Ricci curvature Definition and 11 Threads

Helo, The Lagrangian in general relativity is written in the following form: \begin {aligned} \mathcal {L} & = \frac {1} {2} g ^ {\mu \nu} \nabla \mu \phi \nabla \nu \phi-V (\phi) \\ & = R + \dfrac {16 \pi G} {c ^ {4}} \mathcal {L} _ {\mathcal {M}} \end {aligned} with ## g ^ {\mu \nu}: ## the...

https://arxiv.org/pdf/1812.06239.pdf In this paper,the authors use ricci flow to construct Lifshitz spaces. But it is known that ricci flow is limited by Riemannian manifold, which has a positive metric. but in this paper the author use ricci flow in a lorentz manifold, whose signature...

I want to know if there is some simple metric form for Ricci curvature in dimensions generally. In this paper https://arxiv.org/abs/1402.6334 , formula (5.21), the authers seem had a simple formula for Ricci curvature like this ##R= -\frac{1}{\sqrt{-g}} \partial^\mu \big[\sqrt{-g}(g_{\mu\rho}...

Hello~ For usual Riemann curvature tensors defined: ##R^i_{qkl},## I read in the book of differential geometry that in 3-dimensional space, Ricci curvature tensors, ##R_{ql}=R^i_{qil}## can determine Riemann curvature tensors by the following relation...

Homework Statement A sub-atomic particle is near the event horizon of a black hole. Due to the nearby gravitational field, the Ricci Curvature Tensor is changing rapidly. The particle then performs quantum tunneling. Homework Equations Which version of spacetime does the tunneling particle...

After my studies of metric tensors and Cristoffel symbols, I decided to move on to the Riemann tensor and the Ricci curvature tensor. Now I noticed that the Einstein Field Equations contain the Ricci curvature tensor (R\mu\nu). Some sources say that you can derive this tensor by simply...

I've been studying the Einstein field equations. I learned that the Ricci curvature tensor was expressed as the following commutator: [∇\nu , ∇\mu] I know that these covariant derivatives are being applied to some vector(s). What I don't know however, is whether or not both covariant...

How does Ricci curvature represent "volume deficit"? Hi all, I've been reading some general relativity in my spare time (using Hartle). I'm a bit confused about something. I understand that Riemann curvature is defined in terms of geodesic deviation; the equation of geodesic deviation is...

"Visualizing" Ricci curvature Can someone help me visualize the Ricci curvature? Since it is easier to visualize a surface bending in 3-D, let's try to view this as a sheet with one spatial dimension and one time dimension and embedding into euclidean 3-D. Since the metric can always be...

Homework Statement I'm currently self-studying Carroll's GR book and get stuck by proving the following identity: K^\lambda \nabla _\lambda R = 0 where K is Killing vector and R is the Ricci ScalarHomework Equations Mr.Carroll said that it is suffice to show this by knowing: \nabla _\mu...

Hi, I've been reading through Yau's proof of the Calabi conjecture (1) and I was quite intrigued by the relation R_{i\bar{j}} = - \frac{\partial^2}{\partial z^i \partial \bar{z}^j } [\log \det (g_{s\bar{t}}) ] derived therein. g_{s \bar{t} } is a Kahler metric on a Kahler manifold (I'm...