Back to General Relativity

#### 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

- A

- A freely moving particle moves along a geodesic in spacetime.

#### 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
, the energy-momentum tensor, which represents the density and flow of energy and momentum.*T*_{μν} - You feed
into the*T*_{μν}*Field Equation*and get, the metric tensor of the system*g*_{μν} - You convert
to*g*_{μν}**Γ**^{μ}_{αβ} - You feed
**Γ**into the^{μ}_{αβ}*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
, determines the curvature of spacetime, represented by*T*_{μν}.*g*_{μν} - That is, “Matter tells spacetime how to curve.”

- The density and flow of energy and momentum in a physical system, represented by
- So you figure out
and solve the Field Equation for*T*_{μν}.*g*_{μν} *R*_{μν}*,**T*_{μν}*,**g*_{μν}and*μ*denote the rows and columns of 4×4 tensors.*ν,*- For example, the metric tensor,
, has the form:*g*_{μν}

**Geodesic Postulate**

**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
*x*^{𝝁}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*):

are the usual spherical coordinates.*r, θ, ϕ*

- The first curious thing about the equations is that there are two variables for time:
and 𝝉 (tau).*t*is*t**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.

- A

- Suppose you set up a reference frame using coordinates
**(**, with meter marks and atomic clocks. You track a spacecraft moving along the blue world line. Time*ct,*)*r, θ, ϕ*is the time on your atomic clocks. Time 𝝉 is the time on the spacecraft’s atomic clock.**t**

- 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.