Development of General Relativity

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Problems with Newton’s Theory

  • Mercury’s Orbit
    • In 1859 Urbain Le Verrier pointed out that Newton’s Theory was unable to fully predict the observed precession of Mercury’s orbit around the Sun, even factoring in the gravitational forces by nearby planets. He proposed the existence of another planet, Vulcan, to explain Mercury’s orbit.
  • Newton’s Theory is incompatible with Einstein’s Special Relativity
    • Newton’s Law of Universal Gravitation says:
      • For any pair of physical objects there’s a force on each, toward the other, with magnitude Gm1m2/r2
    • According to Newton the forces on each object act simultaneously. But, per Special Relativity, simultaneity is relative to an inertial reference frame.
      • If an effect is simultaneous with its cause in one frame of reference, there will be another frame of reference where the effect precedes the cause.

Happiest Thought of my Life, 1907

  • Since Newton’s theory of gravitation was incompatible with Special Relativity, Einstein took on the project of developing a theory of gravitation compatible with SR.  For Newton gravity was a force acting instantaneously between objects. Einstein’s initial idea was to make gravity into a field propagating at the speed of light, like electric and magnetic fields. But he couldn’t make it work.
  • In 1907 he had “the happiest thought of my life” (glücklichste Gedanke meines Lebens)
  • Einstein in a lecture in 1922:
    • “The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he would not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity. I continued my thought: A falling man is accelerated. Then what he feels and judges is happening in the accelerated frame of reference. I decided to extend the theory of relativity to the reference frame with acceleration. I felt that in doing so I could solve the problem of gravity at the same time.” 
  • Einstein elsewhere wrote:
    • “For an observer falling freely from the roof of a house there exists—at least in his immediate surroundings— no gravitational field. The observer therefore has the right to interpret his state as at rest.”
    • “The fact, known from experience, that acceleration in free fall is independent of the material is therefore a mighty argument that the postulate of relativity is to be extended to coordinate systems that are moving non‐uniformly relative to one another.”
  • Reconstruction of Einstein’s thought:
    • The Postulate of Relativity in SR says that the laws of physics are the same in all inertial (non-accelerating) reference frames. The falling man’s reference frame is accelerating and is thus not inertial; it’s therefore beyond the scope of SR’s Postulate of Relativity.
    • But to the falling man his reference frame appears inertial, as though he were floating in an inertial reference frame in intergalactic space.  The laws of physics should therefore be true in the falling man’s accelerating reference frame. In general, the laws of physics should be true in all reference frames, accelerating or inertial. Special Relativity needs to become General Relativity.

Thought Experiments and Equivalence Principle, 1907

  • Gravity/Acceleration Thought Experiment
    • Suppose a person is in a windowless room on the Earth’s surface.  He drops a feather and a small steel ball down a vertical tube from which air has been extracted.  The ball and feather fall at the same rate because they undergo the same gravitational acceleration, 32 feet per second per second.
    • Suppose a person is in a windowless room of a spacecraft in intergalactic space accelerating at 32 feet per second, per second. He performs the same experiment, getting the same result.
  • Free Fall/Zero Gravity Thought Experiment
    • Suppose a person defines a reference frame in a windowless room in zero gravity in intergalactic space.  He releases a steel ball, which remains motionless in his reference frame.
    • Suppose a person defines a reference frame in a windowless room in free fall toward Earth. He releases a steel ball, which remains motionless in his reference frame.
  • Based on these thought experiments Einstein proposed his Equivalence Principle, which helped guide the development of GR
    • No experiment can distinguish a uniform acceleration from a uniform gravitational field.

Minkowski’s Flat Spacetime Relativity, 1908

  • In 1908 Hermann Minkowski, Einstein’s former mathematics professor, transformed Einstein’s Special Relativity into a geometry of four-dimensional spacetime, using differential geometry to do the math.
  • 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.”
  • Einstein initially viewed Minkowski’s theory as a mathematical game, but wound up using it as the basis for General Relativity.
  • Minkowski died in 1909 at age 44.

An Eruption of Genius, 1915

  • “Grossmann, you’ve got to help me or I will go crazy.” (Einstein, 1912)
    • Einstein realized he needed heavy-duty mathematics to describe spacetime.
  • Entwurf Paper (1913)
    • In 1913 Einstein and the mathematician Marcel Grossmann published an early version of GR (the Entwurf paper), with Einstein writing the physical part and Grossmann writing the math part (tensors).
  • In the summer of 1915 Einstein realized that his 1913 theory didn’t work, including the prediction of Mercury’s orbit.
  • In November 1915 Einstein had an “eruption of genius”, modifying his equations until he got them right, publishing the correct equations on November 25 

Publication of the Complete Theory, 1916

  • The complete theory of General Relativity was published in May 1916 in the journal Annalen Der Physik with the title “Die Grundlage der allgemeinen Relativitätstheorie” (“The Foundation of General Relativity Theory”). 
  • Excerpts
    • “The Laws of Nature are expressed by means of equations which are valid for all coordinate systems, that is, which are covariant for all possible transformations.” 
      • [Tensor Equations]
    • “The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski.”
      • [Four-dimensional Spacetime]
    • “The mathematical apparatus useful for the general relativity theory, lay already complete in the ‘Absolute Differential Calculus.’” 
      • [Differential Geometry]