Schwarzschild Definition and 324 Threads

Mostly correct. You can observe stuff that is closer to the singularity than you are. It is just that by the time you observe it, you'll be closer to the singularity than the object you observed was when you observed it. The above description is not quite right either. It is just my own...

My project for obtaining my master's degree in computer science involved ray tracing in Schwarzschild spacetime in order to render images of black holes. These light rays had to be computed numerically using the geodesic equation. However, I ran into a problem. The geodesic equation is given as...

Greg Bernhardt submitted a new blog post Fermi-Walker Transport in Schwarzschild Spacetime Continue reading the Original Blog Post.

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...

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?

Let ##u^\alpha## and ##p^\alpha## denote a massive particle's four velocity and four momentum, respectively. Also, let ##\xi^\alpha = (1,0,0,0)## be a time like Killing vector. Since ##g_{00} \xi^0 u^0 = g_{00} p^0 / m = -(1 - 2m / r) E / m## is conserved, if we let ##r \longrightarrow \infty##...

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'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...

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...

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...

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 -...

please interpret this observation. There is a specific radius through a given equation that always gives the correct mass to any star or planet, as well a density. What is the logical explanation for this? Mass = (4π/3) x schwarzschild radius of the star x 4π/3 x (726696460.5 cm.) cube. For...

Are a Schwarzschild singularity and a black hole the same thing?

Hi, I have the following problem: Given the 5-D generalization of the Schwarszschild solution with line element: ds^2=-\Bigg(1-\frac{r^2_+}{r^2}\Bigg)dt^2+\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}dr^2+r^2[d\chi^2+\sin^2(\chi)(d\theta^2+\sin^2(\theta)d\phi^2)] where ##r_+## is a positive constant...

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...

In https://arxiv.org/abs/gr-qc/0309072 Visser starts from Minkowski metric (5), performs a coordinate transformation (6)-(7) and gets Schwarzschild geometry (12). But this should be impossible. Minowski metric has vanishing Riemann curvature tensor, while Schwarzschild geometry hasn't. What do I...

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...

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...

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

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...

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...

Leonard Susskind said "everything that ever fell in, to make the black hole, [..] [is] all contained in [...] progressively thinner and thinner shells that approach the horizon asymptotically, never quite getting there" and from the perspective of someone outside the black hole "a shell, called...

Hi, Page 166, theorem expressed as 7.2, does anybody know it's name? Many thanks

Applying Cartan's first and second structural equations to the vielbein forms \begin{align} e^t = A(r) dt , \ \ \ \ \ e^r = B(r) dr , \ \ \ \ \ e^{\theta} = C(r) d \theta , \ \ \ \ \ e^{\phi} = C(r) \sin \theta d \phi , \end{align} taken from the metric \begin{align} ds^2 = A^2(r) dt^2 - B^2(r)...

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)...

Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 3, A Newtonian Comparison Continue reading the Original PF Insights Post.

Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 2, The Photon Sphere Continue reading the Original PF Insights Post.

Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 1, GPS Satellites Continue reading the Original PF Insights Post.

I would like to ask how rigorous is the statement that Schwarzschild metric has coordinate singularity at Schwarzschild radius. The argument is that singularity at Schwarzschild radius appears because of bad choice of coordinates and can be removed by different choice of coordinates. However...

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...

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...

Consider a non-radial timelike geodesic outside the event horizon. Will it nevertheless cross the horizon radially or are non-radial geodesics also possible inside? I couldn't find any reference regarding a possible angle dependence in this respect.

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?

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}...

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...

Somewhere I ran across a `prescription' for computing the radial locations of the 2 event horizons of a S-dS metric, in which one merely computes where the radial gradient of g00 component vanishes, i.e., dg00/dr = 0. I am wrong, & apparently it's sufficient to merely set g00 = 0 , in order to...

I've been playing around with Maxima and it's ctensor library for tensor manipulation. I decided to have a crack at deriving Schwarzschild's solution for the interior of a constant-density sphere. I've managed to derive a static, spherically symmetric solution, but am struggling a bit with the...

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...

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...

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...

This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf). The Euclidean Schwarzschild black hole $$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{1-\frac{2M}{r}} +...

Hi there guys, Currently writing and comparing two separate Mathematica scripts which can be found here and also here. The first one I've slightly modified to suit my needs and the second one is meant to reproduce the same results. Both scripts are attempting to simulate the trajectory of a...

We can write down the metric of the Schwarzschild black hole in Schwarzschild coordinates. Which aspect of the metric in Schwarzschild coordinates indicates that the coordinates are only valid outside the event horizon?

I'm a little confused about the proper way to find these null geodesics. From the line element, $$c^2 d{\tau}^2=\left(1-\frac{r_s}{r}\right) c^2 dt^2-\left(1-\frac{r_s}{r}\right)^{-1}dr^2-r^2(d{\theta}^2+\sin^2\theta d\phi^2),$$ I think we can set ##d\tau##, ##d\theta## and ##d\phi## to ##0##...

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 ...

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...

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...

Homework Statement Question attached My method was going to be: set ##r=R## and solve for ##n(R)## set ##r=2GM## and solve for ##n(2GM)## I was then going to integrate proper time ##s## over these values of ##r##: ##\int\limits^{n=cos^{-1}(\frac{4GM}{R}-1)}_{n=cos^{-1}(1)=0} s(n) dn ###...

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...

I know black holes are not that well understood but if someone can explain this I'd be grateful. Please correct me if I have anything wrong, I don't know much about this. The Schwarzschild solution of comes from the Einstein field equation I think I have that right. Now I don't understand...