I’ll start with the simple fact: ChatGPT is not a reliable answerer to questions. To try to explain why from scratch would be a heavy lift, but fortunately, Stephen Wolfram has already done the heavy lifting for us in his article, “What is ChatGPT Doing… and Why Does It Work?” [1] In a PF thread…
In this article, we will look at why there are maximum mass limits for objects that are supported against gravity by degeneracy pressure instead of kinetic pressure. We will look at the two known cases of this, white dwarfs and neutron stars; but it should be noted that similar arguments will apply to any postulated…
Part 1: Overview Part 2: Mathematical Details Part 3: Implications In the last article in this series, we finished up with a metric for the Oppenheimer-Snyder collapse: $$ ds^2 = – d\tau^2 + A^2 \left( \eta \right) \left( \frac{dR^2}{1 – 2M \frac{R_-^2}{R_b^2} \frac{1}{R_+}} + R^2 d\Omega^2 \right) $$ Now we will look at some of…
Part 1: Overview Part 2: Mathematical Details Part 3: Implications In a previous article, I described in general terms the model of gravitational collapse of a spherically symmetric massive object, first published by Oppenheimer and Snyder in their classic 1939 paper. In this follow-up article, I will give further mathematical details about the model, using…
This article is intended for anyone who wants to start a thread here at Physics Forums on the twin paradox. There are already many, many threads here on this topic, and they tend to cover the same ground over and over again, so it seems useful to put a description of that ground into an…
Part 1: Overview Part 2: Mathematical Details Part 3: Implications Most people who have spent any time at all studying GR are familiar with the Schwarzschild solution. (A series of Insights articles discusses the key properties of that solution.) Much of that familiarity probably derives from the fact that the Schwarzschild solution describes a black…
I’m sure anyone who has hung out long enough here on Physics Forums has encountered threads that go something like this (I’ll use an example based on threads I’ve seen and participated in in the relativity forum, but I’m sure similar things occur in other forums as well): Original Poster: I don’t understand how black…
In this article, we will analyze “centrifugal force reversal” in Kerr spacetime, similar to what was done for Schwarzschild spacetime in a previous Insights article. As our starting point, we will use the formulas for the frame field, proper acceleration, and vorticity of an observer in a circular orbit in the equatorial plane of Kerr…
In a previous series of articles, I posed the question “Does Gravity Gravitate?” and explained how, depending on how you interpreted the terms “gravity” and “gravitate”, one could answer the question, either way, yes or no. This article will similarly treat its title question. :-) To be sure, this case is a bit simpler than…
In a previous article, I discussed why scientists (among others) are rarely interested in considering a new theory proposed by a non-scientist: because it’s so unlikely that a person who is not familiar with our best current knowledge in a given area of science would be able to come up with a useful new theory…
In the first two articles in this series, we looked at the Einstein Field Equation and Maxwell’s Equations in a static, spherically symmetric spacetime. Using formulas from those two previous articles, I now want to consider the question: what is the maximum amount of work that can be extracted by slowly lowering an object into…
In the first article in this series, we looked at the Einstein Field Equations in a static, spherically symmetric spacetime. In this article, we are going to build on what we saw in the first article to show what Maxwell’s Equations in a static, spherically symmetric spacetime look like. The electromagnetic field tensor is given…
This will be the first of several articles which will provide, for reference, useful equations for static, spherically symmetric spacetimes. This is a common special case that is studied in General Relativity, and it has the advantage of having a general solution for the Einstein Field equation that can be expressed in closed form equations….
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,…
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…
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…
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 quantum mechanics can easily turn into long arguments. Sometimes this even reaches the point where proponents…
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…
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…
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…
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…
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 so little interest…
Introduction The purpose of this article is to discuss the title question from several different viewpoints, 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 prevent…
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….
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…
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…
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.)…
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 [itex]r > 2M[/itex]. (Note that this is not the same as being in a stable circular orbit about the hole; we allow for…
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,…