Handwaving

[mtvessel]

Sep 2016
Why does E=mc²? – Brian Cox and Jeff Forshaw - Da Capo Press, 2009 (Kindle edition)
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This is the second book that I have read in my attempt to understand the origins of Einstein's famous equation. The first, e=mc² by David Bodanis, did a generally good job of explaining the concepts but bottled out when the maths got complicated. This is typical of popular science books, where the general consensus is that every line of maths halves the readership. I find this annoying. Perhaps unusually, I am not afraid of maths (I'm not very good at it, but that's another matter). Any train of mathematical thought can, with a bit of effort, be followed, as long as all the terms used are clearly defined and any leaps in logic are explained.

So my alarms started ringing when the authors promise to derive E=mc² with nothing more complex than Pythagoras' Theorem. Instead they attempt to use closely-argued logic, starting from the impossibility of detecting absolute motion and hence the non-existence of an absolute coordinate system for space. From there they go on to describe the appearance of the speed of light (c) as the ratio of the strengths of the electric and magnetic fields in Faraday's experiments with magnets and wires, which equals the speed of the wave in Maxwell's wave equation mathematical model of the same thing. At this point I was already beginning to become a little annoyed. The authors describe what Maxwell's equations do and say that they are beautiful, but they don't actually tell us what they are. We just have to take their word for it.

After that we approach special relativity via the Michelsen-Morley failure to detect the luminiferous aether and Einstein's subsequent rejection of absolute time. Pythagoras comes in to calculate a formula for the amount by which a clock on a moving train appears to slow for an observer on a station platform, leading to the prediction of a 29-fold increase in the observed lifetimes of muons when they are accelerated close to the speed of light in a particle accelerator compared to when they are not, which has been observed. Again, the incorrigible skeptic might observe a certain element of "trust us, it's true", but the reasoning is clearly explained.

That brings us to the idea of spacetime, approached via a nod to Emmy Noether's massively underappreciated discovery that continuous symmetries (invariant transformations) in nature map to laws of conservation of quantities. This implies that in order to conserve causality for all observers, there must be a transformation involving movement of an object in space and time that is invariant. The connection between the two is of course speed, and it is here that the argument becomes particularly hand-wavey. The authors derive a formula for an invariant "distance" s in spacetime that has either a positive or a negative component, but do not explain where the negative version comes from other than with a vague reference to Occam's Razor¹. Instead, there is a completely unhelpful paean to the Ionian Enchantment which distracts the reader and should have been excised by an alert editor. What makes this even worse is that the positive version which readers will intuitively understand turns out to be the wrong answer since it violates causality. The negative version leads to light cones, Minkowski spacetime and a cosmic speed limit that, via a clever reformulation of the speeding train problem, turns out to be the speed of light.

There is more handwaving in the description of spacetime vectors (where two vectors are described as having the same length but are clearly different in the accompanying diagram), and in the subsequent descriptions of the conservation of momentum and of energy. We finally get somewhere when Cox and Forshaw point out that a vector representing momentum, which traditionally has a direction in space but not in time, has to have a spacetime equivalent if everyone is to agree on its (conserved) value. By constructing such a momentum vector for a moving object of mass m, and considering its co-ordinate in the time direction (ymc, where y is the time dilation factor calculated for the speeding train problem, m is the mass and c is the cosmic speed limit), you can derive a conserved value (call it E) equalling mc²+(1/2)mv². Since the second component is the equation for kinetic energy and will be zero for an object at rest in relation to the observer, you can say that such an object has an intrinsic energy equal to mc².

This derivation requires a positive gale of handwaving; y has to be simplified down to an approximation that only holds for speeds up to around 10% of the speed of light, the c² term has to be introduced by saying that if mc is conserved, then mc² will be too, and the idea that the conserved quantity ymc in the spacetime momentum vector is in fact the object's energy comes out of nowhere. This really isn't a satisfactory explanation of where the equation comes from, and I am pretty sure that the average reader won't be able to follow it. But it's the best we're going to get; the rest of the book looks at the ramifications of the equation, including fission and fusion and the Standard Model. It's nice to see how relativity, quantum mechanics and particle physics all tie together, but they haven't answered the question in the title of their book.

The extremely ad-hoc nature of the mathematical transformations involved does make me wonder whether this is the way in which theoretical physicists actually work. Perhaps Einstein really did get to his formula by tinkering with equations until he got something that he recognised as a representation of another physical quantity. But if so, he, and other great physicists, are a lot less rational than I thought.

The thing is, derivation of E=mc² is really quite simple. It can be done in half a page. This version, which uses nothing more than calculus and Newtonian mechanics, is my favourite. My integral calculus is too rusty to follow it properly, but I feel I could understand it with a bit of help. Cox and Forshaw take 150 pages to get to their derivation, and their attempts to avoid advanced mathematics by handwaving are ultimately more baffling than enlightening.

¹ The equation concerned is s² = x² +/- (ct)², where s is the displacement in spacetime, x is the displacement in space, t is the displacement in time and c is a constant. Observers will agree on the value of s but differ on the values of x and t if their speeds (inertial frames) are different. The derivation for the negative version is in an appendix but is rather opaque - as I understand it, it is the simplest conceivable equation that doesn't assume a Euclidean geometry for spacetime.