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This post is an unofficial sequel to one of my first blog posts from 2007, which was entitled “Quantum mechanics and Tomb Raider“.

One of the oldest and most famous allegories is Plato’s allegory of the cave. This allegory centers around a group of people chained to a wall in a cave that cannot see themselves or each other, but only the two-dimensional shadows of themselves cast on the wall in front of them by some light source they cannot directly see. Because of this, they identify reality with this two-dimensional representation, and have significant conceptual difficulties in trying to view themselves (or the world as a whole) as three-dimensional, until they are freed from the cave and able to venture into the sunlight.

There is a similar conceptual difficulty when trying to understand Einstein’s theory of special relativity (and more so for general relativity, but let us focus on special relativity for now). We are very much accustomed to thinking of reality as a three-dimensional space endowed with a Euclidean geometry that we traverse through in time, but in order to have the clearest view of the universe of special relativity it is better to think of reality instead as a four-dimensional spacetime that is endowed instead with a Minkowski geometry, which mathematically is similar to a (four-dimensional) Euclidean space but with a crucial change of sign in the underlying metric. Indeed, whereas the distance ${ds}$ between two points in Euclidean space ${{\bf R}^3}$ is given by the three-dimensional Pythagorean theorem $\displaystyle ds^2 = dx^2 + dy^2 + dz^2$

under some standard Cartesian coordinate system ${(x,y,z)}$ of that space, and the distance ${ds}$ in a four-dimensional Euclidean space ${{\bf R}^4}$ would be similarly given by $\displaystyle ds^2 = dx^2 + dy^2 + dz^2 + du^2$

under a standard four-dimensional Cartesian coordinate system ${(x,y,z,u)}$, the spacetime interval ${ds}$ in Minkowski space is given by $\displaystyle ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2$

(though in many texts the opposite sign convention ${ds^2 = -dx^2 -dy^2 - dz^2 + c^2dt^2}$ is preferred) in spacetime coordinates ${(x,y,z,t)}$, where ${c}$ is the speed of light. The geometry of Minkowski space is then quite similar algebraically to the geometry of Euclidean space (with the sign change replacing the traditional trigonometric functions ${\sin, \cos, \tan}$, etc. by their hyperbolic counterparts ${\sinh, \cosh, \tanh}$, and with various factors involving “ ${c}$” inserted in the formulae), but also has some qualitative differences to Euclidean space, most notably a causality structure connected to light cones that has no obvious counterpart in Euclidean space.

That said, the analogy between Minkowski space and four-dimensional Euclidean space is strong enough that it serves as a useful conceptual aid when first learning special relativity; for instance the excellent introductory text “Spacetime physics” by Taylor and Wheeler very much adopts this view. On the other hand, this analogy doesn’t directly address the conceptual problem mentioned earlier of viewing reality as a four-dimensional spacetime in the first place, rather than as a three-dimensional space that objects move around in as time progresses. Of course, part of the issue is that we aren’t good at directly visualizing four dimensions in the first place. This latter problem can at least be easily addressed by removing one or two spatial dimensions from this framework – and indeed many relativity texts start with the simplified setting of only having one spatial dimension, so that spacetime becomes two-dimensional and can be depicted with relative ease by spacetime diagrams – but still there is conceptual resistance to the idea of treating time as another spatial dimension, since we clearly cannot “move around” in time as freely as we can in space, nor do we seem able to easily “rotate” between the spatial and temporal axes, the way that we can between the three coordinate axes of Euclidean space.

With this in mind, I thought it might be worth attempting a Plato-type allegory to reconcile the spatial and spacetime views of reality, in a way that can be used to describe (analogues of) some of the less intuitive features of relativity, such as time dilation, length contraction, and the relativity of simultaneity. I have (somewhat whimsically) decided to place this allegory in a Tolkienesque fantasy world (similarly to how my previous allegory to describe quantum mechanics was phrased in a world based on the computer game “Tomb Raider”). This is something of an experiment, and (like any other analogy) the allegory will not be able to perfectly capture every aspect of the phenomenon it is trying to represent, so any feedback to improve the allegory would be appreciated.

Things are pretty quiet here during the holiday season, but one small thing I have been working on recently is a set of notes on special relativity that I will be working through in a few weeks with some bright high school students here at our local math circle.  I have only two hours to spend with this group, and it is unlikely that we will reach the end of the notes (in which I derive the famous mass-energy equivalence relation E=mc^2, largely following Einstein’s original derivation as discussed in this previous blog post); instead we will probably spend a fair chunk of time on related topics which do not actually require special relativity per se, such as spacetime diagrams, the Doppler shift effect, and an analysis of my airport puzzle.  This will be my first time doing something of this sort (in which I will be spending as much time interacting directly with the students as I would lecturing);  I’m not sure exactly how it will play out, being a little outside of my usual comfort zone of undergraduate and graduate teaching, but am looking forward to finding out how it goes.   (In particular, it may end up that the discussion deviates somewhat from my prepared notes.)

The material covered in my notes is certainly not new, but I ultimately decided that it was worth putting up here in case some readers here had any corrections or other feedback to contribute (which, as always, would be greatly appreciated).

[Dec 24 and then Jan 21: notes updated, in response to comments.]

Way back in 2007, I wrote a blog post giving Einstein’s derivation of his famous equation ${E=mc^2}$ for the rest energy of a body with mass ${m}$. (Throughout this post, mass is used to refer to the invariant mass (also known as rest mass) of an object.) This derivation used a number of physical assumptions, including the following:

1. The two postulates of special relativity: firstly, that the laws of physics are the same in every inertial reference frame, and secondly that the speed of light in vacuum is equal ${c}$ in every such inertial frame.
2. Planck’s relation and de Broglie’s law for photons, relating the frequency, energy, and momentum of such photons together.
3. The law of conservation of energy, and the law of conservation of momentum, as well as the additivity of these quantities (i.e. the energy of a system is the sum of the energy of its components, and similarly for momentum).
4. The Newtonian approximations ${E \approx E_0 + \frac{1}{2} m|v|^2}$, ${p \approx m v}$ to energy and momentum at low velocities.

The argument was one-dimensional in nature, in the sense that only one of the three spatial dimensions was actually used in the proof.

As was pointed out in comments in the previous post by Laurens Gunnarsen, this derivation has the curious feature of needing some laws from quantum mechanics (specifically, the Planck and de Broglie laws) in order to derive an equation in special relativity (which does not ostensibly require quantum mechanics). One can then ask whether one can give a derivation that does not require such laws. As pointed out in previous comments, one can use the representation theory of the Lorentz group ${SO(d,1)}$ to give a nice derivation that avoids any quantum mechanics, but it now needs at least two spatial dimensions instead of just one. I decided to work out this derivation in a way that does not explicitly use representation theory (although it is certainly lurking beneath the surface). The concept of momentum is only barely used in this derivation, and the main ingredients are now reduced to the following:

1. The two postulates of special relativity;
2. The law of conservation of energy (and the additivity of energy);
3. The Newtonian approximation ${E \approx E_0 + \frac{1}{2} m|v|^2}$ at low velocities.

The argument (which uses a little bit of calculus, but is otherwise elementary) is given below the fold. Whereas Einstein’s original argument considers a mass emitting two photons in several different reference frames, the argument here considers a large mass breaking up into two equal smaller masses. Viewing this situation in different reference frames gives a functional equation for the relationship between energy, mass, and velocity, which can then be solved using some calculus, using the Newtonian approximation as a boundary condition, to give the famous ${E=mc^2}$ formula.

Disclaimer: As with the previous post, the arguments here are physical arguments rather than purely mathematical ones, and thus do not really qualify as a rigorous mathematical argument, due to the implicit use of a number of physical and metaphysical hypotheses beyond the ones explicitly listed above. (But it would be difficult to say anything non-tautological at all about the physical world if one could rely solely on ${100\%}$ rigorous mathematical reasoning.)

I was recently at an international airport, trying to get from one end of a very long terminal to another.  It inspired in me the following simple maths puzzle, which I thought I would share here:

Suppose you are trying to get from one end A of a terminal to the other end B.  (For simplicity, assume the terminal is a one-dimensional line segment.)  Some portions of the terminal have moving walkways (in both directions); other portions do not.  Your walking speed is a constant $v$, but while on a walkway, it is boosted by the speed $u$ of the walkway for a net speed of $v+u$.  (Obviously, given a choice, one would only take those walkways that are going in the direction one wishes to travel in.)  Your objective is to get from A to B in the shortest time possible.

1. Suppose you need to pause for some period of time, say to tie your shoe.  Is it more efficient to do so while on a walkway, or off the walkway?  Assume the period of time required is the same in both cases.
2. Suppose you have a limited amount of energy available to run and increase your speed to a higher quantity $v'$ (or $v'+u$, if you are on a walkway).  Is it more efficient to run while on a walkway, or off the walkway?  Assume that the energy expenditure is the same in both cases.
3. Do the answers to the above questions change if one takes into account the various effects of special relativity?  (This is of course an academic question rather than a practical one.  But presumably it should be the time in the airport frame that one wants to minimise, not time in one’s personal frame.)

It is not too difficult to answer these questions on both a rigorous mathematical level and a physically intuitive level, but ideally one should be able to come up with a satisfying mathematical explanation that also corresponds well with one’s intuition.

[Update, Dec 11: Hints deleted, as they were based on an incorrect calculation of mine.]

Einstein’s equation $E=mc^2$ describing the equivalence of mass and energy is arguably the most famous equation in physics. But his beautifully elegant derivation of this formula (here is the English translation) from previously understood laws of physics is considerably less famous. (There is an amusing Far Side cartoon in this regard, with the punchline “squared away”, which you can find on-line by searching hard enough, though I will not link to it directly.)

This topic had come up in recent discussion on this blog, so I thought I would present Einstein’s derivation here. Actually, to be precise, in the paper mentioned above, Einstein uses the postulates of special relativity and other known laws of physics to show the following:

Proposition. (Mass-energy equivalence) If a body at rest emits a total energy of E while remaining at rest, then the mass of that body decreases by $E/c^2$.

Assuming that bodies at rest with zero mass necessarily have zero energy, this implies the famous formula $E = mc^2$ – but only for bodies which are at rest. For moving bodies, there is a similar formula, but one has to first decide what the correct definition of mass is for moving bodies; I will not discuss this issue here, but see for instance the Wikipedia entry on this topic.

Broadly speaking, the derivation of the above proposition proceeds via the following five steps:

1. Using the postulates of special relativity, determine how space and time coordinates transform under changes of reference frame (i.e. derive the Lorentz transformations).
2. Using 1., determine how the temporal frequency $\nu$ (and wave number k) of photons transform under changes of reference frame (i.e. derive the formulae for relativistic Doppler shift).
3. Using Planck’s relation $E = h\nu$ (and de Broglie’s law $p = \hbar k$) and 2., determine how the energy E (and momentum p) of photons transform under changes of reference frame.
4. Using the law of conservation of energy (and momentum) and 3., determine how the energy (and momentum) of bodies transform under changes of reference frame.
5. Comparing the results of 4. with the classical Newtonian approximations $KE \approx \frac{1}{2} m|v|^2$ (and $p \approx mv$), deduce the relativistic relationship between mass and energy for bodies at rest (and more generally between mass, velocity, energy, and momentum for moving bodies).

Actually, as it turns out, Einstein’s analysis for bodies at rest only needs to understand changes of reference frame at infinitesimally low velocity, $|v| \ll c$. However, in order to see enough relativistic effects to deduce the mass-energy equivalence, one needs to obtain formulae which are accurate to second order in v (or more precisely, $v/c$), as opposed to those in Newtonian physics which are accurate to first order in v. Also, to understand the relationship between mass, velocity, energy, and momentum for moving bodies rather than bodies at rest, one needs to consider non-infinitesimal changes of reference frame.

Important note: Einstein’s argument is, of course, a physical argument rather than a mathematical one. While I will use the language and formalism of pure mathematics here, it should be emphasised that I am not exactly giving a formal proof of the above Proposition in the sense of modern mathematics; these arguments are instead more like the classical proofs of Euclid, in that numerous “self evident” assumptions about space, time, velocity, etc. will be made along the way. (Indeed, there is a very strong analogy between Euclidean geometry and the Minkowskian geometry of special relativity.) One can of course make these assumptions more explicit, and this has been done in many other places, but I will avoid doing so here in order not to overly obscure Einstein’s original argument. Read the rest of this entry » Dan Asimov on Inequalities, convergence, and… Dan Asimov on Inequalities, convergence, and… Anonymous on AI to Assist Mathematical Reas… Terence Tao on AI to Assist Mathematical Reas… Mathur Maynard Amadi on Inequalities, convergence, and… Alex Kontorovich on Inequalities, convergence, and… Gabe Khan on Inequalities, convergence, and… "不等式、收敛与连续性作为“特… on Inequalities, convergence, and… Some interesting thi… on PCAST Working Group on Generat… jack morava on Inequalities, convergence, and… Wednesday assorted l… on Inequalities, convergence, and… Wednesday assorted l… on Inequalities, convergence, and… Wednesday assorted l… on Inequalities, convergence, and… Mark Lewko on Inequalities, convergence, and… Wednesday assorted l… on Inequalities, convergence, and…