General Relativity

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Table of Contents

  1. Introduction
  2. Scientific Theories
  3. Big Picture
  4. Background
    1. Newton’s Theory of Gravitation (1687)
    2. Special Relativity (1905)
    3. Minkowski Spacetime (1908)
  5. Spacetime, Flat and Curved
  6. Metric Tensor of General Relativity
  7. Line Elements and Metric Tensors
  8. Metric Tensor of General Relativity in Spherical Coordinates
  9. The Postulates of General Relativity
  10. Deriving Predictions
  11. Predictions
  12. Einstein’s Development of General Relativity
  13. Mathematics of General Relativity


  • 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.

Scientific Theories

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

Big Picture


Newton’s Theory of Gravitation (1687)
  • 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.

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Special Relativity (1905)
  • 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.

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Minkowski Spacetime (1908)
  • 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

Spacetime, Flat and Curved

  • 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.

Metric Tensor of General Relativity

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

    • “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, gμν.”
      • “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.”

Line Elements and Metric Tensors

  • 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

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Metric Tensor of General Relativity
in Spherical Coordinates

The Postulates of General Relativity

  • 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

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Deriving Predictions

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  • Orbit of Mercury 1859
  • Gravitational Deflection of Light 1919
  • Gravitational Redshift 1959
  • Gravitational Time Delay of Light 1964 and 1976
  • Gravitational Time Dilation 1971
  • Gravitational Lensing 1979
  • Frame Dragging and Geodetic Effect 2005
  • Gravity Waves 2015

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Einstein’s Development of General Relativity

  • Problems with Newton’s Theory
  • Happiest Thought of my Life, 1907
  • Thought Experiments and Equivalence Principle, 1907
  • Minkowski’s Flat Spacetime Relativity, 1908
  • An Eruption of Genius, 1915
  • Publication of the Complete Theory, 1916

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Mathematics of General Relativity

  • Differential Equations
  • Differential Geometry
  • Tensors

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