Postulates of General Relativity

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The Postulates, Informally Stated

• 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

The Postulates, per 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.

The Postulates

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.