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

    I How to Measure Speed of Light & Is It Constant?

    The question constantly arises how the speed of light is measured and what does it mean that the speed is constant, including at remote points for the observer, including at points beyond the local frame of reference, as you understand it in general relativity (GR). First of all, it should be...
  5. Sciencemaster

    I Adapting Schwarzschild Metric for Nonzero Λ

    So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A...
  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?

    Why does the Schwarzschild metric have a singularity at r=0 if it is only valid outside the spherically symmetric static mass?
  8. Arman777

    Python Packages to Calculate orbits in Schwarzschild Metric

    I am looking for a Python Code/Package to calculate the orbits of the time-like and null-like particles in Schwarzschild metric (in spherical coordinates) Does anyone know such package ? Note: I am mostly looking for packages to calculate the RIGHT side of the given images (i.e the orbits...
  9. Arman777

    Wave equation for Schwarzschild metric

    I am trying to find the $$\nabla_{\mu}\nabla^{\mu} \Phi$$ for $$ds^2 = (1 - \frac{2M}{r})dt^2 + (1 - \frac{2M}{r})^{-1}dr^2 + r^2d\Omega^2$$ I have did some calculations by using $$\nabla_{\mu}\nabla^{\mu}\Phi = \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu}\partial_{\nu}\Phi)$$...
  10. snoopies622

    I Interpreting Schwarzschild Metric: Photon Falls Toward Black Hole

    As a photon falls radially toward the surface of a Schwarzschild black hole, dr/dt approaches zero. Does this mean that, from the viewpoint of a distant (Schwarzschild) observer, the photon slows down or that the distance covered by successive dr's is getting larger?
  11. 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 ...
  12. 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...
  13. 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...
  14. 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?
  15. 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...
  16. 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...
  17. E

    Space With Schwarzschild Metric

    This is a problem from Tensor Calculus:Barry Spain on # 69 Prove that a space with Schwarzschild's metric is an Einstein space, but not a space of constant curvature. The metric as given in the book is $$d\sigma^2=-\bigg(1-\frac{2m}{c^2r}\bigg)^{-1}dr^2-r^2d\theta^2-r^2\sin^2 \theta...
  18. saadhusayn

    Finding the Ricci tensor for the Schwarzschild metric

    I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor. The given distance element is $$ ds^2 = e^{2 \lambda} dt^2 -...
  19. S

    I Time Dilation: Does Gravitational Field Strength Matter?

    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...
  20. MathematicalPhysicist

    I Calculating Killing vectors of Schwarzschild metric

    I am trying to understand the solution to exercise 7.10(e) on pages 175-176 of Robert Scott's student's manual to Schutz's textbook. He writes the following: I don't understand how to find ##S^x, S^z## or ##T^x,T^z## from the metric or from the cartesian representation of the rotation...
  21. 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...
  22. laudprim

    I Modified Schwarzschild Metric: Length Contraction Consequences

    Hello. I am looking for help in establishing all the consequences of a modified Scwazschild metric where the length contraction is removed. ds^2=(1-rs/r)c^2dt^2-dr^2-r^2(... ) Thanks
  23. P

    Natural units in the Schwarzschild Metric

    Homework Statement My Teacher says that in the Schwarzschild metric he uses natural units, where he writes ##g_{rr}=1-2M/R## He says that for one neutron star ##R=5## corresponds to approx 13 KM. Homework Equations ##1l_p=1,616 \cdot 10^{-35}m## The Attempt at a Solution Unfortunately he does...
  24. Buzz Bloom

    I Some geometry questions re Schwarzschild metric

    I would like to ask what I hope are two simple questions about what I recognize to be a complicated subject. I did make an effort to search the Internet for the answers, but the two most promising looking sources I found did not help...
  25. vibhuav

    I Curvature and Schwarzschild metric

    Many textbooks use the space (spacetime, actually, but for now only space is good enough) around a spherically symmetrical Schwarzschild object to demonstrate curvature of space due to gravity. Let’s consider two shells around such a Schwarzschild object (say a neutron star of 1 solar mass)...
  26. 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.
  27. 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.
  28. 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.
  29. P

    I Non-zero components of Riemann curvature tensor with Schwarzschild metric

    I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does...
  30. W

    Proper distance in Schwarzschild metric

    Homework Statement Let the line element be defined as ##ds^2 = -(1-\frac{2m}{r})dt^2+\frac{dr^2}{1-\frac{2m}{r}}+r^2 d\theta^2 + r^2 \sin^2{\theta} d\phi^2## a) Find a formula for proper distance between nearby spherical shells, assuming only the radius changes, and ## r > 2m ## b) Now look...
  31. T

    I Schwarzschild metric radius

    If the Schwarzschild metric is, by construction, valid for ##r > r_S##, where ##r_S## is the Schwarzschild radius, so it does not make sense to talk about what happens at ##r \leq r_S##, because there will be no vacuum anymore. What am I getting wrong?
  32. S

    A Interior Schwarzschild Metric: Pressure Dependence

    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}...
  33. V

    I Difference between Schwarzschild metric and Gravity well.

    I would like to know the difference between this two concepts, specially the difference between the geometry deformations of space-time that they descript. As far as I know the Schawrzschild metric can be represent by Flamm’s paraboloid, but this shape is not the same that the deformation of...
  34. davidge

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

    Why one uses Schwarzschild metric instead of FLRW metric when deriving things such - deflection of light by the sun - precession of perihelia of planets Also, as our solar system is not isotropic nor static, it seems that by using the Schwarzschild metric we would get only an approximation on...
  35. binbagsss

    Conserving Quantity in Schwarzschild Metric

    Homework Statement Conserved quantity Schwarzschild metric. Homework EquationsThe Attempt at a Solution [/B] ##\partial_u=\delta^u_i=k^u## is the KVF ##i=1,2,3## We have that along a geodesic ##K=k^uV_u## is constant , where ##V^u ## is the tangent vector to some affinely parameterised...
  36. davidge

    I Deriving Schwarzschild Solution: Easier Strategies?

    Is there a less boring way of deriving the Schwarzschild solution? The derivation itself is easy to going with; what I don't like is computing all the Christoffel symbols and Ricci tensor components --there are so many possible combinations of indices. I know that by using some constraint...
  37. binbagsss

    I Asymptotically Flat Schwarzschild Metric

    This is probably a stupid question but so as ##r \to \infty ## it is clear that ##-(1-GM/r)dt^2+(1-GM/r)^{-1}dr^2 \to -dt^2 +dr^2 ## However how do you consider ## \lim r \to \infty (r^2d\Omega^2 )##..? Schwarschild metric: ##-(1-GM/r)dt^2+(1-GM/r)^{-1}dr^2+r^2 d\Omega^2## flat metric ...
  38. M

    Schwarzschild metric with angular momentum

    Homework Statement Given the Schwarzschild metric generalisation for a mass M rotating with angular momentum J ##ds^2 = -(1-\frac{2 M}{r}) \; dt^2 +(1-\frac{2 M}{r})^{-1} \;(dr^2 +r^2 \;d\theta ^2 +r^2 \sin ^2 \theta \; d\phi ^2) -\frac{4J}{r} \sin ^2 \theta \; dt d\phi ## a) Write the...
  39. N

    I Deriving Weak-Field Schwarzschild Metric from LEFEs

    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...
  40. M

    A Show Spherical Symmetry of Schwarzschild Metric

    In one of the lectures I was watching it was stated without proof that the Schwarzschild metric is spherically symmetric. I thought it would be a good exercise in getting acquainted with the machinery of GR to show this for at least one of the vector fields in the algebra. The Schwarzschild...
  41. Spinnor

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

    Two masses, m and M, are a fixed distance R apart. One of the masses is much larger then the other. At time t the masses start to fall towards each other. Using Newton's Law of Gravitation we can determine the acceleration of the small mass. Can one use the Schwarzschild metric in the...
  42. V

    B Solving Field Equations & Schwarszchild Metric

    I have read that Albert Einstein was quite (pleasantly) surprised to read Schwarzschild's solution to his field equation because he did not think that any complete analytic solution existed. However, of all the possible scenarios to consider, a point mass in a spherically symmetric field (ie, a...
  43. Ibix

    I Null geodesics in Schwarzschild spacetime

    I was looking at null geodesics in Schwarzschild spacetime. Carroll's lecture notes cover them here: https://preposterousuniverse.com/wp-content/uploads/grnotes-seven.pdf He notes (and justifies) that orbits lie in a plane and chooses coordinates so it's the equatorial plane, then uses Killing...
  44. Stella.Physics

    I Christoffel symbols of Schwarzschild metric with Lagrangian

    So the Schwarzschild metric is given by ds2= -(1-2M/r)dt2 + (1-2M/r)-1dr2+r2dθ2+r2sin2θ dφ2 and the Lagragian is ##{\frac{d}{dσ}}[{\frac{1}{L}}{\frac{dx^α}{dσ}}] + {\frac{∂L}{∂x^α}}=0## with L = dτ/dσ. So for each α=0,1,2,3 we have ##{\frac{d^2 x^1}{dτ^2}}=0## for Minkowski spacetime also...
  45. 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...
  46. 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 =...
  47. 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...
  48. 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.
  49. e2m2a

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

    How did Einstein compute the amount of light deviation due to the Earth's gravitational field when the Schwarzschild metric was not known yet?