The proof relies on two insights. The first is that a right triangle can be decomposed into two smaller copies of itself (Steps 1 and 3). That’s a peculiarity of right triangles. If you try instead, for example, to decompose an equilateral triangle into two smaller equilateral triangles, you’ll find that you can’t. So Einstein’s proof reveals why the Pythagorean theorem applies only to right triangles: they’re the only kind made up of smaller copies of themselves.
.. What we’re seeing here is a quintessential use of a symmetry argument. In science and math, we say that something is symmetrical if some aspect of it stays the same despite a change. A sphere, for instance, has rotational symmetry; rotate it about its center and its appearance stays the same.
.. Throughout his career, Einstein would continue to deploy symmetry arguments like a scalpel, getting to the hidden heart of things. He opened his revolutionary 1905 paper on the special theory of relativity by noting an asymmetry in the existing theories of electricity and magnetism: “It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.” Those asymmetries, Einstein sensed, must be a clue to something rotten at the core of physics as it was then formulated. In his mind, everything else—space, time, matter, energy—was up for grabs, but not symmetry.
.. In general relativity, where space-time itself becomes warped and curved by the matter and energy within it, the Pythagorean theorem still has a part to play; it morphs into a quantity called the metric, which measures the space-time separation between infinitesimally close events, for which curvature can temporarily be overlooked. In a sense, Einstein continued his love affair with the Pythagorean theorem all his life.
.. Incredibly, in the part of his special-relativity paper where he revolutionized our notions of space and time, he used no math beyond high-school algebra and geometry.
Randall Munroe: New Yorker Essay about Relativity
The problem was light. A few dozen years before the space doctor’s time, someone explained with numbers how waves of light and radio move through space. Everyone checked those numbers every way they could, and they seemed to be right. But there was trouble. The numbers said that the wave moved through space a certain distance every second. (The distance is about seven times around Earth.) They didn’t say what was sitting still. They just said a certain distance every second.
It took people a while to realize what a huge problem this was. The numbers said that everyone will see light going that same distance every second, but what happens if you go really fast in the same direction as the light? If someone drove next to a light wave in a really fast car, wouldn’t they see the light going past them slowly? The numbers said no—they would see the light going past them just as fast as if they were standing still.
.. He said that if our ideas about light were right, then our ideas about distance and seconds must be wrong. He said that time doesn’t pass the same for everyone. When you go fast, he said, the world around you changes shape, and time outside starts moving slower.
.. The space doctor figured out that to explain how weight pulls things like light, we have to play around with time again. He showed that if time itself goes slower near heavy things, then the side of the light near the heavy thing won’t go as far every second.
.. But the space doctor figured out that heavy things change the shape of space as well as time. This changes how circles work. If you draw a circle around something heavy, he said, the distance around the edge will be a little shorter than the usual three times the distance across it.
.. To know exactly what time it is on a space boat, they have to change the watches a little to make up for both of these problems. If the space doctor’s ideas were wrong, your phone wouldn’t be able to tell where it was.
IOT: The Sun
How was the Sun formed, and what do we know about its structure and the processes going on inside our nearest star? With Carolin Crawford, Gresham Professor of Astronomy and Fellow of Emmanuel College, Cambridge; Yvonne Elsworth, Professor of Helioseismology at the University of Birmingham; and Louise Harra, Professor of Solar Physics at UCL Mullard Space Science Laboratory.30 min: Sun Spots and Storms: we are much more vulnerable to this type of electrical disruption
The Photon: Slit experiments: Cancelling each other out
It is possible to send two streams of light through slits and get darkness (cancelling each other out). (29-30 min)