Schwarzschild solution Definition and 31 Threads

In the classical Newtonian theory of gravity, the shell theorem holds. ( https://en.wikipedia.org/wiki/Shell_theorem ) In the beginning of the derivation of the Schwarzschild solution, the spherically symmetric object is replaced by a point mass. The proof that this can be done in curved...

I have a theoretical question regarding the validity of Birkhoff's theorem in flat Minkowski spacetime (noting that this is a hypothetical scenario, as mass curves spacetime). Common derivations of Birkhoff's theorem are based on the assumption that the Ricci tensor R_ij = R_kij^k (Einstein...

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...

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...

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...

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...

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...

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?

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}##...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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%)...

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...

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...

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...

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...

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...

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...

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