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. 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...
  2. D

    I Newtonian limit of Schwarzschild metric

    If I am asked to show that the tt-component of the Einstein equation for the static metric ##ds^2 = (1-2\phi(r)) dt^2 - (1+2\phi(r)) dr^2 - r^2(d\theta^2 + sin^2(\theta) d\phi^2)##, where ##|\phi(r)| \ll1## reduces to the Newton's equation, what exactly am I supposed to prove?
  3. 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...
  4. 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?
  5. Rlam90

    Two-mass Schwarzschild metric instead of Kerr metric?

    Just a thought... Would there be any implicit differences between (A) a two-body metric where the two central masses are drawn ever further together, with angular momentum included, and (B) the Kerr metric? Angular momentum would still be part of the system, but it would be explained by a more...
  6. 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...
  7. MattRob

    Validity of Schwarzschild Metric in Real BHs

    So, I've been reading through "Exploring Black Holes: Introduction to General Relativity" by Wheeler and Taylor, and I've had some ideas I wanted to pursue and do some research in regarding trajectories within the event horizon. In this, I'd like to have the mathematical tools to investigate...
  8. U

    Frequency of Photon in Schwarzschild Metric?

    Homework Statement The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##. (a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it...
  9. Z

    Relative speed of orbiting rockets in Schwarzschild metric

    Homework Statement Two rockets are orbiting a Schwarzschild black hole of mass M, in a circular path at some location R in the equatorial plane θ=π/2. The first (rocket A) is orbiting with an angular velocity Ω=dΦ/dt and the second (rocket B) with -Ω (they orbit in opposite directions). Find...
  10. F

    Angular velocity calculation from Schwarzschild metric

    Hello, I need to find the angular velocity using Schwarzschild metric. At first I wrote the metric in general form and omitted the co-latitude: ds2=T*dt2+R*dr2+Φ*dφ2 and wrote a Lagrangian over t variable: L = √(T+R*(dr/dt)2+Φ*(dφ/dt)2) now I can use the Euler–Lagrange equations for φ...
  11. D

    Elliptical Orbits In The Schwarzschild Metric

    I was just wondering how you would go about calculating the proper time for an observer following a freely falling elliptical orbit in a Schwarzschild metric. I am happy with how to calculate the proper time for a circular orbit and was wondering whether if you had two observers start and end...
  12. binbagsss

    General Relativity: Manifold/Sub-Manifold Metric Theorem Q-Schwarzschild

    I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997. I don't understand eq 7.4 from the theorem 7.2. As I understand, theorem 7.2 is used when you have submanifold that foilate the manifold, and the submanifold must be maximally symmetric. I know that 2-spheres are...
  13. T

    The Schwarzschild Metric: Obtaining Equation M=Gm/c^2 & Newton Law at Infinity

    How do you obtain this equation M=Gm/c^2. What does M stand for? Is is Newton law at infinity? Again what is this Newton law at infinity?
  14. P

    Ricci tensor of schwarzschild metric

    In schwarzschild metric: $$ds^2 = e^{v}dt^2 - e^{u}dr^2 - r^2(d\theta^2 +sin^2\theta d\phi^2)$$ where v and u are functions of r only when we calculate the Ricci tensor $R_{\mu\nu}$ the non vanishing ones will only be $$R_{tt}$$,$$R_{rr}$$, $$R_{\theta\theta}$$,$$R_{\phi\phi}$$ But when u and v...
  15. V

    Killing vecotrs of Schwarzschild metric

    Hello, I have a question, whether is possible to looking for Killing Vectors (KV) in this way (I know about general solution): From Schwarzschild metric I can see two KV \frac{\partial}{\partial t} and \frac{\partial}{\partial\phi} . Then I see that other trivial KV arent there. Metric...
  16. Y

    Schwarzschild metric in terms of refractive index

    This is a spin off from another thread: First there are a couple of mathpages http://mathpages.com/rr/s8-04/8-04.htm and http://mathpages.com/rr/s8-03/8-03.htm that discuss the refractive index model and highlights the differences. The first obvious objection is that the 'medium' must have...
  17. M

    Is Schwarzschild metric more intuitive?

    The textbooks claim that the weak field (Newtonian) metric is more intuitive than the Schwarzschild metric, but I don’t agree.The time correction factor for the weak field metric is the same as that for the Schwarzschild metric. But for the length correction factor for the weak field metric is...
  18. N

    Questions about the Schwarzschild metric

    Hello everybody! I have some questions concerning the structure of the Schwarzschild metric, which is given by $$ ds^2=-(1- \frac{2GM}{r})dt^2+(1-\frac{2GM}{r})^{-1}dr^2+ r^2(d\theta^2+ \sin^2(\theta) d\phi^2) $$ where we set $c=1$. I would like to know the following: \\ \\ 1. Why is it...
  19. H

    Is the Schwarzschild metric dimensionless?

    Homework Statement The problem is I am wanting to know if the expression on the right hand side is dimensionless. Homework Equations ds^2 = (1 - \frac{2GM}{c^2 r})c^2 dt^2 The Attempt at a Solution Since the Schwarzschild radius is r = \frac{2GM}{c^2} would I be right in saying that...
  20. grav-universe

    Problem with Schwarzschild metric derivation

    In this Wiki link for the derivation of the Schwarzschild metric, in the section "simplifying the components", g_22 and g_33 are derived. The problem is that upon deriving them, they first set those local measurements of the components for the metric upon a 2_sphere (on the left side) equal to...
  21. grav-universe

    Schwarzschild metric and spherical symmetry

    In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.
  22. E

    Solve Schwarzschild Metric: Transformation & Acceleration

    Hi! Given the schwarzschild metric ds^2=-e^{2\phi}dt^2+\frac{dr^2}{1-\frac{b}{r}} I can make this coordinate transformation \hat e_t'=e^{-\phi}\hat e_t \\ \hat e_r'=(1-b/r)^{1/2}\hat e_r and I will get a flat metric. Is this correct? Another thing I'm a lot confused about: if I am at...
  23. B

    General Relativity - Schwarzschild Metric

    Homework Statement A spaceship is moving without power in a circular orbit about an object with mass M. The radius of the orbit is R = 7GM/c^2 (1) Find the relation between the rate of change of angular position of the spaceship and the proper time and radius of the orbit. Homework...
  24. N

    Timelike geodesics in Schwarzschild Metric.

    Please explain me how to derive the Timelike geodesics in Schwarzschild Metric. Thank you.
  25. A

    Schwarzschild Metric - Rindler coordinates

    Hello, well I just read a paper by Atish Dabholkar and Ashoke Sen, titled "Quantum Black Holes", pp. 4-5 as shown below and I tried to find d\xi^{2}\frac{2GM}{\xi}=d\rho^{2} like this which is different from the eq. in the paper. So, could somebody please help me to find my...
  26. J

    Deriving the Schwarzschild Metric

    I've worked through a common-sense argumenthttp://www.mathpages.com/rr/s8-09/8-09.htm" showing the time-time component of the Schwarzschild metric g_{tt} = \left (\frac{\partial \tau}{\partial r} \right )^2\approx 1-\frac{2 G M}{r c^2} On the other hand, I've not worked through any...
  27. L

    Schwarzschild Metric - Need help understanding

    Alright, first things first, I'm a grade 12 student residing in Ontario, Canada and I'm relatively new to these forums and to the world of physics. I'm doing my grade 12 ISU and I've taught myself how to work around spherical coordinate systems, however, the schwarzschild metric confuses me...
  28. WannabeNewton

    General Question on Schwarzschild Metric

    Hi guys I have a quick question on the Schwarzschild Metric: Since the metric is a solution to the EFEs does it intrinsically have the curvature of the gravitational field embedded in the metric? If so is it represented by the time and spatial components of the metric? If not could you please...
  29. D

    Timelike geodesic equations for the Schwarzschild metric

    I'm following a slightly confusing set of notes in which I can't tell what exactly the timelike geodesic equations for the Schwarzschild metric are (seems to have about 3 different equations for them). How are these derived, or alternatively, does anyone have a link to a site in which they...
  30. Z

    Schwarzschild metric in 3-dimensional space-time

    I'm trying to find Schwarzschild solution for 3-dimensional space-time (i.e. time\otimes space^2). The problem is, I can't take the 4-dimensional solution \[ds^2=\left(1-\frac{r_g}{r}\right) dt^2-\left(1-\frac{r_g}{r}\right)^{-1} dr^2-r^2\left(d\theta^2+sin^2\theta d\phi^2\right)\] and...
  31. P

    The Schwarzschild Metric - A Simple Case

    The Schwarzschild Metric - A Simple Case of How to Calculate! There is thread open at https://www.physicsforums.com/showthread.php?t=431407 about tidal effects but there may be too many question or the chunk asked is simply to large to handle. At any rate, perhaps it is better to have a very...
  32. P

    Tidal Effects in the Schwarzschild Metric

    Assume we have a non rigid ball. When this ball is radially free falling in a Schwarzschild metric the height increases while the width decreases due to tidal effects. How do we calculate the ruler width and height in terms of R and m? When we have two of those balls radially free falling...
  33. bcrowell

    Schwarzschild metric not stationary inside the horizon?

    The Schwarzschild metric, described in Schwarzschild coordinates, has a Killing vector \partial_t. This vector is timelike outside the horizon, but spacelike inside it. Therefore I would think that a Schwarzschild spacetime should not be considered stationary (which also means it can't be...
  34. P

    Schwarzschild metric as induced metric

    According to Nash theorem http://en.wikipedia.org/wiki/Nash_embedding_theorem" every Riemannian manifold can be isometrically embedded into some Euclidean space. I wonder if it's true also in case of pseudoremanninan manifolds. In particular is it possible to find a submanifold in...
  35. H

    Deriving the Schwarzschild metric just by using the equivalence principle

    I've read a few papers about derivation of the Schwarzschild metric by using the equivalence principle ( http://cdsweb.cern.ch/record/1000100/files/0611104.pdf" )... but I couldn't understand them completely they assume , According to Einstein’s equivalence principle, that the influence of...
  36. D

    Physical meanings of universal coordinates in schwarzschild metric

    The metric due to the gravitational field of a spherical mass is described by the schwarzschild metric ds2 = c2 (1 - R/r) dt2 - (1 - R/r)-1 dr2 - r2 d\Omega 2 Where \Omega is the solid angle, and R is the schwarzschild radius. What are the physical meanings of the coordinates t and r...
  37. BWV

    Exploring CTensor & Schwarzschild Metric in Maxima

    playing around with ctensor & the Schwarzschild metric in Maxima what is the difference between interior and exterior Schwarzschild metrics? also when with the exterior Schwarzschild metric the scalar curvature is zero - this cannot be right, can it?
  38. W

    Christoffel Symbols for Schwarzschild Metric (?)

    ROUGH DRAFT I have a beginner's basic question: 1. Schwarzschild Metric components Let \epsilon = rs / r, where rs is the Schwarzschild Radius. Then, as is is well-known: g_{00} = 1 - \epsilon g_{11} = - \left( 1 - \epsilon \right)^{-1} g_{22} = - r^{2} g_{33} = - r^{2} \; sin^{2}(\theta)...
  39. D

    Schwarzschild Metric: The Sun

    In 1916 Schwarzschild wrote down his famous metric to solve (or re-solve using a polar coordinate system) the precession of the perihelion of Mercury. The curvature of spacetime described by the Metric is for any non-rotating spherically symmetric mass. ds^2 = -(1-\frac{2M}{r})dt^2 +...
  40. D

    A different take on the Schwarzschild Metric

    I have written a report on the Schwarzschild Metric, where I derive a version of it that I have never seen before in the literature. I have no idea whether it is correct or not. I would like to submit it for publication except that I would first like someone much more competent than I to...
  41. R

    Schwarzschild metric in Kruskal coord's

    Homework Statement This question is very simple, but it is driving me mad. Show that the Schwarzschild metric in Kruskal coordinates takes the form ds2 = (32M3/r)e-r/2M(-dv2+du2) +r2(d(theta)2 + sin2(theta)*d(phi)2)Homework Equations The equations are just those defining the Kruskal...
  42. Philosophaie

    Ricci Tensor from Schwarzschild Metric

    Looking for the Schwarzschild Solution for this equation: ds^2 = -A(r) / c^2 * dr^2 - r^2 / c^2 *(d\\theta^2 +(sin(\\theta))^2 *d\\phi^2) + B(r) * dt^2 where A(r) = 1 / (1-2*m/r) And B(r) = (1-2*m/r) From this can be calculated the co- and contra-varient metric tensors and...
  43. L

    Perturbing the Schwarzschild metric

    http://camoo.freeshell.org/30.5quest.pdf" Latex source below, please click on link above, though. I've been working through the exercises in the Penrose book "The Road to Reality". There's one that I'm really puzzled about. He's talking about an "eternal" black hole - never created...
  44. K

    Schwarzschild Metric: Combining Special & General Relativity

    Hi, I am new to this forum so I apologise if a similar thread already exists. I am trying to resolve the implications on space and time as you approach a black hole event horizon with respect a distant observer and an onboard observer. My issue relates to combining the effects of both...
  45. Orion1

    Is the Schwarzschild Metric Solution Correct?

    The solution for the Schwarzschild metric is stated from reference 1 as: ds^2=- \left(1-\frac{r_s}{r}\right) c^2 dt^2 + \left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2(d \theta^2 +\sin^2 \theta d \phi^2) The solution for the Schwarzschild metric is stated from references 2 as: ds^2 = \left(1 -...
  46. P

    What is the Inner Product in the Schwarzschild Metric?

    I need to write Schwarzschild Metric, that is in spherical coordinates, into the form that has the metric tensor. Now, if the first the term of the metric is: \Large (ds)^2=f(r)c^2dt^2-... and x0=ct, then the first component gij of the metric tensor g is supposed to be: \Large...
  47. T

    Precession of the perihelion - Schwarzschild metric

    Hi everyone! I was trying to solve this question following the Hartle's book (Gravity: an introduction to Einstein’s general relativity) , exercise 9.15, but I don't know how to do the expansion of (1-2GM/c^2r) in powers of 1/c^2... I know this sounds easy, but I couldn't get the expression...
  48. L

    Schwarzschild metric, cosmological constant

    From an https://www.physicsforums.com/showthread.php?t=140501", a new question comes to me. Is there a known generalisation of the Schwarzschild geometry when the cosmological constant is positive? Are there still black-holes in this case? Are there small modifications to the Newtonian...
  49. T

    Searching for Ricci Scalar in Schwarzschild Metric

    I searched the net for the Ricci scalar for the Schwarzschild metric but in vain. Can anyone tell me what's the Ricci scalar? Are there any standard list or tables that records down the properties of any metric for GR?
  50. D

    Schwarzschild Metric with multiple masses

    I know that it's possible to calculate the rate at which time flows when in the gravitational field of a single spherical mass. But how do you calculate the rate when there are two masses or more? How do they add together?
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