Why Does E=mc2_ - Brian Cox [32]
Flipping the sign means that the mathematics is not much of an extension over the by now familiar equation of Pythagoras. Our task is to figure out whether we should stick with the plus-sign version of Pythagoras, or shift to the minus-sign version of the distance equation. This may look at first sight to be a rather odd thing to investigate. What possible reason could there be for even considering Pythagoras with a minus sign? But that is not the right way to think about things. The formula for distances on a sphere looks nothing like Pythagoras either, so all we are doing is entertaining the idea that spacetime might not be flat in the sense of Euclid. Indeed, since the minus-sign version is the only option other than the plus-sign version (given our assumptions), we have no logical reason to throw it out at this stage. We should therefore keep it and investigate the consequences. If neither the plus- nor the minus-sign versions do the job, and we fail in constructing a workable distance measure in spacetime, then we must go back to the drawing board.
We are now about to plunge into a very elegant but perhaps tricky piece of reasoning. We will stick to our promise of using nothing more complicated than Pythagoras, but you might find that you have to read it twice. It should be worth it, because if you follow closely you might experience a feeling described by biologist Edward O. Wilson as the Ionian Enchantment. It derives from the work of Thales of Miletus, who is credited by Aristotle, two centuries later, as laying the foundations of the physical sciences in Ionia in the sixth century BCE. This poetic term describes the belief that the complexity of the world can be explained by a small number of simple natural laws because at its heart it is orderly and simple (we are reminded of Wigner’s essay). The scientist’s job is to strip away the complexity we see around us and to uncover this underlying simplicity. When the process works out, and the simplicity and unity of the world are revealed, we experience the Ionian Enchantment. Imagine for a moment cradling a snowflake in the palm of your hand. It is an elegant and beautiful structure, possessed of a jagged crystalline symmetry. No two snowflakes are alike, and at first sight this chaotic state of affairs seems to defy a simple explanation. Science has taught us that the apparent complexity of snowflakes hides an exquisite underlying simplicity; each is a configuration of billions of molecules of water, H2O. There is nothing more to a snowflake than that, and yet an overwhelming complex of structure and form emerges when those H2O molecules get together in the atmosphere of our planet on a cold winter’s night.
To settle the question of the plus or minus sign, we need to turn our attention to causality. Let us first suppose that Pythagoras’ is the right equation for distances in spacetime—i.e., s2 = (ct) 2 + x2. Yet again we return to our two events: waking up in bed at 7 a.m. and finishing breakfast in the kitchen at 8 a.m. We’ll do something that may send shivers up your spine as you remember sitting in mathematics classes at school and gazing out the window across the football fields, pristine and inviting in the spring afternoon sunlight—let the waking-up event be called O and the finishing-breakfast event be called A. We do this purely for reasons of brevity, without wishing to don tweed and cover ourselves in chalk dust.
We know that the spatial distance between O and