# What is Schwarzchild: Definition and 42 Discussions

Schwarzschild ([ˈʃvaʁtsʃɪlt]) is a German surname meaning "black sign" or "black shield".
Those bearing the name include:

Karl Schwarzschild (1873–1916), physicist and astronomer
Steven Schwarzschild (1924–1989), philosopher and rabbi
Henry Schwarzschild (1926–1996), civil rights activist
Martin Schwarzschild (1912–1997), astronomer
Shimon Schwarzschild (1925–), environmental activist
Luise Hercus (née Schwarzschild) (1926–), linguist

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1. ### Schwarzschild orbiting problem

So the line element is given by $$ds^2 = (1- \frac{R_s}{r})dt^2 - (1- \frac{R_s}{r})^{-1}dr^2 - r^2d\Omega ^2$$ The object is orbiting at constant radius ##r## in the plane ## \theta = \frac{\pi}{2}##. I am supposed to find the values of ##a## and ##b## in the 4-velocity given by: U =...
2. ### I Is the Schwarzchild Radius Relativistic or Newtonian?

I noticed that the Schwarzschild Radius Formula and the Escape Velocity Formula are actually identical. The Schwarzschild Radius is supposed to be one of the great equations generated from Relativity, while the Escape Velocity is something that was generated just using Newtonian gravity. All you...
3. ### Find the Riemannian curvature tensor component

Given the metric of the gravitational field of a central gravitational body: ds2 = -ev(r)dt2 + eμ(r)dr2 + r2 (dθ2 + sin2θdΦ2) And the Chritofell connection components: Find the Riemannian curvature tensor component R0110 (which is non-zero). I believe the answer uses the Ricci tensor...
4. ### A Four velocity with the Schwarzchild metric

I am trying to solve the following problem but have gotten stuck. Consider a massive particle moving in the radial direction above the Earth, not necessarily on a geodesic, with instantaneous velocity v = dr/dt Both θ and φ can be taken as constant. Calculate the components of the...
5. ### I Exterior Schwarzschild Solution & Gravitational Time Dilation

I've been looking in depth into the exterior Schwarzschild solution of the Einstein field equations. I decided to play with the line element by creating some example scenarios for myself and calculating the space-time interval between two events in these scenarios. Here is the line element: ds2...
6. ### Kepler's Law in Schwarzchild metric

Homework Statement Show Kepler's Third Law holds for circular Schwarzschild orbits. Homework Equations The Attempt at a Solution Setting r' = 0 , \theta' = 0 and \theta = \pi / 2 , where the derivatives are with respect to the variable \lambda and setting c = 1 the Lagrangian is...
7. ### Impact parameter of a photon in Schwarzchild metric

Hi, I'm having trouble answering Question 9.20 in Hobson's book (Link: http://tinyurl.com/pjsymtd). This asks to prove that a photon will just graze the surface of a massive sphere if the impact parameter is b = r(\frac{r}{r-2\mu})^\frac{1}{2} So far I have used the geodeisic equations...
8. ### 4 velocity in Schwarzchild metric

How do we calculate the 4 velocity of a particle that is projected radially downwards at velocity u at a radius ra? The condition on 4 velocity is that gμνvμvν = 1 which implies that at radius ra we have ga00(v0)2 + ga11(v1)2 = 1 (eq 1) So if we start from xμ = (t,r) we get vμ = (1/√g00 ...
9. ### B Smallest Schwarzschild Radius: Theory & Facts

Is there theoretically a smallest possible Schwarzschild radius?
10. ### Meaning of r in Schwarzchild coordinates

I'm trying to understand *quote unquote thread title* by performing some simple (heuristic) analysis on my own. Before beginning, I'd like to present what I've been given to understand here at PF: -r is not the distance from the center of a spherical shell to an arbitrary spatial coordinate on...
11. ### Minimum Orbital Radius Around Black Holes

Hi there, I was reading one of my textbooks and I had a thought. For a black hole, there is minimum orbiting radius of ##R_{min}=3R_s## where ##R_s## is the Schwarzschild Radius. This minimum orbit is created by the fact that in order to obtain an orbit of that radius around a black hole, you...
12. ### Quick question on Geodesic Equation

Starting with the geodesic equation with non-relativistic approximation: \frac{d^2 x^{\mu}}{d \tau^2} + \Gamma_{00}^{\mu} \left( \frac{dx^0}{d\tau} \right)^2 = 0 I know that ## \Gamma_{\alpha \beta}^{\mu} = \frac{\partial x^{\mu}}{\partial y^{\lambda}} \frac{\partial^2 y^{\lambda}}{\partial...
13. ### General Relativity - Deflection of light

Homework Statement Find the deflection of light given this metric, along null geodesics. Homework EquationsThe Attempt at a Solution [/B] Conserved quantities are: e \equiv -\zeta \cdot u = \left( 1 - \frac{2GM}{c^2r} \right) c \frac{dt}{d\lambda} l \equiv \eta \cdot u = r^2 \left( 1 -...
14. ### General Relativity - Circular Orbit around Earth

Homework Statement (a) Find the proper time in the rest frame of particle (b) Find the proper time in the laboratory frame (c) Find the proper time in a photon that travels from A to B in time P Homework EquationsThe Attempt at a Solution Part(a) [/B] The metric is given by: ds^2 =...
15. ### Schwarzchild metric spherically symmetric space or s-t?

This is probably a stupid question, but, is the Schwarzschild metric spherically symmetric just with respect to space or space-time? Looking at the derivation, my thoughts are that it is just wrt space because the derivation is use of 3 space-like Killing vectors , these describe 2-spheres, and...
16. ### Photon "escaping" from photon sphere in Schwarzchild space

Homework Statement Close to a Schwarzschild black hole, a photon is emitted between r = 2(mu) and 3(mu), where \mu = \frac{GM}{c^2} . The photon is emitted at an angle (alpha) to the radial direction. At r = 2(mu), the highest angle that the photon can escape at is (alpha) = 0; at r = 3(mu)...
17. ### Deriving the Schwarzchild radius?

I'm a bit confused about the derivation of the Schwarzschild radius. I can do it quite easily using Newton's Law of gravitation, but this law is only an approximation, so I am wondering whether the result I obtain, r_{s}=\frac{2GM}{c^{2}}, is an approximation or not. It seems to me that it...
18. ### Most proper time in Schwarzchild metric

Hi In the Schwarzschild metric, the proper time is given by c^{2}dτ^{2} = (1- \frac{2\Phi}{c^2})c^2 dt^2 - r^2 dθ^2 with where \Phi is the gravitational potential. I have left out the d\phi and dr terms. If there is a particle moving in a circle of radius R at constant angular velocity ω...
19. ### Schwarzchild Radius: By definition must be dependent on distance.

The Schwarzschild Radius of an object is the length such that if the object is shrunk down that small, the escape velocity becomes equal to the speed of light. That being said, however, the escape velocity of any gravitational body matters where it is measured relative to the center of mass...

The Schwarzschild spacetime can be foliated by 2-sphere, which are spacelike hypersurfaces of constant t and r (Schwarzschild coordinates) with a normal vector ##\partial_t## (outside the horizon). Because a 2-sphere has no center, the coordinate r is not the radius of the sphere and we consider...
21. ### Schwarzchild radius and escape velocity?

We know that the escape velocity at the schwarzchild radius is c. Since the escape velocity is defined as the velocity needed to escape from the gravitational field, to reach a total energy of 0 at infinity: Doesn't this mean that an object falling from infinity starting at rest, to the...
22. ### Schwarzchild Radii for Various Particles and Planck's Mass

Electron 1.35286150173888E-57 Alpha Particle 9.86817701940734E-54 Deuteron 4.96565509720363E-54 Helion 7.43517160660937E-57 Muon 2.79728851560698E-55 Neutron 2.48748433759427E-54 Proton 2.48406026119433E-54 Tau 4.70410373924314E-54 Triton 7.43657350963157E-54 Planck's Mass...
23. ### Mass of Schwarzchild geometry

Homework Statement Find the Mass of the Schwarzschild geometry by calculating, \frac{1}{4\pi}\int_{S}n^{\alpha}\sigma_{\beta}\nabla_{\alpha}\xi^{\beta}dA in a Schwarzschild spacetime and for S a large sphere of coordinate radius R. Here ζ is the Killing vector corresponding to time...
24. ### General Relativity: Particle velocity travelling on Schwarzchild orbit

Homework Statement What is the speed of a particle in the smallest possible circular orbit in the Schwarzschild geometry as measured by a stationary observer at that orbit? Note: The orbit in question happens to be unstable. Homework Equations Normalization condition...
25. ### Escaping the Schwarzchild Radius

There's something bothering me about the event horizons of black holes. The Schwarzschild radius (as I see it) is basically the distance from a center of mass at which the escape velocity is the speed of light. The way escape velocity is defined though is the speed a body must have to "reach...
26. ### Metric for an observer in free fall two schwarzchild radii from black hole.

Hi all, I have a GR exam on tuesday and getting a bit confused as to how to find the metric for an observer in free fall a distance two schwarzchild radii from a black hole. I know this is a bit of a basic question but I am just wondering if I am correct to substitute r=2rs and dt=d(tau)...
27. ### Question about time dilation in the schwarzchild metric?

My first question is the following. Does the radial component of the schwarzchild metric account for just the radius of the body in study or is it the distance between the body and the observer, where the body is treated as a singularity (Point mass particle)? My second question is about how...
28. ### Plotting Planets Orbits around Sun using Schwarzchild Metric

Hello, I'm currently studying general relatively and am trying to plot orbits of planets around the sun using the schwarzchild metric. I've worked out the geodesic equations, working with c=1 to simplify things, and written a MATLAB script to plot trajectories, but I'm struggling to work out...
29. ### Electron schwarzchild radius problem

The schwarszchild radius of an electron=1.353*10^-57m, and to work out the volume of a particle assuming it is spherical is 4/3*pi*radius^3 so the volume of an electron at its schwarzchild radius is 4/3*pi*1.353*10^-57^3 = 0??! WHAT DOES THIS MEAN :S
30. ### Generality of theta=pi/2 in Schwarzchild

I'm doing some work on schwarzchild orbits, and everything assumes that \theta=\pi/2 and claims that this doesn't compromise generality. It seems pretty obvious (since all the geodesics are planar or radial and the metric is spherically symmetric), but can anyone prove that \theta=\pi/2 is fully...
31. ### How Do the Gauges of 1PN and Schwarzschild Metrics Differ?

I'm sure there is a simple answer to this question, but I have been looking at the first Post-Newtonian (1PN) metric (for my own research) and noticed that the time-time component of the GR metric is: g_{00} = -1 + 2U - 2 U^2 where U is the Newtonian potential. The time-time component...
32. ### Schwarzchild metric - rescaled coordinates

Schwarzschild metric - rescaled coordinates Hi, I've been working through a problem (no. 14 in ch. 9) of Alan Lightman's book of GR problems. I can't understand one of the results that are stated without proof. Basically it amounts to a rescaling of coordinates. I know that to first order...
33. ### Schwarzchild spacetime singularity

Hi all! I'm studying black holes and there's a point that I cannot understand. The book I'm reading is Modeling black hole evaporation, by Fabbri and Navarro Salas. The path is the following. After introducing the Schwarzschild metric ds^2 = \left(1 - \frac{2M}{r} \right) \ dt^2 - \left(1 -...
34. ### How does the event horizon of a Schwarzchild black hole nucleate and develop?

I would be grateful if someone could point me to a description of how a sphere of freely infalling matter ---- say equivalent to that of a collapsing massive star --- generates an finite-sized event horizon, as observed far from the incipient black hole. I can only imagine that the event...
35. ### Calculating mass and Schwarzchild radius of Black Hole

Homework Statement The temperature of a black hole is given by T = \frac{\hbar c^3}{8 \pi k G M} where h is Planck's constant, k is Boltzmann constant, G is the universal gravitation constant, and M is mass. Calculate (A) the mass, and (B) the Schwarzschild radius of a black hole at room...
36. ### Schwarzchild solution and orbit precession

In the Schwarzschild geometry of a static spacetime, elliptical test-particle orbits precess at a rate that (famously) agrees with observations of the inner solar system. Yet the model system considered is isolated, spherically symmetric with only the radial coordinate non-Euclidean. I...
37. ### Schwarzchild solution with cosmological constant

How do we solve for it? I still don't know much about non linear equations. Unfortunately, this reduces to R=-4(cosmos constant) which is not a system thus making simplicfication difficult. I'm assuming that we can still use the previous arguments and assume that the metric coompontents Gtt and...

EDIT: Is the Schwarzschild coordinate a radius of curvature in the geodesic? And also, in physics, what do I make of a negative radius of curvature?
39. ### Schwarzchild Mass Explained - What is it?

Can someone briefly explain to me what the schwarzchild mass is?
40. ### Schwarzchild and Reissner-Nordstrom singularities

The Schwarzschild metric has a spacelike singularity, while the R-N metric has a timelike one. The difference between the two physical systems is charge. Obviously you've a very slightly charged black hole, the SC metric is a good approximation because Q/M is too small to really be worried...
41. ### Entropy of a Schwarzchild black hole

Hi, I'm looking for some help on where to start with this question: The surface area of a Schwarzschild black hole is A=16 \pi R^2_c where R_c is the distance of the event horizon from the centre of the black hole. Show that for such a hole containing quantized matter, its entropy can be...
42. ### Is the Schwarzchild Metric Accurate in Predicting Black Holes?

hi recently i attended a lecture where a current researcher from my university was talking about black holes and the schwarzchild metric. basically he was saying no current theory predicts black holes and the schwarzchild solution is not actually correct, his solution was accepted because...