Warped Passages - Lisa Randall [54]
In curved four-dimensional spacetime, we can also define a geodesic. For two events separated in time, a geodesic is the natural path things would take in spacetime to connect one event to the other. Einstein realized that free fall, which is the path of least resistance, is motion along the spacetime geodesic. He concluded that, in the absence of external forces, dropped objects will fall along a geodesic, as with the path of the person on the falling elevator who doesn’t feel his weight or see a ball drop.
However, even when things are following their geodesics through spacetime and there are no external forces, gravity has noticeable effects. We’ve already seen that the local equivalence between gravity and acceleration was one of the critical insights that led Einstein to develop an entirely new way of thinking about gravity. He deduced that, because the acceleration induced by a gravitational force is locally the same for all masses, gravity must be a property of spacetime itself. That’s because “freely falling” means different things in different places, and it is only locally that gravity can be replaced by a single acceleration. My Chinese counterpart and I fall in different directions, even if we are both in our local version of Einstein’s elevator. The fact that the direction of free fall is not the same everywhere is a reflection of the curvature of spacetime. There isn’t a single acceleration that can cancel the effects of gravity everywhere. In curved spacetime, the geodesics of different observers will in general be different. So globally, gravity has observable consequences.
General relativity goes much further than Newtonian gravity because it allows us to calculate the relativistic gravitational field of any distribution of energy and matter. Moreover, the revelation that the geometry of spacetime encodes the effects of gravity permitted Einstein to close a major gap in his original formulation of gravity. Although physicists at the time knew how objects would respond to a gravitational field, they did not know what gravity was. Now they understood that the gravitational field is the distortion of the spacetime fabric caused by matter and energy. This distortion extends throughout the cosmos itself, or, as we will see shortly, throughout a higher-dimensional spacetime that might include branes. All of the gravitational effects of these more complicated situations can be embedded in the ripples and curves of a spacetime surface.
A picture gives perhaps the best description of how matter and energy distort the spacetime fabric to create a gravitational field. Figure 39 shows a sphere of matter sitting in space. The space surrounding the sphere is distorted: the ball makes a depression in the spatial surface whose depth reflects the ball’s mass or energy. A ball passing nearby will roll towards the central depression, where the mass is located. According to general relativity, the spacetime fabric warps in an analogous fashion. Another ball passing through would be accelerated towards the center of the sphere. In this case, the result would agree with what Newton’s law would predict, but the interpretation and calculation of the motion would be very different. According to general relativity, a ball follows the undulations of the spacetime surface, and thereby implements the motion induced by the gravitational field.
Figure 39. A massive object distorts the surrounding space, thereby creating a gravitational field.
Figure 39 is a bit misleading, so you should keep in mind several caveats. First of all, I’ve shown the space surrounding the ball as two-dimensional. But really, the full three-dimensional space and the full four-dimensional spacetime are warped. Time is warped because it too is a dimension from the vantage point of special and general relativity. Warped time is how special relativity tells