Gravity Acceleration through a Hole in Earth

[CutterMcCool]

Consider this a thought experiment.

Suppose the Earth had a stable hole through it to its center wide enough to drop a golf ball down. Assuming that golf ball freefalls (no friction with hole wall), would its acceleration rate continually increase as it fell to the center? If it increases, at what distance from the center (if any) would the golf ball reach the speed of light?

Since g=GM/R^2, as the golf ball falls closer to the Earth's center, R shortens and acceleration increases--unless the effective mass of earth, M, also decreases as the golf ball approaches the center.

A correlated question: would any particle close enough to the center of the Earth to be within its Schwarzschild radius (about 8.8 millimeters) be unable to leave that radius because its escape velocity would be greater than the speed of light?

 

[Doc Al]

Suppose the Earth had a stable hole through it to its center wide enough to drop a golf ball down. Assuming that golf ball freefalls (no friction with hole wall), would its acceleration rate continually increase as it fell to the center?

No.

Since g=GM/R^2, as the golf ball falls closer to the Earth's center, R shortens and acceleration increases--unless the effective mass of earth, M, also decreases as the golf ball approaches the center.

When you are a distance r from the center, only that part of the Earth's mass that is < r from the center contributes to the gravitational acceleration at that point. (Making the simplifying assumption of spherical symmetry.) If you further assume the Earth to be of uniform density, then the acceleration due to gravity will vary linearly from zero at the center to the maximum at the Earth's surface.

A correlated question: would any particle close enough to the center of the Earth to be within its Schwarzschild radius (about 8.8 millimeters) be unable to leave that radius because its escape velocity would be greater than the speed of light?

No.