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# 5: Curvature

• • Contributed by Benjamin Crowell
• Professor (Physics) at Fullerton College

General relativity describes gravitation as a curvature of spacetime, with matter acting as the source of the curvature in the same way that electric charge acts as the source of electric fields. Our goal is to arrive at Einstein’s field equations, which relate the local intrinsic curvature to the locally ambient matter in the same way that Gauss’s law relates the local divergence of the electric field to the charge density. The locality of the equations is necessary because relativity has no action at a distance; cause and effect propagate at a maximum velocity of $$c$$.

• 5.0: Introduction to Curvature
It is hard to define curvature and it can be tricky to distinguish intrinsic curvature (real), from extrinsic curvature (never produce observable effects). The manifestly intrinsic tensor notation protects us from being misled in this respect. If we can formulate a definition of curvature expressed using only tensors that are expressed without reference to any preordained coordinate system, then we know it is physically observable, and not just a superficial feature of a particular model.
• 5.1: Tidal Curvature Versus Curvature Caused by Local Sources
A further complication is the need to distinguish tidal curvature from curvature caused by local sources.
• 5.2: The Stress-energy Tensor
In general, the curvature of spacetime will contain contributions from both tidal forces and local sources, superimposed on one another. To develop the right formulation for the Einstein field equations, we need to eliminate the tidal part. Roughly speaking, we will do this by averaging the sectional curvature over all three of the planes t−x, t−y, and t−z, giving a measure of curvature called the Ricci curvature.
• 5.3: Curvature in Two Spacelike Dimensions
Since the curvature tensors in 3+1 dimensions are complicated, let’s start by considering lower dimensions. The lowest interesting dimension is therefore two.
• 5.4: Curvature Tensors
If we want to express curvature as a tensor, it should have even rank. Also, in a coordinate system in which the coordinates have units of distance (they are not angles, for instance, as in spherical coordinates), we expect that the units of curvature will always be inverse distance squared.
• 5.5: Some Order-of-magnitude Estimates
As a general proposition, calculating an order-of-magnitude estimate of a physical effect requires an understanding of 50% of the physics, while an exact calculation requires about 75%. We’ve reached the point where it’s reasonable to attempt a variety of order-of-magnitude estimates.
• 5.6: The Covariant Derivative
The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly, that is, linearly via the Jacobian matrix of the coordinate transformation.
• 5.7: The Geodesic Equation
In this section, which can be skipped at a first reading, we show how the Christoffel symbols can be used to find differential equations that describe geodesics. A geodesic can be defined as a world-line that preserves tangency under parallel transport.
• 5.8: Torsion
This section describes the concept of gravitational torsion. It can be skipped without loss of continuity, provided that you accept the symmetry property without worrying about what it means physically or what empirical evidence supports it.
• 5.9: From Metric to Curvature
• 5.10: Manifolds (Part 1)
General relativity doesn’t assume a predefined background metric, and this creates a chicken-and-egg problem. We want to define a metric on some space, but how do we even specify the set of points that make up that space? The usual way to define a set of points would be by their coordinates. For example, in two dimensions we could define the space as the set of all ordered pairs (x, y). This doesn’t work in general relativity, because space is not guaranteed to have this structure.
• 5.10: Manifolds (Part 2)
An alternative way of characterizing an n-manifold is as an object that can locally be described by n real coordinates. That is, any sufficiently small neighborhood is homeomorphic to an open set in the space of real-valued n-tuples of the form (x1, x2, . . . , xn). For example, a closed half-plane is not a 2-manifold because no neighborhood of a point on its edge is homeomorphic to any open set in the Cartesian plane.
• 5.11: Units in General Relativity
Analyzing units, also known as dimensional analysis, is one of the first things we learn in freshman physics. It’s a useful way of checking our math, and it seems as though it ought to be straightforward to extend the technique to relativity. It certainly can be done, but it isn’t quite as trivial as might be imagined, and it leads to some surprising new physical ideas.
• 5.E: Curvature (Exercises)

Thumbnail: The overall geometry of the universe is determined by whether the Omega cosmological parameter is less than, equal to or greater than 1. Shown from top to bottom are a closed universe with positive curvature, a hyperbolic universe with negative curvature and a flat universe with zero curvature. Image used with permission (Public Domain; NASA).