Schwarzschild geometry Definition and 30 Threads

In the classical Newtonian theory of gravity, the shell theorem holds. ( https://en.wikipedia.org/wiki/Shell_theorem ) In the beginning of the derivation of the Schwarzschild solution, the spherically symmetric object is replaced by a point mass. The proof that this can be done in curved...

In A short course in general relativity, Foster and Nightingale write: So in modern astronomy, how is this apparent paradox resolved?

Hi all, I am working through Sean Carroll's Textbook, particularly Chapter 5 regarding the Schwarzschild Solution. In this chapter, Energy and Angular Momentum are defined as follows: $$ \begin{align} E &= (1-\frac{2GM}{r})\frac{dt}{d\lambda} \Rightarrow \frac{dt}{d\lambda} = (1-...

Hi, I was keep reading the interesting book Exploring Black Holes - second edition from Taylor, Wheeler, Bertschinger. I'd like to better understand some points they made. In Box 3 section 3-6 an example of coordinate singularity at point O in Euclidean plane in polar coordinates centered there...

I have been learning a bit about Fermi normal coordinates in Eric Poisson's "A Relativist's Toolkit". Problem 1.10 in this book is to express the Schwarzschild metric in Fermi normal coordinates about a radially infalling, timelike geodesic. I know that in the Fermi normal coordinates (denoted...

The Einstein tensors for the Schwarzschild Geometry equal zero. Why do they not equal something that has to do with the central mass, given that the Einstein equations are of the form: Curvature Measure = Measure of Energy/Matter Density?

Schwarzschild Geometry-proper distance. From what I have studied when the Schwarzschild line element is evaluated at constant time and at a constant radius , proper distance becomes a Euclidean distance on the surface of a sphere. What I don't understand is how to evaluate the integral...

Hi, I was reading this insight schwarzschild-geometry-part-1 about the transformation employed to rescale the Schwarzschild coordinate time ##t## to reflect the proper time ##T## of radially infalling objects (Gullstrand-Painleve coordinate time ##T##). As far as I understand it, the vector...

Computing timelike geodesics in the Schwarzschild geometry is pretty straightforward using conserved quantities. You can treat the problem as a variational problem with an effective Lagrangian of ##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2...

Can anyone help me get started with this problem? What should I use for Gni? I've tried to produce Tni by working out Rni (using methods developed in an earlier chapter) but the results don't lead me anywhere. I'm really stuck for a way forward on this problem so if anyone can help, it...

I'm on to section 5.4 of Carroll's book on Schwarzschild geodesics and he says stuff in it which, I think, enlightens me on the use of Killing vectors. I had to go back to section 3.8 on Symmetries and Killing vectors. I now understand the following: A Killing vector satisfies $$...

please interpret this observation. There is a specific radius through a given equation that always gives the correct mass to any star or planet, as well a density. What is the logical explanation for this? Mass = (4π/3) x schwarzschild radius of the star x 4π/3 x (726696460.5 cm.) cube. For...

Hi there guys, Currently writing and comparing two separate Mathematica scripts which can be found here and also here. The first one I've slightly modified to suit my needs and the second one is meant to reproduce the same results. Both scripts are attempting to simulate the trajectory of a...

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 4 Continue reading the Original PF Insights Post.

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 3 Continue reading the Original PF Insights Post.

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 2 Continue reading the Original PF Insights Post.

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 1 Continue reading the Original PF Insights Post.

Homework Statement Hartle, Gravity, P9.8 A spaceship is moving without power in a circular orbit about a black hole of mass M, with Schwarzschild radius 7M. (a) What is the period as measured by an observer at infinity? (b) What is the period as measured by a clock on the spaceship...

"A" starts a journey from a massive body in Schwarzschild geometry in a radial path and returns back to the starting point while "B" stays at rest. Please explain how to find the proper time of "A".

I have that \left( \frac{dR}{d \tau} \right)^2 = ( 1 - \epsilon)^2 ( \frac{R_{\text{max}}}{R}-1) describes the radius of the surface of a collapsing star in Schwarzschild geometry. I need to show it falls to R=0 in time \tau = \frac{\pi M}{(1-\epsilon)^{3/2}} So far I have rearranged to get...

The Schwarzschild solution can be obtained by using Newton for the weak field. However it turns out that this in fact is the exact solution. Is this coincidence or is there more to it? Opinions? http://arxiv.org/pdf/gr-qc/0309072v3

From In Wheeler and Taylor's 'Exploring Black Holes', on pages 3-12, the equation for energy in Schwarzschild geometry for an object in free fall is- \frac{E}{m}=\left(1-\frac{2M}{r}\right)\frac{dt}{d\tau}=1 where \tau is proper time conventionally expressed in Schwarzschild geometry as...

Hello, There are 3 main coordinate systems for a Schwarzschild geometry : Lemaitre-Rylov (LR), Eddington-Finkelstein (EF), Kruskal-Szekeres (KS). Thanks to my readings, I know thaht KS coordinates are better than EF coordinates and that EF coordinates are better than LR coordinates. But, I...

Homework Statement I'm having problems seeing how the transformation to Eddington-Finkelstein in the Schwarzschild geometry works. Any help would be great! Homework Equations So we have the Schwarzschild Geometry given by: ds^2 = -(1-2M/r)dt^2 + (1-2M/r)^-^1 dr^2 +...

To make a long story short I'm suppose to be learning how to "obtain non-zero curvature components of Schwarzschild geometry". However, I'm not sure what all that entails (tensors? differential geometry?). So any advice on what level of math/physics will be needed would be great!

Homework Statement An observer in a rocket is in a circular equatorial orbit arounda planet and the period of the orbit is the same as the period of revolution of the planet. The planet has mass M = 1033kg and radius R = 1000km. The observer sends a signal every 20 seconds according to its...

The Rindler geometry and its horizon can be obtained by a simple succession of Poincaré transformations to match the frame of an accelerated observer. By combining this SR result and the equivalence principle it follows that a uniform gravitational field is represented by the Rindler metric and...

Today, as I guess, there are good indications that black-holes are a reality. But let us go back in time and pretend we are physicists in 1916 or a few years later. Schwarzschild lectures us about its static and spherical solution to the Einstein's equation. The consequence is striking: any...

Dear all, I would like to know about a possible plane Schwarzschild geometry. This would be the analog of a uniform and infinite gravitational field, or a uniform acceleration. I would like to compare it with the solution of the uniformly accelerated motion in SR. Thanks,

Hey folks, working problems in Hartle's GR book and having trouble with this one. Chapter 9 discusses the simplest physically relavent curved geometry, that of Mr. Swarzschild ds^s = -(1 - \frac{2 G M}{r}) dt^2 + (1 - \frac{2 M}{r})^{-1} dr^2 +r^2(d\theta^2 + sin^2\theta d\phi^2) In this...