Gravity and the curvature of spacetime

TITLE: A moment of science, with Brian Greene. What is General Relativity?

BRIAN GREENE: The general theory of relativity is Einstein's pinnacle achievement. It is one of the deepest scientific discoveries—one of the deepest scientific insights of all time. And it all has to do with the force of gravity. So as you probably remember from high school, way back in the 1600s Isaac Newton gave the world some insight into gravity. He wrote down a formula: 

F = G(m₁m₂/r²)

that tells us that every object in the universe pulls on every other with this force called gravity, and you can see that force at work right here. And if you use that math to predict, for example, how the planets move, and goodness gracious, the planets do just what Newton's mathematics says that they should do, which is a spectacular discovery.

But even so, Einstein, when he started to think about these ideas, was left unsatisfied by Newton's ideas of gravity, for a number of reasons. But one concrete one, that's easy to grasp, is: Newton never told us how gravity actually exerts the influence that we associate with it. How is it that the Sun pulls on the Earth or pulls on Saturn? We say it's gravity, but the question is, what's the underlying mechanism by which that force is exerted from the Sun to the Earth, from the Earth to the Sun? And so on.

So Einstein spent 10 long years trying to figure this out, and by 1915 he did indeed come up with the answer. And the answer is that the medium, the mechanism for transmitting the force of gravity, is nothing but space, in fact, space and time. It's nothing but warps and curves in space and time. It's the geometry of the universe that communicates the force of gravity.

So just to get a feel for what this all means, if this is 3D space, the environment that's all around us, it's kind of hard to visualise with that. So let's go to a 2 dimensional version that captures all of the essential ideas. And here it is. Space is flat when there's no matter present, but when you bring in the Sun, Einstein says the fabric of space curves. And if you look in the vicinity of the Earth, Earth also warps the environment. And now if you focus your attention on the Moon, you see exactly what I'm talking about. Einstein is saying that the Moon is kept in orbit because it's rolling along a valley in a curved environment that the Earth creates. And if you pull back and look at the motion of the Earth, it is kept in orbit for exactly the same reason. The Earth, rolling along the valley in the warped environment that the Sun creates. 

Now this, of course, is just an animation, but what Einstein worked on for that marathon decade of research was finding the mathematical equations behind this picture. And, indeed, he found those equations.

R_μν-(1/2)Rg_μν + Λg_μν = (8πG/c⁴)T_μν

A beautiful, compact mathematical formula that describes everything in this imagery. And that formula, together with these profound ideas, constitute Einstein's general theory of relativity.