# What is Time?

## The various meanings and interpretations of time in theoretical physics

One of the questions that I seem to get a lot, as a physics person, is “what is time?”. So in this article I would like to explain the various roles of time in theoretical physics by talking about concepts, and without getting into any complicated math.

The Role of Time in Classical Mechanics

In classical mechanics, a fundamental role is played by the concept of a particle. For a single particle that can move in three dimensional space, the position of the particle is a vector, r = (x, y, z).

Given a system of particles, the goal of classical mechanics is to obtain the equation of motion for each particle in the system. The equation of motion of a single particle describes it’s path through space, and mathematically, the equation of motion of a single particle is a function of time, r(t), whose value gives the (x, y, z) coordinates of the particle at time t.

And so we see clearly that in classical mechanics, time is simply a parameter that is used to describe the path of a particle.

Time Reversibility

Newton’s laws of motion provide a procedure to obtain the equations of motion for a system of particles, by translating a system of particles with given initial positions and velocities into a system of differential equations. The solution of this system are the equations of motion.

It is not hard to show, using the chain rule, that if r(t) is a valid solution of Newton’s second law, then r(-t) is also a solution. In other words, according to Newton’s laws of motion, there is a symmetry between the directions of time, in the sense that if Newton’s laws say that a given path is physically possible, then the path obtained by starting at the end and going backwards is also physically possible.

The Arrow of Time

It might be thought that as the number of particles in a physical system increases, so too does the complexity of the behavior of the entire system. It turns out however, that when dealing with systems that contain an extremely large number of particles, such as the molecules in a gas, new laws emerge which are quite regular in nature. These laws constitute the subjects of thermodynamics and statistical physics.

In thermodynamics, a fundamental role is played by the concept of temperature. The temperature of a system is defined as the average value of the kinetic energy of the particles in the system.

A fundamental role is also played by the concept of thermal equilibrium. If we are given two physical systems at different temperatures and put them next to each other so they can interact, then “over time” the particles in the system will move in such a way so that the temperature evens out throughout the entire system.

The nature of thermal equilibrium tells us that there is a distinct asymmetry between the directions of time. This asymmetry is what is referred to as “The Arrow of Time”. Although the individual particles which make up a thermodynamic system themselves obey Newton’s laws of motion, for which time is reversible, systems which contain a very large number of particles evolve over time in a way that is irreversible. This is often considered a paradox. The solution of the paradox is to realize that in theoretical physics, the meaning of the word “time” depends on the context in which it is used.

The Relationship between Energy and Time

It is no exaggeration to say that the most important concept in all of theoretical physics is symmetry. Corresponding to every symmetry in nature there is a conservation law. The most important and most famous of all conservation laws is the law of conservation of energy, which is the first law of thermodynamics.

It is known from observation that if I do an experiment today, then all other things being equal, if I repeat the same experiment at any time in the future I will obtain the same result. This observation shows that the laws of physics do not change.

The invariance of the laws of physics throughout time is called a symmetry, and this symmetry leads directly to the conservation of energy. The mathematical derivation of this relationship is outside the scope of this article unfortunately.

The Role of Measurement in Quantum Mechanics

The theory of quantum mechanics demands a radical shift in our understanding of the laws of nature, and correspondingly in our understanding of the concept of time.

In quantum mechanics, the concept of a path of a particle has no physical meaning. Instead we are only allowed to make statements about the probability that a particle is at a given position (x, y, z), if we make a measurement at time t.

Particles are described by giving their wave-function, and the wave function evolves in time according to a differential equation, which being similar to the case of classical mechanics, is completely time reversible.

However, the process of measurement plays a fundamental role in quantum mechanics, and introduces a fundamental asymmetry between the directions of time.

When speaking of a measuring device, what is meant is a device which to a sufficient approximation obeys the laws of classical mechanics.

According to the formulation of quantum mechanics, a measurement divides time into three distinct and separate periods: the time before the measurement, the period of time during which the measurement takes place, and the time afterwards.

If a particle is described by a certain wave function before a measurement, then the process of measuring the particle is said to “collapse” the wave function into one of it’s fundamental quantum states, which has a definite measurable value. The collapse of a wave function is, in general, irreversible.

The Role of Time in the Uncertainty Principle

The uncertainty principle states that for a measurement of a quantum system which takes place over a time △t, the change in energy of the system must be at least h /t, where h is a fundamental constant of nature called “Plank’s constant”.

Thus we see that in quantum mechanics, as in classical mechanics, there is a deep and profound relationship between the concepts of energy and time.

Up until now, it has been assumed that the period of time during which a measurement is taken could be made arbitrarily short. In quantum field theory, which is the most accurate theory of physics that currently exists, this is no longer the case.

If a measurement is made of the momentum of a particle over a time △t, and if the change in momentum is △p, then △t must always be greater than h/cp; furthermore, the change in momentum is bounded by the fact that nothing can move faster than the speed of light, c. Using the fact that △p = mv, and substituting △v = 2c, we see that it is impossible to make a measurement of the momentum of a particle of mass m over an period of time shorter than h/2mc².

Temporal Ordering in Quantum Field Theory

In quantum field theory, it is not possible to describe a physical process as a sequence of events which takes place ordered in time.

If for example there are two photon detectors located very close to each other which each detect a photon during a very short time interval, then there is in principle no sequence of events which can be said with certainty to have taken place in between the two measurements.

Instead we are only able to know the probability of measuring certain physical quantities, such as the energy or momentum. The way these probabilities are calculated takes into consideration the contribution from all possible temporal orderings of events that can lead to a given measured value.

The Charge-Parity-Time-Reversal Theorem

Quantum field theory predicts the existence of anti-particles. An anti-particle is similar to a normal particle, except that it has a negative charge, and the notions of left and right handed are reversed.

Anti-particles are often described as “normal particles moving backwards through time” due to a very deep result known as the Charge-Parity-Time-Reversal theorem. This theorem states that reversing the sign of the charge of all the particles in a system, and reversing the meaning of left and right handed (or parity) would have the same effect as reversing the direction of time.

In quantum field theory therefore, we are not allowed to speak about a definite temporal ordering of events, but we are allowed to speak about particles moving backwards through time.

Simultaneous Events in Special Relativity

When particles travel at speeds that approach the speed of light, the laws of classical mechanics are no longer valid, and the laws of special relativity apply instead.

In classical mechanics, there is no issue with making a statement such as event X takes place before event Y; but in special relativity, this is no longer the case. In fact, the special theory of relativity forces us to reconsider the concept of what it means for two things to happen simultaneously.

This was first shown by Einstein in 1905: suppose that observer A is at rest, and perceives that event X and event Y take place at the same time. Now suppose that observer B is travelling close to the speed of light relative to observer A, and is travelling away from X and towards Y. Then since observer B is moving at almost the speed of light away from the light coming from X, and towards the light coming from Y, observer B will perceive that Y actually takes place before X.

Therefore, in special relativity it is required that when describing a physical system, one must also specify a frame of reference, because in this case, the statement that two events take place at the same time is well defined.

The special theory of relativity is founded on two principles: that the laws of physics are symmetrical with respect to the choice of frame of reference, and that the speed of light is constant for all observers.

It turns out that the rate of the passage of time is not the same for all frames of reference, it depends on the relative velocity of the frames.

For example: imagine two twins, A and B; if twin A stays on earth while twin B travels in a ship at close to the speed of light, then time will pass at a slower rate for B than for A, and when B returns he will be younger than his twin. This means that according to the special theory of relativity, time travel into the future is theoretically possible.

The Space-Time Continuum

In classical mechanics, the position of a particle along its path is given by four numbers, the three spatial coordinates and the time. In special relativity the same thing is true. Therefore in classical mechanics, as well as in special relativity, time can be thought of as “the fourth dimension”.

The drastic change in our conception of the nature of time demanded by the theory of relativity comes from the way in which time and space are mixed together, and put on a more equal footing. There is a famous set of equations called the “Lorentz Contraction” which show clearly how when changing from one frame of reference to another, time and space actually become mixed together.

One consequence of this is that the spatial distance between two particles is not the same in all frames of reference, and actually depends on the relative velocity of the frames. Therefore in the theory of relativity, rather than measuring the spatial distance between two particles, it is customary to measure the space-time distance between two events. The function which defines the space-time distance must satisfy the condition that it does not depend in any way on the velocity of the frame of reference. In the special theory of relativity there is a particular choice made for the distance function; but in the general theory the gravitational field defines, and is defined by, the choice of distance function.

The Passage of Time in a Gravitational Field

According to the the general theory of relativity, a gravitational field is equivalent, in a sufficiently small region of space, to frame of reference that is accelerating at a constant rate.

This principle alone is sufficient to demonstrate that clocks which are located at different points in a gravitational field, will appear to run at different speeds.

To see this, let us consider two clocks A and B which are both in a gravitational field, with clock A accelerating towards clock B. Furthermore, let us assume that every second, clock B emits a pulse of light. Then since A is accelerating towards the pulse of light coming from B, it will appear to A that the pulses arrive in less time than one second; and thus it will appear to an observer at A that clock B is running fast. By a similar argument, an observer at B would perceive that clock A is running slow.

This means, for example, that a clock on a satellite that is orbiting the earth will appear to run slow compared to a clock on the surface. For the case of satellite GPS systems, taking this into account has been shown to improve the accuracy of the results, demonstrating that the effect is indeed very real.

The Principle of Maximum Proper Time

An important role is played in general relativity by the concept of proper time. If we consider a frame of reference which is moving and accelerating in space in an arbitrary manner, then the proper time for the motion is defined as the amount of time that has passed on a clock that moves with the frame.

As the velocity of the motion increases, the rate of the passage of proper time decreases; therefore the amount of proper time which is associated with the motion depends not only on it’s position in the gravitational field, but also on how fast it is moving.

The principle of maximum proper time defines the way that particles move in a gravitational field, and it states that free particles in a gravitational field will move along a path that maximizes the proper time for the motion.

The Passage of Time Inside of a Black Hole

A black hole is a region of space in which the gravitational field is so strong that light cannot escape if it travels past a certain critical event horizon. In these regions of space, according to general relativity, the strength of the gravitational field is infinite, and this would imply that inside a black hole the passage of time has ceased.

It has been predicted that quantum effects from inside of a black hole actually cause it to evaporate over time. In general, the presence of an infinity in a physical theory indicates a problem with the theory, and at the energy scales present inside of a black hole, the theory of general relativity is no longer valid.

The Beginning of Time

The theory of the creation of the universe is based on the assumption that at the beginning of time everything was pure energy, the same everywhere and in all directions, and furthermore that the strength of the gravitational field was infinite.

10^-36 seconds after the beginning of time, there is a period of rapid inflation which lasts about 10^-33 seconds, followed by a slower expansion at a rate similar to what we observe experimentally today. But whatever happened before the first 10^-36 second remains unknown.

tl;dr

The laws of physics describe the process of change, but these laws themselves are timeless and unchanging. If nothing ever changed, then the concept of time would have no meaning. Time is something that is used in physics to describe change, and the thing that changes is energy. The way energy moves around and changes form depends on the scales at which we are looking, and therefore the meaning of the word “time” depends on which part of physics we are talking about.