# A Pedagogical, Philosophical, and Mathematical Exploration of General Relativity

Zachary Hall

Here I present a suite of materials produced as the result of a project in which I undertook learning General Relativity (GR), partially under the guidance of Professor Emeritus John Stachel, to whom I am most grateful for his insights and with whom I coauthored one of the draft papers here included, from an historical, analytical, philosophical perspective. While I would like to publish these materials eventually, I have yet to have been able to do so, and would like to make them available to share in the meantime. I thank Professor Dahlstrom for allowing me to host them here.

The first paper - A Gravitomagnetic Extension of Newton-Cartan Theory - follows upon the earlier work of Professor Stachel's on the logical independence of the Gravitational Equivalence Principle (EP) from the postulates of Special Relativity (SR), introducing mathematical objects fundamental to GR but only constructed in response to the conceptual requirements of the equivalence principle, and using the resulting mathematical apparatus to predict a near-field, low-velocity limit of a General-Relativistic spacetime in which a simplified model of the Earth is embedded. This solution includes the frame-dragging, or gravitomagnetic, effects of GR. I also include a user-interactive, visual simulation to help in intuitively understanding this effect. What is shown is the rotation that ideal rods would undergo for a uniform sphere of the mass and radius of Earth rotating at the rate at which Earth does rotate. The relative rotation rate of Earth to the rods shown is sped up by a rate of about 3 million as compared with the solution found. You can move your perspective by clicking and dragging with the mouse, and, if you have a mouse with a rolling wheel, can zoom in and out with that.
Numerous other calculations are presented, including a derivation of the centrifigual and Coriolis forces in terms of an affine-flat spacetime, the amount of splitting between the equatorial radii at which there are stable orbits for orbits in the direction of Earth's rotation vs against it owing to the frame-dragging effects (this is analogous to the Zeeman effect) and an order-of-magnitude estimate of the amount of rotation Stanford's Gravity Probe B should have undergone from these effects.

Second, I include a webpage, inspired by Hermann Bondi's 'Relativity and Common Sense', that seeks to explain the postulates of Special Relativity and their basic conseqences, including three user-interactive, visual Java apps, introduced and explained throughout the lesson and made available for download.

Finally, I include a draft of a paper that attempts to synthesize the insights of the above-detailed works, and give the reader an understanding of the basic statements of GR. While numerous pedagogical resources for GR exist, my aim here is to present a suite of materials that stand on their own as introductions to the two postulates and their consequences, separately considered, and then their synthesis. This resource is also unique in that, while historically, Special Relativity preceded Einstein's postulating the gravitational equivalence principle and reconciling it with the logical appartus inherent to SR, we here afford the ability reader to go either direction -- Gravitational EP first and then SR, or the other way around -- in attempting to comprehend General Relativity.

• A Gravitomagnetic Extension of Newton-Cartan Theory (.pdf)

• Frame Dragging Simulation for Earth (Javascript)

• A User-Interactive Introduction to Special Relativity (Webpage - .jar files for apps available here)

• An Introduction to General Relativity (.pdf)