The Nature of Time: Geometry, Physics and Perception

It has proven difficult to determine the nature of time. Of course physicists, mathematicians, psychologists, and philosophers have generated many ideas thereupon, as sampled in this volume. While these views may be debatable, they reflect the mind — consciousness indeed — and creativity of living observers of life.

Time perception is an important component of causality in the physical and social world. In our everyday life, and in our highly technological culture, we experience the common perception of time, as dictated by the clock, which allows us to clock-watch, and also to judge and estimate the duration of a given span of time, i.e., how much time seems to have gone by since a certain event.

We can regard an organism as an instrument for measuring physical stimuli, comparable to, for instance, a balance. Stimuli produce sensory experiences (“sensations”) and these sensations can be scaled, that is, measured. However, most often our sensory (subjective) experiences do not agree with the corresponding physical measures. Not only is the scatter much greater than in physical measurements; there are also systematic differences. The area in psychology that deals with the relation of subjective, psychological, to physical measures is called psychophysics. Though the main topic of this paper is the measurement of experienced time (duration), the presentation has to begin with an account of psychophysics in general.

Living systems are cognitive systems. Their cognitive processes involve the internal propagation of information, acquired from their environment, in order to produce action in the external world.

Efforts to understand psychological time have taken a range of forms, as revealed by many of the chapters of this book. A. Eisler (this volume) noted the wide variety of temporal experiences and time-research avenues in psychology, including the distinction between retrospective and prospective timing. Time is fundamental to event perception and conception [1], rhythm and music cognition [2] and intersensory integration [3], and the brain adapts accordingly to the temporal requirements of the environment. The brain integrates or segregates sensory information in an timely manner, linking events and forming representations that serve to situate past events and anticipate future ones. The brain adapts so well, and time is so ubiquitous, that meeting these requirements may be taken for granted. However, there are mechanisms for solving temporal requirements, and studying these mechanisms is fundamental for understanding behavior and adaptation. This could explain why so much effort has been devoted over the past 10 years to research on psychological time, either from a psychophysical, animal behavior, neuroscience or cognitive perspective, all of which are non-mutually exclusive.

The purpose of this paper is to propose how to include time into our understanding of consciousness and its possible physical underpinnings. Time and consciousness will be mentioned together twice. Firstly, time will be the part of an attempt to define phenomenal properties of conscious experience, qualia, in physical terms. Secondly, time will appear in the definition of consciousness as the subjective present.

“Temporal displacement” (Zeitverschiebung) is a term coined by Wilhelm Wundt [1] to denote the fact that in some cases perceived simultaneity does not correspond to physical contemporaneousness, and that sequences of very brief stimuli may be perceived as simultaneity or as reversed successions. He thought that in such cases there occurred a “displacement” of physical stimuli in phenomenal time (positive, when they are perceived after their physical occurrence; negative when they are perceived in advance). He investigated the phenomenon with the Complicationspendel shown in Figure 1.

There are different conceptions of time in different scientific disciplines. Apart from millennia of philosophical discussions of time [1], a particularly outstanding discrepancy persists between the concepts of time used in fundamental physical theories on the one hand and psychological theories of perception and cognition on the other.

What is time? St. Augustine remarked that when no one asked him, he knew what time was; however when someone asked him, he did not. Is time a process which flows? Is time a dimension in which processes occur? Does time actually exist?

Time is a basic ingredient of the universe. It is the dimension of change. Change has two meanings, however. There is real change in the sense that world states differing in date also differ in structure or function; and there is temporal change in the sense that states having been future become present and then past. In physics, only real change is acknowledged. Accordingly, time in physics exclusively is the dimension of real change.

Most scientists think that reality can be accurately described while ignoring human subjectivity, as Buccheri points out in this volume. The observer’s sense of being, the so-called ‘hard problem’ of consciousness [1] is considered not just irrelevant but an impossible problem [2,3]. Even those who think consciousness can be explained but only after the discovery of some ‘new physics’ [4], believe that reality can be objectively analysed without taking account of the mind of the observer or thinker. But they are wrong. Here, I show that failure to take the human mind into account in physics causes confusion about time and makes us attribute properties to the physical world that really arise from consciousness.

Recent works [1], [2], [3], [4] have raised the possibility that a quantum formalism can describe both unconscious mechanical brain processes connected with the complex patterns of neuronal activity as well as the conscious activity of the brain. A Hilbert structure has been assumed in order to describe an evolution equation like the Schroedinger equation.

In the introductory paper, Metod Saniga reviews his algebraic geometrical theory of pencil-spacetimes based on Cremona transformations between two projective spaces of three dimensions. These spacetimes exhibit an intimate connection between the extrinsic geometry of the time dimension and the dimensionality of space. Moreover, they seem to provide us with a promising conceptual basis for the possible reconciliation between two extreme concepts of time, viz. physical and psychological.

It goes without saying that a profound mystery lies behind the conventional notions of space and time. Thus, for example, the fact that there are three macroscopic dimensions of space was rigorously proved as early as the great Ptolemy some thousand years ago, yet contemporary science is still lacking any deeper and theoretically well-founded insight into the origin of this puzzling number. Deeper than the enigma of (the dimensionality of) space seems to be that of (the nature of) time. Here, there even exists a sharp contradiction between the way we perceive time and what modern physical theories tell us about the concept. To our senses, time appears to “flow,” “pass,” proceed inexorably from the past through (the unique moment of) the present into the future — the fact commonly known as the arrow of time. Yet, almost all the fundamental equations of physics are strictly timereversible and, in addition, they do not leave any proper place for the concept of the present, the “now.” This failure of current physical theories to properly account for the observed macroscopic dimensionality of space and the intricate nature of time is, in our opinion, asking for a serious revision of the generally adopted physical paradigms about the concepts in question. We think that there is a strong need for the representation of space and time that is more adherent to our perceptions and which includes, in particular, the irreversibility of change.

Traditionally, time has been modelled as a basic variable taking its values from an interval on a real axis. Although special relativity introduced Lorentz transformations mixing rectilinear time and space axes, while general relativity introduced curved spacetimes, the concept of a single underlying time dimension parametrised by a real interval remained. The pervasiveness of this concept was certainly due in large measure to the success of the models it supported, in particular to the expression of physical laws by differential equations which ultimately relied on the limiting process inherent in the notion of a (total or partial) derivativc. Despite this success at the computational level, it has long been clear that the truly ramified nature of time cannot be captured by what amounts to a mathematical convention. The current paper sets out to recall some of the perspectives on time and space that have been emerging from the study of biology and complex systems. Although these examples are still rather isolated and underdeveloped, they are already leading to some new insights. Summarising briefly, it is becoming apparent that each part of a complex system is equipped with its own intrinsic spacetime. When the system functions, the spacetimes of its constituent parts interact in various ways. As biological systems are able to insulate their component parts from environmental influences to a greater or lesser extent, one may propose an answer to Schrödinger’s question “What is life?” [1], characterising biological systems as those systems complex enough to isolate their component spacetimes. By contrast, the success of the single traditional “universal,” “clock,” or “calendar” time in physics is seen to be due to the way in which the component spacetimes of low-level systems are mutually coupled. This presents a different approach to universal time. Rather than being built in to our models ab initio, universal time should be seen as a phenomenon that emerges from the closely coupled interactions of low-level components.

It is generally accepted that events have position in time and in space, but time is not space. The time we experience is qualitatively very different to the physical space in which we exist. For instance, we can move back and forth in space but not in time. Although this observation is predicated on a number of unstated and possibly unwarranted assumptions which we cannot go into here, it is significant. One of the great paradoxes of twentieth century physics was that relativity, in both its special (SR) and general (GR) forms, became very successful precisely because it ignored this and other basic differences. Moreover, in a number of scenarios discussed below, advances in physics were obtained by going further and turning time into a pure imaginary spatial co-ordinate. In this article we review some of the situations where this occurs, starting with special relativity.

In the usual theory of relativity there is no evolution. Worldlines are fixed, everything is frozen once for all in a 4-dimensional “ block universe” V₄. This is in contradiction with our subjective experience of the passage of time. It is in contradiction with what we actually observe.

Since the inception of string theory there has been an incessant strive to find the underlying fundamental physical principle behind string theory, not unlike the principle of equivalence and general covariance in Einstein’s general relativity. This principle might well be related to the existence of an invariant minimal length scale (Planck scale) attainable in Nature. A scale relativistic theory involving spacetime resolultions was developed long ago by Nottale where the Planck scale was postulated as the minimum observer independent invariant resolution in Nature [1]. In [2] we applied this principle to the quantum mechanics of p-branes which led to the construction of C-space (a dimension category) where all p-branes were taken to be on the same footing; i.e. transformations in C-space reshuffled a string history for a five-brane history, a membrane history for a string history, for example. It turned out that Clifford algebras contained the appropriate algebro-geometric features to implement this principle of polydimensional transformations [5].

Many complex systems from physics, biology, society… exhibit a 1/f power spectrum in their time variability so that it is tempting to regard 1/f noise as a unifying principle in the study of time. The principle may be useful in reconciling two opposite views of time, the cyclic and the linear one, the philosophic view of eternity as opposed to that of time and death. The temporal experience of such complex systems may only be obtained thanks to clocks which are continuously or occasionally slaved. Here time is discrete with a unit equal to the averaging time of each experience. Its structure is reflected into the measured arithmetical sequence. They are resets in the frequencies and couplings of the clocks, like in any human made calendar. The statistics of the resets shows about constant variability whatever the averaging time: this is characteristic of the flicker (1/f) noise. In a number of electronic experiments we related the variability in the oscillators to number theory, and time to prime numbers. In such a context, time (and 1/f noise) has to do with Riemann hypothesis that all zeros of the Riemann zeta function are located on the critical line, a mathematical conjecture still open after 150 years.

Consider a physical system that may be observed through time-varying quantities x _t , where t stands for time that may be discrete or continuous. The set x _t may be a realization of a deterministic system, e.g. a unique solution of a differential equation, or a stochastic process. In the latter case each x _t is a random variable. We are interested in the global evolution of the system, not particular realizations x _t , from the point of view of innovation. We call the evolution innovative if the dynamics of the system is such that there is a gain of information about the system as time increases. Our purpose is to associate the concept of internal time with such systems. The internal time will reflect the stage of evolution of the system.

The perception of time is given by the happening of some events that determines a variation in the state of the observed system. In this sense a computation, i.e. a set of well defined transformations that, starting from an initial state (the input) brings to a final state (the output), can be considered a time generator. Each ticking of the clock corresponds to the computer changes of its states. The speed of computation leads to a different perception of time as well as traveling by airplanes changed the perception of spatial distances.

The present paper is intended to discuss how to develop the statistical approach proposed in [1, 2]. According to the relational (or relative) principle, the properties of a model clock represent the properties of physical time. (This relational view goes back to physicists and philosophers such as Leibniz, Mach, Einstein, Poincare, and others) To construct a model of physical time implies constructing a model instrument for its measurement, namely, a clock. In the postulated basic equation, the increment of time is expressed through the average value of increments of spatial coordinates of particles of the system under consideration, and time and space are closely connected. The relationships correspond to conservation laws, and the standard motion equations are derived from this basic correspondence.

Asymmetry between past and future is not usually considered to be inherent to modern science. Instead, such asymmetry is interpreted as an asymmetry in the initial conditions imposed on prediction and reconstruction problems. In the present work the authors analyze the time evolution of systems with memory, using cellular automaton models.

As is readily apparent from chapters 1 and 2, the notion of time per theories of physics, i.e., physical time, seems to lack crucial subjective qualities such as a preferred present and a direction/arrow. The papers in the first part of this chapter do make strides in reconciling psychological and physical time. However, subsequent papers indicate the nature of time per physics differs profoundly from the prima facie subjective experience. Thus, in seeking a theory of quantum gravity, physicists are exploring programs that will render the disparity between physical and psychological time even more pertinacious.

In this article, a topological perspective will be used to establish the long sought for, non-statistical, connection between dynamic mechanical systems, thermodynamic irreversibility, and the arrow of time1. In effect, it will be demonstrated that the Boltzmann paradox can be resolved in terms of Continuous Topological Evolution [1], which differs from classical geometric theories of Continuous Evolution without topological change. Consider the definitions: 1.Causal evolution is defined as a map of C1 functions from a domain of base variables to a unique range of base variables. The maps may be many to one and are not necessarily homeomorphisms.2.Prediction implies that well behaved functional forms (not just numeric point data) on the range of base variables can be deduced from functional forms defined on the domain of base variables.3.Retrodiction implies that functional forms on the domain can be deduced from functional forms on the range.

The author attended an earlier Workshop of this group, in Palermo, 1999, presenting a paper outlining the notion that time, or least time’s arrow, arises from the repeated collapse of quantum mechanical uncertainties in positions and momenta into specific and precise values of one molecule under impact from another [1].

My purpose in this report is to shortly describe how to deal with the arrow of time in quantum theories.

Recent observational results (cf. Refs. [1, 2, 3, 4, 5, 6], in particular results of studies of distant supernovae as standard candles [2, 4, 6] and studies of anisotropy of cosmic microwave background [3], set forth a problem of the cosmological constant once again. The most probable values of Ω_M and Ω_V, i. e. the normalized ratios (with respect to the critical density) of matter density and cosmological-constant energy density, turned out to be most probably equal to 0.3–0.4 and 0.6–0.7 respectively. By usual definition, Ω_M = P_M/P _cr , Ω_v = P_v/P _cr , where P_M is the density of matter, P_M is the density of vacuum; the critical density p _cr = 3H2/8πG; H is the Hubble parameter, G is the gravitational constant. According to the observational tests, the sum of Ω_M and Ω_V seems to be greater than one; this implies that the Universe is closed.

Causality is a very important ingredient of physical theories and is considered to be a fundamental principle. However strict definitions cannot be substituted by slogans. If causality is true in its strictest possible sense as used everywhere in physics outside of General Relativity (GR), then retrocausality is impossible, so either retrocausal observations are untrue or physics is. However we do not have to take this standpoint here, since GR does not seem to support it. For example, Novikov’s paper [1] discusses world lines which are nonmonotonic functions of time within orthodox GR. We argue that GR does not support free and unrestricted travel backward in time. GR claims that spacetime has a unique, objective structure, i.e., geometry, and that this is valid for any observer. Retro motions should be possible for anyone if they are at all possible on spacetime, otherwise for no one. (We are not speaking here about facts; anyway, exactly what are facts and what are not facts is a matter for debate.) This does not mean that retro travel would be equally easy for everybody.

It seems to be extremely difficult to give a precise definition of Time, this mysterious ingredient of the Universe. Intuitively, we have the notion of time as something that flows. Ancient religions have registered it as something unusual, and many myths are built into their dogmas.

The problem associated with time’s nature is well known. It stems from two aspects of time that cannot be reconciled: 1.Time sharply differs from space in that in space, you can either move or stay put, and if you move you can do it in either direction. Not so with time: You cannot remain at the same moment, neither return to earlier moments. Time seems, then, to constantly move. 2.The last sentence in nonsensical. Time cannot move neither can anything move in time, as the very notion of movement (passage, flow, etc.) entails time. Just ask ”what is the speed of time’s movement?” and the absurdity of the statement will become apparent. You can, of course, assume another time parameter of a higher order, but that will necessitate a yet higher time dimension and so on ad infinitum.

In general the problem of causality is closely connected with basic philosophical problems. The main philosophical problem concerns a relation between matter and spirit. There are two extreme points of view on this problem: 1.matter is primary and spirit is secondary;2.spirit is primary and matter is secondary.

Time in Lorentz-invariant physics is connected with a metric signature. The spacetime metric for any space can be written as (1)$$d{{s}^{2}} = {{g}_{{\mu \nu }}}d{{x}^{\mu }}d{{x}^{\nu }} = {{e}^{{\bar{a}}}}{{e}^{{\bar{b}}}}{{\eta }_{{{{{\bar{a}}}_{{\bar{b}}}}}}} = \left( {h_{\mu }^{{\bar{a}}}d{{x}^{\mu }}} \right)\left( {h_{\nu }^{{\bar{b}}}d{{x}^{\nu }}} \right){{\eta }_{{\bar{a}\bar{b}}}}$$ where $$ g_{\mu \nu } = h\begin{array}{*{20}c} {\bar a} \\ \mu \\ \end{array} h\begin{array}{*{20}c} {\bar b} \\ \nu \\ \end{array} \eta _{\bar a\bar b} $$ is the metric, $$ e^{\bar a} = h_\mu ^{\bar a} dx^\mu $$ is a 1-form, $$ h_\mu ^{\bar a} $$ is a vierbein, a is a vier-beiu index, μ is the coordinate index and $$ \eta _{\bar a\bar b} $$ is the metric signature $$ \eta _{\bar a\bar b} = diag\left\{ {\sigma ,1,1,1} \right\} $$. An undefined number σ can be +1 for Euclidean space and -1 for Lorentzian spacetime. We see that the difference between Euclidean and Lorentzian spacetimes is connected to the sign of $$ \sigma = \eta _{\bar 0\bar 0} = \pm 1 $$. For $$ \eta _{\bar 0\bar 0} = + 1 $$ we have Euclidean space and for $$ \eta _{\bar 0\bar 0} = - 1 $$, Lorentzian spacetime.

In 1935 Einstein, Podolsky and Rosen proposed a thought experiment designed to highlight the incompleteness of quantum mechanics. There have since been a number of variants on the original EPR theme, but the set-up considered in this report consists of a neutral pion decaying to an electron-positron pair, π0 → e- + e+. Electrons and positrons possess spin components of ±½ (denoted by ‘up’ and ‘down’ in some frame) whereas pions are spinless, so by conservation of angular momentum a spin-up electron is partnered by a spin-down positron, and vice versa. Given that Hidden Variables types of interpretation have been empirically ruled out, quantum mechanics states that before the spin of one of the particles is measured, both it and its partner are in both spin states simultaneously, and the outgoing wavefunction | Ψ〉 is described by a linear superposition: (1)$$ \left| \psi \right\rangle = \frac{1} {{\sqrt 2 }}\left( {\left| \uparrow \right\rangle e \otimes \left| \downarrow \right\rangle _p - \left| \downarrow \right\rangle e \otimes \left| \uparrow \right\rangle _p } \right). $$

Recent progress in quantum gravity [1] and string theory [2] has raised interest among scientists as to whether or not nature “behaves discretely” at the Planck scale. However, it is not clear what this metaphor means or how it should be implemented into systematic study concerning physics and mathematics in the Planck regime.

In Wheeler’s pregeometry, one attempts to derive properties of the spacetime manifold, such as metric, continuity, dimensionality, topology, locality, symmetry, and causality from an otherwise structureless set [1]. Requardt & Roy refer to this methodologically reductionist attempt to model the properties of spacetime as “bottom up pregeometry” [2]. In an early attempt to derive spacetime dimensionality, Wheeler assigned probability amplitudes to the members of a Borel (structureless) set to stochastically establish spacetime adjacency [3]. Wheeler abandoned this idea, in part, because “too much geometric structure is presupposed to lead to a believable theory of geometric structure” [4]. In particular, he considered the manner in which probability amplitudes were assigned, and a metric introduced, to be ad hoc. However, recent models by Nagels [5], Antonsen [6], and Nowotny & Requardt [7] employing graph theory have, arguably, surmounted these objections.

In the previous three chapters, the concept of time has been discussed by psychologists, mathematicians and physicists almost exclusively within their own domains of interest. In the first chapter, the perception of time and its elaboration within our consciousness has been analysed. The analysis has been carried out on the basis of: a) experimental and theoretical results of the on-going research activity in psychology and b) implications from new consciousness studies in physics and psychology, with little or no resort to ad hoc assumptions foreign to the rigour of science. The second chapter focussed primarily on the mathematical formalisms employed to approach and describe both the subjective and physical aspects of time, and their envisaged connections as well, with a firm adherence to the universally adopted scientific paradigms. The physicists’ points of view have been the contents of chapter three. Though rich in interpretative news, arising mainly from the current debate about the interpretation of Quantum Mechanics, they could not result, due to the traditional methodological rigour always adopted in physics, in sic et simpliciter acceptance of notions foreign to the language of physics. As a result, the three chapters provide the reader with an extensive and variegated, yet firmly scientific, discussion of the concept in question.

“What is time?” That is, what is its nature or essence? “Is it a real thing or not real?” Aristotle raises these ontological questions in Physics (217b).1 He endeavors to provide answers intended to grasp as firmly as possible the elusive entity of chronos (time), which constitutes one of his ten categories, or generic modes of being, in the sense of something “being in time” or “the when” (pote), the “at what time” an event takes place.2 He maintains that natural beings are distinguished from artificial and man-made things by having “within themselves the capacity to move” and to change in various ways. These are determined by the categories involved: e.g. substance in the case of generation and destruction, quantity in the case of increase and decrease, quality in the case of alteration, and place in the case of locomotion. Time, like motion, with which it is ontologically and psychologically connected, as we will see, is a “continuous quantity” (syneches poson) or “magnitude” (megethos). This means that, like all continua, time is characterized by divisibility ad infinitum and, therefore, by the fact that there is not a minimum of time, although time has a limit, the point-like “now” (nun) which divides past from future time.

The concept of endo-physics was introduced [1–2] in the context of selfreference in quantum physics and artificial intelligence (AI) research. Now, regarding quantum physics, it implicitly relies on something existing outer to itself, namely, the existence of an observer related to consciousness. In order to evaluate the relation of consciousness to (quantum) physics, it is necessary to recognise the fact that physical behaviour follows the principle of least action (action principle) which requires the behaviour to be directed towards the physical equilibrium. Now since biology follows a principle which acts against the physical principle, attempting to preserve the distance from physical equilibrium, and consciousness allegedly develops only in biological organisms (at least in the observer concept of quantum physics), the observer lives on an ontologically different level from the physical realm [3–5]. Therefore, the epistemological structure of the observer-observed relation contains a two-levelled ontological structure. Regarding AI, it relies on “creators” of hardware and software, therefore its epistemology is that of the deistic materialism [6], i.e. another ontologically two-levelled structure. Now self-modification, the concept closely related to endophysics [7], is characteristic for a biological and/or psychological level of ontology [4–5]. Therefore, it is an important recognition that self-governing systems follow developmental paths different from physical entities. Therefore endo-physics, putting the observer into a physical context, compresses the two-levelled ontological structure into a monistic ontology and creates epistemic paradoxes with Klein-bottle structures. Nevertheless, when we are aware that there is an at least two-levelled ontology behind the concept of endophysics, it is easy to remove certain types of paradox epistemic structures, like Klein-bottles, since self-reference becomes simply reference of one existential level to another.

The most fascinating of the riddles of time is the so-called passage of time which, however, is not accounted for by natural science. In this contribution, we suggest an approach to the phenomena of time which aims at explaining why time passes for us.1

One of the main activities of human beings has always been to understand the laws ruling the external world, in order to keep it under control as much as possible. A theoretical scenario for an overall representation of the world provides a practical tool for this purpose. Such a scenario is usually built by combining knowledge obtained from everyday observations into a general framework containing all those a priori beliefs necessary for a complete and satisfactory global view of the world. New knowledge, however, is not always compatible with existing scenarios, and gives rise to questions requiring answers capable of eliminating any recognized contradictions. New and careful observations do provide the desired answers, but at the cost of modifying, sometimes drastically, existing scenarios. In the long run this results in a reduction of our a priori beliefs, with the advantage of achieving a more precise and complete representation of the universe.

The approach of philosophy to time differs from that of physics. Whenever these differences are disregarded there is a danger to use the same term for differently perceived phenomena and then false conclusions may be drawn. In general, physics operates with actual time concepts, whilst philosophy generally considers time as a potential form of existence. The two can be brought in accordance, although we should respect the limits how far we can use this or that time concept in the realm of the other discipline. In this article I investigate the temporal sequence as it occurs in an example, which follows the ontological appearance of a phenomenon, namely the sequence of symmetry breaking in the evolution of matter. This process starts with the physical appearance of matter and continues beyond the organic forms of being, up to the development of the human mind. This special outlook beyond physics throws a new light on the comparison between the particular time concept applied by physics and the general time concept of philosophy applicable to all the three fundamental ontological levels. First we restate the general laws of symmetry breaking and then we investigate what role time can play in defining and arranging ontological levels along broken symmetries. I focus on the following issues: • Peculiarities of time and its difference from space,• The actual and potential properties of the time concept,• Time and the direction of the evolution of matter.

One may formulate some implicit premises that dominate in current natural science: • Studies of time are performed by philosophy rather than by natural science.• Time in science is an initial, undefinable concept.• To measure time, it is sufficient to have physical clocks on the basis of gravitational or electromagnetic processes.• The problems of time in natural science are the solved or unsolved problems of relativity theory.• Our Universe is an isolated system.• The conceptual armoury of science has no place for substances like phlogiston, light-bearing ether, entelecheia, etc.