Levi-civita Definition and 79 Threads

I'm building a mental framework for the Levi-Civita connection that is intuitive to me. I start by imagining an arbitrary manifold with arbitrary coordinates embedded in a higher dimensional Euclidean space, then if I take the derivative of an arbitrary coordinate basis vector with respect to...

This is a problem from the textbook Supergravity ( by Daniel Z. Freedman and Antoine Van Proeyen ). I am trying to learn general relativity from this book. I am attempting to do the later part of the Exercise 7.14 ( on page 148 ). Basically it asks us to explicitly show that the Levi-Civita...

We had a thread long time ago concerning the Lie dragging of a vector field ##X## along a given vector field ##V## compared to the Fermi-Walker transport of ##X## along a curve ##C## through a point ##P## that is the integral curve of the vector field ##V## passing through that point. We said...

I have read some text about defining the cross product. It can be defined by both a x b = epsilon_(ijk) a^j b^k e-hat^i and a x b = epsilon^(ijk) a_i b_j e-hat^k why the a and b have opposite indice positions with the epsilon? How to understand that physically?

We know that a metric tensor raises or lowers the indices of a tensor, for e.g. a Levi-Civita tensor. If we are in ##4D## spacetime, then \begin{align} g_{mn}\epsilon^{npqr}=\epsilon_{m}{}^{pqr} \end{align} where ##g_{mn}## is the metric and ##\epsilon^{npqr}## is the Levi-Civita tensor. The...

The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...

How do I write the following expression $$\epsilon_{mnk} J_{1n} \partial_i\left[\frac{x_m J_{2i}}{|\vec{x}-\vec{x}'|}\right]$$ back into vectorial form? Einstein summation convention was used here. Context: The above expression was derived from the derivation of torque on a general current...

I am studying GR and I have these two following unresolved questions up until now: The first one concerns the Levi-Civita connection. There are two conditions which determine the affine connections. The first one is that the connection is torsion-free (which is true for space-time and comes...

Hello all, I was just introduced the Levi-Civita symbol and its utility in vector operations. The textbook I am following claims that, for basis vectors e_1, e_2, e_3 in an orthonormal coordinate system, the symbol can be used to represent the cross product as follows: e_i \times e_j =...

Below is my attempted solution: $$\epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k$$ $$\Rightarrow (\delta_{ik} \, \delta_{jm} - \delta_{im} \, \delta_{jk}) \epsilon_{ijm} \, a_k$$ $$\Rightarrow \delta_{ik} \, \delta_{jm} \, \epsilon_{ijm} \, a_k - \delta_{im} \, \delta_{jk}...

I am trying to understand the following: $$ \epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp} $$ Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices...

εikl εjmngkmMkn = εikl εjknMkn = (in book it changed sign to -εikl εjknMkn - Why? ) By identity εikl εnjkMln = (δinδkj - δijδkn)Mkn = ? I then get .. Mji - δij Mnn ( is this correct ?) There 's more to the question but if can get this part right, I should be able to complete the...

I've got here so far, but first of all I'm not sure if i did it right till the last line and second, if I've been right, i do not know what to do with the rest. should i consider each of levi-civita parentheses in the last line zero? and one additional question about the term in the first line...

I assumed that this would be a straightforward proof, as I could just make the substitution l=j and m=l, but upon doing this, I end up with: δjj δkl - δjl δkj = δkl - δlk Clearly I did not take the right approach in this proof and have no clue as to how to proceed.

Hi all, I'm doing undergraduate research this summer, and a few times I've been told to calculate a term with the following form: ∈abcdpaqbkcsd, where p,q,k and s are four vectors (four-momentum, spin, etc). Now I know this ends up calculating exactly like a 4x4 determinant, I'm just not quite...

Hello! I have learned Riemannian Geometry, so the only connection I have ever worked with is the Levi-Civita connection(covariant derivative of metric tensor vanishes and the Chrystoffel symbols are symmetric). When performing a parallel transport with the L-C connection, angles and lengths are...

Homework Statement Let V be a Levi-Civita connection. a) Let ##f \in F(M)##, (function defined on the manifold M). Show that: ##\nabla_\mu \nabla_\nu f = \nabla_\nu \nabla_\mu f ## b) Let ##\omega \in \Omega^1(M)## (one form on M). Show that ##d \omega = (\nabla_\mu \omega)_\nu dx^\mu \wedge...

Hello everyone, my question concerns the following: Though widely used, there does not seem to be any standard reference where the common symmetrization and anti-symmetrization identities are rigorously proven in the general setting of ##n##-dimensional pseudo-Euclidean spaces. At least I have...

Homework Statement definition of εijk εijk=+1 if ijk = (123, 231, 312) εijk = −1if ijk = (213, 321, 132) , (1.1.1) εijk= 0,otherwise . That is,εijk is nonzero only when all three indices are different. From the definition in Eq. (1.1.1), show that...

If given a position vector defined for a orthogonal curvilinear coordinate system HOW would the matrices that make up the Levi Civita 3x3x3 matrix remain the same? "Levi Civita 3x3x3 is said to be independent of any coordinate system or metric...

Homework Statement i am showing the invariance of the Levi-Civita symbol in 4 dimensions The Attempt at a Solution $$\varepsilon_{ijkl}'=R_{im}R_{jn}R_{kp}R_{lt}\varepsilon_{mnpt}$$...

first of all english is not my mother tongue sorry. I want to ask if you can help me with some of the properties of the levi-civita symbol. I am showing that $$\epsilon_{ijkl}\epsilon_{ijmn}=2!(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$ so i have this...

first of all english is not my mother tongue sorry. I want to ask if you can help me with some of the properties of the levi-civita symbol. I am showing that $$\epsilon_{ijkl}\epsilon_{ijmn}=2!(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$ so i have this...

I set eyes on the next formulas: \begin{align} E_{\alpha \beta \gamma \delta} E_{\rho \sigma \mu \nu} &\equiv g_{\alpha \zeta} g_{\beta \eta} g_{\gamma \theta} g_{\delta \iota} \delta^{\zeta \eta \theta \iota}_{\rho \sigma \mu \nu} \\ E^{\alpha \beta \gamma \delta} E^{\rho \sigma...

Homework Statement Hi,I have got a question as follow: Compute the commutation relations of the position operator R and the angular momentum L.Deduce the commutation relations of R^2 with the angular momentum L Homework EquationsThe Attempt at a Solution In fact I have got the solutions to...

Homework Statement Show that g(d \sigma ^k, \sigma _p \wedge \sigma _q) = \Gamma _{ipq} - \Gamma _{iqp}Homework Equations Given $$\omega_{ij}=\hat e_i \cdot d \hat e _j = \Gamma_{ijk} \sigma^k$$, we can also say that $$d \hat e_j = \omega^i_j \hat e_i$$. Where $$\sigma^k, \sigma_p, \sigma_q$$...

How to proof that εμνρσ Rμνρσ =0 ? Thanks.

Hi all, Can someone explain me the last two steps? I don't know why suddenly there is a term with only two indices, and then in the last step you do something distributive and again three indices. Thanks in advance

1. The problem is: ( a x b )⋅[( b x c ) x ( c x a )] = [a,b,c]^2 = [ a⋅( b x c )]^2 I am supposed to solve this using index notation... and I am having some problems. 2. Homework Equations : I guess I just don't understand the finer points of index notation. Every time I think I am getting...

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...

Hi there, Can two Levi-Civia symbols ## \epsilon^{ckl} \epsilon_{ibj} ## reduced to one with indices ## \epsilon_{kij} ## ? WHERE b and c run from 4 to 5, the other indices run from 1 to 3 and both symbols are multiplied by the following matrices: ## A_{ib}^c ~ B^{kl} ~ C_j ~\epsilon^{ckl}...

Let ##(M,g)## be an ##n##-dimensional pseudo-Riemannian manifold of signature ##(n_+, n_-)## and define the Levi-Civita symbol via $$\varepsilon_{i_1 \dots i_n} \, \theta^{i_1 \dots i_n} = n! \, \theta^{[1 \dots n]} = \theta^1 \wedge \dots \wedge \theta^n$$ where ##\theta^1, \dots, \theta^n##...

Hi, I am going through the derivation of an instanton solution (n=1) in Srednicki Chp. 93. Specifically, I went through eqn.s 93.29-93.38. However the sign of the Levi-Civita Symbol is bugging me: It says that in 4D Euclidean space, \epsilon^{1234}=+1 in Cartesian coordinates implies...

Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...

Hi. Is using the Levi-Civita symbol to calculate cross-product combos like A x (B x C) allot faster than just using the good old determinant method? I ask because my lecturer in electrodynamics 2 told us it is better, but it seems to me that it's going to cost me time to learn to use this...

i really lost with this. i see two possibilities: (1) something like, \epsilon_{abc}\partial_{a}A_{b}e_{c} with a,b,c between 1 and 5 or (2)like that \epsilon_{abcde}\partial_{a}A_{b} one of the options nears correct? thank's a lot

Homework Statement Hi every one, I am really confused on how to calculate the inverse of this Levi Civita symbole's element \epsilon^{tabc}, I tried to used this equation...

Homework Statement Hi all, I'm having trouble evaluating the product g_{αβ}ϵ^{αβγδ}. Where the first term is the metric tensor and the second is the Levi-Civita pseudotensor. I know that it evaluates to 0, but I'm not sure how to arrive at that. The Attempt at a Solution My first thought...

One of the properties of the unique Levi-Civita connection is that it preserves the metric tensor at each point's tangent space, allowing the definition of invariant intervals between points in the manifold. I'd be interested in clarifying: when the metric preserved by the L-C connection is a...

I have been trying to think about the Levi-Civita tensor in the context of Group Theory. Is there a group that it is symmetric to? I'm sorry if this is a double post but I don't think my original identical post submitted correctly. Thanks, Nate

Homework Statement Whats up guys! I've got this question typed up in Word cos I reckon its faster: http://imageshack.com/a/img5/2286/br30.jpg Homework Equations I don't know of any The Attempt at a Solution I don't know where to start! can u guys help me out please? Thanks!

Homework Statement Hey everyone, So I've got to prove a couple of equations to do with the Levi-Civita tensor. So we've been given: \epsilon_{ijk}=-\epsilon_{jik}=-\epsilon_{ikj} We need to prove the following: (1) \epsilon_{ijk}=-\epsilon_{kji} (2)...

I am having trouble establishing a process to verify various identities for problems in index notation. Description of Problem Verify that \epsilon_{ijk}\epsilon_{iqr}=\delta_{jq}\delta_{kr}-\delta_{kq}\delta_{jr} Attempt at Solution I know that the term is only positive if there is an...

Hello, I consider the permutations \sigma_i, where i\in \{1,\ldots,n\}, of the following kind: \sigma_i is obtained by choosing the i-th element from (1,..,n) and by shifting it to the first position; for instance \sigma_3 = (3,1,2,\ldots,n). The parity of \sigma_i is clearly (-1)^{i-1}. For...

Can someone please explain to me why \epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial A_k}{\partial x_j} = 0 where A is a constant vector field.

In many physics literature I have encountered, one of the properties of Levi-Civita tensor is that ε_{ijk}ε_{lmn}is equivalent to a determinant of Kronecker symbols. However this is only taken as a given theorem and is never proved. Is there any source which has proven this property?

Homework Statement I'm trying to understand a proof of the LC-KD identity involving determinants (see attachment), from the book Introduction to Tensor Calculus and Continuum Mechanics by Herinbockel. What is the author saying in the last line of text? How can we sum the deltas in the upper...

I am reading Landau and Lifgarbagez's Classical Theory of Fields, 4th edition. In the beginning of page 18, the completely antisymmetric unit tensor is said to be a pseudotensor, because none of it components changes sign when we change the sign of one or three of the coordinates. Then, in...

Hi all, To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric. Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics...

Can somebody show me how \epsilon_{mni}a_{n}(\epsilon_{ijk}b_j c_{k}) Turns in to \epsilon_{imn}\epsilon_{ijk}a_{n}b_j c_{k} Something about the first \epsilon I'm not seeing here when the terms are moved around.