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.

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  1. D

    I Ricci Scalar For Astronomical Body

    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...
  2. Arman777

    Python Using np.einsum to calculate Ricci scalar

    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...
  3. J

    I Ricci scalar for FRW metric with lapse function

    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?
  4. J

    I Variation of Ricci scalar wrt derivative of metric

    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
  5. A

    B Ricci scalar calculation

    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??
  6. F

    I Covariant derivative of Ricci scalar causing me grief

    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...
  7. J

    A On the dependence of the curvature tensor on the metric

    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...
  8. J

    Curvature of a group manifold

    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...
  9. loops496

    Directional Derivative of Ricci Scalar: Lev-Civita Connection?

    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...
  10. darida

    Integration of Ricci Scalar Over Surface

    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?
  11. binbagsss

    Ricci scalar computation quick question

    Homework Statement [/B] 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 [/B] 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...
  12. S

    Why is Scalar Cam Built with Second-Order Derivative of Metric Ricci Scalar?

    hi why only scalar cam build with second order of derivative of metric is Ricci scalar? thanks
  13. WannabeNewton

    Direction derivative of Ricci scalar w.r.t. killing field

    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...
  14. S

    Ricci scalar and curveture of FRW metric

    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...
  15. G

    Double contraction of curvature tensor -> Ricci scalar times metric

    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...
  16. C

    Need to find the Ricci scalar curvature of this metric

    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...
  17. C

    Need to find the Ricci scalar curvature of this metric

    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...
  18. U

    EFE's question regarding Ricci scalar

    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...
  19. S

    I need to calculate δR: R is Ricci scalar

    I need to calculate δR: R is Ricci scalar
  20. A

    Finding Ricci Scalar Constant Nonzero Line Elements

    i need examplees where the ricci scalar is constant but nonzero . Particulary i search examples of line element. Pd: this is not a homework,
  21. B

    Vanishing Ricci scalar always implies vacuum?

    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...
  22. Fra

    Logic of E-H action, ricci scalar, cosmological constant?

    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...
  23. D

    Exploring the Possibility of Variable Ricci Scalar in Spacetime Calculations

    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
  24. G

    Surface with Ricci scalar equal to two

    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...
  25. G

    Surface with Ricci scalar equal to two

    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...
  26. C

    From the scalar of curvature (Newman-Penrose formalism) to the Ricci scalar

    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...
  27. T

    Searching for Ricci Scalar in Schwarzschild Metric

    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?
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