General Relativity

Table of Contents

General Relativity, Einstein’s theory of gravity, is “the most beautiful physical theory ever invented.”
- Sean Carroll, from his textbook Spacetime and Geometry
For Newton, gravity was an attractive force among objects, acting instantaneously no matter the distance. But instantaneously-acting forces are incompatible with Einstein’s Special Relativity (1905).
Einstein thus sought a theory of gravitation compatible with Special Relativity. The result was General Relativity (1915), the theory that gravity is the curvature of spacetime.

A scientific theory is an axiom system
- designed to explain certain kinds of phenomena
- defined by its postulates
- supported or disproved by its predictions

Newton’s theory is that, for any pair of physical bodies, there’s a force on each in the direction of the other. The force acts instantaneously, no matter the distance. The Sun’s gravitational pull on Earth, for example, is faster than light.

Special Relativity was designed to answer fundamental questions about the speed of light:
- What is the speed of light relative to?
- Why is the speed of light the same, regardless of the observer’s motion?
Einstein’s simple, counterintuitive answer was that the speed of light is the same in every non-accelerating reference frame.
Among the counterintuitive consequences:
- The same event may be longer in one reference frame than in another.
- The same rod may be longer in one reference frame than in another.

In 1908 Hermann Minkowski, Einstein’s former mathematics professor, transformed Einstein’s Special Relativity into a geometry of four-dimensional spacetime

Raum und Zeit, a lecture Minkowski gave in 1908, begins:
- “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”
In Classical Physics a moving object traces out a path through 3D-space and time from point (x, y, z) at a time t to point (x′, y′, z′) at a later time t′.
- The path consists of infinitesimal elements ds² = dx² + dy² + dz²

In Minkowski Spacetime a moving object traces out a path through 4D-spacetime from point (ct, x, y, z) to point (ct′, x′, y′, z′), where c is the speed of light.
The path consists of infinitesimal elements ds² = -c² dt² + dx² + dy² + dz²

In Minkowski’s theory spacetime is flat, with no gravity.
To accommodate gravity, Einstein made spacetime malleable, able to be curved (warped, bent, distorted) by matter and energy.
Gravity is this curving of spacetime.

The Metric Tensor of General Relativity completely characterizes the curvature of spacetime

wikipedia.org/wiki/Metric_tensor_(general_relativity)
- “In general relativity the metric tensor is the fundamental object of study. It captures all the geometric and causal structure of spacetime, being used to define notions such as distance, volume, curvature, angle, future, and past.”
Sean Carroll
- “The metric tensor is such an important object in curved space that it is given a new symbol, g_μν.”
  - “The metric supplies a notion of “past” and “future.”
  - It allows the computation of path length and proper time
  - It determines the “shortest distance” between two points (and therefore the motion of test particles).
  - It replaces the Newtonian gravitational field.
  - It determines causality, by defining the speed of light faster than which no signal can travel.”

The line element ds for a curve or a space is the shortest distance between two neighboring points. Using calculus, ds can be used to calculate distances, lengths, angles, areas, volumes, and geodesics (the shortest line between two points).
The line element is determined by the Metric Tensor and coordinates.
So:
- Metric Tensor + Coordinates ➞ Line Element
- Line Element + Calculus ➞ Distances, Lengths, Angles, Areas, Volumes, Geodesics

General Relativity’s two postulates have been famously articulated by John Archibald Wheeler:
- “Matter tells spacetime how to curve, and curved spacetime tells matter how to move.”
Field Equation
- Matter and energy determine the curvature of spacetime.
Geodesic Postulate
- A freely moving particle moves along a geodesic in spacetime.
  - A geodesic is the shortest line between points