Schwarzschild Definition and 324 Threads

Two masses, m and M, are a fixed distance R apart. One of the masses is much larger then the other. At time t the masses start to fall towards each other. Using Newton's Law of Gravitation we can determine the acceleration of the small mass. Can one use the Schwarzschild metric in the...

Homework Statement The most general form: ##ds^2=e^{2A(r)}dt^2-e^{2B(r)}dr^2-r^2(d\theta_1^2 +sin^2\theta_1(d\theta^2_2+sin^2\theta_2d\phi^2))## Ricci tensors: ##R_{tt}=e^{2(A-B)}(\frac{3A'}{r}+A'^2-B'A'+A'')## ##R_{rr}=-A''+\frac{3B'}{r}+A'(B'-A')## ##R_{\theta_1...

Correct if I'm wrong. V(esc.)=(2GM/R)^1/2 that is equal to R=2GM/V^2 putting v=c,we get R=2GM/c^2 by putting the value of G,M,C we get schwarzschild radius=1.46*10^-27 m/kg

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 4 Continue reading the Original PF Insights Post.

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 3 Continue reading the Original PF Insights Post.

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 2 Continue reading the Original PF Insights Post.

Hello dear friends, today's question is: In a non static and spherically simetric solution for Einstein field equation, will i get a non diagonal term on Ricci tensor ? A R[r][/t] term ? I'm getting it, but not sure if it is right. Thanks.

PeterDonis submitted a new PF Insights post The Schwarzschild Geometry: Part 1 Continue reading the Original PF Insights Post.

I have read that Albert Einstein was quite (pleasantly) surprised to read Schwarzschild's solution to his field equation because he did not think that any complete analytic solution existed. However, of all the possible scenarios to consider, a point mass in a spherically symmetric field (ie, a...

Homework Statement Hartle, Gravity, P9.8 A spaceship is moving without power in a circular orbit about a black hole of mass M, with Schwarzschild radius 7M. (a) What is the period as measured by an observer at infinity? (b) What is the period as measured by a clock on the spaceship...

I was looking at null geodesics in Schwarzschild spacetime. Carroll's lecture notes cover them here: https://preposterousuniverse.com/wp-content/uploads/grnotes-seven.pdf He notes (and justifies) that orbits lie in a plane and chooses coordinates so it's the equatorial plane, then uses Killing...

Learning about Schwarzschild radius from Wikipedia: Is it accurate to say any object of mass crossing the event horizon of a black hole is compressed sufficiently to have its own Schwarzschild radius, becoming a black hole itside of a black hole?

So the Schwarzschild metric is given by ds2= -(1-2M/r)dt2 + (1-2M/r)-1dr2+r2dθ2+r2sin2θ dφ2 and the Lagragian is ##{\frac{d}{dσ}}[{\frac{1}{L}}{\frac{dx^α}{dσ}}] + {\frac{∂L}{∂x^α}}=0## with L = dτ/dσ. So for each α=0,1,2,3 we have ##{\frac{d^2 x^1}{dτ^2}}=0## for Minkowski spacetime also...

Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric: So we have Ricci flow equation,∂tgμν=-2Rμν. And we have metric tensor for schwarzschild metric: Diag((1-rs/r),(1-rs]/r)-1,(r2),(sin2Θ) and ∂tgμν=0 so 0=-2Rμν and we get that Rμν=0.But Rμν should not equal to...

Bill_K submitted a new PF Insights post Orbital Precession in the Schwarzschild and Kerr Metrics Continue reading the Original PF Insights Post.

The Schwarzschild Metric (with ##c=1##), $$ds^2 = -\Big(1-\frac{2GM}{r}\Big)dt^2+\Big(1-\frac{2GM}{r}\Big)^{-1}dr^2+r^2d\Omega^2$$ can be adjusted to a form involving three rectangular coordinates ##x##, ##y##, and ##z##: $$ds^2 =...

Hi, I was wondering if anybody could help me understand the derivation of the Schwarzschild metric developed by the author of mathpages website. Rather than reproduce all the equations via latex, I have attached a 2-page pdf summary that also points to the mathpages article and explains my...

Including dark matter but not including dark energy, what's the Schwarzschild radius of the known universe? Actually, let me put it another way. What's the SR of all the matter and energy thought to be created at the Big Bang? So that would include not just all the matter we see but also all 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.

Where can I find a derivation of the vacuum solution for GR directly from the Riemann tensor of zero trace, i.e., Weyl tensor, instead of the more traditional Schwarzschild derivation?

Anybody have a stupid/intuitive/clever trick/mnemonic/baby-story to remember the Christoffel symbols and the Einstein Field equations for the Schwarzschild metric $$ds^2 = A(r)dt^2 - B(r)dr^2 - r^2 d \theta^2 - r^2 \sin^2(\theta) d\varphi^2$$ without even using a pen? I can derive it all...

[Moderator's note: this thread is spun off from another thread since it was a subthread dealing with a separate topic.] There is definitely a maximally extended spacetime but there is no maximally extended spacelike surface of constant Schwarzschild coordinate time t. The spatial curvature...

How did Einstein compute the amount of light deviation due to the Earth's gravitational field when the Schwarzschild metric was not known yet?

The Schwarzschild equation of motion, where coordinate length is differentiated by proper time is as far as I know r''(t) = -\frac{G\cdot M}{r(t)^2} + r(t)\cdot{\theta}'(t)^2 - \frac{3\cdot G\cdot M\cdot{\theta}'(t)^2}{c^2} {\theta}''(t) = -2\cdot r'(t)/r(t)\cdot{\theta}'(t) Now the question...

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 full power series for the Schwarzschild portion of perihelion shift is given in Mathpages as: where L = a(1-\epsilon^2), a the semi-minor axis and \epsilon the eccentricity. This implies that as \epsilon tends to zero, the perihelion shift tends to a non-vanishing 6\pi m/a + some much...

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?

Is this the proper formula for calculating the Schwarzschild radius of a black hole? rs = 2GM / c2 If it is not, or if anyone has one that might work better, could you refer it to me?

If a metric admits a Killing vector field ##V ## it is possible to define conserved quantities: ## V^{\mu} u_{\mu}=const## where ## u^{\mu}## is the 4 velocity of a particle. For example, Schwarzschild metric admits a timelike Killing vector field. This means that the quantity ##g_{\mu 0}...

Just a thought... Would there be any implicit differences between (A) a two-body metric where the two central masses are drawn ever further together, with angular momentum included, and (B) the Kerr metric? Angular momentum would still be part of the system, but it would be explained by a more...

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...

Hello. In oral exams my professor likes to ask if Alice and Bob can communicate, if Alice ist just above the event horizon of a schwarzschild black hole and Bob ist just below. He wants to hear: Communication is possible, because the event horizon is observer dependent. Only an observer...

I have read that the Schwarzschild radius of a black hole with the mass-energy of the observable universe is roughly equal to the actual Hubble radius of 13.8 billion light years. And I have read that contrary to some popular esoteric interpretations such as "the universe is a black hole", "we...

It seems to me that the Schwarzschild singularity is generated by the metric function which is an invariant and so has the same value in any coordinate system, so why is it not equally valid?

I've never heard or seen it stated this way before, so I'm asking this question just to check my intuition and to see if my understanding is correct: The Planck distance IS the Planck distance PRECISELY BECAUSE its the Schwarzschild Radius (or diameter, I suppose) of a single quantum of energy...

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...

I have a problem (this is not homework) Based on covariant Lagrangian ## \mathcal {L} = \frac {m}{2} \frac{dx^{\mu}}{ds} \frac {dx _ {\mu}}{ds} ## record the equations of motion in Hamiltonian form for a particle in the Schwarzschild metric (SM). Based on Legandre transformations...

Dear PF Forum, I have a question about the size of "singularity" This question has already been asked here, Question about Schwarzschild radius But what I want to know is this density thing that I'd like a confirmation. Actually it's 2.9511896078372906 KM according to his...

Homework Statement The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##. (a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it...

Let me begin with the fact I am a rube in the field of Quantum Physics. I seem to have an innate grasp of certain concepts but if it comes to proving theory with math, I’m out. That being said, I am completely fascinating with Miguel Alcubierre’s theory on collapsing, or “warping”, space...

Homework Statement Two rockets are orbiting a Schwarzschild black hole of mass M, in a circular path at some location R in the equatorial plane θ=π/2. The first (rocket A) is orbiting with an angular velocity Ω=dΦ/dt and the second (rocket B) with -Ω (they orbit in opposite directions). Find...

I want to produce some realistic figures showing the spatial trajectories of test particles in a Schwarzschild spacetime. For instance, I'd like to start a massive test particle at aponegricon (how often do you get to use that word!?) in an orbit that Kepler and Newton would have predicted to be...

So I have the metric as ##ds^{2}=-(1-\frac{2m}{r})dt^{2}+(1-\frac{2m}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##* I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2), where ##r*=r+2m In(\frac{r}{2m}-1)## and to the coordinate system ##v,r,\phi, \theta ##, where...

The coordinates ##u## and ##v## are defined as ##u=t+r*##, ##v=t-r*##, where ##r*=r+2M In(\frac{r}{2M}-1)##. In ##u,r,\theta,\phi ## coordinates the radially null geodesics are given by: ##\frac{du}{dr}=0 ## for infalling, ##\frac{du}{dr}=2(1-\frac{2M}{r})^{-1} ## for outgoing. In the...

Is it possible to derive r=GM/c^2 classically(as in Newtonian gravity) without using eeinstein field equation?

Hello, I need to find the angular velocity using Schwarzschild metric. At first I wrote the metric in general form and omitted the co-latitude: ds2=T*dt2+R*dr2+Φ*dφ2 and wrote a Lagrangian over t variable: L = √(T+R*(dr/dt)2+Φ*(dφ/dt)2) now I can use the Euler–Lagrange equations for φ...

I was just wondering how you would go about calculating the proper time for an observer following a freely falling elliptical orbit in a Schwarzschild metric. I am happy with how to calculate the proper time for a circular orbit and was wondering whether if you had two observers start and end...

I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997. I don't understand eq 7.4 from the theorem 7.2. As I understand, theorem 7.2 is used when you have submanifold that foilate the manifold, and the submanifold must be maximally symmetric. I know that 2-spheres are...

Hello :) I had a question , i recently red a lot about black holes and i had a question about the Schwarzschild Radius . How does that come that π is not in the equation ? The black hole is a sphere so instinctly one could think that Pi should be in the equations.. ? P.S: I'm french and 16 so...

How do you obtain this equation M=Gm/c^2. What does M stand for? Is is Newton law at infinity? Again what is this Newton law at infinity?