# What is Schwarzschild geometry: Definition and 26 Discussions

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916, and around the same time independently by Johannes Droste, who published his much more complete and modern-looking discussion only four months after Schwarzschild.
According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.
The Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. The boundary is not a physical surface, and a person who fell through the event horizon (before being torn apart by tidal forces), would not notice any physical surface at that position; it is a mathematical surface which is significant in determining the black hole's properties. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass M, so in principle (according to general relativity theory) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation.

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1. ### A Schwarzschild metric in Fermi normal coordinates about a radially infalling timelike geodesic

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...
2. ### I Schwarzschild Geometry: Einstein Tensor & Mass Density

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?
3. ### A Schwarzschild Geometry: Evaluating Proper Distance

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...
4. ### I Coord. Time Vector Field: Schwarzschild vs Gullstrand-Painleve

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...
5. ### A Computing Null Geodesics in Schwarzschild Geometry

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...
6. ### A poor man's way to Schwarzschild Geometry

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...
7. ### I Understanding Killing Vectors & Schwarzschild Geodesics

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 ...
8. ### B Question on Schwarzschild Geometry

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...
9. ### A EoM in Schwarzschild geometry: geodesic v Hamilton formalism

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...
10. ### Insights The Schwarzschild Geometry: Part 4 - Comments

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 4 Continue reading the Original PF Insights Post.
11. ### Insights The Schwarzschild Geometry: Part 3 - Comments

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 3 Continue reading the Original PF Insights Post.
12. ### Insights The Schwarzschild Geometry: Part 2 - Comments

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 2 Continue reading the Original PF Insights Post.
13. ### Insights The Schwarzschild Geometry: Part 1 - Comments

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 1 Continue reading the Original PF Insights Post.
14. ### Circular orbits in Schwarzschild geometry

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...
15. ### Calculating Proper Time of "A" in Schwarzschild Geometry

"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".
16. ### Solving the Radius of a Collapsing Star in Schwarzschild Geometry

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...
17. ### Heuristic approach to the Schwarzschild geometry - Accident of deeper meaning?

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
18. ### Energy in Schwarzschild geometry

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...
19. ### Understanding the 3 coordinate systems for a Schwarzschild geometry

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...
20. ### Solving the Transformation to Eddington-Finkelstein in Schwarzschild Geometry

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 +...
21. ### Understanding Schwarzschild Geometry: Math & Physics Requirements

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!
22. ### GR problem - satelite in orbit (Schwarzschild geometry)

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...
23. ### How much is the Schwarzschild geometry a characteristic feature of GR

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...
24. ### Was the Schwarzschild geometry acceptable

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...
25. ### Plane Schwarzschild geometry

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,
26. ### Schwarzschild geometry unstable orbit GR question

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...