What is Schwarzschild solution: Definition and 29 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. D.S.Beyer

    I Discussing Interior Schwarzschild Proper Lengths & Gaussian Curvature

    I'd love have a little discussion about the Interior Schwarzschild Solution. Here's a diagram I slapped together to illustrate the key points. (I assume everyone reading this familiar with embedding diagrams, and using an axis to 'project' a value, in this case the spatial z-axis is replaced by...
  2. E

    B Understanding the Stress-Energy Tensor & Solar Mass in General Relativity

    In the test of General Relativity by perihelion motion of mercury, the stress-energy tensor is set to 0 in Schwarzschild solution. Then, is the curvature caused by solar mass, or by the 0 stress-energy? Or, do we consider solar mass as the gravitating mass? Or the 0 stress-energy the gravitating...
  3. F

    I Is there a zero-mass, vacuum Schwarzschild GR solution?

    The Schwarzschild solution of the Einstein Equation of GR is said to be the only time-independent matter-free solution of that equation. In this usage, does “matter-free solution” mean without matter everywhere except at the singularity of the solution? I thought that the only solutions of the...
  4. T

    I Invariance of timelike Killing vector of Schwarzschild sol.

    I use the ##(-,+,+,+)## signature. In the Schwarzschild solution $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2+r^2d\Omega^2$$ with coordinates $$(t,r,\theta,\phi)$$ the timelike Killing vector $$K^a=\delta^a_0=\partial_0=(1,0,0,0)$$ has a norm squared of...
  5. C

    Coordinate singularity in Schwarzschild solution

    Hi! I have the following problem I don't really know how to approach. Could someone give me a hand? The line element of a black hole is given by: ds^2=\Bigg(1-\frac{2m}{r}\Bigg)d\tau ^2+\Bigg(1-\frac{2m}{r}\Bigg)^{-1} dr^2+r^2\Big(d\theta ^2+\sin^2(\theta)d\phi ^2\Big) It has an apparent...
  6. 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?
  7. Philosophaie

    Schwarzschild Solution for Planetary Motion: Find x'i from xi

    Schwarzschild solution for Planetary Motion: ##g_{ij}= \left( \begin{array}{cccc} \frac{1}{(1-(\frac{2*m}{r}))} & 0 & 0 & 0 \\ 0 & r^2 & 0 & 0 \\ 0 & 0 & r^2*(sin\theta)^2 & 0 \\ 0 & 0 & 0 & c^2*(1-\frac{2*m}{r}) \end{array} \right) ## where ##m=\frac{G*(Mass of Sun)}{c^2}##...
  8. A

    Meaning of Schwarzschild solution in isotropic/anisotropic coordinates

    According to the Schwarzschild solution in the most common anisotropic (Schwarzschild?) coordinates the proper time and the coordinate time are related as...
  9. P

    D'Inverno derivation of Schwarzschild solution

    If you happen to have D'Inverno's Introducing Einstein's Relativity, this is on page 187. He has reduced the metric to non-zero components: g_{00}= e^{h(t)}(1-2m/r) g_{11}=-(1-2m/r)^{-1} g_{22}=-r^2 g_{33}=-r^2\sin^2\theta The final step is a time coordinate transformation that reduces...
  10. P

    Understanding proper distance in Schwarzschild solution.

    I'm trying to understand the Schwarzschild solution concept of proper distance. Given the proper distance equation d\sigma=\frac{dr}{\left(1-\frac{R_{S}}{r}\right)^{1/2}} how would I calculate the coordinate distance. For example - assuming the distance from the Earth to the Sun is...
  11. A

    The Schwarzschild Solution and how it led to the discovery of Black Holes

    I'm currently researching for the term paper of my class on black holes and the topic I selected is the one in the title. I've found some good information so far, but I don't feel like I have enough to get an A without a bit more. I'm curious if anyone has any resources that would aid this...
  12. Demon117

    What is the Lagrangian for geodesics in Schwarzschild coordinates?

    For pure interest I have been trying to solve for the geodesics of the Schwarzschild metric. To do so I know I need to find the explicit Lagrangian for the variational principle for geodesics in this spacetime in Schwarzschild coordinates. How do I derive this lagrangian? I know that the...
  13. D

    Deriving the Schwarzschild solution

    Right so there's this part in my notes where we begin to derive the schwarzchild solution. There's a substitution part I don't understand fully (I think) but I'll start from the beginning...The Schwarzschild Solution. The solution corresponds to the metric corresponding to a static, spherically...
  14. P

    Schwarzschild solution without matter

    Is it theoretically possible that the metric in the whole universe would be described by Kruskal extension of Schwarzschild solution of Einstein equation and in the universe would be no matter at all (vaccum solution everywhere). What's the interpretation of parameter m characterising...
  15. D

    Schwarzschild solution and velocity of stationary observer

    Hey all, I suddenly find myself very confused about velocity and coordinate systems. I have a feeling this is very simple, but sometimes the mind just curls up, you know? ;) When you ask what an observer observe, you need to see things from his point of view - his reference frame. And his...
  16. N

    Does Schwarzschild Solution work inside a massive body?

    The Schwarzschild Solution to Einstein's Field Equations for gravity are said to be exact when outside a spherically symmetric massive body. My question is, can the Schwarzschild Solution also be used inside the massive body, such as a neutron star. In Newtonian gravity we can find the...
  17. H

    Understanding maximally extended Schwarzschild solution

    I'm going to use the notation in this Wiki article, and refer to the diagram therein. I assume the information here is essentially correct, aside from the fact the axes on the diagram should be T and R. Firstly, I wonder what sort of uniqueness properties this space-time is supposed to have...
  18. Jonathan Scott

    Schwarzschild solution radial coordinate and black holes

    When people appear to be getting very confused about the weird nature of black holes, I normally respond with answers based on standard black hole theory, but I sometimes feel I should also call attention to the point that some people now think that the "black hole" solutions to the...
  19. Jonathan Scott

    Apparent distances as seen from within Schwarzschild solution

    I've recently been wondering what the exact relationship is between apparent distances to objects as seen by an observer within a spherical gravitational potential well (described by the Schwarzschild solution) compared with distance as seen by an outside observer. I've decided that the...
  20. S

    Do All Masses Exhibit Schwarzschild Singularities?

    From the Schwarzschild solution.This solution has two singularities,one at r = 0 and one at r = 2M.I have a questions 1.Every Mass has these singularities ? 2.For singularity at r = 0 it is the center of mass ? thank you
  21. Jonathan Scott

    Schwarzschild solution and Machian variable G

    If one rewrites the Schwarzschild solution in terms of a radial coordinate R = r - 2GM where r is the Schwarzschild radial coordinate (as for example is done by Marcel Brillouin in his 1923 paper where he explains why he considers that r = 2GM is effectively the origin), then all of the factors...
  22. K

    Asymmetric Schwarzschild solution

    The Schwarzschild solution to the EFE has the (possibly not physical) 'two sided' view (aka wormhole). Anyone know what would happen in a thought experiment if you added mass to one side of the wormhole? So say mass was M (which is seen from both sides). If you add dM to one side, (say 50%)...
  23. Oxymoron

    Schwarzschild Solution using Cartan's Formalism

    I posted a thread in the Homework section on my attempt to find the Schwarzschild solution using Cartan's method instead of the orthodox Christoffel symbol method. Unfortunately I wasn't getting any help :redface: Then I was asked to move the thread to this section because I may get more...
  24. Oxymoron

    Schwarzschild Solution using Cartan's Equations

    Im having some trouble coming up with my six independent connection 1-forms. I have been given a metric: g = -H_0(r)^2dt\otimes dt + H_1(r)^2 dr\otimes dr + r^2 d\theta\otimes d\theta + r^2\sin^2\theta d\phi \otimes d\phi. I need to find H_0(r) and H_1(r), which are functions of r and...
  25. P

    Circular orbit in schwarzschild solution

    Hello everybody, I've been working on a problem about circular orbits in schwarschild spacetime. Recently a infrared flare has been detected from SgrA*m and the lightcurve during the flare has shown some quasiperiodic oscillations with a period of about 17 minutes. Some astronomers interpreted...
  26. P

    Geodesics in schwarzschild solution

    Hello everybody, I was studying the lecture notes about the schwarzschild solution for general relativity. In a particular example they calculate the equations of motion of a particle falling straight into a black hole. But there are some things about the calculation I really don't get...
  27. R

    Ricci components in Schwarzschild solution

    Attached is a word document where I am setting out the Ricci tensor components to solve for the terms in the line element equation and get to the Schwarzschild solution. I am overlooking something as I have a sign off in the Ricci components in comparison to the many texts and literature I have...
  28. R

    Help Needed for Schwarzschild Solution Model - Romeo

    Consider a model in which we let a body fall (radially) into a star, being the simplest example of the schwarzschild solution, in which the angular parts of the solution may be ignored, so that we consider: ds^2 = -(1- GM/r) dt^2 + (1- GM/r)^{-1}dr^2. I have been told that this may be...
  29. B

    Schwarzschild Solution: Find Rigorous Inside Body

    Where can I found the rigorous Schwarzschild solution INSIDE a body?