What is Ricci tensor: Definition and 58 Discussions

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.
The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe.
Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bilinear form (Besse 1987, p. 43). Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function. However, there are other ways to draw the same analogy.
In three-dimensional topology, the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated Riemann curvature tensor. In part, this simplicity allows for the application of many geometric and analytic tools, which led to the solution of the Poincaré conjecture through the work of Richard S. Hamilton and Grigory Perelman.
In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. This is since lower bounds on the Ricci tensor can be successfully used in studying the length functional in Riemannian geometry, as first shown in 1941 via Myers's theorem.
One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. This, for instance, explains its presence in the Bochner formula, which is used ubiquitously in Riemannian geometry. For example, this formula explains why the gradient estimates due to Shing-Tung Yau (and their developments such as the Cheng-Yau and Li-Yau inequalities) nearly always depend on a lower bound for the Ricci curvature.
In 2007, John Lott, Karl-Theodor Sturm, and Cedric Villani demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a Riemannian manifold, together with its volume form. This established a deep link between Ricci curvature and Wasserstein geometry and optimal transport, which is presently the subject of much research.

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  1. physicsuniverse02

    Does anyone know which are Ricci and Riemann Tensors of FRW metric?

    I just need to compare my results of the Ricci and Riemann Tensors of FRW metric, but only considering the spatial coordinates.
  2. BiGyElLoWhAt

    I A couple questions about the Riemann Tensor, definition and convention

    According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...
  3. snypehype46

    I Computing Ricci Tensor Coefficients w/ Tetrad Formalism

    I'm reading "Differentiable manifolds: A Theoretical Physics Approach" by Castillo and on page 170 of the book a calculation of the Ricci tensor coefficients for a metric is illustrated. In the book the starting point for this method is the equation given by: $$d\theta^i = \Gamma^i_{[jk]}...
  4. D

    I Ricci Tensor: Covariant Derivative & Its Significance

    I read recently that Einstein initially tried the Ricci tensor alone as the left hand side his field equation but the covariant derivative wasn't zero as the energy tensor was. What is the covariant derivative of the Ricci tensor if not zero?
  5. B

    A The derivation of the volume form in Ricci tensor

    In this derivation,i am not sure why the second derivative of the vector ## S_j '' ## is equal to ## R^{u_j}{}_{xyz} s^y_j v^z y^x## could anyone explain this bit to me thank you it seems ## S_j '' ## is just the "ordinary derivative" part but it is not actually equal to ## R^{u_j}{}_{xyz}...
  6. saadhusayn

    Finding the Ricci tensor for the Schwarzschild metric

    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 -...
  7. D

    I Problem: perturbation of Ricci tensor

    I am trying to calculate the Ricci tensor in terms of small perturbation hμν over arbitrary background metric gμν whit the restriction \left| \dfrac{h_{\mu\nu}}{g_{\mu\nu}} \right| << 1 Following Michele Maggiore Gravitational Waves vol 1 I correctly expressed the Chirstoffel symbol in terms...
  8. Z

    I Calculating the Ricci tensor on the surface of a 3D sphere

    Hello, I'm trying to calculate Christoffel symbols on 2D surface of 3D sphere, the metric tensor is gij = diag {1/(1 − k*r2), r2}, where k is the curvature. I derived it using the formula for symbols of second kind, but I think I've made mistake somewhere. Then I would like to know which of the...
  9. H

    How Do You Calculate the Ricci Tensor for the AdS Metric in 4 Dimensions?

    Consider the AdS metric in D+1 dimensions ds^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right) I wanted to calculate the Ricci tensor for this metric for D=3. ([\eta_{\mu\nu} is the Minkowski metric in D dimensions) I have found the following Christoffel symbols...
  10. H

    A Calculating Ricci tensor in AdS space

    Consider the AdS metric in D+1 dimensions ds^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right) I wanted to calculate the Ricci tensor for this metric for D=3. (\eta_{\mu\nu} is the Minkowski metric in D dimensions) I have found the following Christoffel symbols...
  11. binbagsss

    General Relativity, identity isotropic, Ricci tensor

    Homework Statement Attached Homework EquationsThe Attempt at a Solution So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold? I don't quite understand by what is meant by 'these vectors give preferred directions'. Can...
  12. Leonardo Machado

    A Non static and isotropic solution for Einstein Field Eq

    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.
  13. A

    I Ricci Tensor in Vacuum Inside Earth - Ideas Appreciated

    Imagine a hole drilled through the Earth from which all air has been removed thus creating a vacuum. Let a cluster of test particles in the shape of a sphere be dropped into this hole. The volume of the balls should start to decrease. However, in his article "The Meaning of Einstein's Equation"...
  14. A

    I Ricci tensor for Schwarzschild metric

    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...
  15. J

    How do I properly use Ricci calculus in this example?

    Do I substitute A_\mu + \partial_\mu \lambda everywhere A_\mu appears, then expand out? Do I substitute a contravariant form of the substitution for A^\mu as well? (If so, do I use a metric to convert it first?) I’m new to Ricci calculus; an explanation as to the meaning of raised and lowered...
  16. Victor Alencar

    A Geometrical interpretation of Ricci and Riemann tensors?

    I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information? The Riemann tensor contains all the informations about your space. Riemann tensor appears when you compare the change of the sabe...
  17. A

    I Exploring the Ricci Tensor: Einstein Field Equations

    Hello I've been have been done some research about Einstein Field Equations and I want to get great perspective of Ricci tensor so can somebody explain me what Ricci tensor does and what's the mathmatical value of Ricci tensor.
  18. S

    Example Application Ricci Tensor & Scalar for 3D Understanding

    Can anyone show me an example of applying the Ricci curvature tensor to something other than GR? I also ask the same for the curvature scalar. Lately I've been trying to truly increase my understanding of curvature, so that I can see exactly how solutions of the EFE's predict the existence and...
  19. 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...
  20. E

    Baez's vizualisation of Ricci tensor

    I am reading Baez's article http://arxiv.org/pdf/gr-qc/0103044v5.pdf and I do not understand paragraph before equation (10), page 18. Equation (9) will be true if anyone component holds in all local inertial coordinate systems. This is a bit like the observation that all of Maxwell’s equations...
  21. A

    Ricci tensor equals zero implies flat splace?

    Hi, my question is the title, if Ricci tensor equals zero implies flat space? Thanks for your help
  22. U

    Raising and lowering Ricci Tensor

    Taken from Hobson's book: How is this done? Starting from: R_{abcd} = -R_{bacd} Apply ##g^{aa}## followed by ##g^{ab}## g^{aa}g^{aa} R_{abcd} = -g^{ab}g^{aa}R_{bacd} g^{ab}R^a_{bcd} = -g^{ab}g^{aa}R_{bacd} R^{aa}_{cd} = - g^{ab}g^{aa} R_{bacd} Applying ##g_{aa}## to both sides...
  23. Y

    Help with the variation of the Ricci tensor to the metric

    I should calculate the variation of the Ricci scalar to the metric ##\delta R/\delta g^{\mu\nu}##. According to ##\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}##, ##\delta R_{\mu\nu}## should be calculated. I have referred to the wiki page...
  24. L

    Ricci Tensor Equation in Zee's "Einstein's Gravity in a Nutshell" Explained

    In Zee's "Einstein's Gravity in a Nutshell" on page 363, while deriving the Schwarzschild solution, we have How does it work? How are the rhs and lhs equal? Where does the factor 2 come from, why just one derivative left? thanks for any replies!
  25. P

    Ricci tensor of schwarzschild metric

    In schwarzschild metric: $$ds^2 = e^{v}dt^2 - e^{u}dr^2 - r^2(d\theta^2 +sin^2\theta d\phi^2)$$ where v and u are functions of r only when we calculate the Ricci tensor $R_{\mu\nu}$ the non vanishing ones will only be $$R_{tt}$$,$$R_{rr}$$, $$R_{\theta\theta}$$,$$R_{\phi\phi}$$ But when u and v...
  26. S

    Can the Ricci Tensor be Simplified Further? Suggestions Needed!

    In my studies of methods to simplify the Einstein field equations, I first decided to go about expanding the Ricci tensor in terms of the metric tensor. I have been mostly successful in doing this, but there are a couple of complications that I would like your opinions on. At the bottom of...
  27. C

    Ricci Tensor Proportional to Divergence of Christoffel Symbol?

    I'm reading an old article published by Kaluza "On the Unity Problem of Physics" where i encounter an expression for the Ricci tensor given by $$R_{\mu \nu} = \Gamma^\rho_{\ \mu \nu, \rho}$$ where he has used the weak field approximation ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}## where...
  28. F

    Quantitative Meaning of Ricci Tensor

    Hello, I am studying general relativity right now and I am very curious about the Ricci tensor and its meaning. I keep running into definitions that explain how the Ricci tensor describes the deviation in volume as a space is affected by gravity. However, I have yet to find any quantitative...
  29. andrewkirk

    Intuitive description of what the Ricci tensor & scalar represent?

    Is there a simple intuitive description of what the Ricci tensor and scalar represent? I have what seems to me a straightforward understanding of what the Riemann tensor Rabcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined...
  30. P

    What is the Symmetry of the Ricci Tensor?

    Hey, I have been doing a few proofs and stumbled across this little problem. Trying to show the symmetry of the Ricci tensor by using the Riemann tensor definition ##R^m_{\ ikp} = \partial_k \Gamma^m_{\ ip} - \partial_p \Gamma^m_{\ ki} + \Gamma^a_{\ ip} \Gamma^m_{\ ak} - \Gamma^a_{\ ik}...
  31. G

    What does vanishing Ricci tensor signify ?

    Are Ricci flat manifolds analogous to flat space-time ? Further for Ricci flat manifolds does the Riemann tensor vanish ?
  32. WannabeNewton

    Killing fields as eigenvectors of Ricci tensor

    Hi guys! I need help on a problem from one of my GR texts. Suppose that ##\xi^a## is a killing vector field and consider its twist ##\omega_a = \epsilon_{abcd}\xi^b \nabla^c \xi^d##. I must show that ##\omega_a = \nabla_a \omega## for some scalar field ##\omega##, which is equivalent to showing...
  33. P

    Ricci tensor of the orthogonal space

    While reading this article I got stuck with Eq.(54). I've been trying to derive it but I can't get their result. I believe my problem is in understanding their hints. They say that they get the result from the Gauss embedding equation and the Ricci identities for the 4-velocity, u^a. Is the...
  34. E

    Wrong signs Ricci tensor components RW metric tric

    Hi, I am working through GR by myself and decided to derive the Friedmann equations from the RW metric w. ( +,-,-,-) signature. I succeeded except that I get right value but the opposite sign for each of the Ricci tensor components and the Ricci scalar e.g. For R00 I get +3R../R not -3R../R . I...
  35. P

    Why Are Off-Diagonal Ricci Tensors Zero in Symmetric FRW Metrics?

    Hi does anyone know a formal definition for why off diagonal ricci tensors are equal to zero in a symmetric standard FRW metric?
  36. A

    Spaces of Constant Curvature and the Ricci Tensor

    Hi all, I was just interested in verification of a concept. If we are given the full Riemann tensor in the form which implies constant curvature (i.e. lambda multiplying metric components) does this imply that the Ricci tensor vanishes? The question stems from why the vacuum equations cannot be...
  37. A

    Vanishing Ricci Tensor in 3 Dimensions

    In my general relativity course my professor recommended that it would be useful to convince ourselves that in 3 dimensions the vacuum field equations are trivial because the vanishing of the Ricci tensor implies the vanishing of the full Riemann tensor. However, I am unsure of how to show this...
  38. M

    Ricci Tensor: Understanding the Mathematics & Concepts

    Hi everyone, I am new to PF but truly appreciate the kind assistance from all people in this forum. I am very interested in learning Relativity as I really want to know it essence either in its Physics or Mathematics. I have a little basics on integration and some vector calculus. Amazed by...
  39. J

    Deriving the meaning of the Ricci Tensor

    I am trying to understand the meaning of the Ricci Tensor. I tried to work it out in a way that was meaningful to me based on ideas from Baez and Loveridge. Unfortumately, the forum tool won't allow me to include the URLs to those documents in this post. Anyway, I get the wrong answer. Can...
  40. N

    Ricci tensor for electromagnetic field

    Electromagnetic fields mostly have a stress-energy tensor in which the trace is zero. Is traceless stress energy tensor always implies Ricci scalar is zero? If yes how to prove that?
  41. N

    Covariant Derivation of the Ricci Tensor: Einstein's Method Now Online

    The full derivation of the covariant derivative of the Ricci Tensor as Einstein did it, is now available on line at https://sites.google.com/site/generalrelativity101/appendix-c-the-covariant-derivative-of-the-ricci-tensor For those who wish to study it.
  42. P

    Understanding Ricci Tensor of FRW Universe: Equation 74 Explained

    I am trying to understand FRW universe. To do so I am following the link below: http://www.phys.washington.edu/users/dbkaplan/555/lecture_04.pdf I am confused at equation 74. I got R00 but for Rij part I am always getting a\ddot{a}. I am trying to solve it for k =0. Can some please...
  43. N

    Help Covariant Derivative of Ricci Tensor the hard way.

    I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-2\Gamma^{α}_{μ\gamma}R_{αβ} or \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-\Gamma^{α}_{μ\gamma}R_{αβ}-\Gamma^{β}_{μ\gamma}R_{αβ}...
  44. N

    Help Covariant Derivative of Ricci Tensor the hard way.

    I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-2\Gamma^{α}_{μ\gamma}R_{αβ} or \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-\Gamma^{α}_{μ\gamma}R_{αβ}-\Gamma^{β}_{μ\gamma}R_{αβ}...
  45. D

    Riemann tensor, Ricci tensor of a 3 sphere

    Homework Statement I have the metric of a three sphere: g_{\mu \nu} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\theta \end{pmatrix} Find Riemann tensor, Ricci tensor and Ricci scalar for the given metric. Homework Equations I have all the formulas I need, and I...
  46. G

    Ricci tensor from Ricci 1-form

    Hi all, once again I'm stuck on something I am quite certain is silly, but here it goes. My confusion pertains to the equation Ric=R^{a}\otimes e_{a} where Ric is the Ricci tensor, R^{a} is the Ricci 1-form and e_{a} are the elements of an orthonormal basis. Now, let's say for...
  47. M

    Ricci tensor: symmetric or not?

    I am really confused and the question can appear to be trivial or stupid: Is the Ricci tensor symmetric or anti-symmetric in a torsion-free affine connection? I am full of troubles since two different references gives two different answers (sorry no one is in english language but one of...
  48. M

    Ricci Tensor and Metric Restrictions

    I've been wondering two things lately. Why did Einstein make the assumption that the Ricci tensor is 0 in empty space. Is there a physical/mathematical reason? I know later he set it equal to another tensor...which leads to all the cosmological constant business, but I'm just curious why he...
  49. A

    Calculating the Ricci Tensor in 5D Using GRTensor

    grtensor 5d? I want to calculate the Ricci tensor for a 5-D metric. For example , the randall sundrum metric. ds^2=dw^2+exp(-2A(w))*(-dt^2+dx^2+dy^2+dz^2) there is any computer program to calculate ricci tensor in 5d spacetime? In 4d , using grtensor for the metric...