schwarzschild metric Definition and Topics - 17 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. M

    A Orbit velocity in Schwarzschild metric?

    Hi, I'm trying to deduce orbit velocity of a particle with mass from Schwarzschild metric. I know for Newtonian gravity it is: $$v^2=GM\left(\frac{2}{r}-\frac{1}{a}\right)$$ The so called vis-viva equation. Where ##a## is the length of the semi-major axis of the orbit. For Schwarzschild metric...
  2. Ibix

    I Maximally extended Schwarzschild spacetime

    I have a very quick question about the maximally extended Schwarzschild spacetime. I know you can't reach regions III and IV from I and II, and vice versa. But can you see in? If I'm in region I and I look down, the null paths reaching me originated in the white hole singularity. Likewise in...
  3. S

    I Does gravitational field strength cause time dilation?

    Hi I have 2 questions. There are 2 planets and one clock on each of them. One of them has a bigger gravitational field strength. And two clock have same distance from the core. 1-) Does time dilation occur between two? Which clock ticks slower? 2-) If time dilation occurs, which formula...
  4. T

    Volume of a sphere in Schwarzschild metric

    Homework Statement Calculate the volume of a sphere of radius ##r## in the Schwarzschild metric. Homework Equations I know that \begin{align} dV&=\sqrt{g_\text{11}g_\text{22}g_\text{33}}dx^1dx^2dx^3 \nonumber \\ &= \sqrt{(1-r_s/r)^{-1}(r^2)(r^2\sin^2\theta)} \nonumber \end{align} in the...
  5. RUTA

    Insights The Schwarzschild Metric: Part 3, A Newtonian Comparison - Comments

    Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 3, A Newtonian Comparison Continue reading the Original PF Insights Post.
  6. RUTA

    Insights The Schwarzschild Metric: Part 2, The Photon Sphere - Comments

    Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 2, The Photon Sphere Continue reading the Original PF Insights Post.
  7. RUTA

    Insights The Schwarzschild Metric: Part 1, GPS Satellites - Comments

    Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 1, GPS Satellites Continue reading the Original PF Insights Post.
  8. S

    A Pressure dependence in interior Schwarzschild metric

    I'm looking influence of pressure on the general interior Schwarzschild metric (see for example the book by Weinberg, eq. 11.1.11 and 11.1.16. The radial component of the metric (usually called A(r)) depends only on the mass included up to radius r A(r) = \left(1-\frac{ 2G M(r)}{r}\right)^{-1}...
  9. N

    I How to derive weak-field Schwarzschild metric from LEFE's?

    I am trying to derive weak-field Schwarzschild metric using Linearized Einstein's field equations of gravity: []hμν – 1/2 ημν []hγγ = -16πG/ c4 Tμν For static, spherically symmetrical case, the Energy- momentum tensor: Tμν = diag { ρc2 , 0, 0, 0 } Corresponding metric perturbations for...
  10. Ibix

    I Null geodesics in Schwarzschild spacetime

    I was looking at null geodesics in Schwarzschild spacetime. Carroll's lecture notes cover them here: He notes (and justifies) that orbits lie in a plane and chooses coordinates so it's the equatorial plane, then uses Killing...
  11. A

    I Ricci tensor for Schwarzschild metric

    Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric: So we have Ricci flow equation,∂tgμν=-2Rμν. And we have metric tensor for schwarzschild metric: Diag((1-rs/r),(1-rs]/r)-1,(r2),(sin2Θ) and ∂tgμν=0 so 0=-2Rμν and we get that Rμν=0.But Rμν should not equal to...
  12. A

    I Moving Schwarzschild Black Hole

    The Schwarzschild Metric (with ##c=1##), $$ds^2 = -\Big(1-\frac{2GM}{r}\Big)dt^2+\Big(1-\frac{2GM}{r}\Big)^{-1}dr^2+r^2d\Omega^2$$ can be adjusted to a form involving three rectangular coordinates ##x##, ##y##, and ##z##: $$ds^2 =...
  13. M

    B Schwarzschild Metric Derivation?

    Hi, I was wondering if anybody could help me understand the derivation of the Schwarzschild metric developed by the author of mathpages website. Rather than reproduce all the equations via latex, I have attached a 2-page pdf summary that also points to the mathpages article and explains my...
  14. Yukterez

    I Schwarzschild equation of motion: initial conditions

    The Schwarzschild equation of motion, where coordinate length is differentiated by proper time is as far as I know r''(t) = -\frac{G\cdot M}{r(t)^2} + r(t)\cdot{\theta}'(t)^2 - \frac{3\cdot G\cdot M\cdot{\theta}'(t)^2}{c^2} {\theta}''(t) = -2\cdot r'(t)/r(t)\cdot{\theta}'(t) Now the question...
  15. L

    Momentum of a photon heading towards a spherical mass

    Homework Statement A distant observer is at rest relative to a spherical mass and at a distance where the effects of gravity are negligible. The distant observer sends a photon radially towards the mass. At the distant observer, the photon's frequency is f. What is the momentum relative to...
  16. redtree

    A Assumptions of the Schwarzschild solution to the EFE

    Given that no assumption is of a point energy is necessary to derive the vacuum (Schwarzschild) solution to the EFE, why is the solution assumed to apply to spacetime surrounding a point energy?
  17. sevenperforce

    Stellar-mass black hole formation sequence

    I feel like this could go in quite a few of the Physics subforums (Quantum Physics, Beyond the Standard Model, Special and General Relativity, or High Energy, Nuclear, Particle Physics) instead of Astronomy and Cosmology, but hopefully this will work. This is my first question I've posed here...