Jim’s Brain Teaser

Jim Al-Khalili posted a question on Twitter recently relating to information and mass. It’s an interesting set of questions (and mistakes!) which I hope to make an interesting post about. Hopefully I’ll get the physics correct. If not, I’m open to corrections.

Since Twitter is rather poor for following conversations you’re not involved in, here’s the gist (paraphrased);

Jim: If I shuffle a pack of cards, does it weigh more before or after it’s been shuffled?

Jim: OK confusing question, what about an iPod; if the memory is full, does it weigh more than if the memory is empty?

Jim: The point of this question is: Does information have mass? Answer being yes.

Let’s make some things clear from the get-go:

  1. By “mass” we mean “energy which is equivalent to mass via E=m(c^2)
  2. The first two phrasings of the same question are confusing (and the third is misleading, reasons for which I will describe later.)
  3. I don’t actually disagree with him, or at least not necessarily.

So why is this interesting? Well, firstly I think that it’s an interesting idea that information has mass, and just simply by knowing stuff I might get heavier! As a student, this is obviously why I’m overweight (not the doughnuts and Warcraft, oh no!) Another reason it’s interesting is that the phrasing of the questions is misleading to the point of being confusing so many people that Jim was forced to rephrase several times! Lastly, the arguments that Jim seems to have had on Twitter with people disbelieving him attest to the unwillingness of people to embrace abstract concepts if they don’t fit with what they think constitutes real life.

Now for some Physics! Here we’ll take Einstein’s equation for mass-energy and classical Thermodynamics to be true simply for the sake of my sanity. That means if we can prove by thermodynamic arguments that a physical state gains energy by storing information, then we have proved that it gains mass and that Jim is vindicated. I will outline the simplest argument by which I believe Jim is vindicated, I’m aware that there are many other principles that Jim cited and they can be found at the end of this post.

So first we define our concept of information. For these purposes, information is a quantity stored by a physical system in terms of observable quantities, the organization of which is the information. This organization must be maintained for the information to be stored. The observables must act thermodynamically in nature, which is to say that electron spins, magnetic bits, chemical potentials, neuron states etc are allowed but (for example) a set of beer mats is not.

Now let me introduce the Helmholtz equation for free energy;

F = U – TS

Where F is the free energy, U is the ‘internal’ energy, T the temperature, and S the  entropy of the system. Entropy is a quantity which describes the organizational state of a thermodynamic system; a state which is highly ordered has a lower entropy than the same system with less order. A good example might be the difference between a crystal of ice and the same amount of water; ice is crystalline and is more ordered, so it has a lower entropy than liquid water because water is less well structured.* In a simple information storage system, the information of the thermodynamic system is stored by changing the structure of the system to reflect the information’s content in some predefined way. Usually, this increases the order (i.e. decreases the entropy of a system) by a thermodynamically irreversible process. This is where Helmholtz’s equation comes into play!

Internal energy, U, relies on the thermodynamic state of the system and is constant since it shares its internal energy equally with its surroundings. This means that if we decrease the entropy of the system then we will increase its free energy! Therefore we have increased the energy of the system by storing information.

I like to think of it in a different way – namely that changing the entropy of the system takes a certain amount of energy input, which then means that the system has more energy.

Both ways of thinking result in the conclusion that the mass increases. This is why I believe that Jim was correct in saying that “the mass of my iPad when its memory is full is more than when it’s completely blank”**.

Now for the bit where the Physics becomes background and I state my opinion on the matter***. What I do not agree with is that this is equivalent to the statement that “information has mass”. My interpretation of the argument which I have put forward above is that storing information intrinsically increases the mass of the system in which it is stored. In other words, the information that is stored has no mass but the storage medium itself gains mass because it is storing information. That means that I do not believe that “information has mass”. There’s a subtle difference in terms of interpretation but it does mean that my understanding of information, mass and thermodynamic systems do not change. This is important considering the following example I dreamt up;

Consider some photons flying through space in a straight line****. They are randomly positioned and separated by a significant distance. Their total energy is some value, call it E. Now let us change the order of the photons – let’s say they now represent the Morse Code message “S O S”. They are still a significant distance apart but the information is there simply as a function of whether there is a photon in a given length along their shared axis in their frame of rest. Has their total energy, E, increased due to the information which they now carry?

If I assume that the number of photons does not change between the two cases, then my answer to this would be that the energy, E, does not increase. My reasoning is thus;

  1. Since the photons are not interacting, they do not constitute a thermodynamic system.
  2. This means that their entropy cannot be considered as a whole. A lone photon’s entropy is constant when it is in isolation.
  3. This means that each individual photon’s energy is unchanging with respect to its position on the “line” on which the photons sit.
  4. Therefore the energy of the system does not change when it carries information.
  5. So the information itself does not have any intrinsic mass.

There are many complexities to this thought experiment in reality, but I think it demonstrates what I think – that it is the interaction of entities carrying information which increases the mass of the system by virtue of it carrying information. The information itself doesn’t actually have a mass, it just causes a change in energy distribution which results in a change in mass as the information is stored in a system.

In fact, as Jim pointed out to me, an experiment to test this has already been done to verify whether information can be converted directly back to energy! This is directly equivalent to showing that information and mass are commutable quantities. In other words, information can be converted into mass.

Thanks very much to Jim Al-Khalili for his help supplying feedback on the subject.

* – Actually water has some interesting structure on the nanoscale, see http://www.lsbu.ac.uk/water/ for further reading! It is, however, still of higher entropy than ice.

** – Again, paraphrased but accurate to the sentiment.

*** – This means that if you disagree you must at least recognize that my argument does not necessarily preclude you from having a valid opinion. I have no wish to rile someone up with no real cause. This set of arguments is not based on “solid” Physics and should you know better, please persuade me rather than berate me.

**** –  Their colour and medium are irrelevant, their source is perfect and they are not coherent, and the line is straight or an approximation thereof according to general relativity, let’s not overcomplicate this! If you prefer, think instead of neutrinos since they do not interact at any distance significant to this thought experiment.

Sources/Further Reading:







About stoove

A physicist, researcher, and gamesman. Likes to think about the mathematics and mechanics behind all sorts of different things, and writing up the thoughts for you to read. A competent programmer, enjoys public speaking and mechanical keyboards. Has opinions which might even change from time to time.
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