## The Arrow of Time

We are all aware of an intuitive "flow" of time from past to future. Not only do we feel this flow of time, but we
also see it manifested in the behaviour of objects which change over time. Many objects seem to behave differently in
the forward time direction when compared to the backward time direction. For example, we don't see a spilt glass of
water jumping up and going back into the glass, we don't see a broken egg reforming itself. These effects all add
to the impression that there is some sort of "forward direction" in the time dimension. This directionality is called
the *Arrow of Time*.

However, this "arrow of time" is something of a mystery to physicists because, at the microscopic level, all fundamental physical processes appear to be time-reversible (we'll consider this later). Also, as shown on the Time and the Block Universe page, our universe appears to have a spacetime structure in which all of time is laid-out in a "block universe", i.e., there is no actual "flow" of time, no movement of a "now" point.

So on this page we will investigate the cause of this mysterious "Arrow of Time".

### Entropy

Entropy can be considered the amount of disorder in a system. For example, a car that has rusted could be said to have a greater entropy value than a new car: bits of the car may have fallen off, the paint may be flaking. Basically, the molecules of the car have become more disordered over time: entropy has increased.

As has just been just discussed, all microscopic processes appear to be time-reversible. The question of why we see
an "arrow of time" in macroscopic processes has therefore presented physics with a long-standing conundrum. For this
reason, much attention has focussed on the fact that the entropy of a closed system increases with time, i.e., a system
will gradually become more disordered with time. Eventually the system (gas in a closed container, for example) will
reach a state when all its molecules are completely randomly orientated. This state is called *thermal equilibrium*.
The rule that entropy increases with time is called the *second law of thermodynamics*.

The reason for this increase in entropy can be seen from a purely probabilistic argument: a system will have many more possible disordered states than ordered states, so a system which changes state randomly will most likely move to a more disordered state. It's really just a matter of likelihood. For this reason, the second "law" of thermodynamics is not really a "law" at all, certainly not an unbreakable law on the same basis as other physical laws - it is a statistical principle. In fact, it might be possible for a room full of randomly-distributed particles to re-order itself quite by chance so that all the particles end up in one corner of the room - it would just be incredibly unlikely!

While the second "law" of thermodynamics is "just" a statistical principle, it is a mightily powerful statistical principle! This is because the basis of the second law - that "disorder will increase" - seems so obvious, and seems to appeal to a fundamental, platonic principle of mathematics. For this reason, the second law manages to appear even more fundamental and unbreakable than the other physical laws, some of which (for example, the amount of electric charge on an electron) seem rather arbitrary in comparison. This fundamental strength of the second law is described well by the astrophysicist Sir Arthur Eddington:

"If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations - then so much the worse for Maxwell's equations. If it is found to be contradicted by observation - well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can offer you no hope; there is nothing for it but to collapse in deepest humiliation."

- Sir Arthur Eddington

One of the most inexplicable features of the early universe is that it had an incredibly low value of entropy. This value of entropy was so low that even now - 13 billion years later - we still find ourselves living in a world of relatively low entropy. As a result, many of the objects we surround ourselves with have low entropy: new cars, and perfect unbroken eggs in egg cups. And these objects are basically falling apart around us as they inevitably move to higher entropy states: cars rust, eggs fall on the floor and break. Hence, the increase in entropy in our ordered world is one reason why we detect an apparent "arrow of time".

### But change of entropy is fundamentally time-symmetrical!!

However, this is a good time to clear-up a very widely-held misconception about the change of entropy: that change of
entropy is in some way fundamentally time-asymmetric, that entropy change behaves fundamentally differently in the forward
time direction to the backward time direction. This is absolutely not the case. **In the general case, entropy increases
in the backward time direction in just the same way as it increases in the forward time direction: change of entropy is
symmetrical with time.** (However, a very small minority of physicists might still believe change of entropy is
time-asymmetric - see my comments at the bottom of this discussion with the notoriously tetchy physicist Luboš Motl
here).

The probabilistic basis of the second law of thermodynamics simply says that a system will have many more possible disordered states than ordered states, so a system which changes state randomly will most likely move to a more disordered state. This seems very clear and obvious - such a simple statement is never going to be the cause of something so mysterious as fundamental time-asymmetry. Indeed, this change to a more disordered state is just as applicable in the reverse time direction as in the forward time direction: it's just a change of state, independent of time.

But what about the second law of thermodynamics which states that "entropy increases with time"? This seems to imply
a fundamental time-asymmetry to entropy. But we have to realise that the second law only applies to special-case systems:
objects with low entropy, the sort of objects we generally encounter in everyday life (rusting cars, etc.). In fact,
if we consider general-case objects (i.e., objects in thermal equilibrium), objects which have never been arranged into any
sort of order, then their entropy is at a maximum already so their entropy can only **decrease** with time -
completely at odds with the second law!

This generally-held misconception that change of entropy is fundamentally time-asymmetrical is revealed by the Loschmidt paradox. The Loschmidt paradox considers the apparently fundamental time-asymmetry of entropy implied by the second law and states that this is at odds with the known time symmetry of fundamental processes. It is only when we realise that the second law is frequently badly stated and hence contains unstated assumptions (which have been just considered) that the Loschmidt paradox is resolved. (Wikipedia describes this resolution of the paradox, showing how one of the key assumptions of Boltzmann's version of the second law of thermodynamics was flawed - see here).

But if change of entropy is time-symmetric, why do we see the entropy of the universe as only increasing?
Roger Penrose considers this question in his book *The Road to Reality*. Penrose considers what we might expect to
happen if we trace the entropy of the universe back in time from the state it is in now. If change of entropy is really
time-symmetrical, then we should expect to see entropy **increasing** as we trace the universe into the past, just as we
will see entropy increasing into the future. But we know, in fact, that the universe had a **lower** entropy in
the past: i.e., the entropy of the universe actually reduces in the past. So where does this asymmetry come from?

As Roger Penrose goes on to reveal, the time-asymmetry of change of entropy within the universe is explained by the extraordinarily low entropy of the universe at its origin:

Basically, the low-entropy past of the universe "fixes" the experiment. If we want to get a symmetrical answer then we have to be careful to conduct a symmetrical experiment. Rather than starting with a special-case low entropy universe, we have to imagine a universe which started in thermal equilibrium and has reached its current state unaided, purely by chance:

After that low-entropy point is reached, we then see entropy starting to increase according to the second law.
But the key thing is that if we trace the entropy of the universe back in time past the low-entropy point we
now see that symmetry that Roger Penrose sought. Hence, change of entropy is **fundamentally symmetrical**.

In fact, throughout this discussion on the arrow of time we will find that the arrow of time is caused by the time-symmetric second law of thermodynamics, together with the very special, low-entropy initial conditions of the universe.

(This discussion on time-symmetric entropy change is based on an example by J. Richard Gott in his book *Time Travel in
Einstein's Universe* in which the role of the universe is played by an ice cube - see
here. The ice cube example is considered in detail in Chapter 6 of Brian Greene's book
*The Fabric of the Cosmos*.)

### Causality

We all have a very strong feeling of a directionality of time, which has a flow in a forwards direction. As Michael Lockwood says in
his book *The Labyrinth of Time*: *"We regard the forward direction in time, in stark contrast to the backward direction,
as the direction in which causality is permitted to operate. Causes, we assume, can precede their effects, but
cannot follow them."*

But we have just seen how physical processes appear to be time-symmetrical, with no distinction between the forward and backward
directions. So where does that leave causality? As Michael Lockwood again says about the passage of time: *"We find no hint of this
in the formalism of Newtonian physics. Not only is there no explicit reference to a passage or flow of time; there is not even
any reference to cause and effect. Indeed, there is not even any directionality".*

"But", you might protest, "surely causality works in only one direction: forwards in time? I kick a football - the football doesn't kick me." Well, let's consider the example immediately below of forward causality. We see a snooker cue coming in from the left, hitting the white ball, which then causes the white ball to hit the red ball:

However, if you shoot a movie of that sequence, and then play it backwards, it still makes perfect physical sense. As you can see below, we then have the red ball coming in from the right, hitting the white ball, which then causes the white ball to hit the cue backwards. So, because of the symmetry of the laws of physics, this process of causality - which we thought only applied to the forward direction of time - in fact applies equally to the backward direction of time as well:

The reason why we don't see causality happening in the backward direction is purely because of a bias in our psychological systems: something about the complexity of our psychological system (our brains!) causes our thought processes to work only in the forward direction of time (this will be considered below). The great advantage of recording the sequence on a movie and then playing the movie backwards (to reveal the time symmetry of causality) is that a movie camera works in a much more simple fashion than our brains and thus has no such psychological bias in the forward direction: it works in exactly the same way forward as backward.

So if causality is time-symmetrical, we could in fact think of our current situations are being caused by time-reversed future events as much as by past events! For example, as I sit here by my desk in work this morning, I could consider my position as being caused by me being in my apartment this evening, and driving my car from there backward in time, backward down the road the work, to put me in work this morning! It's a bit brain-bending, but it's equally valid as saying "I got up this morning, and drove forwards to work". It seems strange, but that's only because of our psychological bias. The movie of my complete day at work would tell the correct (time-reversible) story.

### The Quantum Mechanical Arrow of Time

As has just been explained, almost all known physical principles (from Newtonian mechanics through to Einstein's relativity) have a completely symmetric treatment of past and future. Nowhere in any of these equations is there anything which distinguishes a forward direction of time from a backward direction of time. The exception to this rule appears to be quantum mechanics. On the page on The Quantum Casino it was explained how, when we make a measurement of a quantum observable, there is a "collapse of the wavefunction" in which a probability wave collapses to generate a single observed value from a range of possible values. This process appears to work in the forward time direction only, i.e., it is irreversible.

An explanation for this apparent "collapse of the wavefunction" is presented in detail on the page on Quantum Decoherence, so I don't want to repeat it here. Suffice to say that the coherent phase relationships of the interference terms are destroyed when a particle interacts with the environment. The dissipation of these terms into the wider environment can be interpreted in terms of increasing entropy (again, see the section on "Decoherence and Entropy" on the Quantum Decoherence page for full details). Quantum decoherence can then be understood as a thermodynamic process: after decoherence, the process is said to be thermodynamically irreversible.

So once again the underlying physical principles appear to be time symmetric, with no fundamental preference for either the forward or backward time
direction. The apparent arrow of time produced by the "collapse of the wavefunction" is once again shown to be a result of increasing entropy.
As Andreas Albrecht explains in his paper
Cosmic Inflation and the Arrow of Time (when considering decoherence in the double-slit experiment):
*"A double-slit electron striking a photographic plate is only a good quantum measurement to the extent that the photographic plate is well constructed,
and has a very low probability of re-emitting the electron in the coherent 'double slit' state. Good photographic plates are possible because of the
thermodynamic arrow of time: the electron striking the plate puts the internal degrees of freedom of the plate into a higher entropy state, which is
essentially impossible to reverse. Furthermore, different electron positions on the plate become entangled with different states of the internal degrees
of freedom, so there is essentially no interference between positions of the electron. From this point of view, the quantum mechanical arrow of time is
none other than the thermodynamic arrow of time."*

### Why can't we remember the future?

If physical processes all appear to be time-reversible at a fundamental level, we might ask the question "Why can't we remember the future?" After all, we can remember the past, and physics seems to make no distinction between past, present, and future. So why don't we already have prior knowledge of what is going to happen in the future?

In order to answer this question, we shall consider the reasoning of James Hartle which is based around
the *radiative arrow of time*:

#### The Radiative Arrow of Time

In his paper The Physics of "Now", James Hartle makes the point that the reason we can't remember the future is because we have not yet received any information about future events. This thinking is based on the idea of a "light cone", the shape of which is defined by the speed of light:

At first glance, this might seem a very straightforward explanation of why we are unable to remember the future: it takes time for a light ray (photons) carrying information to reach us from a distant event. Basically, in the future we will have more information about distant events than we have at present. It is hard to imagine a situation in which light behaves differently - it would appear that light will always take time to travel from a point A to a point B:

This principle - that light will always take time, travelling forwards in time between two points -
is called the *radiative arrow of time* (also known as the *electromagnetic arrow of time*).
But this apparently clear-cut principle is not as clear-cut as it first appears. It turns out that the
"world line" of the photon is the same for a photon travelling forwards in time from point A to B as it is
for a photon travelling backward in time from point B to point A:

In fact, if we temporarily forget about the little arrows on the world lines (which indicate "cause" and "effect") then we see that the world lines of both the forward and backward photons are precisely identical:

This principle is clearly illustrated by a Feynman diagram of particle interactions which can be rotated at will, showing particle interactions work exactly the same backward in time as forward in time:

It makes no sense to talk about the entropy of a single photon (entropy is a statistical property of a large group of particles), so a single photon has no arrow of time. However, we do not receive our information about distant events in the form of single photons. Rather, it appears we receive information in the form of light rays which are composed of billions of photons (bosons are quite happy to congregate in the same state, and gather together in a cooperative fashion to create light rays). For this reason, studies of the radiative arrow of time have concentrated on studying the Maxwell electromagnetic field equations which treats light as a field with a wave nature (rather than considering the path of individual particles).

It is often quoted that Maxwell's electromagnetic field equations are time-reversible and so allow for
*advanced* (backward-in-time) waves as well as *retarded* (forward-in-time) waves. However, in
practice it is much easier to produce a retarded wave than an advanced wave, and this reveals the limitations
of Maxwell's equations as a full description of the behaviour of light. We need to combine Maxwell's equations
with something else in order to derive a radiative arrow of time.

James Hartle attempts to use Maxwell's equations to deduce the radiative arrow of time in Appendix A of his
aforementioned paper
The Physics of "Now" which is called *The Cosmological Origin of Time's Arrow*.
His approach (based on principles described in H. Dieter Zeh's book *The Physical Basis for the Direction of Time*)
combines the time-symmetric Maxwell's equations with the time-asymmetric boundary conditions of
the universe as a whole (he considers the asymmetrical total amount of electromagnetic radiation). The approach
suggests that because there were no free electromagnetic fields at the start of the universe, but there are fields
in the future, those fields must all be caused by retarded waves that have their sources in the past. However, I
don't see how the radiative arrow of time can depend on the total of electromagnetic fields in the
universe in this way. There's no equivalent of the second law of thermodynamics (increasing entropy) for electromagnetic
fields. The total of electromagnetic field in an isolated system does not tend to increase (as is the case with entropy).
The radiative arrow of time must surely depend on the increasing sum total of entropy in the universe, not the total of
electromagnetic field. Surely the radiative arrow of time must have the same cause as the thermodynamic arrow of
time.

At the beginning of the last century, Walter Ritz proposed that only retarded (forward-in-time) waves were physically
possible (i.e., the process was fundamentally time-asymmetric). In 1908 and 1909 he had a famous argument with Einstein
over this matter, as Einstein believed the process was fundamentally symmetric and could be explained by thermodynamic
arguments (see
here). It turns out that it is easier
to create a light ray in the forward time direction as the behaviour of the billions of photons as they are produced
(by an ordered source such as a light bulb) and scattered (when they reach a target) can be understood in turns of
increasing entropy: *"This arrow has been reversed in carefully-worked experiments which have created convergent
waves, so this arrow probably follows from the thermodynamic arrow in that meeting the conditions to produce a
convergent wave requires more order than the conditions for a radiative wave. Put differently, the probability
for initial conditions that produce a convergent wave is much lower than the probability for initial conditions
that produce a radiative wave. In fact, normally a radiative wave increases entropy, while a convergent wave decreases
it."* (see the
Wikipedia article on the Arrow of Time). Hence, the
reason we do not see convergent, advanced waves can be explained in terms of entropy.

When I turn on an electric light, for example, the photons leave the bulb in a relatively ordered form. The
photons then radiate away from the bulb, redistributing themselves around the room (i.e., a radiative wave), creating
a state of greater disorder - increased entropy. As Andreas Albrecht explains in his paper
Cosmic Inflation and the Arrow of Time: *"The complete absence of the time-reverse
of radiation absorption is understood to be one feature of the thermodynamic arrow of time in our world. A hillside
absorbing an evening news broadcast is entering a higher entropy state, and the entropy would have to decrease
for any of the troublesome time-reversed cases to take place. So in the end, the radiation arrow of time is none
other than the thermodynamic arrow of time."*

#### Could it be possible to remember the future?

If we consider the hypothetical situation in which we have found a way to circumvent the limitations imposed by the radiative arrow of time, it is interesting to ask if it could ever be possible to remember the future. And, if so, what would our "memories" by like?

In this respect, the Scottish philosopher Donald Mackay suggested an interesting "thought experiment". Mackay wondered if it could ever be possible to predict how someone will behave in the future, and, if so, what would be the consequences for human free will. If we had complete knowledge of the current state of a person's brain, would we be able to accurately predict a person's actions in the short-term future? Basically, if we are able to predict how a person will behave - and the decisions they will make - in the future then human free will is shown to be a fallacy, an illusion.

However, Mackay suggested that it would be impossible to predict a person's future decisions **if
that predicted future was made known to the person**. This is because the person could then choose
to act in a different way from how you have told him he will behave. This is described by John D. Barrow in
his book *Impossibility*: *"Consider a person who is asked to choose between soup or salad for lunch.
If we introduce a brain scientist who not only knows the complete state of this person's brain, but that of the
entire universe as well at present, we could ask whether this scientist can infallibly announce what the choice
of lunch will be. The answer is 'No'. The subject can always be stubborn, and adopt a strategy that says
'If you say that I will choose soup, then I will choose salad, and vice versa'. Under these conditions it is
logically impossible for the scientist to predict infallibly what the person will choose if the
scientist makes his prediction known."*

So if a person gains access to knowledge about his future behaviour, it would appear that it becomes impossible to predict that future. But this knowledge about future behaviour is precisely what a person will gain if he is able to remember the future. So if a person is able to remember the future, he could then choose to act in a different way to how his memory of the future tells him he will act! There would appear to be a logical inconsistency here: if a person is able to remember the future, then those memories of the future instantly become unreliable. Therefore, it would appear to be impossible to "remember the future".

As an example, here's how Dilbert might behave if he could remember the future: