Borehole Oscillators
11 hours ago
- #Schwarzschild Solutions
- #Newtonian Gravity
- #General Relativity
- A test particle dropped into a radial borehole through a uniform density ball undergoes simple harmonic motion with the same period as a particle in a circular orbit grazing the surface.
- Newtonian gravity calculations show that the radial acceleration inside the ball is proportional to the displacement, leading to harmonic motion.
- The shell theorem in Newtonian physics allows treating the mass inside any radius as concentrated at the center for gravitational calculations.
- In General Relativity, the Schwarzschild solutions describe the spacetime around a point mass (first solution) and a uniform density ball (second solution).
- The second Schwarzschild solution involves a ball of incompressible fluid, with pressure balancing gravitational attraction internally.
- Radial motion in the vacuum outside the ball (first Schwarzschild metric) shows that a particle falling from rest follows a path similar to Newtonian predictions but with relativistic corrections.
- For a particle in a borehole through the ball, the coordinate-time angular velocity is slower than for a grazing circular orbit, leading to different periods in GR.
- Visualizations of the spacetime geometry can be achieved through embeddings in higher-dimensional spaces, illustrating the curvature effects.
- The pressure at the center of the ball becomes infinite if the radius approaches (9/4)M, setting a limit on the ball's compactness.
- Light traveling radially through the borehole reaches the center in a finite time, with specific conditions relating to the ball's radius and mass.