The concepts of time and space are indivisible and fundamental in physics. We might even say that without them, physics as we know it wouldn’t exist. The main goal of physics is to describe all the natural processes in our Universe with general rules with predicting power that we call laws. More specifically, physics aims to give an answer to the question of how changes we observe in our world happen and not why they happen. A first step to analyze these changes consists in defining a four-dimensional coordinate system that allows us to track these variations, where three of the dimensions are used to determine the position of bodies in space and the fourth is used to set the instant when they do. Space is, thus, the stage where all the analyzed events happen and time is a parameter that allows us to follow their evolution. If the world around us didn’t change, if it were static like a photograph, there wouldn’t even be a necessity for the concept of time. So the ideas of space and time are inseparable notions. (Heraclit, according to the translation B. Haxton, 2001).
The simplest change that we can consider is the one produced by the variation of the spatial position of the bodies around us. You could also consider changes in their state or chemical composition. But the changes in the state and composition of bodies correspond essentially to changes in the positions and velocities – that is to say, in the rates of position change – of the particles constituting those bodies. Therefore, any change can be eventually reduced to a variation of the 4 space-time coordinates of particles. Evidently, if the four coordinates always shifted simultaneously, we wouldn’t be able to distinguish the three spatial dimensions from the temporal one. All four dimensions would be interchangeable. But the variations we observe in the world around us show that a coordinate, which we call time, changes constantly while each of the other three can experience a variation or not, depending on the case. This subtle difference is fundamental to distinguishing space and time.
Our perception of space and time
The three spatial dimensions, as we perceive them (fig. 1), are independent from each other and from the time dimension. They are all continuous, meaning that their variation can be subdivided in arbitrarily small increments, and homogeneous, meaning that all of their points behave equivalently. Now, while the spatial dimensions are all isotropic – all directions behave in the same way – the time dimension presents a preferred direction. Just as with space we can choose to move upwards or downwards, right or left, forward or backwards and the pace of our liking, with time we can only move towards the future and not back to the past.
Figure 1. Representation of the three continuous independent spatial dimensions and the also independent time dimension as we perceive them.
We can’t even choose the pace at which we move in the time dimension. This rate seems to be imposed by an inescapable force. The origin of these two particular features of time, the arrow of time and the pace at which it is trodden have intrigued physicists of all times.
The arrow of time
As we were saying, all the changes in nature can be reduced essentially to variations in the positions of particles. Classical mechanics – the Branch of physics that studies the motion of bodies that are experiencing a force (or interaction), shows us that these variations are described by time-reversible second order differential equations. That is to say that if we change the sign of time, the trajectory of the system will remain the same, only travelled backwards. That implies that the motion of individual bodies doesn’t make a distinction between the past and the future. There is no arrow of time. Where, then, do we get the intuition of time passing in a certain direction, from the past to the future and never the other way around?
The answer to this question is provided not by classical mechanics but by thermodynamics. While it's true that the motion of a few bodies doesn’t make a distinction in the sign of time, the motion of many bodies (or particles) at the same time does (fig. 2). This motion tends towards an increasing disarray in its positions and velocities if they are not messy to begin with. The reason for this tendency is purely statistical: it’s more probable to find the positions and velocities of a system in disarray than in perfect order. Systems tend to evolve towards an increasing disorder. For instance, the evolution of a gas inside a closed box parting from an initial situation where all the molecules are concentrated in a small corner is such that gradually the molecules tend to spread throughout all the available space and with the mean energy of the lot. This is the state with maximum disorder. If we were shown this clip backwards, we would immediately notice that the final result is not natural, even though from a pure mechanical point of view it is possible. This tendency of closed particle systems towards an increasing randomness is known in thermodynamics as an entropy increase.
The entropy increase on natural systems is, thus, responsible for the arrow of time (Eddington, 1928). Any system will evolve towards the state of maximum entropy. We say that the system relaxes because, when it achieves the final state of maximum disorder, it automatically loses all memory of the order of the initial conditions. At this point, the system stops evolving as if its time has stopped.
Figure 2.In physics, the arrow of time appears with the evolution of many-particle systems and it is given by the time direction of the system towards an increase in the disorder of the positions and velocities of its particles.
The speed of time
Now we may ask, at which pace does time run? The answer is simple: at the pace imposed by the increase of entropy. Living beings are physical systems that constantly exchange matter and energy with their surroundings through complex biochemical reactions which produce a continuous increase of entropy. This is why life implies a passage of time, meaning, the evolution of the system in a certain time direction. When living organisms die, the closed system that contains them rapidly tends to a maximum disorder and time stops existing for them. It only continues for the rest of the Universe that goes on evolving towards a crescent entropy (Blum, 2015).
Human beings, apart from being alive, have memories and are aware of the changes they perceive within themselves and specially around them, most of which represent an increase of the entropy of the Universe. This creates the illusion of the passing of time. But the speed at which each individual perceives time passing is very subjective. It proceeds from their perception of change. Among these changes, there are cyclic phenomena, for instance day and night and the variation in light that they cause, which allow us to get a relatively objective perception of time passing. Internal changes are way harder to perceive so, if it weren’t for the former, we would be at a loss. It would be like standing in front of a static picture; it’d be hard to calibrate the time. This is clearly shown by the fact that people who have been buried under avalanches or due to an earthquake or isolated in a mine lose, in great measure, the notion of the time passed. The difference in the degree of memory or awareness of changes in their surroundings is the reason why kids and adults perceive time passing at different speeds, even though the biochemical processes involved in both kinds of subjects are very similar. When one is bored because not much is happening or is waiting for an expected change, one gets the feeling that time goes by slowly.
To study the changes in our surroundings in the most objective way possible, that is to say, to be able to compare the descriptions from different observers, physicists use clocks, cyclic machines that are easily replicable and in which nothing seems to change from one cycle to the next. This allows us to assume that the time lapse passed in each cycle is the same and adopt it as a yardstick or unit of time. Some of the examples of cyclic motions used as clocks through history are the motion of the sun around the earth, the oscillation of a pendulum of a certain length, the frequency of the light emitted by a certain excited atom, etc.
Relativistic space-time
The comparison among several observers makes us wonder if time runs at the same speed for all of them. Since we perceive space as homogeneous, the pace at which time goes by cannot depend on the position of the clock - or of the observer carrying it - but, could it depend on their velocity? This potential dependence is harder to detect using our senses. The only reason for rejecting it is rather a reasoning: the principle of maximum that always goes by the theory - the famous Ockham’s razor -. Since nothing seemed to contradict this, physicists assumed from the beginning that time didn’t depend on the velocity of the clock. This assumption is precisely what leads us to believe that time and space are independent from one another.
Galileo proved that, if this independence is preserved, the velocity of an object relative to an observer moving towards it equals the sum of the two velocities - with respect to an inertial reference frame - of the object and the observer. If both objects are moving away from each other, the relative velocity will equal the difference of the two absolute velocities. This way of adding up or subtracting velocities is known as Galileo’s transformation group, in honor of its finder. Velocities seem to agree with this behaviour. For instance, due to this when two cars crash the damages will be more severe if both cars were moving in opposite directions than if they moved in the same direction. Experiments performed in the laboratory confirm with great accuracy this relation between velocities, which proves that time does not depend on the velocity of the observer.
Time and velocity: special relativity
But the velocities that are often involved in lab experiments and in our daily lives are not exceptionally large when compared to the speed of light. There was nothing that actually prevented Galileo's transformations from failing at sufficiently high velocities. This became apparent with Maxwell's discovery of the laws of electromagnetism. In these laws, the vacuum speed of light is a constant independent of the velocities of the emitter and the receptor. In other words, light didn't verify Galileo's relation between velocities. Einstein realized that the only possible explanation to this mind-boggling phenomenon was that, contrary to what had been assumed so far, the time indicated in a clock does depend on its velocity (Mould, 2002).
Space and time are, then, not independent from each other; they are linked by the velocity of the observer. We have therefore to abandon the intuitive idea of a three-dimensional space with a plane Euclidean metric and an independent time dimension. Space and time are interconnected in a combined entity named space-time which contains an 4-dimensional also plane metric called Minkowski metric, which causes the speed of light in vacuum to always be 300.000 km per second, independently of the observer. A surprising consequence of this strong link between space and time is that the time of an observer stretches or contracts depending on their relative velocity to the speed of light. And that is not all. The space of a system in motion also dilates or contracts depending on its relative speed to the observer.
Observers than compare their measurements of space and time often move at small velocities. For this "common" observers, it is like the speed of light is infinite and space and time are not linked. For observers moving at high speeds, the differences can become remarkable. So, the faster a system moves relative to the speed of light, the slower its time passes and the more its space contracts, even though it is not aware of all this. This deceleration or acceleration of time predicted by Einstein's special relativity can be empirically tested using clocks that have been briefly subjected to extremely high velocities (fig. 3). We have the example where one of the clocks is located inside a satellite orbiting the Earth at a high speed. When we retrieve the clock we observe that it is slow. An even more spectacular consequence of this paradigm shift in our conception of space-time is the famous law E=mc2 discovered by Einstein that, as it is commonly known, is at heart of the vastly used nuclear energy.
Figure 3. When the velocity of an observer approaches the speed of light, their time slows down and their space contracts with respect to an observer that is still. Neither notices any effect unless they compare their measurements, though.
The concepts of space and time are relative - and not absolute as people used to believe - since they depend on the velocity of the observer relative to the speed of light. Why? We don't know; What we know is that this is just how the Universe works. As I mentioned before, physics doesn't tell us why things happen but how they happen. Nevertheless, in order to not disappoint the reader, I want to remark that even if the assumption of space and time being absolute and independent were correct, we couldn't provide any other explanation than the fact that they appear more natural and intuitive to us because in our daily lives we only experiment with small velocities.
Time and gravity: general relativity
This hasn't been the only surprise that Einstein kept for us. While trying to derive a law of gravitation that didn't depend of the acceleration of the reference frame as is the case for Newton's gravity, Einstein discovered that, contrary to what was believed at the moment, space wasn't homogeneous and therefore, not only does time depend on the velocity of the observer but also on its position! The feature that modifies the homogeneity of space, in the frame of this new theory called general relativity, is gravity itself. (Ellis y Williams, 2000) (fig. 4).
Figure 4. The gravity caused by the presence of a mass-energy source curves space-time in such a way that the geodesics (shortest lines between two points) that particles follow around it also curve.
For Newton, space was absolute and homogeneous. If it contained gravity, it'd appear as a mass that curved trajectories of other massive particles. In turn, for Einstein, gravity wasn't a mass added to space-time but it was inherent to it. In presence of a source of mass or energy that causes gravity - bear in mind that in relativity mass and energy are equivalent through the relation E=mc2-, space-time curves and particles -with mass or without it- follow curved lines. That is to say, the metric of the space-time stops being minkowskian. Due to the more or less pointed curvature of space-time, which depends on how much mass-energy there's at some point, space stops being homogeneous. The more mass-energy or gravity there is, the slower time runs and the more space contracts (Wheeler, 1990) (fig. 5).
Figure 5. The curvature of space-time created by gravity causes space to contract and time to dilate where gravity is stronger in comparison to where gravity is weaker. Clearly, an observer located at each point won’t notice anything unless they make a comparison between the measurements at both points.
So the time indicated in a clock- or the rate of the biochemical reactions in our body- stretches not only when it moves at a great speed but when gravity is more intense in its position as well, and space experiments a similar alteration. For weak gravitational fields, time dilation and space contraction are so tiny that there's almost no difference at all. In the case of intense gravitational fields, instead, the difference can be quite remarkable. For example, near a black hole, space-time is extremely curved and time runs way slowlier than for an observer far away. Actually, if the observer is located at a certain distance from the center of the black hole which is known as the event horizon, time will stretch infinitely for them, and that is why nothing can escape, not even light, to the outside. On the other hand, inside the event horizon, moving forward in space is equivalent to moving towards the future and vice versa. This is why in the famous science-fiction film - and not so much fiction- named Interstellar (Thorne, 2014), after the astronaut crosses the event horizon of the black hole Gargantua (fig. 6) when he moves inwards or outwards he is also moving forth and back in her daughter's time on the outside.
Figure 6. Image of Gargantua, the black hole from the film Interstellar. In its nearby regions, time dilates, and this can only be noticed relative to what happens far away. In its interior, space and time appear inverted and discretized.
Quantum space-time
Black hole physics still has some more surprises in store. When matter falls inside a black hole and disappears for good from the Universe, the mass, energy and angular momentum of the black hole increase so these quantities are conserved, as usual. Still, other quantities present in quantum physics- quantum numbers- that are conserved in any interaction as well, don't anymore when matter falls into a black hole. This is the case for the baryonic number (number of protons, neutrons, etc.) or the leptonic number (number of electrons, positions, neutrinos, etc.) for instance.
Since black holes don't have a property that indicates their total number of baryons or leptons, when matter falls in them, this information is lost, as if these quantities were not conserved. There's an inconsistency between quantum physics and gravitation that is manifested here.
This issue, which has today's physicists concerned, shows that gravity and therefore space-time are deeply connected to quantum physics in such a way that we cannot fully understand one without understanding the other. This concept is also masterfully shown in Interstellar. We cannot forget the fact that the scientific consultant of this film, professor Kip Thorne, has been awarded the Physics Nobel Prize this year for all his works on general relativity that led to the detection of gravitational waves.
So it seems that quantum physics has something to say about the structure of space-time as well. All we have discussed so far answers to the classical view of physics, inspired in our perception of the macroscopic world. It's true that relativity's space-time strays a little from the picture created by our perception. Time and space are neither independent nor homogeneous. But this happens only because in our daily lives we don't experiment with high enough velocities or sufficiently intense gravitational fields. And still, we don't have any doubts about the basic property of space-time: it is a continuous structure. How could it no be? You will think. Well, as we will see now even this basic property is wobbling in the quantum vision of physics.
Our perception of the reality around us distinguishes between particles and waves. In the former, the physical magnitudes that describe them such as mass, energy or angular momentum, take discrete values. In turn, in the case of waves, like for instance light, these magnitudes are continuous which causes phenomena like diffraction of interference patterns which are impossible to achieve with particles. And even so, the so-called photoelectric effect discovered by Planck made Einstein admit that light is also quantized, meaning that it is made by little energy "quanta" which are called photons, that behave as particles. The discovery of photons opened up a brand-new field in physics, quantum physics, that has allowed us to dive into the astonishing world of the microcosmos (Liboff, 1980; McEvoy y Zarate, 2004).
Admittedly, in the microcosmos everything behaves as light does. Sometimes as a continuous wave and others as a set of discrete particles. The microcosmos is also characterized by the famous uncertainty of Heisenberg. The different physical magnitudes are grouped in pairs in a way that, within each pair, one cannot simultaneously determine the values of both magnitudes with complete accuracy. For example, if we measure the velocity of a particle with precision, then it's position remains completely uncertain, and the other way around. Or, if we determine the energy of a particle with precision, we won't be able to accurately determine the instant in which it took that value. This uncertainty is at heart of the wave-particle duality. These apparently "unnatural" behaviours of the microcosmos escape our understanding that has adapted to the perception of the macroscopic world. Still, the detailed calculations performed in quantum physics have been rigorously tested in the lab (or at particle accelerator).
Due to this, quantum physics is considered to be extraordinarily successful. Thanks to it we are able to describe three out of the four fundamental interactions known in nature with the properties of its carrying particles. These interactions are: electromagnetism, the weak nuclear force and the strong nuclear force. The photon is the carrying particle of the former and other particles which are out of the matter at hand, are the carrying particles of the latter. The only interaction that we don't know how to describe yet is precisely gravity. There's supposed to exist a carrying particle for it as well, the graviton, but it hasn't been detected yet and we don't know its characteristics.
General Relativity tells us that space-time is determined by gravity. Therefore, looking at gravity from a quantum perspective should imply the quantization of space-time! So it seems that, contrary to general belief, space-time is not a continuous structure, it only appeared as such because we perceive it in a macroscopic way, at least in regions outside of black holes. That's why inside of Gargantua, space and time not only appear inverted but also discretized, since gravitation of quantum physics are deeply linked.
At the Big Bang, the origin of the Universe, space and time should have been quantized. Even more so, since there still weren't any particles there that could interact, the arrow of time could not have existed and the three spatial dimensions and the fourth time dimension should have been interchangeable. This would explain why we cannot go back in time to before the initial singularity: regressing further than that would be equivalent to going upwards or downwards, right or left, forwards or backwards. Only with the birth of the first particles should the arrow of time have shown up.
Currently, all of these are mere assumptions. The quantum structure of spcae-time and its relation with the expansion of the Universe are still to be unraveled. What's for sure is that, once we understand the quantum nature of gravity, we will see how different the concepts of space and time are from what is suggested to us by our perception. They will probably even escape our logic as it happens in many other aspects of quantum physics. Still, we will know how they really work and what happened at the Big Bang.
Bibliography
Blum, H.F. 2015. Time’s arrow and evolution. Princeton University Press.
Eddington, A. 1928. The nature of the physical world. Dent, London.
Ellis, G.F.R. i Williams, R.M. 2000. Flat and curved space-times. Oxford University Press.
Heràclit. 2001. Fragments: The collected wisdom of Heraclitus. Haxton, B. (traducció). Viking, The Penguin Group, New York. ISBN 0-670-89195-9
Liboff, R. 1980. Introductory quantum mechanics. Addison-Wesley, Reading. ISBN 978-0805387148
McEvoy, J.P. i Zarate, O. 2004. Introducing quantum theory. Totem Books. ISBN 1-84046-577-8.
Mould, R.A. 2002. Basic relativity. Springer-Verlag. ISBN 0-387-95210-1
Thorne, K. 2014. The science of interstellar. W. W. Norton & Company, ISBN 978-0393351378
Wheeler, J.A. 1990. A journey into gravity and spacetime. Scientific American Library. W. H. Freeman, San Francisco. ISBN 0-7167-6034-7
About the Author
Eduard Salvador is an astronomer and professor at the Department of Quantum Physics and Astrophysics at the University of Barcelona. He is a researcher in observational cosmology and leads the research group on Formation and Evolution of Galaxies at the Institute of Cosmos Sciences, where he studies, among others, the physical processes that lead to the formation of galaxies and galactic systems, the influence of the environment on their properties and the gravitational clustering of dark matter.