Geodesic Equation & Orbital Surface Area Around the Sun

[Philosophaie]

The "s" in the geodesic equation refers to the "surface area" for that portion of the orbit around a star or black hole.

For a small enough "delta t" the surface areas are the same.

Around a small star the orbital surface area (without the other interfering gravitational sources) would look something like an sphere or an ellipsoid.

How would you describe what the surface area and the geodesic equation around the Sun?

 

[Dale]

The "s" in the geodesic equation refers to the "surface area" for that portion of the orbit around a star or black hole.

No. It is the spacetime interval along the geodesic, not a surface area. More generally, s is any affine parameter.

 

[Philosophaie]

No. It is the spacetime interval along the geodesic, not a surface area. More generally, s is any affine parameter.

The Affine parameter is along a straight or parallel path in a gravitationally curved area.

Is "s" like possible paths along that curved area? How is it measured?

 

[Dale]

The Affine parameter is along a straight or parallel path in a gravitationally curved area.

The affine parameter is defined along any curve. If additionally the affine parameter satisfies the geodesic equation then the curve is a geodesic and we can reasonably call it "straight".

Is "s" like possible paths along that curved area? How is it measured?

Again, s has nothing whatsoever to do with area. Please get that mistaken idea out of your head. It is not an area.

If the path is timelike then s is measured with a clock, if the path is spacelike s is measured with a rod (possibly a curved rod), if the path is lightlike or mixed then s is measured with both clocks and rods.