Covariant Definition and 361 Threads

Hi all, I'm trying to figure out the link between the connection coefficients (Christoffel symbols), the propagator, and the coordinate description of the covariant derivative with the connection coefficients. As in...

Homework Statement My teacher solved this in class but I'm not understanding some parts of tis solution. Show that \nabla_i V^i is scalar. Homework Equations \nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \Gamma^{i}_{ik} V^{k} The Attempt at a Solution To start this...

torsion --> covariant deriv of det(g) non-zero? I am missing something here. This paper makes the case (on page 5) that for non-vanishing torsion, the usual invariant volume element \sqrt{-g}d^4x is not appropriate because the covariant derivative of sqrt(-g) is non-zero. This perplexes me...

Quick question. Suppose we have a manifold with a metric and a metric compatible symmetric connection. Suppose further that we have a smooth vector field V on this manifold. I see two ways to take the derivative of this vector field. I can regard my vector field as a vector-valued 0-form...

Homework Statement In Wald's text on General Relativity he makes an assertion that I'm not sure why it is allowed mathematically. Here's the basic setup: Let \omega_{b} be a dual vector, \nabla_{b} and \tilde{\nabla}_{b} be two covariant derivatives and f\in\mathscr{F}. Then we may let...

Can the covariant components of a vector, v, be thought of as v multiplied by a matrix of linearly independent vectors that span the vector space, and the contravariant components of the same vector, v, the vector v multiplied by the *inverse* of that same matrix? thinking about it like that...

Can the covariant components of a vector, v, be thought of as v multiplied by a matrix of linearly independent vectors that span the vector space, and the contravariant components of the same vector, v, the vector v multiplied by the *inverse* of that same matrix? thinking about it like that...

Hi there. This is my first time working with tensors, so I have to break the ice I think. I have this exercise, which I don't know how to solve, which says: If V=V_1...V_n is a first order covariant tensor, prove that: T_{ik}=\frac{\partial V_i}{\partial x^k}-\frac{\partial V_k}{\partial x^i}...

Hi, I have some more questions about this stuff, which, as always, confuses me. I am (extremely slowly) working through Biship and Crittenden, and I'm pretty much at the point where I don't think I can understand it much at all, so, I think instead of trying to go through it all not knowing...

Homework Statement So my question is related somehow to the Fierz Identities. I'm taking a course on QFT. My teacher explained in class that instead of using the traces method one could use another, more intuitive, method. He said that we could use the fact that if we garante that we have the...

Hi I am trying to learn about covariant and contravariant vectors and derivatives. The videos I have been watching talk about displacement vector as the basis for contravariant vectors and gradient as the basis for covariant vectors. Can somone tlel me the difference between displacemement...

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.

Hi there! I saw this exercise that we have to calculate the covariant derivative of a vector field (in polar coordinates). Most of them equals zero, but two of them are non-zero, sugesting that this vector field is not constant. What i want to understand is the physical meaning of this values...

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}...

It recently came to my attention that there exists two "kinds" of Lorentz invariants: the covariant and the noncovariant ones. The covariant ones would be Lorentz scalars e.g. fully contracted Lorentz tensors. If one applies the Lorentz transformation to a covariant Lorentz scalar, one would...

The connection \nabla is defined in terms of its action on tensor fields. For example, acting on a vector field Y with respect to another vector field X we get \nabla_X Y = X^\mu ({Y^\alpha}_{,\mu} + Y^\nu {\Gamma^\alpha}_{\mu\nu})e_\alpha = X^\mu {Y^\alpha}_{;\mu}e_\alpha and we call...

Homework Statement Using the definition of divergence d(i_{X}dV) = (div X)dV where X:M\rightarrow TM is a vector field, dV is a volume element and i_X is a contraction operator e.g. i_{X}T = X^{k}T^{i_{1}...i_{r}}_{kj_{2}...j_{s}}, prove that if we use Levi-Civita connection then the...

My book defines the covariant derivative of a tangent vector field as the directional derivative of each component, and then we subtract out the normal component to the surface. I am a little confused about proving some properties. One of them states: If x(u, v) is an orthogonal patch, x_u...

Hi In ch84, Srednicki is considering the gauge group SU(N) with a real scalar field \Phi^a in the adjoint rep. He then says it will prove more convienient to work with the matrix valued field \Phi=\Phi^a T^a and says the covariant derivative of this is...

Hi, I am familiar with the covariant derivative of the tangent vector to a path, \nabla_{\alpha}u^{\beta} and some interesting ways to use it. I am wondering about \nabla_{\alpha}x^{\beta}=\frac{\partial x^\beta}{\partial...

In their article [Integrals in the theory of electron correlations, Annalen der Physik 7, 71] L.Onsager at el. write: By resolving the vector \vec{s} into its contravariant components in the oblique coordinate system formed by the vectors \vec{q} and \vec{Q} it is possible to reduce the region...

Hi there, I was doing some calculations with tensors and ran into a result which seems a bit odd to me. I hope someone can validate this or tell me where my mistake is. So I have a normal orthonormal frame field \{E_i\} in the neighbourhood of a point p in a Riemannian manifold (M,g), i.e...

Homework Statement Show that \frac{{\partial}g^c{^d}}{{\partial}g_a{_b}}=-\frac{1}{2}(g^a{^c}g^b{^d}+g^b{^c}g^a{^d}) Homework Equations The Attempt at a Solution It seems like it should be simple, but I just do not see how to come up with the above solution. This is what I am coming...

Is general relativity REALLY generally covariant?

Why is it that a contravariant tensor must be contracted with a covariant tensor, and vice versa? Why is this so?

On a 2 dimensional Riemannian manifold how does one derive the covariant derivative from the connection 1 form on the tangent unit circle bundle?

The question is in the pdf file,thank you!:smile: M is a Riemannian manifold, $\vdash$ is a global connection on M compatible with the Riemannian metric.In terms of local coordinates $u^1,...,u^n$ defined on a coordinate neighborhood $U \subset M$, the connection $\vdash$ is...

The covariant derivative is different in form for different tensors, depending on their rank. What about other mathematical entities? The electromagnetic field A is a vector, but it has complex values. Is the covariant derivative different for complex valued vectors? And what about...

I'm having trouble understanding the proof/solution below (please see photo, I also wrote out the problem below). I highlighted the part of my problem in red (in the picture attached). Basically I'm not sure what identity they use to get the Kronecker delta after differentiating or whether they...

http://arxiv.org/abs/1109.1290 [B]Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint[/B Authors: Emanuele Alesci, Thomas Thiemann, Antonia Zipfel (Submitted on 6 Sep 2011) Abstract: It is often emphasized that spin-foam models could realize a projection on...

We know that in GR it is not possible for arbitrary spacetimes to define a conserved energy by using a 3-integral. There are some obstacles like the covariant conservation law DT = 0 (D = covariant derivative; T = energy-momentum-tensor) does not allow for the usual dV integration (like dj...

If we consider have an unique external world therefor all people who want to describe this, should portray one thing. it seems logically that all people with different situation, either they are stationary or they move with constant speed or variable speed, describe world uniquely. this...

I am aware that the following operation: mathbf{M}_{ij} \delta_{ij} produces mathbf{M}_{ii} or mathbf{M}_jj However, if we have the following operation: mathbf{M}_{ij} \delta^i{}_j will the tensor M be transformed at all? Thank you for your time.

Homework Statement The problem is from Mathematical Methods in the Physical Sciences, 3rd Ed. Ch10, Sec. 10, Q4. My question is a bit subtle as I have actually figured out the problem, just that I don't understand my solution. The problem reads: 4) What are the physical components...

hi, I understand that Tab,b=0 because the change in density equals the negative divergence, but why do the christoffel symbols vanish for Tab;b=0?

what would Rabcd;e look like in terms of it's christoffels? or Rab;c

How can the derivative of a basis vector at a point be the linear combination of tangent vectors at that point? For example, if you take a sphere, then the derivative of the polar basis vector with respect to the polar coordinate is in the radial direction. How can something in the radial...

every people know that covariance principle is important in physics. before Lorentz transformations and special relativity, how we can check covariance principle about Maxwell 's equations?

I'm pretty comfortable with special relativity, and at least familiar with the principles of the general theory, but recently I've tried to learn SR using tensors. It is my first foray into this branch of mathematics. I understand they're handy because they represent invariant objects, but the...

As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial...

As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial...

Hey everyone, I am reading a Schaum's Outline on Tensor Calculus and came to something I can't seem to understand. I'm admittedly young to be reading this but so far I've understood everything except this. My question is: what is the difference between a contravariant tensor and a covariant...

What is the difference between covariant and manifestly covariant? And is this correct? The equation for covariant differention: \nabla_\lambda T^\mu=\frac{\partial{T^\mu}}{\partial{x^{\lambda}}}+{\sum}_{\rho}{\Gamma}^{\mu}_{\rho \lambda}T^{\rho} And equation is manifestly coverint if I...

In my relativity notes, I have several remarks like the following one: "The Lorentz condition on the potentials can be written in manifestly covariant form in this way: \partial_i A^i = 0 , where the A^i are the components of the 4-potential." This made me realize I probably have not...

I have a question about Garrity's formula at the top of p. 125, here, for a function from the set of 2-form fields to the set of tangent vector fields, together with the formula on p. 123 for the exterior derivative of a 1-form field and Theorem 6.3.1 on p. 125 (Garrity: All the Mathematics you...

Covariant Uncertainty Principle Hi, is it possible to write the uncertainty principle as a dot product like: \eta^{\alpha\beta}\Delta x_{\alpha}\Delta p_{\beta}\geq\hbar or even to generalize it as g^{\alpha\beta}\Delta x_{\alpha}\Delta p_{\beta}\geq\hbar ?

My apologies about lack of precision in nomenclature. So I wanted to know how to express a certain idea about choice of basis on a manifold... Let's suppose I am solving a reaction-diffusion equation with finite elements. If I consider a surface that is constrained to lie in a flat plane or...

Homework Statement Given a matrix {latex] A_11 =A_22 = 0 A_12 =A_21 = x/y +y/x [ /latex] Find the contravariant components in polar coordinates. Answer: [itex] A_11 = 2 A_22 = -2/r^2 A_12 = 2cot(2 /theta)/r [ /latex] Homework Equations I used the polar coordinates metric to raise...

And if so, How? From the post 15 and 16 of the thread https://www.physicsforums.com/showthread.php?t=474719" But total charge and total current, Q and I, do form a 4-vector, don't they? There seem to be two ways to solve this, but I can't figure out which one is right. Properly...