We are all very familiar with time. It is ingrained in everything. It can rarely be disregarded as an unimportant factor. Organization, doing things, meeting people, travelling, sports matches, pieces of music, history, nature, all these things must be contextualised in time. When did it happen? How long did it take? What time will it be over? In art and culture, science and politics; time is a theme and a metaphor, a saviour and a constraint. But why does it have to go forward?
The fundamental laws of physics that govern mechanics, electricity, magnetism, gravity and quantum mechanics do not require that time only moves forward. In fact many pieces of theory are as valid with time moving forward as they are with time reversing, Benjamin Button style.
For example, a solution to the wave equation which describes the dynamics of electric and magnetic fields is given by a Green’s function which has two forms. One is known as the Retarded Green’s Function, which is affected by an electromagnetic disturbance which occurred in the past and now influences the present field. The other is the Advanced Green’s function, which is affected by an electromagnetic disturbance from the future. A state which has not yet been realised but influences the present field in the same way the disturbance from the past did, mathematically speaking.
In Mechanics, whether it is the Newtonian, Lagrangian or Hamiltonian formalism, one can take the limit as time goes to infinity of a time dependent system. That is to say we have a system which exists in the present, for which we have theory and equations describing it’s dynamics: position, velocity, momentum, etc. And we can use these time dependent equations to see what the system will do as time evolves infinitely. What state the system will be in an infinite length of time into the future. But there exists a completely valid canonical transformation, the time reversal transformation, whereby we can reverse the direction of time. Then the limit goes infinitely into the past. The system evolves in reverse from the present moment. Time does not have to go forward for the theory to remain valid.
The Schrodinger Picture of Quantum Mechanics involves time dependent states, and time independent observables. This theory does not require that the observables, the physically measurable quantities, even depend on time!
The answer to our question comes from Thermodynamics.
“In the elementary equations of the world, the arrow of time appears only where there is heat.”
The basic idea is that heat cannot be transferred from a cold body to a hot one, provided the external environment remains the same. Things do no naturally heat up of their own accord if no external energy is supplied. But heat can be transferred from a hot body to a cold one. If a tablespoon of boiling water is added to a glass of cold water, the hot water will not stay together. It will diffuse and disperse, and will slightly increase the temperature of the glass until an equilibrium temperature is reached. The heated water molecules are “shuffled” throughout the glass, until they are evenly dispersed.
Consider then that the tablespoon of hot water is dyed some colour. When it is added to the glass the system is in a very specific state. With all the dye concentrated in one location, and all the undyed water surrounding it. As time passes, the dye disperses, the way heat does, and dyes the rest of the water in the glass. This state is much less ordered than the initial state. Equivalently, there are much more ways to realise the mixed dispersed state, than the one where all the dye is concentrated together. All systems which are affected by heat are thus governed by this disordering of states. All systems gradually tend towards disorder. There is only one way to arrange the 26 letters of the Roman alphabet in alphabetical order, but there are 403,291,461,126,605,635,584,000,000 (~400 septillion or 26 factorial) ways to list the 26 letters in no particular order.
This idea was first introduced by Clausius when he defined entropy to describe the irreversible flow of heat. A system’s entropy can increase or remain constant, but it cannot decrease. . It was then Ludwig Boltzmann who grasped the intuition that entropy was related to the order or disorder of a system. The increase of entropy is directly linked to the increase of disorder. This is the Second Law of Thermodynamics and it is the difference between past and future.
If time were to run backward, heat would have to flow from a cold body to a hot one, a disordered state would become an ordered one, a shuffled deck of cards would arrange itself by number and suit, a pile of sand would build itself into a sandcastle. A glass of Ribena would unmix itself and separate into water and concentrated Ribena. This is not physically realisable, and thus time must go forward. And entropy is the only quantity which demands it.