![]() The Einstein field equations are not just one equation, then, but rather a suite of 16 different equations: one for each of the “4 × 4” combinations. The “Weyl” part is shape distorting, and, counterintuitively enough, plays no role in the Einstein field equations. ![]() The “Ricci” part is volume distorting, and that plays a role in the Einstein tensor, as the Einstein tensor is made up of the Ricci tensor and the Ricci scalar, with some constants and the metric thrown in. In Riemannian geometry, where manifolds are not required to be straight and rigid but can be arbitrarily curved, you can break that curvature up into two parts: parts that distort the volume of an object and parts that distort the shape of an object. Our universe, with three spatial dimensions and one time dimension, means the geometry of our universe can be mathematically treated as a four-dimensional manifold. Why would we need so many equations just to describe gravitation, whereas Newton only needed one?īecause geometry is a complicated beast, because we are working in four dimensions, and because what happens in one dimension, or even in one location, can propagate outward and affect every location in the universe, if only you allow enough time to pass. Credit: Christopher Vitale of Networkologies and The Pratt Institute In general relativity, space and time are continuous, with all forms of energy contributing to spacetime’s curvature. Instead of an empty, blank, three-dimensional grid, putting a mass down causes what would have been “straight” lines to instead become curved by a specific amount. ![]() Instead, we have each of the four dimensions (t, x, y, z) affecting each of the other four (t, x, y, z), for a total of 4 × 4, or 16, equations. In general relativity, the fact that we have four dimensions (three space and one time) as well as two subscripts, which physicists know as indices, means that there is not one equation, nor even three or four. But Newton’s F = m a is not a single equation but rather three separate equations: F x = ma x for the “x” direction, F y = ma y for the “y” direction, and F z = ma z for the “z” direction. But we can also write down systems of equations and represent them with a single simple formulation that encodes these relationships.Į = mc² is a scalar equation because energy (E), mass (m), and the speed of light (c) all have only single, unique values. Most often, when we write down an equation, we are writing down a scalar equation, that is, an equation that only represents a single equality, where the sum of everything on the left-hand side equals everything on the right. You might be wondering what is with all those subscripts - those weird “μν” combinations of Greek letters you see at the bottom of the Einstein tensor, the metric, and the stress-energy tensor. The Einstein tensor is shown decomposed, at left, into the Ricci tensor and Ricci scalar. Here, in plain English, is what it truly means.Ī mural of the Einstein field equations, with an illustration of light bending around the eclipsed sun, the observations that first validated general relativity back in 1919. Yet despite its success over more than 100 years, almost no one understands what the one equation that governs general relativity is actually about. Put forth in 1915 by Einstein and validated four years later during a total solar eclipse - when the bending of starlight coming from light sources behind the sun agreed with Einstein’s predictions and not Newton’s - general relativity has passed every observational and experimental test we have ever concocted. This fundamental idea - that matter and energy tells spacetime how to curve, and that curved spacetime, in turn, tells matter and energy how to move - represented a revolutionary new view of the universe. But Einstein’s conception was entirely different, based on the idea that space and time were unified into a fabric, spacetime, and that the curvature of spacetime told not only matter but also energy how to move within it. Before Einstein, we thought of gravitation in Newtonian terms: that everything in the universe that has a mass instantaneously attracts every other mass, dependent on the value of their masses, the gravitational constant, and the square of the distance between them. Although Einstein is a legendary figure in science for a large number of reasons - E = mc², the photoelectric effect, and the notion that the speed of light is a constant for everyone - his most enduring discovery is also the least understood: his theory of gravitation, general relativity.
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