## Fermi-Walker Transport in Kerr Spacetime

In the last two posts in this series, we developed some tools for looking at Fermi-Walker transport in Minkowski spacetime and then applied them in Schwarzschild spacetime. In this post, we’ll look at Kerr spacetime, which will introduce some additional complexities. The first congruence we will look at in Kerr spacetime is the hovering congruence, […]

## Fermi-Walker Transport in Schwarzschild Spacetime

In the first post in this series, we introduced the concepts of frame field, Fermi-Walker transport, and the “Fermi derivative” of a frame field, and developed some basic machinery for dealing with them. In this post, we will use that machinery to look at two congruences in Schwarzschild spacetime, to see how things differ from […]

## How to Study Fermi-Walker Transport in Minkowski Spacetime

This is the first of several posts that will develop some mathematical machinery for studying Fermi-Walker transport. In this first post, we focus on Minkowski spacetime in order to introduce the basic concepts without having to deal with the complications introduced by spacetime curvature. Before looking at Fermi-Walker transport, we first need to introduce the […]

## The Fundamental Difference in Interpretations of Quantum Mechanics

A topic that continually comes up in discussions of quantum mechanics is the existence of many different interpretations. Not only are there different interpretations, but people often get quite emphatic about the one they favor, so that discussions of QM can easily turn into long arguments. Sometimes this even reaches the point where proponents of […]

## The Schwarzschild Geometry: Physically Reasonable?

In the last article, we looked at various counterintuitive features of the Schwarzschild spacetime geometry, as illustrated in the Kruskal-Szekeres spacetime diagram. But counterintuitive, in itself, does not mean physically unreasonable or unlikely. So the obvious next question is, how much of the entire spacetime geometry we have been looking at is actually believed […]

## The Schwarzschild Geometry: Spacetime Diagrams

When we left off in part 2, we were looking at the metric for the Schwarzschild geometry in Kruskal-Szekeres coordinates: $$ds^2 = \frac{32 M^3}{r} \left( – dT^2 + dX^2 \right) + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)$$ And we saw that we can draw a spacetime diagram using these coordinates […]

## The Schwarzschild Geometry: Coordinates

At the end of part 1, we looked at the form the metric of the Schwarzschild geometry takes in Gullstrand-Painleve coordinates: $$ds^2 = – \left( 1 – \frac{2M}{r} \right) dT^2 + 2 \sqrt{\frac{2M}{r}} dT dr + dr^2 + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)$$ We left off with two open […]

## The Schwarzschild Geometry: Key Properties

Not long after Einstein published his Field Equation, the first exact solution was found by Karl Schwarzschild. This solution is one of the best known and most often discussed, and its properties have played a significant role in the development of General Relativity as a theory, as well as in efforts to find a […]

## Why Won’t You Look at My New Theory?

In any forum where science is discussed, there will always be people who have a great new personal theory and can’t understand why no one else is interested in it. Here at PF we have rules about this, but I want to look at the more general question of why there is apparently so little […]

## Do Black Holes Really Exist?

The purpose of this article is to discuss the title question from several different viewpoints, in order to show that it isn’t as simple as it looks. We will look at some common misconceptions that lead people to think the answer must be “no”, and we will look at some of the issues involved that […]

## The Block Universe – Refuting a Common Argument

The “block universe” interpretation of SR has come up repeatedly in threads here on PF. Rather than link to them, I want to summarize a common argument that is made for the “block universe” being necessary, and then summarize the arguments I made in those threads to show why the common argument is not valid. […]

## Does Gravity Gravitate: The Wave

In the first two posts in this series, we looked at different ways of interpreting the question “does gravity gravitate?” We left off at the end of the last post with an open question: what do the various “mass integrals” look like in a spacetime where gravitational waves are being emitted? Let’s look at […]

## Does Gravity Gravitate? Part 2

In the first post of this series, I talked about two ways to answer the title question, one leading to the answer “no” and the other leading to the answer “yes”. However, this will leave a lot of people who ask our title question unsatisfied, because the usual motivations for asking the question have […]

## Does Gravity Gravitate?

The title question of this article is one that often comes up in PF threads, and I would like to give my take on it. This will be the first of several posts on this subject. Short answer: mu. (The terms of the question are not well-defined, so it doesn’t have a well-defined answer.) […]

## Centrifugal Force Reversal Near A Black Hole

My goal in this article is to derive a simple equation for the proper acceleration of an observer traveling on a circular path around a Schwarzschild black hole at some constant radius $r > 2M$. (Note that this is not the same as being in a stable circular orbit about the hole; we allow for […]

## A Short Proof of Birkhoff’s Theorem

Birkhoff’s theorem is a very useful result in General Relativity, and pretty much any textbook has a proof of it. The one I first read was in Misner, Thorne, & Wheeler (MTW), many years ago, but it was only much later that I realized that MTW’s statement of the proof does something that, strictly speaking, […]