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.

 

[sardar]

The geodesic equation is a fundamental equation in the study of general relativity, which describes the motion of particles in a curved spacetime. In the context of orbital dynamics, the "s" in the geodesic equation refers to the surface area of the orbit around a star or black hole. This surface area is a measure of the amount of space that is enclosed by the orbiting body as it moves around the central object.

The geodesic equation tells us how the surface area changes as the orbiting body moves through spacetime. It takes into account the curvature of spacetime caused by the mass of the central object, as well as any other interfering gravitational sources. This equation is important because it allows us to accurately predict the path of an orbiting body and understand how it is affected by the gravitational forces at play.

In the case of the Sun, the orbital surface area would appear as a slightly flattened sphere or ellipsoid, due to the Sun's relatively large mass compared to the orbiting body. This surface area would also be affected by the gravitational pull of other planets and celestial bodies in the solar system. By using the geodesic equation, we can model the precise trajectory of a planet or satellite as it orbits the Sun and account for these various gravitational influences.

Overall, the geodesic equation and the concept of orbital surface area provide a powerful tool for understanding the complex dynamics of celestial bodies in our universe. By studying and applying these principles, we can gain a deeper understanding of the fundamental laws of nature and how they shape the world around us.