What is Schwarzschild metric: Definition and 106 Discussions
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916, and around the same time independently by Johannes Droste, who published his much more complete and modern-looking discussion only four months after Schwarzschild.
According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.
The Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. The boundary is not a physical surface, and a person who fell through the event horizon (before being torn apart by tidal forces), would not notice any physical surface at that position; it is a mathematical surface which is significant in determining the black hole's properties. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass M, so in principle (according to general relativity theory) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation.
I'm studying Eddington-Finkelstein coordinates for Schwarzschild metric. Adopting the coordinate set ##(t,r,\theta,\phi)##, the line element assumes the form:
$$
ds^2 = \left(1 - \frac{R_S}{r}\right)dt^2 - \left(1 - \frac{R_S}{r}\right)^{-1}dr^2 - r^2 [d\theta^2 + (\sin{\theta})^2d\phi^2],
$$...
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'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...
The question constantly arises how the speed of light is measured and what does it mean that the speed is constant, including at remote points for the observer, including at points beyond the local frame of reference, as you understand it in general relativity (GR).
First of all, it should be...
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...
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 $$V(r)=\frac {L^2}{2r^2}-\frac...
I am looking for a Python Code/Package to calculate the orbits of the time-like and null-like particles in Schwarzschild metric (in spherical coordinates)
Does anyone know such package ?
Note: I am mostly looking for packages to calculate the RIGHT side of the given images (i.e the orbits...
I am trying to find the $$\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)$$...
As a photon falls radially toward the surface of a Schwarzschild black hole, dr/dt approaches zero. Does this mean that, from the viewpoint of a distant (Schwarzschild) observer, the photon slows down or that the distance covered by successive dr's is getting larger?
I have no idea if this is an “A” level question, but I will put that down.
From the Schwarzschild metric, and with the help of the Maxima program, one of the geodesic equations is:
(I will have to attach a pdf for the equations...)
I believe this integrates to the following, with ...
I am trying to understand active diffeomorphism by looking at Schwarzschild metric as an example.
Consider the Schwarzschild metric given by the metric
$$g(r,t) = (1-\frac{r_s}{r}) dt^2 - \frac{1}{(1-\frac{r_s}{r})} dr^2 - r^2 d\Omega^2 $$
We identify the metric new metric at r with the old...
Hello! I am a bit confused about the definition of the radius in Schwarzschild metric. In the Schutz book on GR (pg. 264, General rules for integrating the equations) he says: "A tiny sphere of radius ##r = \epsilon## has circumference ##2\pi\epsilon##, and proper radius...
Since it's possible to choose a coordinate chart where the Schwarzschild metric components are dependent on time, why that's not done? Would'nt there be a scenario where such a choice would be useful?
Hi, I'm trying to deduce orbit velocity of a particle with mass from Schwarzschild metric. I know for Newtonian gravity it is:
$$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 -...
Hi
I have 2 questions.
There are 2 planets and one clock on each of them. One of them has a bigger gravitational field strength. And two clock have same distance from the core.
1-) Does time dilation occur between two? Which clock ticks slower?
2-) If time dilation occurs, which formula...
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...
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.
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...
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...
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 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 ...
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...
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...
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...
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...
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...
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...
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.