## Thursday, July 23, 2015

### Comparing special and general relativity

This year is celebrated as the centenary of general relativity, as ten years ago was the centenary of special relativity. What is the difference? Special relativity is the theory of flat spacetime, including Lorentz transformations and electromagnetism. General relativity is the theory of curved spacetime, and gravity.

Einstein fans disagree over which is the greater accomplishment. Special relativity changed thinking about space and time in a way that permeates XX century physics. General relativity is always lauded as a great theory, but its effects are barely measurable and it has had almost no influence on other branches of physics. It has some influence on cosmology, but not much.

So special relativity is the more influential theory, by far. But some Einstein fans prefer to praise general relativity, because that was a conceptually much more difficult accomplishment. Special relativity can be easily explained with some undergraduate linear algebra, but general relativity requires tensors and differential geometry.

Einstein's role was also different. His 1905 special relativity paper was written on his own, building on published papers. His 1915 general relativity was a collaboration with mathematicians. Some people see one as more credit-worthy than the other.

People often say that GPS requires special and general relativity clock corrections, but it is really just special relativity corrections. There is an effect due to satellite speed and special relativity, and an effect due to gravity that is often called general relativity. But the necessary gravity formula was actually derived by Einstein in 1907 from special relativity, using what he called "the happiest thought of my life". This was before he understood relativity as a spacetime theory, and many years before he knew anything about tensors or curvature.

Sometimes people say that special relativity is just about constant velocity inertial motion, but that is not how it was viewed in the early days, say 1895-1910. It was often applied to accelerating electrons and other particles. Gravitational time dilation can be calculated by comparing to acceleration in a flat spacetime. Viewed this way, the only truly measurable general relativity effects are things like precession of Mercury's orbit, and that is a very tiny effect that took centuries to notice.

Even if relativity had never been discovered, we would probably still have GPS. Nobody would understand why the satellite clocks had to be re-synchronized so often, but they could have figured out some heuristics for resetting the clocks.