Schwarzchild metric Definition and 24 Threads

[Mentors' note: This thread is spun off from https://www.physicsforums.com/threads/proper-time-of-the-observer-resting-in-cmb-reference-frame.1080211/] @Nugatory said What does it correspond to in-between the infinite distance and the Schwarzschild radius?

I have this problem: First of all, I would like to have an equation for the proper time. I was using the Eddington-Finkelstein coordinates in the Schwarzschild metric : $$ds^2 = - (1-\frac{2M}{r}) dt^2 + 2dtdr +r^2 d\Omega^2$$ Knowing that the observer falls radially, thus ##d\phi = d\theta =...

As personal curiosity, I want to calculate which is the difference in "travelled height" between a photon that goes across the width of an elevator - which is more or less 2[m] in my country - and a tiny mass particle that free-falls starting at the same "height" as the photon origin, and is...

Let's say we have some observer in some curved spacetime, and we have another observer moving relative to them with some velocity ##v## that is a significant fraction of ##c##. How would coordinates in this curved spacetime change between the two reference frames? For example, imagine a...

So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A...

I'm not sure how to approach this question. So I start off with the fact the path taken is space-like, $$ds^2>0$$ Input the Schwarzschild metric, $$−(1−\frac{2GM}{r})dt^2+(1−\frac{2GM}{r})^{−1}dr^2>0$$ Where I assume the mass doesn't move in angular direction. How should I continue?

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 have tried integration by parts where, ##c dt = -\frac{1}{\sqrt{r*}} \frac{r^{3/2} dr}{r - r*} = \frac{1}{\sqrt{(r*)^3}} \frac{r^{3/2} dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##u = r^{3/2} \quad \quad dv = \frac{dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##du = \frac{3}{2} r^{1/2} dr...

In a circular orbit, the 4-velocity is given by (I have already normalized it) $$ u^{\mu} = \left(1-\frac{3M}{r}\right)^{-\frac{1}{2}} (1,0,0,\Omega) $$Now, taking the covariant derivative, the only non vanishing term will be $$ a^{1} = \Gamma^{1}_{00}u^{0}u^{0} + \Gamma^{1}_{33}u^{3}u^{3} $$...

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 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...

Hello I have been reading about Schwarzschild metric and scources what I read said that Schwarzschild metric is used to describe a non-rotating black holes. And what I can not understand is what can you calculate with it? It would be good if you give some examples where you can use it.

OK, so it's time to start a new thread. I heard many times that there exists only one black hole solution for a given mass and angular momentum, but I know already that this is not true. We all know that if we throw something into an existing black hole, its event horizon starts to ripple. So...

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?

The Schwarzschild spacetime is defined by the following line element \begin{equation*} ds^2 = - \left( 1 - \frac{2m}{r} \right)dt^2 + \frac{1}{1-\frac{2m}{r}}dr^2 + r^2 d\theta^2 + r^2\sin \theta^2 d\phi^2. \end{equation*} We can use the isotropic coordinates, obtained from the Schwarzschild...

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 ...

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...

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...

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...

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...

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 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...