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+---
+title: "General Relativity"
+---
+
+# 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 <cite>Gravitation</cite>, 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 <em>relativity</em> 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 {{<katex>}}r(t) = r_0 + 30t{{</katex>}}.
+
+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
+{{<katex>}}\pi{{</katex>}} 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 <em>spacetime</em>, 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 details - 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
+<cite>Spacetime and Geometry</cite>, and the gigantic
+<cite>Gravitation</cite> of Misner, Thorne, and Wheeler.