# Postulates of General Relativity

Back to General Relativity

#### 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.