Covariant Definition and 361 Threads

Homework Statement The problem concerns how to transform a covariant differentiation. Using this formula for covariant differentiation and demanding that it is a (1,1) tensor: \nabla_cX^a=\partial_cX^a+\Gamma^a_{bc}X^b it should be proven that \Gamma'^a_{bc}=...

A little stuck while working through a derivation. Hope someone can help. Homework Statement Starting from -\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad})+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0 I need to obtain the Killing equations, i.e...

I'm teaching myself about connections and came across something I am not completely sure about. The text I am using defines a connection as taking in two vector fields and outputting a vector field. However, later when discussing covariant derivatives along a curve I see this equation...

Hi, I'm having problems following a derivation for the covariant derivative. I've shown the line where I'm having trouble: http://img15.imageshack.us/img15/49/covariantderivative.jpg The general argument being used is that if the covariant derivative must follow the product rule it can...

Rovelli has given his team and himself just seven months to advance LQG to a new stage. Why do I call this "throwing the eagle"? Because a Roman general would on occasion hurl his legion's eagle standard into the opposing army's midst, confident his side could rout their foes the recover it...

Homework Statement For a perfect fluid verify that the spatial components of T^{\mu \nu};_{\nu} = 0 in the Newtonian limit reduce to (\rho v^{i}),_{t} + (\rho v^{i} v^{j}),_{j} + P,_{i} + \rho \phi ,_{i} Homework Equations Metric ds^{2} = -(1+2 \phi )dt^{2} + (1-2 \phi) (dx^{2} +...

Hello once again. I'm trying to understand the relation between the superspace representation of the SUSY generators Q_\alpha,\overline Q_{\dot\beta} and the covariant derivatives on superspaces D_\alpha, \overline D_{\dot\beta}: Q_\alpha = \frac{\partial}{\partial\theta^\alpha} -...

Could anyone tell me about the covariant derivative with respect to the connection?

Is this correct? \nabla _{\vec{p}} \vec{p} = (\nabla_a \vec{p} ) p^a =< (\nabla_a p^0 ) p^a, (\nabla_a p^1 ) p^a , (\nabla_a p^2 ) p^a, (\nabla_a p^3 ) p^a > (where the a's are summed from 0 to 3)

If I use the following Lagrangian: \mathcal{L} = \frac{1}{2} m v^2 + e A \cdot v/c = \frac{1}{2} m \dot{x}_\mu \dot{x}^\mu + e A_\nu \dot{x}^\nu /c I can arrive at the Lorentz force equation in tensor form: m \ddot{x}_\mu &= (q/c) F_{\mu\beta} \dot{x}^\beta details offline...

in my GR book, it claims that integral of a covariant divergence reduces to a surface term. I'm not sure if I see this... So, is it true that: \int_{\Sigma}\sqrt{-g}\nabla_{\mu} V^{\mu} d^nx= \int_{\partial\Sigma}\sqrt{-g} V^{\mu} d^{n-1}x if so, how do I make sense of the d^{n-1}x term? would...

...of the four-momentum vector. Why is the energy of a particle identified with p0 instead of p0? Is there a theoretical basis for this, or was it simply observed that p0 is conserved in a larger set of circumstances?

In developing the Yang-Mills Lagrangian, Wikipedia defines the covariant derivative as \ D_ \mu = \partial _\mu + A _\mu (x) . Is A_mu to be taken as a 1-form, so that \ D _\mu \Phi = \partial _\mu \Phi + A _\mu (x) or an operator on \Phi, such that \ D _\mu \Phi = \partial...

Homework Statement Let e_{i} with i=1,2 be an orthonormal basis in two-dimensional Euclidean space ie. the metric is g_{ij} = \delta _{ij}. In the this basis the vector v has contravariant components v^{i} = (1,2). Consider the new basis e_{1}^{'} = 5e_{1} - 2e_{2} e_{2}^{'} = 3e_{1} - e_{2}...

If there is a contravariant vector v=aa+bb+cc with a reciprocal vector system where [abc]v=xb×c+ya×c+za×b would the vector expressed in the reciprocal vector system be a covariant vector? Is there any connection between the reciprocal vector system of a covariant vector and a...

Can you give me the definition of exterior covariant derivative or any reference web page ? Wiki does not involve enough info.I am not able to do calculation with respect to given definition there. Thanks in advance

Here are some papers on the covariant entropy bound conjectured by Raphael Bousso http://arxiv.org/abs/hep-th/9905177 http://arxiv.org/abs/hep-th/9908070 http://arxiv.org/abs/hep-th/0305149 It would be a significant development if the conjectured bound could be proven to hold in LQC...

Hi. I am attempting to gain some intuition for what the covariant derivative of a tensor field is. I have a good intuition about the covariant derivative of vector fields (measuring how the vector changes as you move in a particular direction), and I understand how to extend the covariant...

When we derive equation of motion by variation of the action, we use rules of ordinary differentiation and integration. So only ordinary derivatives can appear in the equation. Now in general relativity we are supposed to replace all those ordinary derivatives by covariant derivatives. Is that...

Sorry, but some of you may think this is a stupid question. (I'm only 16 years old.) I have just now gotten into the field of tensors and topology, after studying vector calculus and differential equations and I have two questions: a) What exactly is a covector? b) What is the...

Hi, I'm trying to verify that the covariant derivative of the metric tensor is D(g) = 0. But I have a few questions: 1) This is a scalar 0 or a tensorial 0? Because it is suposed that the covariant derivative of a (m,n) tensor is a (m,n+1) tensor, and g is a (0,2) tensor so I think this 0...

Homework Statement Help! I wish to prove the following important statements: (1) The presence of Christoffel symbols in the covariant derivative of a tensor assures that this covariant derivative can transform like a tensor. (2) The reason for this is because, under transformation, the...

Homework Statement Show that \frac{\partial\phi}{\partial x^{\mu}} is a covariant four vector . Homework Equations All covariant four vector transformations . The Attempt at a Solution I really didn't understand what question implies . How can this vector be showed as being a...

Homework Statement I am trying to show that the components of the covariant derivative [tex] \del_b v^a are the mixed components of a rank-2 tensor. If I scan in my calculations, will someone have a look at them? Homework Equations The Attempt at a Solution

Hi. I'm considering the covariant derivative \nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma_{\mu\nu}^\lambda V^\lambda in spherical coordinates in flat 3D space (x = r cos sin, y = r sin sin, z = r cos; usual stuff). Now I wrote down the gradient of a scalar function f, for which I got...

Homework Statement Can someone explain/help me prove the formulas \vec{e_a}' = \frac{ \partial{x^b}}{\partial x'^a} \vec{e_b} \vec{e^a}' = \frac{ \partial{x'^a}}{\partial x^b} \vec{e^b} I do not understand why the partial derivative flip? Homework Equations The Attempt at a Solution

just a quick query, I know that, \nabla_0 A_{\alpha}= \partial_0 A_{\alpha} - \Gamma^{\beta}_{0 \alpha} A_{\beta} But what does \nabla^0 A_{\alpha} equal?

I am not as well read as most of the people in here so I thought I would ask you guys first. What work has been done in the way of developing a covariant field theory? I'm going to ramble for just a little bit so try to follow. It seems to me that QFT is built on two principles Poincare...

In anglo-american literature -a physical quantity is invariant if it has the same magnitude in all inertial reference frame, -an expression relating more physical quantities is covariant if it has the same algebraic structure in all inertial reference frames (rr-cctt) ? Thanks in advance

If we work in cartesian coordinates, we say for instance, that D_x \phi = \left( \frac{\partial}{\partial x} + i g \sum_a T_a A^a_x \right) \phi where g is the gauge coupling, and \{T^a\} are the generators of the gauge group, and \{A^a_\mu\} is the gauge vector field. But what happens when...

I'm not really sure where to put this, so I thought it post it here! I'm reading through my GR lecture notes, and have come across a comment that has confused me. I quote Now, I don't really see how this is true. For example, consider a scalar field f. The covariant derivative of this is...

I've heard of something called a covariant derivative. what motivates it and what is it?

Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...

Is there any relationship between the Lie (\pounds) and covariant derivative (\nabla)? Say I have 2 vector fields V, W and a metric g, the Lie and covariant derivative of W along V are: \pounds_{V}W = [V,W] V^\alpha \nabla_\alpha W^\mu = V^\alpha \partial_\alpha W^\mu + V^\alpha...

Dear Fellows, Do anyone have an idea of whether there must be a system tensor in order to be able to transform from the covariant form of a certain tensor to its contravariant one? This is a bit important to get rigid basics about tensors. Schwartz Vandslire...

Does anyone know the physical (or historical) basis for the terms covariant and contravariant? I'm guessing a particular class of mapping always tranforms components (of ..?) in exactly two different ways, so I'm wondering what the mappings are (Change of coordinate charts? Lorentz...

It's physics based but actually a maths question so I'm asking it here rather than the physics forums. I = \int \mathcal{L}\; d^{4}x I is invariant under some transformation \delta_{\epsilon} if \delta_{\epsilon}\mathcal{L} = \partial_{\mu}X^{\mu} for some function/tensor/field thingy X^{\mu}...

I find the covariant version of loop quantum gravity http://arxiv.org/abs/gr-qc?papernum=0608135 more appealing than the usual LQG approach. What the experts think?

That's a crucial point of GR ! And I have always problems with that. Back to the basics, with your help. Thanks Michel

Why we have to definte covariant derivative?

I read some books and see that the definition of covariant tensor and contravariant tensor. Covariant tensor(rank 2) A'_ab=(&x_u/&x'_a)(&x_v/&x'_b)A_uv Where A_uv=(&x_u/&x_p)(&x_u/&x_p) Where p is a scalar Contravariant tensor(rank 2) A'^uv=(&x'^u/&x^a)(&x'^v/&x^b)A^ab Where A^ab=dx_a...

If we define the Gradient of a function: \uparrow u= Gra(f) wich is a vector then what would be the covariant derivative: \nabla _{u}u where the vector u has been defined above...i know the covariant derivative is a vector but i don,t know well how to calculate it...thank you.

The covariant derivative is A^\mu_{\sigma} = \frac{\partial A^\mu}{\partial x_{\sigma}} + \Gamma^\mu_{\sigma \alpha}A^\alpha ... why?

How is the covariant derivative derived?

Can anyone explain to me what is contravariant and covariant? I just know that they are tensors with specific transformation properties (from website of MathWorld), i also know that the relation between two is the -ve sign. Then dose it mean that: given a 4-velocity of a particle is the...

I am trying to solve an exercise from MTW Gravitation and the following issue has come up: Let D denote uppercase delta (covariant derivative operator) [ _ , _ ] denotes the commutator f is a scalar field, and A and B are vector fields Question: Is it true that [D_A,D_B]f = D_[A,B]f ?

We all know that time-ordering depends on the choice of Lorentz frame. So my question is somewhat obvious... Please give me a hint on where to look up that problem, eg. why the S-matrix theory is covariant. I guess the time-ordering prescription is implicitly defined for each single...

Hello! I am trying t solution Navier-Stokes equation and I cannot find something about Laplacian. I would like to solution Laplace’a equation for each component.I am trying to transform cylindrical coordinate. I would like to search equation for covariant derivative. For divergence of a...

I've already handed in my (I can only assume) incorrect solution, but I just felt like posting, though I'm not sure if anyone will be able to help. I have a rank-2 covariant tensor, T sub i,j. This can be written in the form of t sub i,j + alpha*metric tensor*T super k, sub k (I hope my...

Since there are some equations in my question. I write my question in the following attachment. It is about the covariant derivative of a contravariant vector. Thank you so much!