Notes on General Relativity

Einstein’s theory of general relativity is the geometric theory of gravitation as inertial motion in curved spacetime. Because of its geometric nature, it is possibly physics’ most elegant theory.

General relativity can easily fill a 1,200 page tome - that tome is Misner, Thorne, and Wheeler’s Gravitation, which I own and recommend. (Thorne shared a well-deserved Nobel in 2017 with Barish and Weiss for the detection of gravitational waves, which are predicted by general relativity.)

I am not a physicist, but I want to write down the simplistic “layman’s” explanation of gravitation I often give, as well as any notes or observations I come up with while studying general relativity or working problems in it.

A Simplistic Explanation of Gravitation

There is no shortage of oversimplified explanations of gravitation in the theory of general relativity, but I feel that my particular one is useful because it oversimplifies different topics from most. In particular, I tend to gloss over the relativity parts of relativity to focus on the fact that free fall is inertial motion. If you read this, you will by no means have the whole story - just a small piece of it with many crucial details missing. If you find it interesting, you may enjoy delving into the full theories of special and general relativity.

Newtonian inertial motion

Newtonian kinematics tells us that the ordinary state of an object’s motion is inertial motion - motion with a constant velocity vector. Consider a rock moving through empty space at 30 miles per second. If we look at only the one-dimensional subspace along the rock’s direction of motion, then we can describe the rock’s position as a function of time by the equation \(r(t) = r_0 + 30t\).

Now, instead of considering merely one-dimensional space, let’s upgrade to considering two-dimensional spacetime: a plane where one dimension is space and one dimension is time. The equation above is clearly the equation of a line in the spacetime plane. In general, inertial paths in spacetime are straight lines in spacetime.

Manifolds

Another name for a “plane” is “two-dimensional Euclidean space”. That is, the plane is “flat” and its geometry is the familiar plane geometry of Euclid - e.g., the interior angles of a triangle sum to \(\pi\) radians, a straight line segment is the shortest distance between two points, &c. A manifold is a space that approaches Euclidean geometry in the small, but may have a larger structure that deviates from it. Most of us are familiar with manifold life, due to living on the surface of a very large approximately spherical planet. In the small, we can treat the surface of the Earth as flat, and the smaller the area we care about the more accurate this treatment becomes. But on the macro scale, e.g. travel between continents, treating the Earth as flat would be a mistake leading to very wrong results.

One interesting problem that arises in manifold geometry is finding shortest-distance paths between two points. With Euclidean geometry, the shortest path between two points is a segment of a straight line. On our approximately spherical Earth, geodesics are segments of great circles (circles which are diameters of the sphere, e.g. the Equator or any longitude meridian (but not non-Equator latitude circles)).

Inertial motion is geodesic motion

The key to how gravitation works in general relativity is that inertial motion is not merely straight-line paths in Euclidean spacetime - it is geodesic paths in a spacetime manifold. The reason that gravitation falls out of this fact is because mass curves spacetime. (It is important that it curves spacetime, rather than merely curving space.) In the absence of mass to curve spacetime, these geodesics are straight-line geodesics, as with our rock hurtling through empty space. But in curved spacetime, the geodesic path tends toward the massive object that caused the curvature.

Furthermore, if we consider a single unit interval of the path’s arc-length in both cases, we find that this component of the path comes at the expense of its component in the time direction - hence gravitational time dilation. If physicists and geometers will forgive the ghastly oversimplification here: some of the motion that would have been forward in time in flat spacetime is now directed towards the massive body in curved spacetime.

The fact that free-fall reference frames are the inertial reference frames of general relativity means that our familiar reference frames standing on the surface of Earth are in fact non-inertial, accelerated frames - unless we have just jumped off a bridge and are in free fall. Just as in Newtonian mechanics, fictitious forces may appear in non-inertial frames, and the “force of gravity” we appear to experience in a standing-on-Earth reference frame is such a fictitious force.

What has been glossed over?

A vast amount of interesting and important detail - up to and including all of the “relativity” part of relativity. I don’t have any particular book recommendations for special relativity (which comes first), but some books I’ve liked on general relativity are Carroll’s Spacetime and Geometry, and the gigantic Gravitation of Misner, Thorne, and Wheeler.