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.
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...
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:
$$d\theta^i = \Gamma^i_{[jk]}...
I read recently that Einstein initially tried the Ricci tensor alone as the left hand side his field equation but the covariant derivative wasn't zero as the energy tensor was. What is the covariant derivative of the Ricci tensor if not zero?
In this derivation,i am not sure why the second derivative of the vector ## S_j '' ## is equal to ## R^{u_j}{}_{xyz} s^y_j v^z y^x##
could anyone explain this bit to me
thank you
it seems ## S_j '' ## is just the "ordinary derivative" part but it is not actually equal to ## R^{u_j}{}_{xyz}...
I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor.
The given distance element is
$$ ds^2 = e^{2 \lambda} dt^2 -...
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...
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...
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...
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...
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...
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.
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"...
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...
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...
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...
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.
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...
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 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...
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...
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...
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!
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...
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...
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...
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...
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...
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}...
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...
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...
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...
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...
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...
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...
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...
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?
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.
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...
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}...
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}...
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...
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...
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...
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...
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...