Schwarzschild Definition and 319 Threads

Hi, I'm wondering if it is possible to calculate the volume of a black hole using the Schwarzschild metric. After computing the volume I get the follow integral: $$V = 4 \pi \int_0^r \frac{1}{\sqrt{ (1- \frac{r_s}{r'})}} r'^2 dr$$ This integral diverges at the upper bound. The only way I...

What is the orbital period in General Relativity using the Schwarzschild metric? In classical mechanics, it is something like T=2pi(GnM/a3). Where a is the semi-major axis, this is for a small body orbiting a larger one. I think I have an idea but I am not 100% sure. I am interested in an...

Dirac in his "GTR" (Chap 19, page 34-35) finds a coordinate system ##(\tau, \rho)## which has no coordinate singularity at ##r=2m##. Explicitly, the transformation looks like (after some algebra): $$\tau=t + 4m\sqrt{\frac{r}{2m}} + 2m\log{\frac{\sqrt{r/2m}-1}{\sqrt{r/2m}+1}}$$ $$\rho=t +...

Hi. I am looking for an equation for the round trip elapsed proper time of a clock that is initially moving vertically straight up with a given initial velocity, reaches apogee and then returns to the starting location under gravity. I would like to use the external Schwarzschild geometry of a...

The Einstein tensors for the Schwarzschild Geometry equal zero. Why do they not equal something that has to do with the central mass, given that the Einstein equations are of the form: Curvature Measure = Measure of Energy/Matter Density?

In 1916, Karl Schwarzschild was the first person to present a solution to Einstein's field equations. I am using a form of his equation that is presented in Tensors, Relativity and Cosmology by Mirjana Dalarsson and Nils Dalarsson (Chapter 19, p.205). I am approaching what may be the simplest...

Schutz finds that the orbital period for a circular orbit in Schwarzschild is $$ P = 2 \pi \sqrt {\frac { r^3} {M} }$$ He gets this from $$ \frac {dt} {d\phi} = \frac {dt / d\tau} {d\phi/d\tau} $$ Where previously he had ## \frac {d\phi}{d\tau} = \tilde L / r^2## and ## \frac {dt}{d\tau} =...

The Schwarzschild solution could simply be expressed as $$ds^2=-(1-2GM/r)dt^2+(1-2GM/r)^-1dr^2+r^2d\omega^2$$. Is it possible that we could obtained a new metric into the form as $$ds^2=-(1-2GM/r)^-1dt^2+(1-2GM/r)dr^2+r^2d\omega^2$$? If possible, what are the steps and procedures that should be...

According to Schutz, the line element for large r in Schwarzschild is $$ ds^2 \approx - ( 1 - \frac {2M} {r}) dt^2 + (1 + \frac {2M} {r}) dr^2 + r^2 d\Omega^2 $$ and one can find coordinates (x, y, z) such that this becomes $$ ds^2 \approx - ( 1 - \frac {2M} {R}) dt^2 + (1 + \frac {2M} {R})...

https://www.msn.com/en-us/news/technology/the-astonishing-scientific-theory-that-says-the-universe-might-be-inside-a-black-hole/ar-AA17lxtF?ocid=msedgdhp&pc=U531&cvid=272cb184de9c48fbbd3b321120e37dac Michio Kaku has often joked, "If you want to know what it looks like inside of a black hole...

Read the bio / fiction chapter on Karl Schwarzschild in Benjamin Labatut’s great Book, and curious on a little color on how he developed the solution - I had thought finding an exact solution in GR was just math chops, but actually any Lorentzian metric is an exact solution, so the difficulty...

In the Penrose chart for Schwarzschild spacetime, the boundary "at infinity" appears to be connected all the way around. I want to explore what that means physically and whether particular boundary points that appear to be connected on the chart actually are. I am using the following notes as a...

Schwarzschild Geometry-proper distance. From what I have studied when the Schwarzschild line element is evaluated at constant time and at a constant radius , proper distance becomes a Euclidean distance on the surface of a sphere. What I don't understand is how to evaluate the integral...

The Schwarzschild metric implies a potential different from that of Newtonian gravity. Is there a relationship between it and the process by which particles can be absorbed by other particles? (I haven't studied QFT yet)

I understood that the event horizon is a null surface and not a place in space, what is the relationship between it and the Schwarzschild radius? Also, what does the Schwarzschild radius physically represent for example for an object such as a star?

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...

Real quick, are the terms "Schwarzschild Radius" and "Event Horizon" for a black hole interchangeable or is there some subtle difference between the two? Just looking for a ballpark answer here

Hi, I was reading this insight schwarzschild-geometry-part-1 about the transformation employed to rescale the Schwarzschild coordinate time ##t## to reflect the proper time ##T## of radially infalling objects (Gullstrand-Painleve coordinate time ##T##). As far as I understand it, the vector...

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?

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Hello, there. I am learning the chapter, The Schwarzschild Solution, in Spacetime and geometry by Caroll. I could not grasp the idea of circular orbits. It starts from the equations for ##r##, $$\frac 1 2 (\frac {dr}{d \lambda})^2 +V(r) =\mathcal E$$ where $$V(r)=\frac {L^2}{2r^2}-\frac...

Why does the Schwarzschild metric have a singularity at r=0 if it is only valid outside the spherically symmetric static mass?

From "standard" formula we have that the gravity acceleration a = GM/r^2 and that the Schwarzschild radius rs = 2 GM / c^2 Is it possible to compute the gravity acceleration at Schwarzschild radius putting r = rs? In this case we will have a = c^4 / (4GM) This mean that a very very...

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...

Computing timelike geodesics in the Schwarzschild geometry is pretty straightforward using conserved quantities. You can treat the problem as a variational problem with an effective Lagrangian of ##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2...

I am looking for a Python Code/Package to calculate the orbits of the time-like and null-like particles in Schwarzschild metric (in spherical coordinates) Does anyone know such package ? Note: I am mostly looking for packages to calculate the RIGHT side of the given images (i.e the orbits...

I am trying to find the $$\nabla_{\mu}\nabla^{\mu} \Phi$$ for $$ds^2 = (1 - \frac{2M}{r})dt^2 + (1 - \frac{2M}{r})^{-1}dr^2 + r^2d\Omega^2$$ I have did some calculations by using $$\nabla_{\mu}\nabla^{\mu}\Phi = \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu}\partial_{\nu}\Phi)$$...

Does this give solutions which are physically acceptable?

Assuming the line element ##ds^s=e^{2\alpha}dt^2-e^{2\beta}dr^2-r^2{d\Omega}^2 ##as usual into the form ##ds^s=e^{-2\alpha}dt^2-e^{-2\beta}dr^2-r^2{d\Omega}^2##, I found that the ##G_{tt}## tensor component of first expression do not reconcile with the second one though, it fits for ##G_{rr}...

As a photon falls radially toward the surface of a Schwarzschild black hole, dr/dt approaches zero. Does this mean that, from the viewpoint of a distant (Schwarzschild) observer, the photon slows down or that the distance covered by successive dr's is getting larger?

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'm a bit confused about GR : what is more significant about the considered spacetime, the metric, which is time-independent, or the embedding (there are already some posts on PF about it), which describes the shape of a manifold, but is time-dependent ?

When calculating the Schwarzschild radius are we supposed to be using the rest mass of the object or its total energy?

I had a thought that I wanted to share in another thread, but it wandered way off track and quite properly was closed. But I thought the separate idea that I had spawned from the old thread was worthy of posting in a new thread. I do not want to re-open the old thread, though! In flat...

I notice that in a Schwarzschild black hole, at r=r_{s}/2, the c dt and dr terms are exactly the opposite of what they are in external, normal flat space (Minkowski metric). That is, one gets them by multiplying both terms by negative one. I'm having trouble grasping what this means. An...

Recently, I was tasked to find the surface area of the Schwarzschild Black Hole. I have managed to do so using spherical and prolate spheroidal coordinates. However, my lecturer insists on only using Weyl canonical coordinates to directly calculate the surface area. The apparent problem arises...

Can anyone help me get started with this problem? What should I use for Gni? I've tried to produce Tni by working out Rni (using methods developed in an earlier chapter) but the results don't lead me anywhere. I'm really stuck for a way forward on this problem so if anyone can help, it...

For the purpose of this thread the metric is ds2 = - (1-rs/r) c2 dt2 + dr2 / (1-rs/r) where rs = 2GM/c2. (I modified the above from https://jila.colorado.edu/~ajsh/bh/schwp.html .) I assume that the two spherical shells are stationary. Therefore dt = 0. The r coordinate for the radii of the...

For someone who have just finished the study of the (fundamentals) of Schwarzschild spacetime, what would be the next natural topic to study?

According to the theory, every mass has a Schwarzschild radius associated. Any object whose radius is smaller than its Schwarzschild radius is called a black hole. So in principle is possible to create mini-black holes, it is just a fact of matter condensed. Those mini black holes have their...

The concept of that when a photon's trajectory intersects with the Schwarzschild Radius/event horizon, said photon will never exit the Schwarzschild Radius/event horizon. Or any other object besides a photon for that matter. So far what has been the strongest evidence for this prediction?

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...

I have no idea if this is an “A” level question, but I will put that down. From the Schwarzschild metric, and with the help of the Maxima program, one of the geodesic equations is: (I will have to attach a pdf for the equations...) I believe this integrates to the following, with ...

The thing is that this is an exercise that I have to show my teacher but I don´t know how to get the answer.The exercise says: "A body of mass m moving in the Keplerian field V = −M/r (in G = 1 units) has a total conserved energy, Etot = 1 /2( m r˙^2 + r ^2ϕ˙ ^2 )− mM/r. Show that the...

I am trying to understand active diffeomorphism by looking at Schwarzschild metric as an example. Consider the Schwarzschild metric given by the metric $$g(r,t) = (1-\frac{r_s}{r}) dt^2 - \frac{1}{(1-\frac{r_s}{r})} dr^2 - r^2 d\Omega^2 $$ We identify the metric new metric at r with the old...

I had this idea when some people said that LHC can produce black hole. Based on the calculation of Schwarzschild Radius, any mass than 9.375×10^7 kg have a Schwarzschild radius smaller than the plank length. Particles inside LHC or other particle accelerator have clearly radii smaller than that...

Edit: I'm leaving the original post as is, but after discussion I'm not confused over coordinate time having a physical meaning. I was confused over a particular use of a coordinate time difference to solve a problem, in which a particular coordinate time interval for a particular choice of...

A singularity would be: a location in spacetime where the gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system. (wiki) If the threshold to get a singularity is reached then space-time curvature...

Hello! I am a bit confused about the definition of the radius in Schwarzschild metric. In the Schutz book on GR (pg. 264, General rules for integrating the equations) he says: "A tiny sphere of radius ##r = \epsilon## has circumference ##2\pi\epsilon##, and proper radius...