics.