What is Schwarzschild metric: Definition and 106 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

    I Eddington-Finkelstein coordinates for Schwarzschild metric

    I'm studying Eddington-Finkelstein coordinates for Schwarzschild metric. Adopting the coordinate set ##(t,r,\theta,\phi)##, the line element assumes the form: $$ ds^2 = \left(1 - \frac{R_S}{r}\right)dt^2 - \left(1 - \frac{R_S}{r}\right)^{-1}dr^2 - r^2 [d\theta^2 + (\sin{\theta})^2d\phi^2], $$...
  2. A

    I Schwarzschild Metric & Particle Absorption

    The Schwarzschild metric implies a potential different from that of Newtonian gravity. Is there a relationship between it and the process by which particles can be absorbed by other particles? (I haven't studied QFT yet)
  3. BiGyElLoWhAt

    I Equations of motion for the Schwarzschild metric (nonlinear PDE)

    I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...
  4. mef

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  5. Sciencemaster

    I Adapting Schwarzschild Metric for Nonzero Λ

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  6. Haorong Wu

    I Geodesics in Schwarzschild metric

    Hello, there. I am learning the chapter, The Schwarzschild Solution, in Spacetime and geometry by Caroll. I could not grasp the idea of circular orbits. It starts from the equations for ##r##, $$\frac 1 2 (\frac {dr}{d \lambda})^2 +V(r) =\mathcal E$$ where $$V(r)=\frac {L^2}{2r^2}-\frac...
  7. Charles_Xu

    I Schwarzschild Metric Singularity: Why?

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  8. Arman777

    Python Packages to Calculate orbits in Schwarzschild Metric

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  9. Arman777

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  10. V

    I Klein-Gordon: Schwarzschild Metric, Physically Acceptable?

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  11. snoopies622

    I Interpreting Schwarzschild Metric: Photon Falls Toward Black Hole

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  12. E

    A Schwarzschild Metric Geodesic Eq: Qs & Answers

    I have no idea if this is an “A” level question, but I will put that down. From the Schwarzschild metric, and with the help of the Maxima program, one of the geodesic equations is: (I will have to attach a pdf for the equations...) I believe this integrates to the following, with ...
  13. P

    I Active Diffeomorphisms of Schwarzschild Metric

    I am trying to understand active diffeomorphism by looking at Schwarzschild metric as an example. Consider the Schwarzschild metric given by the metric $$g(r,t) = (1-\frac{r_s}{r}) dt^2 - \frac{1}{(1-\frac{r_s}{r})} dr^2 - r^2 d\Omega^2 $$ We identify the metric new metric at r with the old...
  14. K

    I Radius in Schwarzschild Metric: Definition Explained

    Hello! I am a bit confused about the definition of the radius in Schwarzschild metric. In the Schutz book on GR (pg. 264, General rules for integrating the equations) he says: "A tiny sphere of radius ##r = \epsilon## has circumference ##2\pi\epsilon##, and proper radius...
  15. K

    I Schwarzschild metric not dependent on time

    Since it's possible to choose a coordinate chart where the Schwarzschild metric components are dependent on time, why that's not done? Would'nt there be a scenario where such a choice would be useful?
  16. 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...
  17. Ibix

    I Maximally extended Schwarzschild spacetime

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  18. E

    Space With Schwarzschild Metric

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  19. saadhusayn

    Finding the Ricci tensor for the Schwarzschild metric

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  20. S

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  21. MathematicalPhysicist

    I Calculating Killing vectors of Schwarzschild metric

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  22. T

    Volume of a sphere in Schwarzschild metric

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  23. laudprim

    I Modified Schwarzschild Metric: Length Contraction Consequences

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  24. P

    Natural units in the Schwarzschild Metric

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  25. Buzz Bloom

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  26. vibhuav

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  27. RUTA

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  28. RUTA

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  29. RUTA

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  30. P

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  31. W

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  32. T

    I What happens at r ≤ rS in the Schwarzschild metric?

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  33. S

    A Interior Schwarzschild Metric: Pressure Dependence

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  34. V

    I Difference between Schwarzschild metric and Gravity well.

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  35. davidge

    I Why Schwarzschild Metric for Deflection of Light & Precession of Perihelia?

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  36. binbagsss

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  37. davidge

    I Deriving Schwarzschild Solution: Easier Strategies?

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  38. binbagsss

    I Asymptotically Flat Schwarzschild Metric

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  39. M

    Schwarzschild metric with angular momentum

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  40. N

    I Deriving Weak-Field Schwarzschild Metric from LEFEs

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  41. M

    A Show Spherical Symmetry of Schwarzschild Metric

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  42. Spinnor

    I Maximize dτ w/ Schwarzschild Metric: Two Masses, m & M

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  43. V

    B Solving Field Equations & Schwarszchild Metric

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  44. Ibix

    I Null geodesics in Schwarzschild spacetime

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  45. Stella.Physics

    I Christoffel symbols of Schwarzschild metric with Lagrangian

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  46. A

    I Ricci tensor for Schwarzschild metric

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  47. A

    I Moving Schwarzschild Black Hole

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  48. M

    B Schwarzschild Metric Derivation?

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  49. A

    B Schwarzschild Metric: Non-Rotating Black Holes & Examples

    Hello I have been reading about Schwarzschild metric and scources what I read said that Schwarzschild metric is used to describe a non-rotating black holes. And what I can not understand is what can you calculate with it? It would be good if you give some examples where you can use it.
  50. e2m2a

    A Einstein & Light Deviation: Compute w/o Schwarzschild Metric

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