Well I don't, but should surprise nobody here So let me take you at your word; please explain to me what this Schwarzschild metric is. And no, this is not an empty challenge from one who doesn't believe you know what you are talking about - I do. Plus, although I have big problems with the motivation that models any real-world phenomenon on a piece of already well-established mathematics, this is probably philosophical. Speaking as one outside the "magic circle", I often find physics gives mathematical answers when I haven't a clue what the question is! Anyway, you may take it as given that I have a passable knowledge of smooth 4-manifolds with a semi-Riemann metric. In other words, I have no need of an extensive tutorial on the General Theory insofar as it uses this sort of mathematics PS If you do decide to enlighten me, you may as moderator, want to split this into a separate thread. But since Riemann geometry is most certainly non-linear, maybe it would be slightly relevant? Your choice
Why-oh-why would you need such an explanation? You have knowledge of 4 dimensional pseudo-Riemannian geometry, so you understand what the Ricci tensor is, i.e. you understand the statement \(R_{ab}=0\) for such a manifold. This is just a set of equations (as you know) depending on the metric tensor for a given manifold. The Schwarzschild metric corresponds to a solution to these equations obeying certain symmetry conditions. Typing "Schwarzschild" into Google would have told you this immediately.
The Schwarzschild metric is a solution to Einstein's general relativistic field equations for the case of a non-rotating spherical mass (such as a planet). It basically tells you how spacetime is curved near a spherical mass. No doubt there's at least a wikipedia article on it if you want the specific mathematics. The Schwarzschild metric was unknown until after Einstein published his general theory of relativity. How is it that you have studied manifolds and semi-Riemann metrics and yet have apparently not come across the Schwarzschild metric? Have you done an abstract maths course without physics, or what?
So the answer to my question (which was perfectly genuine) seems, by consensus, to be "Google it". This of course has the virtue of keeping the discussion brief, but I confess not quite what I was expecting. Nonetheless, I thank you both for your trouble
Genuinely interested people would have done so. Your post, in which the query was raised, read like "I know all about linear 2nd order ODEs, but I want someone to explain to me what a Bessel function is". I realise you're keen to keep up the "I'm just a mathematician, so I couldn't possibly be familiar with things that crop up in physics..." vibe, but when you display such wilful ignorance, the act gets a little old.
Is it a profound relief when he thinks "God, I'm glad I'm not as thick and deluded as Vkothii, having to lie to people online about subjects he knows nothing about as a way of validating himself to himself, what a sad existence" when he reads your posts? Or maybe I'm just not getting your point....
It must be a profound relief for you, having a place to come to to validate your own self-worth, by thinking you have some profound grasp of mentality. Or reality. It's nice to pretend, isn't it?
Ah, but at the end of the day my sum abilities in physics and contributions to it exist outside of the internet. If all my posts, on every forum I'd ever been a member of, were deleted it'd have zero impact on the physics and maths I've done, I am more than the sum of my forum posts. Unlike you or people like Reiku, I dial down, not up, my physics and maths when I come here. I get paid to do maths. Coming here and ripping on you is entertainment. Please Register or Log in to view the hidden image!