Why Does E=mc2_ - Brian Cox [90]
he can follow without any confusion because at any point on the earth he can assume Euclidean geometry works just fine and, as a result, the idea of a straight line is clear to him. Even so, he ends up walking in a circular path, although we can think of the circle as being build up of lots of little straight lines. Now let’s return to the case of gravity and spacetime. The notion of straight lines through curved spacetime is entirely analogous to the notion of straight lines on the earth’s surface. The complication arises because spacetime is a four-dimensional “surface,” while the earth’s surface is only two-dimensional. But once again the complication is more to do with our limited imagination rather than any increase in mathematical complexity. In fact, the mathematics of geometry on the surface of a sphere is no harder than the mathematics of geometry in spacetime. Armed with the idea of straight lines (they are also known as geodesics) in spacetime we might be so bold as to suggest how gravity works. We have seen that gravity can be banished in exchange for curved spacetime and that locally the spacetime is the “flat” spacetime of Minkowski. We know very well by this point in the book how things move in such an environment. For example, if a particle is at rest it will remain so (unless something comes along and gives it a push or pull). That means it follows a spacetime trajectory that moves only along the time axis. Likewise, objects that are moving with a constant speed will carry on moving in the same direction and at the same speed (again, unless something comes and knocks them off course). In this case they will follow straight lines on the spacetime diagram that are tilted away from the time axis. So, on each tiny patch of spacetime everything should follow a straight line unless acted upon by some external influence. The whole appearance of gravity emerges when we sew all of the little patches together; for only then do the individual straight lines join together into something more interesting, like the orbit of a planet around the sun. We have not said how to join up the patches in order to build the warping of spacetime, and it is Einstein’s equation of 1915 that determines exactly how we are to do that. But the bottom line could not be much simpler—gravity has been banished in exchange for pure geometry.
So gravity is geometry and all things move along straight lines in spacetime unless they are knocked off course. But at any given point in spacetime there is an infinite number of geodesics, just as there is an infinite number of straight lines passing through any point on the earth’s surface (or any other surface, for that matter). So how are we to figure out which spacetime trajectory an object will move along? The answer is simple enough: Circumstances dictate it. For example, the person on the trek around the earth could start out in any number of directions. He chooses which route to take. Likewise, an object dropped from rest near to the earth will start out on one spacetime geodesic while one that is thrown will start out on a different geodesic. By specifying the direction an object moves through spacetime at any particular point, we therefore know its complete trajectory. Moreover, all objects heading off in that particular direction necessarily follow the same trajectory, irrespective of their internal properties (like mass or electric charge). They just follow a straight line, and that’s all there is to it. In this way the curved spacetime view of gravity beautifully expresses the principle of equivalence that so captivated Einstein.
Our musings on the nature of space and time have led us to understand that the earth is doing nothing more than falling in a straight line around the sun. It is just that the straight line is in a curved spacetime, which manifests itself as a (nearly) circular orbit in space. We have not gone ahead and proved that the sun warps spacetime such that the earth falls along a geodesic whose shadow in three-dimensional space is (nearly) a circle. We haven’t done