Postulates

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Postulates

  • 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

John Archibald Wheeler

  • “Matter tells spacetime how to curve, and curved spacetime tells matter how to move.”

Analogy with Spheres on a Surface

  • Analog of Field Equation 
    • The mass and location of the spheres determine the curvature of the blue-checkered surface.
  • Analog of Geodesic Postulate
    • The spheres move along geodesics on the blue-checkered surface.

How the Postulates Generate Predictions

  • For a given physical system you figure out Tμν, the energy-momentum tensor, which represents the density and flow of energy and momentum.
  • You feed Tμν into the Field Equation and get gμν, the metric tensor of the system
  • You convert gμν to Γμαβ
  • You feed Γμαβ into the Geodesic Equation to find the acceleration of a particle x.

Field Equation

  • Matter and energy determine the curvature of spacetime per the Field Equation:
  • The equation says:
    • The density and flow of energy and momentum in a physical system, represented by Tμν, determines the curvature of spacetime, represented by gμν.
    • That is, “Matter tells spacetime how to curve.”
  • So you figure out Tμν and solve the Field Equation for gμν.
  • Rμν, Tμν, gμν represent mathematical entities called tensors.  The indices, μ and ν, denote the rows and columns of 4×4 tensors.
  • For example, the metric tensor, gμν, has the form:

Geodesic Postulate

  • Geodesic Postulate
    • A freely moving particle moves along a geodesic in curved spacetime
    • That is, “curved spacetime tells matter how to move.”
  • A particle is freely moving if it’s free from influences other than spacetime curvature.
  • A geodesic is the shortest line between two points in a space or on a surface. The line PQ in the diagram is a geodesic on the sphere.
  • Geodesic Equation
    • The Geodesic Equation calculates the geodesic of a freely moving particle and thus its movement through spacetime.
  • The superscript 𝝁  in x𝝁 stands for the four coordinates of spacetime. The Geodesic Equation is thus equivalent to four equations, one for each component.  For example, using (ct, r, θ, ϕ):
  • r, θ, ϕ are the usual spherical coordinates.
  • The first curious thing about the equations is that there are two variables for time: t and 𝝉 (tau).
    • t is coordinate time, the time in an observer’s reference frame.
    • 𝝉 (tau) is proper time, the time measured by a clock carried along a world line.
      • A world line is the unique path an object takes through spacetime from one spacetime point to another.
  • Suppose you set up a reference frame using coordinates (ct, r, θ, ϕ), with meter marks and atomic clocks. You track a spacecraft moving along the blue world line. Time t is the time on your atomic clocks. Time 𝝉 is the time on the spacecraft’s atomic clock.
  • Thus the Geodesic Equation computes the acceleration of a particle relative to the proper time of the world line on which it’s moving. 
  • The second curious thing is the acceleration of coordinate time relative to the proper time: 
  • Thus time can flow at different rates in different reference frames.