Warped Passages - Lisa Randall [53]
Because I will almost always consider space and time together from now on, we will generally find the notion of spacetime more useful than the notion of space. Spacetime has one more dimension than space: in addition to “up-down,” “left-right,” and “forwards-backwards,” it includes time. In 1908 the mathematician Hermann Minkowski used geometric notions to develop this idea of an absolute spacetime fabric. Whereas Einstein studied spacetime using time and space coordinates that depended on a frame of reference, Minkowski identified the observer-independent spacetime fabric that can be used to characterize a given physical situation.
In the rest of the book, when I refer to dimensionality I will be giving the number of spacetime dimensions, except where I explicitly state otherwise. For example, when we look around us we see what I will from now on refer to as a four-dimensional universe. Occasionally I will single out time and talk about a “three-plus-one”-dimensional universe, or three spatial dimensions. Bear in mind that all these terms refer to the same setting—one that has three dimensions of space and one of time.
The spacetime fabric is a very important notion. It concisely characterizes the geometry that corresponds to the gravitational field produced by a particular distribution of energy and matter. But Einstein initially disliked the idea, which had seemed to him like an overly fancy way to reformulate the physics that he had already explained. However, he eventually recognized that the spacetime fabric was essential for completely describing general relativity and calculating gravitational fields. (For the record, Minkowski wasn’t overly impressed with Einstein on first acquaintance, either. Based on Einstein’s performance in Minkowski’s calculus class back when Einstein was a student, Minkowski had concluded then that Einstein was a “lazy dog.”)
Einstein wasn’t alone in resisting non-Euclidean geometry. His friend Marcel Grossmann, a Swiss mathematician, also considered it unduly complicated and tried to talk Einstein out of using it. However, they eventually agreed that the only tractable way of explaining gravity was by using non-Euclidean geometry to represent the spacetime fabric. Only then could Einstein interpret and calculate the warping of spacetime that was equivalent to gravity, which turned out to be the key to completing general relativity. After Grossmann conceded defeat, both he and Einstein struggled through the intricacies of differential geometry to simplify their highly complicated earlier attempts to arrive at a formulation of the theory of gravity. In the end, they completed the theory of general relativity and reached a deeper understanding of gravity itself.
Einstein’s Theory of General Relativity
General relativity presented a radical revision of the concept of gravity. We now understand gravity—the force that keeps your feet on the ground and binds together our galaxy and the universe—not as a force acting directly on objects, but as a consequence of the geometry of spacetime, an idea that took Einstein’s view of the union of space and time to its logical conclusion. General relativity exploits the deep connection between inertial and gravitational mass to formulate the effect of gravity solely in terms of the geometry of spacetime. Any distribution of matter or energy curves or warps spacetime. Curved pathways in spacetime determine gravitational motion, and the matter and energy of the universe cause spacetime itself to expand, undulate, or contract.
In flat space the shortest distance between two points, the geodesic, is a straight line. In curved space we still can define a geodesic as the shortest path between two points, but that path won’t necessarily look straight. For example, routes of airplanes that follow great circles on the Earth are geodesics. (A great circle is any