Covariant Definition and 361 Threads

Take R2. Take a function f(x,y) defined on R2 which maps every point to a real number. The gradient of this at any point mean a vector which points in the direction of steepest incline. The magnitude of the vector is the value of the derivative of the function in that direction. Both of these...

Why is it that the covariant derivative of a covariant tensor does not seem to follow the product rule like contravariant tensors do when taking the covariant derivatives of those? Here is a visual of what I mean: This is the covariant derivative of a contravariant vector. As you can see, it...

For someone who does not already know Lie group and bundle theory, the formulation of covariant derivatives through parallel transport in the principal, and associated vector bundles, might seem unnecessarily complicated. In that light, I wondered what the virtues of the principal/associated...

Why is a covariant formalism preferred over a canonical formalism in loop quantum gravity, in simple layman terms

Can someone explain why the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector and the derivative with respect to a covariant coordinate transforms as a contravariant 4-vector.

Hi, please help me .. How can I derivative covariant and contravariant fields? as in the attached picture Thanks.. http://www.gulfup.com/?tNXcaN w.r.t alpha

When we write contravariant and covariant indices, for example for the Lorentz transformation, does it matter if we write \Lambda^\mu\,_\nu or \Lambda^\mu_\nu ? i.e. if the \nu index is to the right of the \mu or they are at the same place with respect to left-right?

Hi guys. I was reading a paper in which a calculation was done to show that in Schwarzschild space-time, if we consider a time-like circular orbit 4-velocity ##u^{\mu} = \gamma(\xi^{\mu} + \omega \eta^{\mu})## where ##\xi^{\mu}## is the time-like Killing field, ##\eta^{\mu}## is the axial...

In the context of tensor calculus, what is the difference between intrinsic derivative and covariant derivative?

I remember I have read somewhere that contravariant/covariant vectors correspond to polar/axial vectors in physics, respectively. Examples for polar/axial vectors are position, velocity,... and angular momentum, torque,..., respectively. Is this right? Can I prove that, say, any axial...

I will take the differential form of position vector r: ##\vec{r}=r\hat{r}## ##d\vec{r}=dr\hat{r}+rd\hat{r}## So, now I need find ##d\hat{r}## ##d\hat{r}=\frac{d\hat{r}}{dr}dr+\frac{d\hat{r}}{d\theta}d\theta## ##\frac{d\hat{r}}{dr}=\Gamma ^{r}_{rr}\hat{r}+\Gamma...

Hellow everybody! A simples question: is it correct the graphic representation for covariant (x₀, y₀) and contravariant (x⁰, y⁰) coordinates of black vector?

Gravitomagnetism is this thing https://en.wikipedia.org/wiki/Gravitomagnetism Like electromagnetism equations, but adapted for gravity The equations just like that are manifestly problematic. If we use four-momentum as the source( appropriate units), does the equation become covariant...

In Theodore Frankel's book, "The Geometry of Physics", he observes at page 248 that the covariant derivative of a vector field can be written as $$\nabla_X v = e_iX^j (v^i_{,j} + \omega^i_{jk} v^k)= e_i(dv^i(X) + \omega^i_k(X) v^k) = e_i (dv^i + \omega^i_k v^k)(X)$$ where ##\omega^i_k =...

Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that $$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$ without going into...

Hi. I'm trying to understand a derivation of the Bianchi idenity which starts from the torsion tensor in a torsion free space; $$ 0 = T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$$ according to the author, covariant differentiation of this identity with respect to a vector Z yields $$$ 0 =...

If the notions of covariant and contravariant tensors were not introduced,what would happen?E.g. what form will the Einstein E.q. Guv=8πTuv be changed into ?

This thread is in part belated appreciation of J.S.Engle's seemingly orderly systematic way of doing research, and good writing style. I'm not kidding: a clear widely understandable writing style helps a lot. If you want a well-written review (from basics to latest) of spinfoams he has one here...

Hello, i try to prove that ∂μFμ\nu + ig[Aμ, Fμ\nu] = [Dμ,Fμ\nu] with the Dμ = ∂μ + igAμ but i have a problem with the term Fμ\nu∂μ ... i try to demonstrate that is nil, but i don't know if it's right... Fμ\nu∂μ \Psi = \int (∂\nuFμ\nu) (∂μ\Psi) + \int Fμ\nu∂μ∂\nu \Psi = (∂\nuFμ\nu) [\Psi ]∞∞...

If I have a 4x4 Covarient Metric Tensor g_{ik}. I can find the determinant: G = det(g_{ik}) How do I find the 4x4 Cofactor of g_ik? G^{ik} then g^{ik}=G^{ik}/G

Hey all, I starting to study QED along with a slew with other materials. (I read in the QED book and when I don't understand a reference I go to Jackson's E&M and work some problems out, it has been beneficial thus far!) Most of the topics are not too far fetched but I am struggling to...

Consider a 2-sphere S2 with coordinates xμ=(θ,\phi) and metric ds2=dθ2+sin2θ d\phi2 and a vector \vec{V} with components Vμ=(0,1). Calculate the following quantities. ∇θ∇\phiVθ ∇\phi∇θVθ

I'm confused about the difference between a contravariant and covariant vector. Some books and articles seem to say that there really is no difference, that a vector is a vector, and can be written in terms of contravariant components associated with a particular basis, or can be written in...

Hello everyone! I'm trying to learn the derivation the covariant derivative for a covector, but I can't seem to find it. I am trying to derive this: \nabla_{α} V_{μ} = \partial_{α} V_{μ} - \Gamma^{β}_{αμ} V_{β} If this is a definition, I want to know why it works with the definition...

Thanks. I started a new thread, because I've seen what seemed to me a contradictory claim in Carroll's GR notes (Eq 5.38) - he says diff invariance is enough to get covariant energy conservation. I've never understood whether Carroll's claims and the ones in these papers are really...

I think this may be a simple yes or no question. I am currently reading a book Vector and Tensor Analysis by Borisenko. In it he introduces a reciprocal basis \vec{e_{i}} (where i=1,2,3) for a basis \vec{e^{i}} (where i is an index, not an exponent) that may or may not be orthogonal...

I recently came across a very cool book called Div, Grad, and Curl are Dead by Burke. This is apparently a bit of a cult classic among mathematicians, not to be confused with Div, Grad, Curl, and All That. Burke was killed in a car accident before he could put the book in final, publishable...

I am studying Riemannian Geometry and General Relativity and feel like I don't have enough practice with covariant vectors. I can convert vector components and basis vectors between contravariant and covariant but I can't do anything else with them in the covariant form. I thought converting the...

Is it enough to see the covariance of the wave equation the fourth-vector potential (\phi, \bar{A}) satisfy? I mean, is this enough to prove the covariance of Maxwell equations? The equation would be ∂_{\mu}∂^{\mu}A^{\nu} =\frac{4\pi}{c} J^{\nu}

Homework Statement Derive L_v(u_a)=v^b \partial_b u_a + u_b \partial_a v^b Homework Equations L_v(w^a)=v^b \partial_b w^a - w^b \partial_b v^a L_v(f)=v^a \partial_a f where f is a scalar. The Attempt at a Solution In the end I get stuck with something like this, L_v(u_a)w^a=v^b...

In f(R) gravity as http://en.wikipedia.org/wiki/F%28R%29_gravity , i have problem with the term [ g_ab □ - ∇_a ∇_b ] F(R) , well actually is [ ∇_b ∇_a - ∇_a ∇_b ] F(R) , but F is a function of Ricci Factor and Ricci Factor is expressed as a(t) ( scale factor ) . for the a = b = 0 i say this...

hello, please see the attached snapshot (taken from 'Problem book in relativity and gravitation'). In the last equation I think there would be no semicolon. Here is why I believe (S is scalar by the way): S;α[βγ] = 1/2 * ( S;αβγ - S;αγβ ) Now from the equation which precedes it, we have ...

Will anyone help me to under stand the covariant and contravariant vector ? And can anyone show me the derivation of dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz

Let $$f:U \to \mathbb{R}^3$$ be a surface with local coordinates $$f_i=\frac{\partial f}{\partial u^i}$$. Let $\omega$ be a one-form. I want to express $$\nabla \omega$$ in terms of local coordinates and Christoffel symboles. Where $$\nabla$$ is the Levi-Civita connection (thus it coincides with...

Hi everyone, I'm trying to prove a relation in which I need do commute covariant derivatives of a bitensor. The equation is quite long but I need to write something like this: Given a bitensor G^{\alpha}_{\beta'}(x,x'), where the unprimed indexes (\alpha,\beta, etc) are assigned to the...

Hello, Can anyone tell me the general formula for commuting covariant derivatives, I mean, given a (r,s)-tensor field what is the formula to commute covariant derivatives? I found a formula http://pt.scribd.com/doc/25834757/21/Commuting-covariant-derivatives page 25, Eq.6.18 but it doesn't...

suppose we have unit vector z=(0,0,0,1), we can use it to form a tensor z^\mu z^\nu , it is easy to check that ∂_\μ( z^\mu z^\nu=0 in Minkowski spacetime, now I want to generalize this equation to general curved spacetime, so that ∇μ ( znew^\mu znew^\nu)=0. But I am not sure how to...

Homework Statement In the oblique coordinate system K' defined in class the position vector r′ can be written as: r'=a\hat{e'}_{1}+b\hat{e'}_{2} Are a and b the covariant (perpendicular) or contravariant (parallel) components of r′? Why? Give an explanation based on vectors’ properties...

Ok, so here's my problem. I just graduated with a mathematics degree and am going full force into a physics graduate program. I'm taking a course called mathematical methods for physicists, in which the first subject is tensors. Everyone else seems to be comfortable with the material, but me...

Homework Statement Using the Leibniz rule and: \nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b \nabla_{a}\Phi=\partial\Phi Show that \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b} . The question is from Ray's Introducing Einsteins relativity, My attempt...

Hi, I am trying to study Quantum Field Theory by myself from Mandl and Shaw second edition and I am having trouble understanding the section on covariant commutation relations. I understand the idea that field at equal times at two different points commute because they cannot "communicate"...

I have been given the following problem: The covariant vector field is: \(v_{i}\) = \begin{matrix} x+y\\ x-y\end{matrix}What are the components for this vector field at (4,1)? \(v_{i}\) = \begin{matrix} 5\\ 3\end{matrix} Now I can use this information to solve the...

"Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge". http://en.wikipedia.org/wiki/Gravitoelectromagnetism Essentially the idea is to take Maxwell's equations and make a substitution...

I posted this question in homeworks section but no one can help, or seems unable to answer my question. I am hoping if I post it here, someone might be able to clear it up for me: https://www.physicsforums.com/showthread.php?t=621238 Thanks in advance.

Hi everyone, I am having a little trouble with the difference between a covariant vector and contravariant vector. The examples that I come across say that an example of a contravariant vector is velocity and that a contravariant must contra-vary with a change of basis to compensate. So...

http://en.wikipedia.org/wiki/Four-force At the bottom of that page, the author provides the generalization of four force in general relativity, where the partial derivative is replaced with the covariant derivative. However if you notice on the second term in the third equality, there is a...

Hi all, I am wondering if it is possible to derive the definition of a Christoffel symbols using the Covariant Derivative of the Metric Tensor. If yes, can I get a step-by-step solution? Thanks! Joe W.

Hi all, I am new to General Relativity and I started with General Relativity Course on Youtube posted by Stanford (Leonard Susskind's lectures on GR). So first thing to understand is transformation of covariant and contravariant vectors. Before I can understand a transformation, I would...

Homework Statement Show that \nabla_a(\sqrt{-det\;h}S^a)=\partial_a(\sqrt{-det\;h}S^a) where h is the metric and S^a a vector. Homework Equations \nabla_a V^b = \partial_a V^b+\Gamma^b_{ac}V^c \Gamma^a_{ab} = \frac{1}{2det\;h}\partial_b\sqrt{det\;h} \nabla_a\sqrt{-det\;h} (is that...

I am trying to familiarize myself with the use of fibre bundles and associated bundles but am having some problems actually making calculations. I would like to show that the covariant derivative along the horizontal lift of a curve in the base space vanishes (which should be just a matter of...