Warped Passages - Lisa Randall [55]
Nonetheless, in all cases matter tells spacetime how to curve, and spacetime tells matter how to move. Curved spacetime sets up the geodesic paths along which, in the absence of other forces, things will travel. Gravity is encoded into the geometry of spacetime. It took Einstein the better part of a decade to deduce this precise connection between spacetime and gravity, and to incorporate the effects of the gravitational field itself—after all, the gravitational field carries energy, and is therefore bending spacetime.* It was a heroic effort.
In his famous equations, Einstein specified how to find the universe’s gravitational field, given the contents of the universe. Although his best-known equation is E = mc2, physicists use the term “Einstein’s equations” to refer to the equations that determine the gravitational field. The equations accomplish this formidable task by showing how to determine the metric of spacetime from a known distribution of matter.9 The metric you calculate determines the spacetime geometry by telling you how to translate numbers associated with arbitrary scale units into physical distances and shapes that determine the geometry.
With the final formulation of general relativity, physicists could determine the gravitational field and calculate its influence. As with previous formulations of gravity, physicists use these equations to figure out how matter moves in a given gravitational field. For example, they can plug in the mass and position of a big spherical body, such as the Sun or the Earth, and calculate the well-known Newtonian gravitational attraction. In this particular example, the results wouldn’t be new—but their meaning would be. Matter and energy bend spacetime, and that bending gives rise to gravity. But general relativity has the further advantage that it incorporates any type of energy—including that of the gravitational field itself—into the distribution of matter and energy. This makes the theory useful even in situations where gravity itself contributes a significant amount of energy.
Because they apply to any distribution of energy, Einstein’s equations changed the outlook for cosmologists—historians of the cosmos. Now, if scientists knew the matter and energy content of the universe, they could calculate its evolution. In an empty universe, space would be completely flat, with no ripples or undulations—no curvature at all. But when energy and matter fill the universe, they distort spacetime, producing interesting possibilities for the universe’s structure and behavior over time.
We most definitely do not live in a static universe: as we will soon see, we just might live in a warped, five-dimensional one. Fortunately, general relativity tells us how to calculate their consequences. Just as there are examples of two-dimensional geometries with positive, zero, and negative curvature, there are four-dimensional geometrical configurations of spacetime with positive, zero, and negative curvature, which could arise from appropriate distributions of matter and energy. Later on, when we discuss cosmology and branes in extra dimensions, the distortions of spacetime arising from matter and energy—both in our visible universe and on the branes and in the bulk—will be of critical importance. We’ll see that the three types of spacetime curvature (positive, negative, and zero) might be realized in higher dimensions as well.
General relativity has lots of consequences that you can’t calculate with Newtonian gravity. Among its many merits, general relativity eliminated the annoying action-at-a-distance