The Relativity of Time- Which Came First, the Boom or the Bang?

“Relativity”- M.C. Escher

I was a young and excitable undergraduate when first I laid eyes on the enchanting theory of relativity. Never will I forget the sound of my jaw dropping when our professor casually stated that simultaneity is relative and that moving clocks run slow, which prompted me to ask “what do you, as a physicist, mean when you speak of time?”¹

The purpose of this question was to gain some intuition for the relative theory of time and thereby gain a deeper understanding of the discovery.  After all, the fact that a cannonball and feather experience the same acceleration in free fall is completely unintuitive. Our initial reaction to witnessing this fact is one of confusion and disbelief which stem from reasoning based on an incomplete picture of the world i.e. “the heavier ball is pulled much harder so it must fall faster”. However, in light of Newton’s second law i.e. “the heavier ball needs much more force to equally accelerate”, the result seems almost obvious.

The aim of this essay is to reveal the common flaws in our logical conception of time in an effort to demystify the results of special relativity. Some may argue that this is not necessary since appealing to correct mathematical manipulation will always lead to the desired results. This is certainly true provided that one is manipulating an expression which precisely captures their desired physical problem and is further capable of interpreting the results. This requires a good understanding of the link between physical concepts and their mathematical representations, which can easily be lost in the trails of symbolic acrobatics:

“Our symbolic mechanism is eminently useful and powerful, but the danger is ever-present that we become drowned in a language which has its well defined grammatical rules but eventually loses all content and becomes a nebulous sham.”- Cornelius Lanczos

Understandably, the classic approach to special relativity begins with the well-justified postulate that the speed of light is constant in all inertial reference frames. While mathematically palatable, this approach is conceptually difficult in that the consequences are not so intuitively appealing. I discovered an alternative and much more satisfying presentation in the exceptionally lucid works of Einstein and Reichenbach which I will follow once more².

The Measurement of Time

First, let us recall that Physics concerns itself purely with measurable quantities. That is, no quantity should exist in the framework of physics if we haven’t agreed on how to measure it.

Measurement of time is fundamentally achieved by the observation of periodic processes, that is, processes which ideally repeat exactly e.g. a pendulum swinging back and forth while maintaining the amplitude of its swing. When we say that we are measuring time, we are in fact only counting the number periods completed by some periodic process.

A crucial question comes to mind: can we tell if successive periods of an ideal clock last the same amount of time?

At first sight, it would appear that they must last the same amount of time. However, Einstein pointed out that unless there exists some experiment through which we can test whether or not this is true, then any answer to this question is valid as a definition since it is empirically undecidable- we have no way of deciding if it is true or false.

In order to measure the duration of each period of a given test clock, we need to make use of another reference clock, perhaps operating by a different mechanism. Suppose that the reference clock shows that successive periods of the test-clock elapse over different times. You may be tempted to deduce that our test-clock is non-uniform however this would only follow if we already knew that our reference clock was uniform. We are not allowed to assume this since we would have to prove it by means of an experiment similar to this one. Therefore, from these observations, we can only deduce that each clock is non-uniform with respect to the other.

Suppose instead that the reference clock shows that successive periods occur in equal times. Even then, it does not necessarily follow that those periods last for the same amount of time. A correct deduction would be that the clock periods are synchronised i.e. if they are both non-uniform, then they are non-uniform in exactly the same way.

This question is, in fact, a pseudo-problem because it takes us into a logical circle: we count periods to tell time, but to tell if two periods last the same amount of time we would have to count periods. You may object, calling the laws of physics to your aid in an attempt to escape this vicious circle.

The problem is that the laws themselves are built on this very definition. That two successive periods of an ideal clock last the same amount of time is not a matter of knowledge but is a matter of arbitrary definition up to the requirement that any period lasts a strictly positive amount of time³.

The price of changing this definition is an unnecessary alteration to the laws of physics which teaches us nothing new and introduces unnecessary complications. Any “new effects” can ultimately be traced back to our unnatural definition of time. More on this in another essay.

Whichever definition we choose to employ, we should be able to come up with laws of physics which make correct predictions simply because we determine the laws after we make this definition. The resulting simplicity of our laws will directly depend on the definitions we choose to employ so, until we run into a local universal force i.e. one whose effect varies from one place or time to another, let us assume that there are no such forces exist (see my previous essay if you missed the discussion on universal forces and coordinative definitions⁴)

The Relativity of Simultaneity

Having dealt with the local measurement of time, we must now examine the comparison of time at different locations. The central concept which gives rise to all the apparent weirdness of special relativity is that of the simultaneity of distant events.

In the past, it was commonly believed that it is always possible to provide an absolute answer to the question “did the spatially-separate events A and B occur simultaneously?”

Absoluteness dictates that error-free measurements of any observer using any set of coordinative definitions would lead to the same answer.

Conversely, relativity dictates that error-free measurements made by different observers can lead to alternative but equally valid answers, each of which depends on some choice of coordinative definition.

Such a proposition is not necessarily illogical: if I asked you to determine the order of three sliding beads on a circular ring without any additional information, you will find that you are able to manipulate them so that any bead is the ‘first’ and any bead is ‘last’.

So how does this relate to our measurements of time? Before we get there, we must precisely define what is meant by the statement “the events A and B occurred simultaneously.”

If an event occurs at a distant location, how do we come to the knowledge that it has occurred?  The answer to this question is summarised by one word: news. When a plate breaks, we receive the news of it breaking by sound or sight or the sudden silence that ensues. In all cases, there is a slight delay between the event actually occurring and the news arriving at our location because the speeds of the messengers, of light and sound, are finite.

The faster the messenger or the closer the scene, the closer we are to knowing the state of that scene in our present moment. The speed of light is so great that the delay is practically unnoticed over relatively short distances. This is why we are inclined to believe that whenever we look around, we are observing the present moment of everything we see when, in fact, the further away we look, the further in the past is the scene we see:

If we receive the news that an event just occurred in the Amazon, and we know that it took the messenger two days to deliver the news, then our news of current events in the Amazon is ironically two days old: we have no idea what is happening there in our present moment!

Physically speaking, time is a means of ordering of events: to say that  “events A and B occurred simultaneously” is to essentially say “we do not know which one happened before the other”. In other words, we cannot order these events in time.

This lack of knowledge could be technical, due to limitations in the precision of our instruments, or it could be logical, due to the possibility that they actually occurred simultaneously implying that they truly cannot be ordered in time. We will concern ourselves only with the latter possibility.

Let us examine a realistic situation of relative simultaneity in a thought experiment. Consider the following accurate depiction of a duel and let us assume that second shooter doesn’t see the bang before he shoots.

Is it possible to tell who fired first?

It is very tempting to jump the gun and say it’s obvious, however, let us pause for a moment and suppose that both actually did pull the trigger simultaneously.

Since our perspective is closer to one shooter, we receive news of the bang happening first so we believe that he shot first. Similarly, an observer positioned closer to the other shooter would believe that they shot first. Finally, a special observer, who lies somewhere between the two shooters, will receive the news of both shots being fired at the same time.

From this information, our individual observers cannot determine the actual order of the two distant events, but they can order the arrival of news from the events. If an investigator tries to work out what really happened, they would find that the task is impossible from the observers’ accounts alone!

If the speed of the messenger is finite, then, as demonstrated above, then it is possible for observers to arrive at alternative, equally valid, accounts of the order arrival of the news from two spatially separate sources.

Without a faster messenger, the only situation where all will necessarily agree on the order is when the news from the first source arrives at the location of second source before it provides a broadcast.

Why? Because this ensures that no observer will hear the second broadcast before the first one, no matter how close they are to the second source.

On the other hand, if the speed of the messenger is infinite, then there will be no delay between any event happening and news of it arriving at any distant location. Therefore, all observers will always agree on the order of arrival of the news implying that the investigator, as well as the observers, will be able to correctly order all events.

One may object and claim that, with knowledge of the speed of the messenger and the distances between the observers and the events, it is possible for the investigator to work out the correct order of the events and thus determine who fired first.

However, as Einstein pointed out, we are not allowed to assume knowledge of the speed of our messenger without first being able to tell if two distant events occurred simultaneously. This may be very surprising, but it can be demonstrated by examining how one determines the speed of a signal.

In principle, there is only one way which consists of sending a signal between two points A and B, separated by a distance , and recording the times of arrival of the signal t_A and t_B as told by clocks placed at points A and B respectively. The speed v is then given by:

Notice that in relating the expression to the time taken to travel the distance l, we are implicitly assuming that the clocks at A and B are synchronised i.e. that at any given moment both clocks indicate the same time. Essentially, we are assuming that if a clock at A shows the time t then B shows the same time t simultaneously.

One may still object and claim that we do not need to know the time at B if we simply place a mirror there and use the clock at A to work out how long it takes the signal to travel from A to B and return. However, this implicitly assumes that the speed of the signal is the same in both directions along l.  We cannot confirm this assumption without knowing its time of arrival at B, which brings us back to the matter of synchronising distant clocks.

Thus we arrive at a circle once more: to decide if two distant events occurred simultaneously, we require knowledge of the speed of the messenger delivering the news, but to determine the speed of the messenger we need to have some means of deciding if two distant events occurred simultaneously.

Einstein escaped this circle by noticing it: the simultaneity of distant events is not a matter of knowledge, but a matter of definition. He went further still and provided a coordinative definition equivalent to the unfalsifiable⁶ assertion that the speed of light is the same in all directions (and in all inertial reference frames but we’ll get there in a later essay). If t_R is the time indicated by clock A when the (light) signal returns from B, then Einstein’s definition is:

In general, one is allowed to make any definition of the following form:

Where ε is any number greater than zero and less than 1. This restriction is due to two facts: that the speed of light is finite so , and so the signal necessarily returns to A after its arrival at B so . Any definition, apart from Einstein’s definition, would be equivalent to asserting that light travels at different speeds in different directions- an assertion which again is essentially unfalsifiable for the reasons outlined above.

The deeper consequence is that the speed of light cannot be directly measured- it must in fact be implicitly defined.

Ultimately we cannot truly tell who shot first if we rely on light as our messenger- two investigators using different coordinative definitions can arrive at contradictory answers which are equally valid.

This suggests the following important generalisation to the definition of simultaneity:

 If two spatially separate events are indeterminate as to time order, then they are said to be simultaneous.

However, one imperative fact holds regardless of the choice of coordinative definition of simultaneity: each shooter believes that they shot first i.e. neither one can claim that they pulled the trigger as a result of the other shooting first i.e. there is no cause-and-effect relationship between these two events.

One last thing: let us suppose that we are given two distant events and are told that one event is the cause of the other. Then we necessarily know that the cause occurred before the effect. If the events occurred simultaneously, then it is not possible for one to cause the other unless if there exists a signal which travels at infinite speed from the location of the cause to the location of the effect c.f. action at a distance.

Contrary to popular belief, this is not logically impossible. It is rejected empirically for one reason: there seems to be a finite limit to the speed of causal propagation equal to that of the speed of light in a vacuum, a fact which has been thoroughly tested by experiment.

At no point in this essay did we need to consider the motion of clocks to arrive at our conclusions. It was demonstrated, I hope, that a relative theory of time is as natural as an absolute theory of time and that the physical choice is ultimately decided by experiment. On this matter Reichenbach said:

“It is a serious mistake to believe that if the state of motion is taken into consideration, the relativity of simultaneity is necessary. Actually, the relativity of simultaneity has nothing to do with the relativity of motion. It rests solely on the existence of a finite limiting velocity for causal propagation.”- Hans Reichenbach

I think that’s enough content for one essay. I will further explore the concept of causality, simultaneity and their relationship with topology in the next essay of this series.

Thanks for reading!

1: Yes, I was that guy and yes I still recall the collective eye roll that swept across the lecture hall when I asked this question. I asked so many questions that QM became my nickname (I think it stands for Question Man).

2: Relativity: the Special and General Theory- Albert Einstein, The Philosophy of Space and Time- Hans Reichenbach. I seriously cannot overemphasise how exceptionally enlightening I found these books- read them if you ever get the chance!

3: The time at a certain point must be a monotonically increasing function: this is our time metric! Incidentally, this is a good place to clarify the physical difference between proper time and coordinate time. Proper time is the total number of periods recorded by the clock between two given events. Coordinate time is what we get when we coordinate a time to each period, and in some cases both are exactly the same.

4: A metrical coordinative definition is a spatial metric. Provided it has the properties of a metric, it is admissible as a definition i.e. it cannot be completely arbitrary. The key point to notice is that a coordinate system does not define the geometry of the space to which the system refers. Rigid rods would correspond to intervals on coordinate curves while a metric coordinates a distance to these intervals, giving rise to the geometry of the space. One of the reasons why physical space is modelled as a three dimensional manifold is to capture the notion that it fundamentally has no geometry until we equip it with a metric.

5: Escher’s lithograph “Relativity” exemplifies this: contrary to popular belief, the structure depicted is not another impossible structure, someone actually built it with Lego!

6: Provided that light is the fastest causal signal in the universe. Even if we do find something faster, we’d arrive at the same problem because we can’t test the fastest causal signal against itself without going round in circles.

Quantum News Channel 1

Last time, we talked a little bit about entropy and classical information theory. If you haven’t already, check that article here. It gives a good introduction to this one, introduction so good that I have serious trouble writing another one here. Hopefully this short introduction will be over soon.

At the end of the previous post I mentioned that classically, probabilities are quite simple and that it gets much more complicated in a quantum case. Why exactly? And what is the “quantum case”?

Quantum mechanics is the language of modern physics. It is a base on which more complicated theories can be built, for example regarding behaviour of particles and fields, describing new phases of matter, interactions of lasers and atoms, or collective motion of solid materials. But all these theories rely on quantum mechanics. Some people, like Scott Aaronson [1], claim that quantum mechanics is at its core a theory about probability and information. It is just a more… complex one. Literally. As he puts it in his book:

“But if quantum mechanics isn’t physics in the usual sense – if it’s not about matter, or energy, or waves, or particles – then what is it about? From my perspective, it’s about information and probabilities and observables, and how they relate to each other.”

Let’s say you have a box, and in that box you have, say, 3 oranges, 5 apples and a pear. You reach inside and you grab one fruit. It is easy to see, because we are used to classic probability theory, that a chance of grabbing an orange is , a chance of getting an apple is , and a pear . We just divide the number of fruit of the requested kind by the total number of all fruit in the box. One particular property of this is that these probabilities sum to . What does that mean? Well, a sum of probabilities of getting an orange, an apple and a pear is a probability that you will get either an orange, an apple or a pear when you reach in and grab a fruit. Provided our assumptions – that you will indeed grab something and there are no other fruit in the box, this will obviously happen. It is called a sure event. This a fairly obvious, but crucial property – the sum of probabilities of all disjoint events, all different possible things that can happen, I mean, really all of them, must be equal to 1, because “something” will always happen.

Scientists started developing quantum mechanics because they encountered a range of peculiar phenomena which their classical theories failed to account for. Now we know that the fundamental reason behind this was that the classical probability theory fails to describe things happening at very small scales. Classically, if a probability of one event is and a probability of a different, disjoint event (think about grabbing an orange vs grabbing an apple) is , the probability that either one of these independent events will happen is the sum of the probability of each one happening on it’s own i.e . This in general doesn’t work in quantum mechanics.

In the quantum world, probabilities are changed to something called probability amplitudes. We can add probability amplitudes of two similar events to get an amplitude of a joint event (either the first or the second thing happens). Amplitudes are different from probabilities – first, they can be negative! What is more, it can actually be shown that for this theory to be consistent, amplitudes will sometimes have to be complex numbers. Second, it’s their squared moduli (if you are not familiar with complex numbers, think about squared modulus as a “square” of a complex number which ends up being a real number) what adds to , not the amplitudes themselves. In other words, if we know that one set of amplitudes describes everything that can happen, a sum of squared moduli of these amplitudes will be exactly , again, because “something” will always happen.

What exactly we are supposed to be adding? Should it be the squared moduli or the amplitudes themselves?

Any quantum system can be described as being in some state. That state is a mixture of several basis states, as described above. We are allowed to play with that state. Change it, make it evolve. We say we apply quantum operations, make the state interact with the enviroment, apply magnetic fields, shine light onto it. The state then changes. In particular, these operations might involve two different states at least partially evolving into the same state, with different amplitudes. Or the same state evoling into some other state in two different ways! In that case, to find out what is the total amplitude of that state in the mixture, we add the amplitudes coming from different processes, but ending up in the same state. Then the resulting amplitude is the amplitude of that state.

After these operations, we usually want to measure the state. By measuring, we ask a quantum system the question: which state are you in? But this question is not an open one! It is a close one. We have to pick all the possible answers we can get. The number of these answers is limited to the dimension of the system. For a single qubit, the dimension is two, so we can only get two different answers, the system can only tell is that it is in one of the two states that we pick. But if the state is somewhere in between, we are forcing it to change by performing the measurement, we are forcing it to become a different state, one of the ones we picked. The probability of that happening is the square of the amplitude associated with that state. If we then prepare another, identical state, and perform the measurement again, we might get a different answer, even if we do everything exactly in the same way. This is because quantum mechanics is instrinsically probabilistic (well, according to most interpretations at least). If we want to then calculate the probability of getting either one or the other states, we enter the classical zone and we first square the amplitudes of each state and then add these squares for the joint probability.

For example, let’s imagine a quantum particle that travels from a point A to one of the points B or C. These are somehow the only two points where it can end up. On its way, it can either travel through point X or point Y, but that is not known – we never learn which path was taken by the particle, but we do know where it ended up in. We thus add amplitudes of particle travelling through X to B and through Y to B too get an amplitude of ending up at point B. And then we do the same for point C, add amplitudes of travelling through X to C and through Y to C, and then we square the whole thing to get the probability. Now, if we add the two squares, this will add up to one. This is because in macroscopic scale probabilities are classical, and because B and C are different states,  easily distinguishable, the classical rules apply.

Because these amplitudes are in general complex numbers, it is possible that the amplitude of getting through X to B and an amplitude of getting through Y to B are opposite numbers and they add up to 0! In which case the existence of two different paths leads to a complete cancellation of any possibility of a particle ending up at B. This is the paradox scientists were struggling with: if you only have one path, the particle can end up at B, if you have only another one, the particle can still end up at B, but when both paths are available, it becomes impossible suddenly.

This is resolved by using an alternative probability theory. In fact, this assumption about probabilities, that they can involve negative numbers is the only one you need to make to recreate quantum mechanics. The fact that probabilities work differently than we previously thought is what underlies most of the modern physical theories.

But okay, we wanted to talk about information, right?

To talk about quantum information, we need a fundamental portion of it. In classical world, it was called a bit, a number, either 0 or 1. Something that either is or isn’t. It was realised by a physical system, which could remain in two separate states, for example magnetized or demagnetized, black or white, up or down. It is important to remember information is and always was physical, needed a physical realization. In the quantum world, we have qubits. Qubits
also are physical objects, that can occupy two independent quantum states, which are traditionally denoted as  and . These strange asymmetric brackets are ‘kets’ from Dirac bra-ket notation, it is just a way physicist write quantum states. But qubits, as quantum objects, can also occupy other states, which are linear compositions of the two states above, or the states “between” them. In other words, it might be in a state   , where and are numbers. Complex numbers, in general. These are exactly the amplitudes I mentioned earlier. A system in this state is called a superposition, exactly because it is a sum of two different states with some amplitudes. Superposition is a kind of subjective concept – a quantum state is represented as a vector in some space, but whether this vector is a basis vector or not depends on the choice of basis, for example, mentioned  and states are called computational basis, but there are many other bases that can be used depending on circumstances. Therefore the same state might be regarded as a superposition or not depending on from which angle we are viewing the system. The important bit is, once you settle for a basis, for a particular angle from which you want to view your quantum system, the superposition states are possible. The basis states, whatever you choose them to be, are not going to be the only states your system might reside in.

 

 

 

A quantum system may remain in such state as long as it is not disrupted in any way. The problem is, we don’t usually know what the state is exactly. And trying to find out – a measurement, is already a disruption of some sort, because it has to involve some physical process that interacts with the state, and such states are really fragile. When we see things, photons need to reflect off them and reach our eyes, when we hear them, air particles vibrate on the way from the source to our ears, when we touch things, electrical signals travel through our body to our brains. The interaction has to be there for anything to be observed, and really what is important is that it happens, not that its products reach our bodies. In other words, the existence of a human or other conscious being is not needed for a measurement to be conducted. Unfortunately, it is a common misconception among laymen and that needs to be stressed.

One of the reasons quantum information is so different from classical is a mysterious quantity that has no equivalent in a classical world, entanglement. Entanglement is a property of a system consisting of many particles, and it is a peculiar way the states of particles this system consists of are correlated. An example of an entangled system is a pair of photons of opposite polarisations. Before measurement, we know that the polarisations are opposite, but we don’t know exactly what are their directions. After we measure only one of these polarisations – we can measure it in whatever way we want, along any axis we choose, and that way will affect the result, the second one becomes defined and exactly opposite to the first one! It turns out that entanglement has many peculiar properties, some of which are not yet completely understood.

Classically, systems that are correlated, can be correlated to different levels, some pairs of variables are correlated very strongly, and some only a little bit. For example, there is a very strong correlation between your age in years and your age in minutes – it is basically the same quantity, but in different units. But there also is correlation between the age and height across children – older kids are going to in general be taller, but people have different heights, because that is partially determined by genetics, so younger children might be taller than older ones if their parents are especially tall. So there are different levels or degrees of correlation even classically. Similarly, we can find that there are different degrees of entanglement. Some pairs of systems can be strongly entangled, and some only to some level. A fascinating result is that it can be extracted from physical systems which are not completely entangled to form a fewer number of systems which are, so it can be treated as a resource. What’s more, entanglement of a single particle is limited – if the particle is completely entangled with another particle, it cannot be entangled with anything else. [3] It is called monogamy of entanglement and we can see that it is completely different from the classical correlation, because classically, you can have as many systems strongly correlated with each other as you manage to create. This is because entanglement is not just statistical correlation. It is much stronger, because in quantum mechanics you can measure the state of the system in many different ways, and entanglement forces correlation in each of these measurements, not just one.

That is why quantum systems can be correlated much more than classical systems because of entanglement. This is how physicists have confirmed that the world cannot be purely classical, that is, levels of correlation which are classically proven to be impossible are actually possible in practical experiments involving highly entangled particles!

For example, think about a following game. There are two players, Alice and Bob, who collaborate to win. Each of them has a standard deck of cards and every turn, draws one card from the deck and doesn’t show it to the other player. Then, basing on the card that they’ve drawn, each of them decides either to push a button or not. They have to make their decisions simultaneously, so that they are not affected by the other’s decision. Now, if they both pushed their buttons or they both haven’t, they win provided at least one of the cards they’ve drawn was red. But if one of them pushed their button and the other have not, they win only if both their cards were black.

It is quite easily seen that they can win in  of games if they just never push their buttons ever, because there is a 75% chance that at least one of their cards is going to be red. It can also be shown that it is the highest overall probability of winning for any strategy they are allowed to come up with.

However, if the choice of a card from the deck is determined not by sheer luck, but by a clever quantum measurement of a half of entangled quantum state Alice and Bob share between each other, the probability can go up to about 85%! That beats the classical result which can be proved to be correct. Both Alice and Bob only measure their own qubit, which has exactly 50% chance to be in one of two states, what can be associated with a card being either red or black, so the probabilities work out. Also, they can be very far away from each other, so that for any information from the other party was to influence their decision, it would have to travel much faster than the speed of light in a vacuum. However, despite all this, the quantum nature of entanglement still bumps up probabilities to 85%! That has been confirmed experimentally, and it is a realization of one of the Bell inequalities. These inequalities are rigorous statements, which reality would have to satisfy if it was purely classical or at least local, what means, there would be some local hidden influences, which cannot be seen directly, but which change the way it behaves. Experimentally, however, it was in most cases confirmed that it doesn’t satisfy those, and actually it comes close to what non-local quantum theory predicts.

Entanglement is just one of the resources that quantum information theory utilizes, and which make it much richer than classical theory. What the quantum information theory tries to do is to find out about what other types of information are there in the quantum world, because there are certainly more than in the classical world, what other types of information processing are there, and then what can we actually do with those and what exactly are the limits we have to obey while performing those processes, how do this different types of information change, are created and then destroyed.

The researchers managed to find many, many more such types, listing them all here would not be possible, so for now try to think about this and come back in two weeks for more!

In the meantime, you can read more about entanglement and quantum mechanics here:

1. An excellent semi-popular book by Scott Aaronson about information and computation “Quantum computing since Democritus” with more details about the approach that I have basically stolen from him.
2. Bennett, Charles H.; Bernstein, Herbert J.; Popescu, Sandu; Schumacher, Benjamin (1996). “Concentrating Partial Entanglement by Local Operations”. Phys. Rev. A. 53: 2046–2052 – the first paper about entanglement distillation.
3. V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, 052306 (2000), a paper that introduced monogamy of entanglement.
4. The “Bible of quantum computing” with chapters about quantum mechanics and entanglement “Quantum computation and quantum information” by Michael Nielsen and Isaac Chuang
5. Definetely also check out Scott’s blog: https://www.scottaaronson.com/blog/
6. Very short description of quantum computing, from one of research group at University of Strathclyde, I am including it because it is short and I took one of the images from this site, I hope they don’t mind because it was really nice.

 

The Relativity of Geometry – Is the Earth Flat?

“Hand with Reflecting Sphere”- M.C. Escher

Does experiment single out a unique geometry as being that of experience?

I asked this rather striking question at the end of my previous essay with the hope that the reader would be led to the unexpected and insightful discovery made by Poincaré¹ at the turn of the century. In answering this question, we will come to understand the imperative and subtle role played by definitions in our models. In particular, we will follow Reichenbach’s beautiful work² in our quest to solve this problem and what we will find is one aspect of the treasure that lies at the heart of relativity.

Like all great stories, let us begin with a thought experiment. Suppose you are a Flatlander³ living in a spherical surface. Would you be able to deduce the fact that you live in a spherical surface by making local measurements i.e. without going all the way around the sphere? (Spoiler for the experienced readers who have been told it is possible: it’s actually not)

“Of course!” answer the astute or geometrically inclined readers:

“We would discover that geometry in this surface is consistent with spherical geometry. For example, if we were to draw a circle and measure the quotient of its circumference and diameter we would find that our answer deviates slightly from π.

To be more precise, notice that if we take a planar slice through a spherical shell, it would be divided into two shells whose boundary is circle. Now recall that a circle, properly defined, is the collection of points which are equidistant from some point called the centre.

With that in mind, one can identify the blue shell as a non-planar circle of radius R as measured in the shell, and circumference C. In other words, a Flatlander would be under the false impression that the shell is a planar circle whose centre is located at the North Pole and has radius R.

We can also identify C as the circumference of the “normal” planar circle of radius a in the plane of our slice whose centre is at a distance h below the North Pole, and perpendicular to the plane as indicated in the diagram.

The ratio of the circumference and diameter of a planar circle is given by:

C is the same for both circles so we translate the radius of the planar circle in terms of non-planar quantities:  and .

For the non-planar circle we end up with a different ratio which we must express in terms of R, the only quantity Flatlanders can measure.

(1)

Where is some constant.

Since the Flatlanders do not see this three-dimesional distortion, they expect that instead of (1) and will indeed be surprised to discover that their measurement of pi depends on the size of the circle they draw!

Of course, one measurement alone is not enough for anyone to make any broad deductions, so the experiment is repeated for a bunch of circles of different sizes. The results are then plotted in a graph of against R.

The resulting curve would suggest the relation (1) for some constant k. A clever Flatlander might then suggest that Flatland is in fact a spherical shell to explain these results. Better still, this model provides a solid interpretation for the constant k as the inverse of the r, thereby discovering the radius of Flatland as a bonus!

Consistency of our results with (1) is what is meant by the statement that the deviation is consistent with the results of spherical geometry.  Thus we can deduce that we occupy a spherical surface, at least locally and on the scale of our measurements⁴.”

If these arguments are carefully examined, one would certainly agree with fact that the measurements will be consistent with spherical geometry. However, it does not immediately follow from our results that we occupy a spherical surface, even locally and on the scale of our measurements. To see why let us consider another thought experiment.

Descartes’ Demon

Suppose once more that you are a Flatlander, except that you now occupy a flat plane in which you have performed these experiments and are about to repeat them in a demonstration for your Flatlander friends (who don’t really care but like you enough to hang about). Before you do this, a three-dimensional Cartesian demon, wishing to cause an existential crisis, produces a mysterious field of force in your space.

This peculiar force is similar to heat in that it causes all materials to expand. However, unlike heat, it has the unique property of affecting all materials in the same way (all materials have the same expansion coefficient at each point in space) and there is no possible way of insulating yourself from its effects.

Such a force was aptly called a universal force by Reichenbach. He also identified another type of force which affects different materials in different ways, which he called a differential force.

It is possible to directly demonstrate the existence of a differential force, such as heat, by observing its effects on different materials: this is how most thermometers work.

For example, the mercury expands far more than glass does under the influence of heat. In a thermometer, we may imagine the body of mercury, which is in the shape of the glass tube, as being magnified under the influence of heat. The same thing happens to glass tube except that the degree of magnification differs.  Therefore, when heated, the mercury ends up travelling up the glass tube in search of more space since the volume contained by the glass has not equally increased.

If heat were a universal force, however, the situation would be completely different. In that case, a mercury (or any other material for that matter⁵) thermometer would not work as an indicator of temperature since the body of mercury would expand in exactly the same way as the body of glass holding it (provided the temperature is the uniform across the glass and mercury). The only effect of heat would be an apparent magnification of the thermometer i.e. a change in the geometry of the space it occupies.

In fact, the unique properties of a universal force critically imply that it is impossible to directly demonstrate its existence (or even distinguish between different types of universal forces which have the same effect).

To directly demonstrate that a measuring rod and some object have the same length, we would have to place the object on a sheet of paper, mark its ends, and then replace it with the rod. If the rods ends line up with the marks then we have demonstrated they have the same length.

The issue is that a universal force would equally stretch the object and the section of the measuring rod indicating its length: if two objects have the same length along an axis, then since the force is material-blind, it does not see a difference between them along that axis. This leads to an important question: how can we tell if the size of the metre rule has changed when we rely on it for our measurements?

You may be inclined to design some clever experiments involving light signals to do this, but you must remember that the force would affect light as well. To make this point clearer, observe the space slightly above a flame and you will see that through the patch of hot air, the geometry of the world appears distorted. A universal force would have exactly the same effect.

“Balcony”- M.C. Escher

It is only possible to indirectly deduce that such a universal force exists through its effect on the geometry of space. For example, suppose we have four measuring rods at our disposal which are all equal in length and that there is one special line in the plane along which the force acts. We now attempt to construct a square with these rods and accidentally place one on that special line.

We construct the square by ensuring that the rods are all at right angles to one another: Euclidean geometry guarantees that we must end up with a square when we are done. Instead, we are surprised to discover that we end up with one of the two following shapes:“How is this possible?” we wonder. “Maybe the construction process is inaccurate because the angles are not exactly right angles” someone suggests.  So we instead attempt to simply ensure that the ends of the rods are the vertices of a closed quadrilateral: Euclidean geometry tells us that an equilateral quadrilateral is a rhombus. Instead, we are shocked to find that we necessarily end up with a trapezium instead. To be precise, as Flatlanders, we would find that there is only one pair of parallel sides if we were to measure the angles at each vertex.We are led to one of two conclusions: either Euclidean geometry is wrong or our rods do not have the same length. The second option is easier to stomach so we attempt to check that the rods are still equal in length: we compare the parallel sides by marking the vertices and switching the rods. We find, to our amazement, that we still end up with exactly the same shape. How can they possibly form a trapezium if the rods are of equal size when we compare them at both locations i.e. the parallel sides of the trapezium? Is Euclidean geometry wrong then?

That is not quite correct either. Our surprise and amazement stem from the fact that we believed that we occupied a planar surface since the results of Euclidean geometry used to apply. Now we are faced with the result that an equilateral quadrilateral forms a trapezium- a Non-Euclidean result. This does not mean that Euclidean geometry is wrong: it means that our assumption that it holds is no longer true. The universal force has changed our geometry so that Euclidean results no longer apply.

When we came up with the idea of measuring rods, we were secretly assuming that the rods are rigid i.e. that their length is constant and unchanging everywhere in space: a Euclidean assumption. However, in the face of a universal force, this assumption breaks down as we have just seen.

The Relativity of Geometry

Let us dig a little deeper into our metrical assumptions. Suppose that two Flatlanders each have measuring rods which are equal in length at point A in space. One decides to travel along some path to a point B and the other follows along a different path. Will the rods necessarily be equal in length at point B?

After a little thought, we would find that it is not logically necessary. It is an empirical fact that if two rods are found to be equal in length in one region in space then they will have equal length at all regions in space, no matter what path we choose to take. Can we deduce from this that their length does not change while they were carried from A to B?

Once more, this is not logically necessary – both have length at some point and both have length  at another point, but this does not imply that . More critically, there is no way of directly demonstrating nor indirectly deducing this relation without prior knowledge of our geometry.

If this knowledge is not available, then the congruence of lengths at distant points becomes a matter of definition, which can be as simple or as complicated as we want it to be. It is not possible to appeal to our physical laws in an attempt to prove or falsify the definition since they implicitly rely on it in their formulation.

To clarify, if we make the definition that measuring rods are rigid and discover later that the speed of light is the same in all directions, then this physical law itself would change and become far more complicated if we use a different definition which states that rods are longer in some regions of space⁶.

A coordinative definition is necessarily made, explicitly or implicitly, before coordinating abstract concepts to physical entities. A metrical coordinative definition coordinates a value of length to a measuring rod at all points in space. For example, when we coordinate the concept “standard unit of length” to some stick in Paris, we implicitly make the coordinative definition that its length is the same and equal to one meter wherever it is placed- otherwise, the concept is worse than useless.

Armed with this knowledge, let us return to Flatland where our plane is now under the influence of the Cartesian demon’s force field. The demon cleverly arranges for the field to alter our measuring rods so that after we perform our circle experiments again, we end up with results consistent with spherical geometry. We are then forced⁷ to one of two conclusions: either the geometry in our region has changed or a universal force is altering our measuring rods, and we further admit that we do not have the ability to distinguish between the two.

This triggers the intended existential crisis: if we do not know what the geometry of our universe is, then how can we tell whether the results of our measurements are influenced by the action of a universal force field on our measuring rods or are due to the actual geometry of the universe?

It is logically impossible for us to locally tell the difference between the case where we live in a spherical shell and the case where we live plane in which a universal force is acting to make it seem that way.

If we choose to make the coordinative definition that a measuring rod has the same length everywhere i.e. that it is rigid, then we would deduce that we live in a spherical surface, where there are no universal forces by definition. However, if we choose to make a more complex coordinative definition such that a measuring rod has a length which varies in such a way that we reproduce the effects of the universal force, then we would deduce that we live on a planar surface where the force is acting. The question “what is the real geometry of our universe” becomes objectively indeterminate as far as Flatlanders are concerned.

The simplest way to go however is to set all universal forces to zero by defining our measuring rods to be rigid. With this definition, our geometry is uniquely determined and the results of our experiments are entirely explained as being due to that geometry.

The statements “this space is a locally spherical surface” or “this space is a flat Euclidean surface” are incomplete and untestable if we do not specify our choice of metrical coordinative definition. It is the logical limitation of our empirical knowledge of the universe that prevents us from singling out one absolute physical geometry as being that of experience.

Asking for an absolute answer to objectively indeterminate questions is bad and can lead to confusion. As Reichenbach beautifully puts it:

“Pseudo-problems arise if we look for truth where definitions are needed.”

We must admit that as Flatlanders, we cannot logically decide if either definition is “the true one”. We are also able to formulate our physical laws with any definition we choose to adopt: the way we choose to picture our universe does not fundamentally change our experience of it.

That being said, we usually find that one definition simplifies our physical laws and enables us to make predictions more easily than the other and therefore choose to adopt it.  “I define, therefore I am (a theoretical physicist)” claims a bold Flatlander in triumph.

Therein lies one aspect of Einstein’s vision when he demonstrated that, instead of regarding the universe as being a Euclidean spacetime⁸ where a universal force known as gravity exists, we may equally regard the universe as being a Non-Euclidean spacetime where no such force exists.

Up until then, the law of gravitation was just an empirical law which led to accurate predictions most of the time, with some notable exceptions. By switching the playing field and thinking of gravity as a universal force, Einstein tied the notion of gravity to a concrete element of the geometry of spacetime known as its intrinsic curvature.

Once his model was complete, he “translated” the results back to the Euclidean spacetime we assume for the purposes of making measurements and found that this model could explain those notable exceptions to the predictions of Newtonian gravity as well as agreeing with all correct Newtonian predictions.

You may be wondering what the difference is and it lies in the fact it was not possible before then to derive the law of Newtonian gravity from some principle: we had to settle for measurements and an empirical formula. Gravity is incredibly weak, which makes it very difficult to make precise measurements: we would not be able to account for small deviations in the law until their effect is astronomically magnified.

Starting with a principle and working out the physical law is far easier provided that you’ve found a principle which reproduces Newtonian results and provides better predictions as well. It took no less than Einstein and Hilbert to do this.

I feel that this is a good place to stop being thought-provoking and will allow you to peacefully digest the heavy contents of this essay. As always, I welcome any and all comments and suggestions.

My next essay will be on the mystery that is Time. Before I get there, I would like to leave you with a question to contemplate: how can an observer decide if two events occur simultaneously if the messenger relaying the news of each event has a speed limit? Be careful in answering this question – classical expositions on special relativity do not really touch upon this matter (even though Einstein was very careful in providing an answer in his exposition).

Oh and disclaimer: this essay does not in any way imply that the Earth can be regarded as being flat unless you change your metrical coordinative definition of length, come up with a new law accounting for it in physics, and change all the others along with it. (Not to mention the fact that gain nothing for your troubles since whatever shape you end up with will necessarily be topologically equivalent to a sphere). Good luck with that.

1. Henri Poincaré: Non-Euclidean Geometries (1891)
2. Hans Reichenbach: The Philosophy of Space and Time (1956)
3. Two –Dimensional beings occupying Flatland, the fictional setting of a charming novel of the same name, written in Victorian times. Disclaimer: this is not the type of novel that would be appreciated by SJWs.
4. It is not inconceivable that we might end up with results which suggest a complex shape that appears spherical on the scale of our measuring rods but ceases to be so upon examination of much smaller scales. Similarly, we can only deduce that the region of space where we performed our experiments is geometrically consistent with spherical geometry. This may seem pedantic but it in modern physics, it is a recurring theme that things are rarely what they seem…
5. Pun intended.
6. Another way to think about this is by imagining how the world would appear if observed through a spherical glass and how the formulae for the laws of physics would change but certain predictions would be independent of this change e.g. how long it would take an apple to fall to the ground vs. the shape of the path it takes and even then, shape is relative to our definition but the fact that the path is a geodesic is not.
7. Pun not intended.
8. Just ignore the complications of bringing time into the picture for now.

The physics of information and uncertainty

They say we live in an information age. That usually refers to the fact that communication between humans residing across the globe is much quicker than it used to be and that value is more than ever associated with knowledge. But what is information actually?

Information is… just a word. An abstract concept! The real question we should be asking here is about things that can be seen or measured. Instead of thinking about what “information” means, we should be thinking about what does it mean to “inform” someone of something, we should focus on a verb, because it has a definite, physical meaning.

So that when you inform someone of something, you send them some pieces of knowledge, change their perception of reality, add some new element. With useful information you can often change your mind about what you think about chances of certain events occurring. With more data, we make our guesses about reality more effective.

Perhaps you need to find a person called Jamie in a crowded room. You don’t know anything about that person, it can be anyone. Maybe someone tells you that Jamie is a man, not a woman, which limits your choice to about half the people in a typical room. Then you might be told Jamie is over 40 years old, which then filters out younger men and boys present, if there are any. Finally you may learn Jamie has a moustache and wears a red t-shirt, which limits your choice to just one person, who you then approach.

Each of these messages contained some information, because it let you change your perception about who is that person you are looking for. A bit more formally, it let you update the probability of finding Jamie by asking a random person if they are him. Or in yet other words, it limited your uncertainty about the stranger’s identity to some degree.

However information might seem a volatile, abstract concept, every transfer of information involves a physical process. You might hear it, which means that particles of air vibrate in a specific fashion that reaches your ear, or see it, when photons enter your eyes and then signals travel to your brain which interprets it, or get it in some other way. Storage of information also requires physics, a message might be written down on a piece of paper, or as a sequence of magnetized and demagnetized segments on a disc, but it has to be physically contained somewhere to exist. We say that information is physical.

You might still think, though, that information is then just a concept humans adopted for themselves, that for physics there is no difference between a sheet of paper with words in English written in black ink and a similar sheet of paper with similar amount of ink on it. And it would be kind of correct, but only kind of.

Two configurations like the ones above can be equivalent in a sense that they are equally probable, if you sprinkled some ink over the paper there is an equal chance that it would form these exact shapes. However, the catch is in the word “exactly”. Surely, sheets with random dots on them constitute a much wider category than sheets of paper with words on them. Therefore the chance that randomly sprinkled ink would form a word – any word for that matter – on a piece of paper is much lower than that it would form something that doesn’t look like a word. So even though a probability of getting one exact configuration is the same as a probability of another exact configuration, thinking about these exact configurations – microstates, as they are called in statistical mechanics – is not want we want to do here. Words appear less random than dots simply because there are much fewer configurations that resemble words! In the same way, if we tossed a coin 6 times, and got a following sequence: H,H,H,H,H,H, it would look to us much less random than a sequence like H,H,T,H,T,H for example.

So far, this might seem like juggling words around. To get more physical, think about gas in a box. Say there are about particles in that box, and so there is a plethora of exact configurations of their positions and momenta, the parameters that are in principle needed to fully, exactly describe the system. In most of them, the particles would appear to be almost evenly spread across the box. Now, let us think about the unusual configurations, when all the particles are in the right half of the box, and are randomly spread there. There are much fewer such configurations, so that is much less probable. How exactly?

If we assume the box is divided into equally probable little spaces the particles can occupy, no matter how big these spaces actually are, then there are configurations in the case when particles can be spread out across the whole box. But in a half of the box there are only spaces, what means only configurations – a square root.

Physicists like to talk about something called an entropy of each category. Because it should not matter how small these spaces actually are, the change in the numerical description corresponding to a change between a container filled with gas particles and the one half filled should be the same, the earliest equation for entropy, written down by Boltzmann, involves a logarithm, which deals with that issue:

Where is entropy, is a number of possible states – arrangements in this case, in general this also involves momenta, and is a Boltzmann constant. This equation shows that the entropy of a half filled container is a half of the entropy of a full container. It however assumes that each microstate has an exact same probability to occur. If the probabilities are different, equal to , the entropy changes to Gibbs’ form, given by

This is a bit different, before we had number of states, a large number, now we have probabilities, which are small. However, we notice that Gibbs’ formula becomes Boltzmann’s formula if we replace all -s by , one over the number of microstates, which is a probability of a single microstate if they are all the same. This is because the minus sign in front inverts the argument of the logarithm.

Entropy is a key concept here. The second law of thermodynamics, known since XIX century, states that in any isolated thermodynamic process, the total entropy of the system can either stay the same or increase. It can never decrease, although entropy of certain parts of the system might, on the expense of greater increase in other parts. If the system evolves via spontaneous processes, each increasing its entropy, it will finally reach the maximum allowed entropy state. In other words, if we place gas in half of the container and release it, it will fill the whole box. We can break an egg, but it is impossible to put it back together. Order is in general harder to achieve than chaos. Entropy is a measure of how much uncertain we are about the system. If we know everything – the entropy should be zero, if we know nothing at all, it should be maximal. Therefore entropy is a measure of information, the more information you have, the smaller uncertainty and the smaller entropy.

Thermodynamics connects entropy to work and heat, and this is how physics distinguishes information from the lack of it. Essentially, if you want to achieve greater order in your system, smaller uncertainty, you need to expend some energy for that purpose, do some work, which will then turn into a form of energy that cannot be collected back in its entirety. To create new information, you need to spend energy, which might not be totally regained after that information is destroyed. Also, information tends to wear out over time, when spontaneous processes increase entropy, gas travels and fills the box by itself, random errors erase bits in a computer storage, quantum system lose coherence due to interaction with the environment – we’ll get to that!, we lose some knowledge about the system that we previously had.

Okay, we’ve got entropy. Entropy changes the game a little bit, we now know that physics cares about information in a sense. Now, let’s move on to answer a question about what information is in more detail.

In the middle of the XXth century, American mathematician Claude Shannon was thinking about communication. He defined communication as an act of transfer of messages, exactly or approximately, from one point to another. He was interested in, generally speaking, the efficiency of such transfer. He was considering something called communication channel, at one end of which there is an information source and on another, there is a receiver. The source generates symbols from some alphabet, arranged in words. The source can be imagined as a human writing down a message for example. Then, the message is encoded into another language, the language of the channel. It might be a series of ones and zeroes, or dots, lines and spaces of a telegraph, or specific sounds, or even clouds of smoke. The point is, we don’t transfer our messages directly over the large distances, we need to encode them first. The person receiving the message has to then decode it to understand it. Usually, the transfer is costly, there is a cost associated with a transfer of each symbol through the channel. Therefore when we encode, we want to be as concise as possible, but still retain the whole message we want to communicate.

The goal of Shannon was to show what are the limits of efficiency of such coding and decoding, to find out how many words can we squeeze through a channel to still be perfectly understood on the other end. The reason we can do this at all is because languages are somehow redundant, letters we use come up in certain patterns called words, and in those patterns only. Words form some grammar, which has to be obeyed for the whole sentence to make sense. A random sequence of letters is very unlikely to mean something for a human. Making use of the correlations might decrease lengths of encoded messages. For example, if you receive letters e,x,p,e,r,i,e,n,c in that order, you are almost sure that the next letter will be e. So when e comes, it doesn’t carry any new information – it might be dropped without influencing the understanding. Also, some letters or words are used more often than others, what means that encoding them with shortest codes will minimize the length of the encoded message. In old telegraph codes, a letter e, the most frequent letter in English texts, was encoded with a single dot, the shortest possible signal that could be sent.  Basically, there is a room for improvement. Shannon’s idea was to find out an objective measure of an amount of information carried by typical messages in a given language, and compare that to the maximum amount of information that can be transferred through the channel, and therefore find out the maximum rate of compression that allows to transfer the message most efficiently.

And what is this measure of information? It is entropy! It is not exactly the same entropy as in thermodynamics, however it is inspired by it and it behaves similarly mathematically. Let’s first paint the initial picture. In the first approximation, we consider a source to generate the letters randomly, following certain probability distribution. That means we are skipping a part of its structure for now, we neglect for example the fact that certain letters follow others more often. We only focus on how often each letter appears in text on average. Of course, the source doesn’t actually generate random letters, as in a popular meaning of a word “random”. However, the frequencies with which letters appear in the text are defined and reflect the given probabilities. That is all we need for now.

If letter number appears with probability , then entropy of a source is:

The same equation as for Gibbs entropy! What this entropy function reflects is a minimal average number of bits needed to encode one letter of a message generated by a source. In other words, if our message is letters long, with the most efficient coding we are going to need on average bits to encode it.

A simple example might be as follows, imagine a source, a language, in which there are only three letters, A, B and C, and A appears twice as often as B, which appears as often as C. That means , . We can calculate the entropy of this source:

That means we need at least $\frac32$ bits per letter to encode a message here. How can we do that? First, we keep in mind that we should try using shorter codes to represent more frequent letters. So code for A should be the shortest. Why not just use one bit to represent it, then? We can set 0 to represent A, then a string 10 to represent B and a string 11 to represent C. Decoding such message will then look as follows. We consider the first bit, if it’s 0, we write A and move on. If it’s 1, we check the next one, if it’s 0, we write B and if it’s 1 we write C and then move on.

How efficient is this? We can see that because A basically takes 50% of the message, and letters B and C take 25% each, an average of bits will be 50% + 25% . Exactly equal to the entropy value! Basically, the number of bits we aim to use for each letter should be as close as possible to , where is the probability of that letter. Then, when we calculate the bitrate, we recreate the entropy function.

Entropy was then again used as a measure of information, however in a slightly different way. We once more had probabilities, reflecting our lack of knowledge about what will come next, this time however the higher entropy of a message, the more information was there in the message. So there are two ways to think about it, either entropy is an average information learnt from a letter, or an average uncertainty about what the letter will be before we know it – which is actually the same thing, but at a different point of time.  A message which is constructed from an alphabet containing only one letter, no spaces, no nothing, doesn’t actually carry any information, and has entropy of 0. But we also don’t need to spend resources to transfer that message anywhere, we already know what it is.

We wanted to reduce redundancy to be more efficient, but sometimes we actually would like to introduce some. For example, if a communication channel we are using is somehow noisy, there can be errors introduced in transfer, and we want to make sure that regardless of these errors we can actually be understood. If during transfer some letters have small chance to change to other letters, you might receive, for example, a word ‘exparience’. There is no such word in English, but you guess it was supposed to be ‘experience’. The redundancy is good, because it lets us recover a distorted message. There are different types of noise and some of them are worse than others, so different levels of redundancy will be needed in each situation. The question is, how much redundancy is enough?

Let us focus on a specific case, of which generalizations we will not bother here, when an encoded message consists of 0-s and 1-s, and a probability that 0 changes randomly to 1, or 1 to 0 is equal to some . Then, 0 staying 0 or 1 staying 1 has a probability . It can be then shown that if we introduce so-called binary entropy function the rate of compression has to be proportional to . The closer is to 1, the more bits we need in order to transfer the message. This makes sense, corresponds to , and that means we can have no idea whatsoever about what the message actually was, as exactly a half of the bits are flipped, so the message looks like a completely random string.

So here entropy is once again a measure of uncertainty, lack of knowledge about the system.

Information has a lot to do with probabilities, chances of certain things happening. The closer are probabilities to 1, the more certain we are, so the more information we have. Even though obviously information theory was developed to enormous degrees, this connection is quite simple, because classically, probability is simple. It is just a number between 0 and 1, which can be interpreted either as a fraction of events that satisfy some special criteria, or our degree of certainty that a single event is going to satisfy them, but mathematically, it is a number between 0 and 1.

This will get tremendously more complicated when we introduce quantum mechanics. But worry not! I will be your guide… in my next essay here.

For now, you can get ahead of me and check out some useful reading:

1. Any thermodynamics textbook to learn about entropy in physics – good one is “Concepts in thermal physics” by Stephen and Katherine Blundell
2. The “Bible of quantum computing” with an introduction about classical information theory “Quantum computation and quantum information” by Michael Nielsen and Isaac Chuang
3. An excellent semi-popular book by Scott Aaronson about information and computation “Quantum computing since Democritus”
4. Definetely also check out Scott’s blog: https://www.scottaaronson.com/blog/
5. A more philosophical treatment in a book “Quantum Information Theory and Foundations of Quantum Mechanics” by Christopher Timpson

On Mathematics, Physics and Truth

What is the role of truth, and by extension philosophy, in physics and in mathematics?

This question was carefully examined at the turn of the 20^th century when, faced with a plethora of strange and counter-intuitive results, it was deemed necessary to come to terms with the role of truth in each field. What is the influence of mathematical results on our picture of reality? Is nature really whatever our physical models say it is? What makes a theory “acceptable” in mathematics or in physics?

On the mathematical front-lines, David Hilbert, following Riemann, Helmholtz, Clifford and Poincaré, championed a view which challenged the concept of truth insofar as mathematics is concerned. He argued that the metaphysical truth of a statement is irrelevant to mathematics, which instead ought to be concerned purely with the logical consequences of that statement.

To clarify what I mean by “metaphysical truth,” let us consider an example: in his famous works on Geometry, Euclid begins with the statement that “a point is that which has no part”.  Is this true? Does a point exist? We may take two different roads to answer this question.

On the first road, we believe that in physical reality there exists such an entity which we are attempting to describe with our definition. In this case, we must somehow show that this entity exists and that our definition captures it adequately.

On the second (modern) road, we assert that, in some hypothetical reality, there exists an entity called a point which is precisely described by our definition. In this case, the point exists in our hypothetical reality by assertion. Whether this hypothetical reality coincides with physical reality is out of the question – a matter for a physicist or philosopher to decide upon.

Each assertion we make within our hypothetical reality is more or less an assumption on which we will build all of our deductions. It’s natural to wonder if we have made a redundant assumption, in which case we simply need to prove that our redundant assumption is actually just a consequence of the others.

In any such hypothetical reality, we will find that the truth of any mathematical statement therein is tantamount to the truth of a set of irreducible statements, the things we must assume which are formally known as axioms. This set of axioms is essentially the backbone of any mathematical construction within that reality.

Within Hilbert’s paradigm, mathematics evolved into a study of consequences which dismissed the notion of metaphysical truth. Any set of axioms is deemed permissible with one caveat – the axioms must be consistent¹. In other words, if we can prove the truth of two contradictory statements with our axioms, then they cannot all be simultaneously true. Such a construction is not allowed since it is demonstrably self-contradictory and therefore logically inconsistent i.e. mathematical heresy.

A famous example is given by Russel’s paradox: imagine there is a big list of lists that don’t list themselves. Does the big list contain itself?

If it does then it must be a list that lists itself so it cannot be on the big list.

If it doesn’t then it is a list which doesn’t list itself meaning it should be on the big list.

Wait a second. Being on the big list means it isn’t on the big list and not being on the big list means it is on the big list: PARADOX.

Clearly, such a construction cannot exist and mathematicians took great care to adjust the axioms of the reality of mathematical lists (set theory) to prevent such constructions from existing.

An important question to ask at this point is, can an axiom be true in one consistent reality but false in another? Well, it turns out that some mathematical constructions may require it to be true, others may require it to be false, and some may even be independent of it altogether (consistent whether it is true or false).

The classical example of this (and indeed the reason why this matter was debated in the first place) is the fact that Euclid’s parallel postulate is not a necessary axiom for consistent geometries. The postulate asserts that, given a straight line and a point not on this line, there is exactly one parallel line which goes through this point. Seems absolutely true, doesn’t it?

In a stroke of genius, Poincaré showed that Non-Euclidean geometries (where the parallel postulate fails) are as consistent as Euclidean geometry by constructing a dictionary allowing one to translate the results of one geometry into the results of the other and vice versa. As a result, if people accept that Euclidean geometries are legitimate then they would have to accept that Non-Euclidean geometries are equally legitimate.

In summary, we see that an axiom is more or less a definition, a rule of a mathematical game: asking if the rules of the game are true is like asking if the rules of Monopoly are true. The mathematician’s job is purely to determine how the game unfolds, and not to concern themselves with the rules of the game, no matter how weird or whacky. Provided that a move cannot be both correct and incorrect according to the rules (imagine the arguments) then it is acceptable as a game. This is the “shut up and prove” ethos.

However, if we wish to apply mathematical results to solve some problem, the approach becomes radically different.  We must first model the problem by making several idealisations and assumptions. We then attempt to understand its internal structure, that is the variables of the problem and the relations between them, determine if it has a well-defined answer or if we require additional assumptions to arrive at a unique answer and so on.

We may find that the problem can be modelled within several mathematical systems allowing us the freedom to choose the most convenient or simple one to arrive at a solution. For example, we may be asked to prove some relation which may be a horrible task when performed analytically while being astonishingly simple within a geometric framework. Perspective is key.

As a result, physicists are not so free to evoke any set of mathematical axioms to describe natural phenomena. The truth of axioms is a real concern to physicists since their veracity is ultimately decided, directly or indirectly, by experiment.

An important point to make here is our models are mainly meant to supply us with useful information requiring as little ink as possible. If we are attempting to model a stone dropped two metres in the air, it makes sense to assume that there is no air resistance since it has such a small effect. However, if it were falling 100m, then our models’ predictions would quickly depart from reality.

We would have to add some fine print to our model saying e.g. “this prediction is valid within a length scale of up to 20m.” Interestingly, one can go so far as to say that it is a law of physics that stones will fall according to this model in a range of up to 20m, however that would be a waste of a law for one important reason: it follows from other laws which hold over a much larger scale of lengths and speeds.

Does this mean that those laws are not “really” true? Such a question is on par with the question posed of mathematicians with regards to constructions with a slight difference. In the realm of physics we have to decide if “really” true means that this model is exactly what nature is or if “really” true means that nature behaves in a manner consistent with this model, where applicable.

As a rule, experiments do not tell us if a model is correct, quite the opposite: they tell us if a model cannot be correct. We cannot know if an electron is truly a spherical entity or some incredibly complex entity that behaves in a manner entirely consistent with spherical entities (as in there is no experiment that we can conceive of or perform that allows us to distinguish between the two). We can, however, know that an electron is not something which behaves in a manner that is inconsistent with spherical entities.

When we say we know what an electron is, we, in fact, mean to say that we have discovered an entity which has certain properties, and by properties we mean relations; expressions of how this entity interacts with other entities. We call this entity the electron. Our model of the electron is a representation of the knowledge we have thus far acquired with regards to its behaviour, and we have no means of determining if this knowledge is complete.

Should physics then be concerned with the true nature of the substance it describes, or should it be content with modelling the behaviour of said substance and not ask if the model itself is metaphysically true?

To answer this rather dramatically, suppose that someone claims that gravity is really a manifestation of mass spirits. They then make claims about the behaviour of mass spirits which makes them entirely consistent with Newtonian gravity as far as predictions are concerned. Suppose further that one of their statements about the theory of mass spirits is a definition that “the spirit is a projection of the living soul of matter”.

As far as a physicist is concerned, this statement is meaningless because it is not testable. It is an irrelevant and vague complication the neither helps nor harms the act of prediction. One could just as easily make any number of such claims which are consistent with the theory of gravity but differ on such arbitrary untestable statements.

We have decided that experiment is the ultimate judge of physical truth. Experiment sheds no light on the nature of the things we study, it sheds light only on the relations between the things we study. It cannot distinguish between theories which agree on all predictions. Such theories are equal in the eyes of physics insofar as their predictions, and ultimately the relations they describe, are all the same. Any other details are completely irrelevant since there is no means of testing their truth or falsehood. As such, they are regarded as being useless embellishments and are therefore discarded.

As Poincaré puts it,

“The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things; outside those relations there is no reality knowable.”

Physicists had to face yet another difficulty in this period. It was previously assumed that it is possible, at least in principle, to measure any observable quantity to arbitrary precision – the only limitation being how well the designed the experiment or how skilful the experimenter.

For example, there are fundamental limitations to how precisely one can simultaneously measure certain observable quantities. In other words, we may know precisely that the electron is located at this point, but the price of this knowledge is a total lack of knowledge as to what its momentum is and vice versa.

Now comes a key question: does the electron actually have a specific momentum and position simultaneously even though there are no known experiments allowing us to observe them?

Should physics be concerned with whatever lies behind the scenes of observable reality, or should it be content with any model which is consistent with our experience of observable reality?

There is one approach which states that we should care insofar as how much we stand to gain in the ability to make predictions or in mathematical simplicity. Otherwise, we shouldn’t really worry about the details. A “shut up and calculate” ethos.

Einstein was among the few who objected vehemently to this ethos for reasons which I think are well captured in the following quote of his:

“Man tries to make for himself, in the fashion that suits him best, a simplified and intelligible picture of the world. He then tries to some extent to substitute this cosmos of his for the world of experience, and thus to overcome it. He makes this cosmos and its construction the pivot of his emotional life in order to find in this way the peace and serenity which he cannot find in the narrow whirlpool of personal experience.”

His pursuit was not exactly one of “doing physics,” as much as it was a desire to make sense out of the universe. That a model is unsatisfactory despite making accurate predictions is deeply subjective. What Einstein was doing is not exactly the cold and calculating physics I have been describing here. His conclusions were not derived from purely physical considerations, but rather a careful balance between physics and philosophy.

This is all I wish to say as far as this essay is concerned. In closing, I would like to leave you with a few questions to contemplate (which I will address in my next essay):

Does experiment single out a geometry as being that of experience or are we free to choose any geometry, provided we make suitable adjustments to the laws of nature?

Hiding behind this question is an exceptionally important consideration; what assumptions do we implicitly make in bridging the gap between the physical reality and geometry? Are these assumptions arbitrary or are they a priori necessary?

An important question: why should we care? Einstein provides a rather eloquent answer here:

“Concepts that have proven useful in ordering things easily achieve such an authority over us that we forget their earthly origins and accept them as unalterable givens. Thus they come to be stamped as “necessities of thought,” “a priori givens,” etc. The path of scientific advance is often made impassable for a long time through such errors. For that reason, it is by no means an idle game if we become practiced in analysing the long commonplace concepts and exhibiting those circumstances upon which their justification and usefulness depend, how they have grown up, individually, out of the givens of experience. By this means, their all-too-great authority will be broken. They will be removed if they cannot be properly legitimated, corrected if their correlation with given things be far too superfluous, replaced by others if a new system can be established that we prefer for whatever reason.”

1- Proving consistency is by no means simple- or even possible in some cases c.f. Gödel’s Incompleteness Theorems.

For further reading, I highly recommend the following resources:

1. “Science and the Method” – Henri Poincaré
2. “Beyond Geometry: Classic Papers from Riemann to Einstein” – Peter Pesic
3. “The Philosophy of Space and Time” – Hans Reichenbach

Finally, the photo was found on Pinterest!

A warm welcome to the blog!

Hello everyone!

This is is the blog that we finally managed to start, after discussing lots of ideas in physics and related subjects since we have met. These are the results of random thoughts that we had about the nature of the reality around us and specific topics that relate to it. The world of science is now much greater than it ever was, and no single person is able to understand or even learn everything. These are the bits that we have managed to catch in the fishnets of our brains due to curiosity, people we have met, experiments we were a part of, and resources we have read and which we try to put in a simple form which should be understood. If you don’t get it, it’s probably our fault!

The blog contains short articles or essays on things which we find incredibly interesting and hope you will find them at least a little bit interesting. Definitely do not hesitate to let us know in the comments what you think, regardless of your level of advancement in science.

About us:

Maher aka M originally from Egypt, graduated with a Bachelor’s degree in Physics at University College London then moved to Cambridge (not Massachusetts) for a Part III Master’s course in mathematics and theoretical physics, which he graduated from recently. Besides physics, he is a passionate theatre actor and he does other things which he should write about here so that he seems more like a real person.

msciwojo – Jakub Mrożek, originally from Poland, studied together with Maher at UCL, but went on to finish a Master’s there, to then move to Oxford to start a DPhil in Condensed Matter Physics, which he is stressing about right now. He is also quite passionate about dancing, mainly latino and swing dances are in his portfolio, and involved in discussing political ideas – but don’t worry, here ain’t no place for that!