Much Ado About Nothing?

- Introduction

Description switch from Newtonian gravity to GR (points "fixed in space" are not accelerated in NG, while freely falling points are accelerated, versus the "points fixed in space" being accelerated in GR, while freely falling points are not accelerated).

Spherical pendulum example:

Lagrangian/Hamiltonian, Jacobi reformulation (force to geometry), embedding geometry, centrifugal force: potential and centripetal acceleration, space plus time geodesics!

centalk1.pdf [6 MEG]

- Nonrotating black hole: spacetime geodesics, Lagrangian/Hamiltonian, effective
potential, reparametrize with conformal transformation, embedding geometry, centripetal
acceleration.

Nonrelativistic accelerated reference frame, relativistic accelerated reference frame, spacetime splitting, inertial forces, centripetal acceleration, circular orbits.

Homework: Lorentz spherical pendulum (rotating black hole).

centalk2.pdf [4.7 MEG]

This is a talk assuming a knowledge of advanced undergraduate mechanics as far as the Lagrangian/Hamiltonian approach to dynamics and some awareness of special relativity. It uses a familiar example, the spherical pendulum (mass on a string under a constant gravitational force downward, therefore moving on a sphere), to show how force can be converted into geometry through the Jacobi reparametrization of the time variable while also providing a typical example of centripetal acceleration and centrifugal force. Not only are the solution curves geodesics on the deformed 2-sphere (embedding diagrams show this), but on the 3-spacetime of the Jacobi time line crossed with this sphere they are actually null geodesics, with the Jacobi time an affine parameter.

The reverse passage takes spacetime geodesics for a nonrotating black hole in the equatorial plane (a 3-d spacetime) to motion under a central "gravitational force field". The conformal transformation to the optical metric converts the solutions to geodesics on the deformed equatorial plane (like the Jacobi reparametrization). The embedding diagrams then show the reversal of centrifugal force.

The final part then shows the general scheme of how a family of reference observers reintroduce the idea of force in GR. Starting with the familiar nonrelativistic accelerated reference frame example of uniform rotation, which leads to Lorentz force law-like inertial forces, the observer splitting process is introduced for a general spacetime, leading first to the inertial forces which are analogous to the Newtonian gravitational force and a new magnetic-like force, and then finally to centripetal acceleration. Circular orbits are then briefly discussed for their centripetal acceleration properties, and contrasted with their optical counterparts.

As homework, one could develop the full analogy for a rotating black hole using the Lorentz spherical pendulum (no gravity but a charged mass on a string, or confined to a sphere in some way, under the influence of an axially symmetric magnetic field and a constant electric field along the axis of symmetry).