Relativity and Black Holes
Curvature and Geometry

Perhaps you have heard it said that according to relativity theory, spacetime is "bent" or "curved" in the presence of mass. Let's explore what this really means.

A curve in 3-dimensional Euclidean space is a vector valued function of a parameter t:
If describes the location of a particle at time t, then is the velocity vector and is the acceleration vector. The speed of the particle is the scalar function defined as the magnitude of the velocity function:

If is parameterized in terms of s such that is a unit vector tangent to curve , then is said to be parameterized in terms of arclength. Notice that with such a parameterization, the speed function is a constant: . We can think of as describing a particle that moves uniformly along the curve .

In general, a particle traveling in space can accelerate in two fundamental ways:
  1. it can go faster or slower in its direction of travel, or
  2. it can change its direction of travel.
This means that if is a particle's position at time t, then its acceleration can be written as
a sum of a vector in the direction of travel and a vector perpendicular to the direction of travel. The component of acceleration in the direction of travel reflects how the particle is speeding up or slowing down. The other component reflects how the particle is changing direction. Therefore if we parameterize in terms of arclength, then the only component of acceleration will be the component which reflects a change in direction. When we discuss curvature, this is what we are interested in.

The curvature k(s) of a curve which is parameterized in terms of arclength is the magnitude of the acceleration vector .
A circle of radius r has a curvature of size 1/r. Therefore, small circles have large curvature and large circles have small curvature. The curvature of a line is 0. In general, an object with zero curvature is "flat."

Informally, we can talk about the curvature of a surface in terms of how the surface interacts with tangent planes.
  1. A surface has positive curvature at a point P if a plane tangent to the surface at point P lies (locally) on one side of the surface and intersects the surface (locally) only at P.
  2. A surface has negative curvature at a point P if a plane tangent to the surface at point P lies (locally) on both sides of the surface.
  3. A surface has zero curvature at a point P if a tangent plane to the surface at point P lies (locally) on one side of the surface and intersects the surface on a set that includes a line segment containing P.
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For example, a plane tangent to a sphere lies entirely on one side of the sphere, and so a sphere is of positive curvature. In fact, a sphere of radius r is of curvature 1/r2.
A saddle shaped surface (or more precisely, a hyperbolic paraboloid) is of negative curvature. A tangent plane lies on both sides of the surface. Here, the point of tangency is red and the points of intersection are blue. Pringles potato chips are familiar examples of sections of a hyperbolic paraboloid.
A cylinder is of zero curvature since a tangent plane lies on one side of the cylinder and the points of intersection (here in blue) are a line containing the point of tangency. Also, of course, a plane is of zero curvature.

If two surfaces have the same curvature, we can smoothly transform one into the other without changing distances (the transformation is called an isometry). For example, a sheet of paper (used here to represent a curvature zero plane) can be rolled up to form a cylinder (which also has zero curvature). However, we cannot role the paper smoothly into a sphere (which is of positive curvature). For example, if we try to giftwrap a basketball, then the paper will overlap itself and have to be crumpled. We also cannot role the paper smoothly over a saddle shaped surface (which is of negative curvature) since this would require ripping the paper.

The curvature of a surface is also related to the geometry of the surface.

For example, on a surface of zero curvature, the geometry is the usual Euclidean geometry in which the Parallel Postulate holds and the sum of the angles of a triangle are 180o.
On a surface of positive curvature, the geometry is called elliptic geometry. In this case, the sum of the angles of a triangle is greater than 180o.
On a surface of negative curvature, the geometry is called hyperbolic geometry and the sum of the angles of a triangle is less than 180o.

An n-manifold is, in a crude sense, an n-dimensional surface. For example, the spatial components of our universe form a 3-manifold and the Minkowski spacetime in which we live is a 4-manifold. The curvature of a manifold is not a single number, though. It is described by an object called a tensor. For an n-dimensional manifold, the curvature is given by the Riemann-Christoffel curvature tensor which has n2(n2-1)/12 independent components (when n>1). Einstein's general theory of relativity gives the components of this tensor (and therefore the curvature of spacetime) in the presence of mass.

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