Undoubtedly, the concept of time is possibly one of the greatest mysteries of nature. The nature of time has always been a subject of debate both from the point of view of philosophy and physics. But this has taken on special relevance as a consequence of the development of the theory of relativity, which has marked a turning point in the perception of space-time.

Throughout history, different philosophical theories have been put forward on the nature of time [1], although it has been from the twentieth century onwards when the greatest development has taken place, mainly due to advances in physics. Thus, it is worth mentioning the argument against the reality of time put forward by McTaggart [2], such that time does not exist and that the perception of a temporal order is simply an appearance, which has had a great influence on philosophical thought.

However, McTaggart’s argument is based on the ordering of events, as we perceive them. From this idea, several philosophical theories have been developed, such as A-theory, B-theory, C-theory and D-theory [3]. However, this philosophical development is based on abstract reasoning, without relying on the knowledge provided by physical models, which raises questions of an ontological nature. 

Thus, both relativity theory and quantum theory show that the emergent reality is an observable reality, which means that in the case of space-time both spatial and temporal coordinates are observable parameters, emerging from an underlying reality. In the case of time this raises the question: Does the fact that something is past, present or future imply that it is something real? Consequently, how does the reality shown by physics connect with the philosophical thesis?

If we focus on an analysis based on physical knowledge, there are two fundamental aspects in the conception of time. The first and most obvious is the perception of the passage of time, on which the idea of past, present and future is based, which Arthur Eddington defined as the arrow of time [4], which highlights its irreversibility. The second aspect is what Carlo Rovelli [5] defines as “loss of unity” and refers to space-time relativity, which makes the concept of past, present and future an arbitrary concept, based on the perception of physical events.

But, in addition to using physical criteria in the analysis of the nature of time, it seems necessary to analyze it from the point of view of information theory  [6], which allows an abstract approach to overcome the limitations derived from the secrets locked in the underlying reality. This is possible since any element of reality must have an abstract representation, i.e. by information, otherwise it cannot be perceived by any means, be it sensory organ or measuring device, so it will not be an element of reality.

The topology of time

From the Newtonian point of view, the dynamics of classical systems develops in the context of space-time of four dimensions, three spatial dimensions (x,y,z) and one temporal dimension (t), so that the state of the system can be expressed as a function of the generalized coordinates q and the generalized moments p as a function f(q,p,t), where q and p are tuples (ordered lists of coordinates and moments) that determine the state of each of the elements that compose the system.

Thus, for a system of point particles, the state of each particle is determined by the coordinates of its position q = (x,y,z) and of its momentum p = (mẋ, mẏ, mż). This representation is very convenient, since it allows the analysis of the systems by calculating continuous time functions. However, this view can lead to a wrong interpretation since identifying time as a mathematical variable makes it conceived as a reversible variable. This becomes clear if the dynamics of the system is represented as a sequence of states, which according to quantum theory has a discrete nature [7] and can be expressed in terms of a classical system (CS) as:

        CS = {… S_i-2(q_i-2,p_i-2), S_i-1(q_i-1,p_i+-), S_i(q_i,p_i), S_i+1(q_i+1,p_i+1), S_i+2(q_i+2,p_i+2),…}

According to this representation, we define the past as the sequence {… S_i-2(q_i-2,p_i-2), S_i-1(q_i-1,p_i+-)},  the future as the sequence {S_i+1(q_i+1,p_i+1), S_i+2(q_i+2,p_i+2),…} and the present as the state S_i(q_i,p_i). The question that arises is: Do the sequences {… S_i-3(q_i-3,p_i-3), S_i-2(q_i-2,p_i-2), S_i-1(q_i-1,p_i+-)} y {S_i+1(q_i+1,p_i+1), S_i+2(q_i+2,p_i+2), S_i+3(q_i+3,p_i+3),…} have real existence? Or on the contrary: Are they the product of the perception of the emergent reality?

In the case of a quantum system its state is represented by its wave function Ψ(q), which is a superposition of the wave functions that compose the system:

        Ψ(q,t) = Ψ(q₁,t) ⊗ Ψ(q₁,t) …⊗ Ψ(q_i,t) …⊗ Ψ(q_n,t)

Thus, the dynamics of the system can be expressed as a discrete sequence of states:

        QS = {… Ψ_i-2(q _i-2), Ψ_i-1(q _i-1), Ψ_i(q _i), Ψ_i+1(q _i+1), Ψ_i+2(q _i+2), …}

As in the case of the classical system Ψ_i(q) would represent the present state, while {… Ψ_i-2(q), ΨY_i-1(q)} represents the past and {Ψ_i+1(q), Ψ_i+2(q), …} the future, although as will be discussed later this interpretation is questionable.

However, it is essential to emphasize that the sequences of the classical system CS and the quantum system QS have, from the point of view of information theory, a characteristic that makes that their nature, and therefore their interpretation, must be different. Thus, quantum systems have a reversible nature, since their dynamics is determined by unitary transformations [8], so that all the states of the sequence contain the same amount of information. In other words, their entropy remains constant throughout the sequence:

        H(Ψ_i(q _i)) = H(Ψ_i(q _i+1)).

In contrast, classical systems are irreversible [9], so the amount of information of the sequence states grows systematically, such that:

        H(S_i(q_i,p_i)) < H(S_i+1(q_i+1,p_i+1)).

Concerning the entropy increase of classical systems, the post “An interpretation of the collapse of the wave function” has dealt with the nature of entropy growth from the “Pauli’s Master Equation” [10], which demonstrates that quantum reality is a source of emergent information towards classical reality. However, this demonstration is abstract in nature and provides no clues as to how this occurs physically, so it remains a mystery. Obviously, the entropy growth of classical systems assumes that there must be a source of information and, as has been justified, this source is quantum reality.

This makes the states of the classical system sequence distinguishable, establishing a directional order. On the contrary, the states of the quantum system are not distinguishable, since they all contain the same information because quantum theory has a reversible nature. And here we must make a crucial point, linked to the process of observation of quantum states, which may lead us to think that this interpretation is not correct. Thus, the classical states emerge as a consequence of the interaction of the quantum components of the system, which may lead to the conclusion that the quantum states are distinguishable, but the truth is that the states that are distinguishable are the emerging classical states.

According to this reasoning the following logical conclusion can be drawn. Time is a property that emerges from quantum reality as a consequence of the fact that the classical states of the system are distinguishable, establishing in addition what has been called the arrow of time, in such a way that the sequence of states has a distinguishable characteristic such as the entropy of the system.

This also makes it possible to hypothesize that time only has an observable existence at the classical level, while at the quantum level the dynamics of the system would not be subject to the concept of time, and would therefore be determined by means of other mechanisms. In principle this may seem contradictory, since according to the formulation of quantum mechanics the time variable appears explicitly. In reality this would be nothing more than a mathematical contraption that allows expressing a quantum model at the boundary that separates the quantum system and the classical system and thus describe the classical reality from the quantum mathematical model. In this sense it should be considered that the quantum model is nothing more than a mathematical model of the emerging reality that arises from an underlying nature, which for the moment is unknown and which tries to be interpreted by new models, such as string theory.

An argument that can support this idea is also found in the theory of loop quantum gravitation (LQG) [11], which is defined as a substrate-independent theory, meaning that it is not embedded in a space-time structure, and which posits that space and time emerge at distances about 10 times the Planck length [12].

The arrow of time

When analyzing the sequences of states CS and QS we have alluded to the past, present and future, which would be an emergent concept determined by the evolution of the entropy of the system. This seems clear in classical reality. But as reasoned, the sequence of quantum states is indistinguishable, so it would not be possible to establish the concept of past, present and future.

A fundamental aspect that must be overcome is the influence of the Newtonian view of the interpretation of time. Thus, in the fundamental equation of dynamics:

        F = m d²x/dt²

the variable time is squared, this indicates that the equation does not distinguish t from -t, i.e., it is the same backward or forward in time, so that the dynamics of the system is reversible. This at the time led to Laplace’s causal determinism, which remained in force until the development of statistical mechanics and Boltzmann’s interpretation of the concept of entropy. To this we must add that throughout the twentieth century scientific development has led without any doubt to the conclusion that physics cannot be completely deterministic, both classical physics and quantum physics [13].

Therefore, it can be said that the development of calculus and the use of the continuous variable time (t) in the determination of dynamical processes has been fundamental and very fruitful for the development of physics. However, it must be concluded that this can be considered a mathematical contraption that does not reflect the true nature of time. Thus, when a trajectory is represented on coordinate axes, the sensation is created that time can be reversed at will, which would be justified by the reversibility of the processes.

However, classical processes are always subject to thermodynamic constraints, which make these processes irreversible, which mean that for an isolated system its state evolves in such a way that its entropy grows steadily and therefore the quantity and information describing the system, so that a future state cannot be reverted to a past state. Consequently, if the state of the system is represented as a function of time, it could be thought that the time variable could be reverted as if a cursor were moved on the time axis, which does not seem to have physical reality, since the growth of entropy is not compatible with this operation.

To further emphasize the idea of the possibility of moving in time as if it were an axis or a cursor, we can consider the evolution of a reversible system, which can reach a certain state S_i and continue to evolve, and after a certain moment it can reach the state S_i again. But this does not mean that time has been reversed, but rather that time always evolves in the direction of the dynamics of the system and the only thing that happens is that the state of the system can return to a past state in a reversible way. However, in classical systems this is only a hypothetical proposal, since reversible systems are ideal systems free of thermodynamic behavior, such as gravitational, electromagnetic and frictionless mechanical systems. To say, ideal models that do not interact with an underlying reality.

In short, the state of a system is a sequence determined by an index that grows systematically. Therefore, the idea of a time axis, although it allows us to visualize and treat systems intuitively, should be something we should discard, since it leads us to a misconception of the nature of time. Therefore, time is not a free variable, but the perception of a sequence of states.

Returning to the concept of past, present and future, it can be assured that according to information theory, the state of present is supported by the state S_i(q_i,p_i), and therefore is part of the classical reality. As for the sequence of past states {… S_i-3(q_i-3,p_i-3), S_i-2(q_i-2,p_i-2), S_i-1(q_i-1,p_i-1)}  to be a classical reality would require that these states continue to exist physically, something totally impossible since it would require an increase of information in the system that is not in accordance with the increase of its entropy, so this concept is also purely perceptual. On the other hand, if this were possible the system would be reversible.

In the case of the future sequence of states {S_i+1(q_i+1,p_i+1), S_i+2(q_i+2,p_i+2),…}  it is a classical reality for occurring with a degree of uncertainty that makes it not predictable. Even supposing this were possible, the states of the present would have to increase the amount of information to hold accurate forecasts of the future, which would increase their entropy, which is at disagreement with observable reality. Therefore, the concept of the future is not a classical reality, being a purely perceptual concept. In short, it can be concluded that the only concept of classical reality is the state of the present.

The relativistic context

Consequently, classical systems offer a vision of reality as a continuous sequence of states, while quantum physics modifies it, establishing that the dynamics of systems is a discrete sequence of states. However, the classical view is no more than an appearance at the macroscopic level. However, the theory of relativity [14] modifies the classical view, such that the description of a system is a sequence of events. If to this we add the quantum view, the description of the system is a discrete sequence of events.

But in addition, the theory of relativity offers a perspective in which the perception of time depends on the reference system and therefore on the observer. Thus, as the following figure shows, clocks in motion are slower than stationary clocks, so that we can no longer speak of a single time sequence, but that it depends on the observer.

However, this does not modify the hypothesis put forward, which is to consider time as the perception of a sequence of states or events. This reinforces the idea that time emerges from an underlying reality and that its perception varies according to how it is observed. Thus, each observer has an independent view of time, determined by a sequence of events.

In addition to the relative perception of time, the theory of relativity has deeper implications, since it establishes a link between space and time, such that the relativistic interval

        ds² = c² dt² – dx² – dy² – dz² = c² dt² – (dx² + dy²+ + dz²)

is invariant and therefore takes the same value in any reference frame.

As a consequence, both the perception of time and space depends on the observer and as the following figure shows, simultaneous events in one reference frame are observed as events occurring at different instants of time in another reference frame, so that in this one they are not simultaneous, giving rise to the concept of relativity of simultaneity.

In spite of this behavior, the view of time as the perception of a sequence of events is not modified, since although the sequences of events in each reference system are correlated, in each reference system there is a sequence of events that will be interpreted as the flow of time corresponding to each observer.

The above arguments are valid for inertial reference frames, i.e. free of acceleration. However, the theory of general relativity [15], based on the principles of covariance and equivalence, establishes the metric of the deformation of space-time in the presence of matter-energy and how this deformation acts as a gravitational field. These principles are defined as:

• The Covariance Principle states that the laws of physics must take the same form in all reference frames.
• The Equivalence Principle states that a system subjected to a gravitational field is indistinguishable from a non-inertial reference frame (subjected to acceleration).

It should be noted that, although the equivalence principle was fundamental in the development of general relativity, it is not a fundamental ingredient, and is not verified in the presence of electromagnetic fields. 

It follows from the theory of general relativity that acceleration bends space-time, paradigmatic examples being the gravitational redshift of photons escaping from the gravitational field, or gravitational lenses. For this reason, it is essential to analyze the concept of time perception from the point of view of this perspective.

Thus, the following figure shows a round trip to Andromeda by a spacecraft propelled with acceleration a = g. It shows the time course in the Earth reference frame t and the proper time in the spacecraft reference frame T, such that the time course on Earth is slower than in the spacecraft by a value determined by g. The fact that the time course is produced by the velocity of the spacecraft in an inertial system or by the acceleration of the spacecraft does not modify the reasoning used throughout the test, since the time course is determined exclusively in each of the reference systems by the sequence of events observed in each of them independently.

Therefore, it can be concluded that the perception of time is produced by the sequence of events occurring in the observing reference system. To avoid possible anthropic interpretations, an entity endowed with the ability to detect events and to develop artificial intelligence (AI) algorithms can be proposed as an observer. As a consequence, it can be concluded that the entity will develop a concept of time based on the sequence of events. Evidently, the developed concept will not be reversible, since this sequence is organized by an index.

However, if the event detection mechanisms were not sufficiently accurate, the entity could deduce that the dynamics of the process could be cyclic and therefore reversible. However, the sequence of events is ordered and will therefore be interpreted as flowing in a single direction.

Thus, identical entities located in different reference systems will perceive a different sequence of events of the dynamics, determined by the laws of relativity. But the underlying reality sets a mark on each of the events that is defined as physical time, and to which the observing entities are inexorably subject in their real time clocks. Therefore, the question that remains to be answered is what the nature of this behavior is.

Physical time

So far, the term perception has been used to sidestep this issue. But it is clear that although real time clocks run at different rates in different reference systems, all clocks are perfectly synchronized. But for this to be possible a total connection of the universe in its underlying reality is necessary. This must be so, since the clocks located in the different reference systems run synchronously, regardless of their location, even though they run at different speeds.

Thus, in the example of the trip to Andromeda, when the ship returns to Earth, the elapsed time of the trip in the Earth’s reference system is T = 153.72 years, while in the ship’s clock it is t = 16.92 years, but both clocks are synchronized by the parameter g, so that they run according to the expression dt = γdT. The question arises: What indications are there that the underlying reality of the universe is a fully connected structure?

There are several physical clues arising from relativistic and quantum physics, such as space-time in the photon reference frame and quantum particle entanglement. Thus, in the case of the photon γ→∞, so that any interval of time and space in the direction of motion in the reference frame of the observer tends to zero in the reference frame of the photon. If we further consider that the state of the photon is a superposition of states in any direction, the universe for a photon is a singular point without space-time dimensions. This suggests that space-time arises from an underlying reality from which time emerges as a completely cosmologically synchronized reality.

In the context of quantum physics, particle entanglement provides another clue to the interconnections in the structure on which classical reality is based. Thus, the measurement of two entangled particles implies the exchange of quantum information between them independently of their position in space and instantaneously, as deduced from the superposition of quantum states and which Schrödinger posed as a thought experiment in “Schrödinger’s cat” [16]. This behavior seems to contradict the impossibility of transferring information faster than the speed of light, which raised a controversy known as the EPR paradox [17], which has been resolved theoretically and experimentally [18],  [19].

Therefore, at the classical scale information cannot travel faster than the speed of light. However, at the quantum scale reality behaves as if there were no space-time constraints. This indicates that space and time are realities that emerge at the classical scale but do not have a quantum reality, whereas space-time at the classical scale emerges from a quantum reality, which is unknown so far.

But perhaps the argument that most clearly supports the global interconnectedness of space-time is the Covariance Principle, which explicitly recognizes this interconnectedness by stating that the laws of physics must take the same form in all reference frames.

Finally, the question that arises is the underlying nature of space-time. In the current state of development of physics, the Standard Particle Model is available, which describes the quantum interactions between particles in the context of space-time. In this theoretical scheme, space-time is identified with the vacuum, which in quantum field theory is identified with the quantum vacuum which is the quantum state with the lowest possible energy, but this model does not seem to allow a theoretical analysis of how space-time emerges. Perhaps, the development of a model of fields that give sense to the physical reality of the vacuum and that integrates the standard model of particles will allow in the future to investigate how the space-time reality emerges from this model.

[1]	N. Emery, N. Markosian y M. Sullivan, «”Time”, The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2020/entries/time/&gt;,» [En línea].
[2]	J. M. E. McTaggart, «The Unreality of Time, http://www.jstor.org/stable/2248314,» Mind, vol. 17, nº 68, pp. 457-474, 1908.
[3]	S. Baron, K. Miller y J. Tallant, Out of Time. A Philosophical Study of Timelessness, Oxford University Press, 2022.
[4]	A. S. Eddington, The nature of the physical world, Cambridge University Press, 1948.
[5]	C. Rovelli, The order of time, Riverhead Books, 2018.
[6]	C. E. Shannon, “A Mathematical Theory of Communication,” The Bell system technical journal, vol. 27, pp. 379-423, 623-656, 1948.
[7]	P. Ball, Designing the Molecular World, Princeton University Press, 1994.
[8]	L. E. Ballentine, Quantum Mechanics. A Modern Development. Chapter 3., World Scientific Publishing Co., 2000.
[9]	A. Ben-Naim, A Farewell to Entropy: Statistical Thermodynamics Based on Information, World Publishing Company, 2008.
[10]	F. Schwabl, Statistical Mechanics, pp. 491-494, Springer, 2006.
[11]	A. Ashtekar y E. Bianchi, «A Short Review of Loop Quantum Gravity URL= <arXiv:2104.04394v1 [gr-qc]>,» 2021.
[12]	L. Smolin, «The case for background independence. URL = < https://arxiv.org/abs/hep-th/0507235v1&gt;,» 2005. [En línea].
[13]	I. Reznikoff, «A class of deductive theories that cannot be deterministic: classical and quantum physics are not deterministic. URL = https://arxiv.org/abs/1203.2945v3,» 2013. [En línea].
[14]	A. Einstein, «On The Electrodynamics Of Moving Bodies,» 1905.
[15]	T. P. Cheng, Relativity, Gravitation and Cosmology, Oxford University Press, 2010.
[16]	E. Schrödinger, «The Present Situation in Quantum Mechanics. (Trans. John Trimmer),» Naturwissenschaften, vol. 23, pp. 844-849, 1935.
[17]	A. Einstein, B. Podolsky and N. Rose, “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?,” Physical Review, vol. 47, pp. 777-780, 1935.
[18]	J. S. Bell, «On the Einstein Podolsky Rosen Paradox,» Physics, vol. 1, nº 3, pp. 195-290, 1964.
[19]	A. Aspect, P. Grangier and G. Roger, “Experimental Tests of Realistic Local Theories via Bell’s Theorem,” Phys. Rev. Lett., vol. 47, pp. 460-463, 1981.