What is Ricci scalar: Definition and 27 Discussions
In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.
In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action. The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics. The scalar curvature of an n-manifold is defined as the trace of the Ricci tensor, and it can be defined as n(n − 1) times the average of the sectional curvatures at a point.
At first sight, the scalar curvature in dimension at least 3 seems to be a weak invariant with little influence on the global geometry of a manifold, but in fact some deep theorems show the power of scalar curvature. One such result is the positive mass theorem of Schoen, Yau and Witten. Related results give an almost complete understanding of which manifolds have a Riemannian metric with positive scalar curvature.
What would be a rough estimate for the Ricci scalar curvature of an astronomical object like the sun? Assuming the sun is a perfect fluid and you are calculating the rest frame of the sun, only the density component would be factored in. Assuming the sun is roughly 2*1030 kg. Please just make...
I was trying to calculate
$$R = g^{ij}R_{ij}$$ bu using einsum but I couldn't not work it out. Anyone can help me ? Here are some of the resources
https://stackoverflow.com/questions/26089893/understanding-numpys-einsum...
I need the Ricci scalar for the FRW metric with a general lapse function ##N##:
$$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$
Could someone put this into Mathematica as I don't have it?
I understand from the wiki entry on the Einstein-Hilbert action that:
$$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$
What is the following?
$$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$
Is there a place I could look up such GR expressions on the internet?
Thanks
Given metric ds2=dr2-r2dθ2
Gamma comes as Γ122=r,Γ212=Γ221=1/r
The Reimann tensor comes as
R11=R2121=∂1Γ212-Γm12Γ21m=0,only non zero terms .
Similary R22=R1212=∂1Γ122-Γm21Γ1m2=0,only non zero terms.
Therefore R(ricci scaler)=0
Is the space flat??
Hi all
I'm having trouble understanding what I'm missing here. Basically, if I write the Ricci scalar as the contracted Ricci tensor, then take the covariant derivative, I get something that disagrees with the Bianchi identity:
\begin{align*}
R&=g^{\mu\nu}R_{\mu\nu}\\
\Rightarrow \nabla...
Hello!
I was thinking about the Riemann curvature tensor(and the torsion tensor) and the way they are defined and it seems to me that they just need a connection(not Levi-Civita) to be defined. They don't need a metric. So, in reality, we can talk about the Riemann curvature tensor of smooth...
So I'm working with some group manifolds.
The part that's getting to me is the Ricci scalar I'm using to describe the curvature.
I have identified the groups that I'm using but that's not really relevant at the moment.
We're using a left-invariant metric ##\mathcal{M}_{ab}##.
Now I've got the...
I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary?
I'm not sure about it but I believe since the Lie derivative is...
Does this integration of Ricci scalar over surface apply in general or just for compact surfaces?
∫RdS = χ(g)
where χ(g) is Euler characteristic.
And could anybody give me some good references to prove the formula?
Homework Statement
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I am trying to compute ##R## from the 3-d metric: ##ds^{2}=d\chi^{2}+f^{2}\chi(d\theta^{2}+sin^{2}\theta d\phi^{2})##Homework Equations
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The space also satisfies the below relationships:
##R=3k##
## R_{abcd}=\frac{1}{6}R(g_{ac}g_{db}-g_{ad}g_{bc})## [1]
The Attempt...
Homework Statement
I didn't really know if this belonged here or in the math section but it is from a physics book so what the heck =D. I have to show that the directional derivative of the ricci scalar along a killing vector field vanishes i.e. \triangledown _{\xi }R = \xi ^{\rho...
hi
we know that our universe is homogenous and isotropic in large scale.
the metric describe these conditions is FRW metric.
In FRW, we have constant,k, that represent the surveture of space.
it can be 1,0,-1.
but the the Einstan Eq, Ricci scalar is obtained as function of time! and this...
Double contraction of curvature tensor --> Ricci scalar times metric
I'm trying to follow the derivation of the Einstein tensor through double contraction of the covariant derivative of the Bianchi identity. (Carroll presentation.) Only one step in this derivation still puzzles me.
What I...
Need to find the Ricci scalar curvature of this metric:
ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:
<The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z...
Homework Statement
Need to find the Ricci scalar curvature of this metric:
ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2
Homework Equations
The Attempt at a Solution
I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:
<The Christoffel...
Quick question about the EFE's. When writing the einstein tensor G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}, and using the definition of the Ricci scalar R=g^{\mu\nu}R_{\mu\nu}, how does this not give you problems when you expand out R so that the second term becomes...
I have been searching through the literature and popular textbooks for this simple answer.
I know that in the absence of soures, i.e. matter fields the Ricci scalar is zero. This is synonymous with saying that the Ricci scalar vanishes in vacuum and that the resulting space is flat. However...
Logic of E-H action, ricci scalar, cosmological constant??
This crazy thread is mean to stimulate some reflections on the logic of Einsteins Equations. It would be interesting if those who have any ideas can join. Maybe it could be enlightning?
The common way of thinking about GR is that we...
Does it make sense for the Ricci Scalar to be a function of the spacetime coordinates?
In previous calculations I have carried out in the past, everytime the Ricci Scalar has been returned as a constant, rather than being explicitly dependent on the coordinates.
Thanks for any replies
A two-dimensional Rienmannian manifold has a metric given by
ds^2=e^f dr^2 + r^2 dTHETA^2
where f=f(r) is a function of the coordinate r
Eventually I calculated that Ricci scalar is R=-1/r* d(e^-f)/dr
if e^-f=1-r^2 what is this surface?
In this case R comes to be equal to 2
I've...
A two-dimensional Rienmannian manifold has a metric given by
ds^2=e^f dr^2 + r^2 dTHETA^2
where f=f(r) is a function of the coordinate r
Eventually I calculated that Ricci scalar is R=-1/r* d(e^-f)/dr
if e^-f=1-r^2 what this surface is?
In this case R comes to be equal to 2
I've...
I calculate trace-free Ricci scalars (Phi00, Phi01,Phi02, etc) and scalar of curvature (Lambda=R/24) in Newman-Penrose formalism using a computer package. How can I find the Ricci scalar out of them? I though R was the Ricci scalar but Lambda comes non-zero for a spacetime whose Ricci scalar is...
I searched the net for the Ricci scalar for the Schwarzschild metric but in vain. Can anyone tell me what's the Ricci scalar?
Are there any standard list or tables that records down the properties of any metric for GR?