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**Deriving Predictions in Newton’s Theory**

**Deriving Predictions in General Relativity**

**Deriving Predictions in General Relativity**

**Example**

**A Single Sphere Universe**

**Example****Deriving Acceleration at the Sphere’s Surface in Newton’s Theory**

**Deriving Acceleration at the Sphere’s Surface in Newton’s Theory****Deriving Acceleration ****at the Sphere’s Surface** in General Relativity

**Deriving Acceleration****at the Sphere’s Surface**in General Relativity- For the (
**c**𝜽, 𝝓) coordinate system the Geodesic Equation is equivalent to the four equations on the right.*t, r,* - Since acceleration at the sphere’s surface is in only one direction, we’ll use coordinates
and*t*, ignoring 𝜽 and 𝝓.*r*

**Schwarzschild Metric**

**Schwarzschild Metric**- The solution of the Field Equation for the single sphere universe is the Schwarzschild Metric:

- where
- Coordinates are (
𝜽, 𝝓)*c t, r,* *M*= Gravitational Constant*G*= radius of sphere*r*= speed of light*c*

- Coordinates are (
- wikipedia.org/wiki/Karl_Schwarzschild (pronounced
**Schvatz**-Shield)- “Karl Schwarzschild provided the first exact solution to the Einstein field equation of general relativity, for the limited case of a single spherical non-rotating mass, which he accomplished in 1915, the same year that Einstein first introduced general relativity.”
- “Schwarzschild accomplished this while serving in the German army during World War I. He died the following year.”

- britannica.com/biography/Karl-Schwarzschild
- “He also laid the foundation of the theory of black holes by using the general equations to demonstrate that bodies of sufficient mass would have an escape velocity exceeding the speed of light and, therefore, would not be directly observable.”