Schwarzchild Definition and 42 Threads

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 =...

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...

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...

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...

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...

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...

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...

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 ...

Is there theoretically a smallest possible Schwarzschild radius?

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...

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...

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...

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 -...

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 =...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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...

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

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...

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...

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...

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 -...

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...

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...

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...

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?

Can someone briefly explain to me what the schwarzchild mass is?

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...

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...

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...