Covariant Definition and 345 Threads

Homework Statement Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)## Homework Equations ##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ## ##D_vU^ρ = ∂_vU^ρ +...

I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions: 1) Wald first defines...

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...

1) I read different texts on Contravariant , Covariant vectors. 2) Contravariant - they say is like vector . Covariant is like gradient From what I see they have those vector spaces because it eventually helps get scalar out of it if we multiply contravariant by covariant Also Contravariant...

I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a connection and how it relates to the covariant derivative. As I understand it a connection ##\nabla...

Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...

What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?

I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922 If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...

Hi! Let ##T^{ik}## be the stress-energy-tensor, and ##v_k## some future-pointing, time-like four vector. How can I see that the object ##T^{ik}v_k## is future-pointing and not space-like? Thank you for your help!

I know if the number of coordinates are same in both the old and new frame then A.B=A`.B` . But if the number of coordinates are not same in both old and new frame then A.B=0 means that both the vectors A and B are perpendicular. Why is it so that if the number of coordinates of both the frames...

I now study general relativity and have a few questions regarding the mathematical formulation: 1) What ist the relation between an connection and a covariant derivative? Can you explain the exact difference? 2) One a lorentzian manifold, what ist the relation between the...

Homework Statement Hi all, I currently have a modified Einstein-Hilbert action, with extra terms coming from some vector field A_\mu = (A_0(t),0,0,0), given by \mathcal{L}_A = -\frac{1}{2} \nabla _\mu A_\nu \nabla ^\mu A ^\nu +\frac{1}{2} R_{\mu \nu} A^\mu A^\nu . The resulting field...

I assume that all the fundamental physics known - as of today - can be reduced to quantum general covariant fields (including spacetime itself to be seen as a field of those...). Now, sorry if my question is quite abstract and based on tomorrow's hypothetical new physics, but would it be against...

Urs Schreiber submitted a new PF Insights post Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively Continue reading the Original PF Insights Post.

If R is Ricci scalar ∇i∇j F(R) = ? , where ∇i is covariant derivative.

Homework Statement This is really 3 questions in one but I figure it can be grouped together: 1. The vector A = i xy + j (2y-z2) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis. 2. Transform the...

Homework Statement Consider the fermionic part of the QCD Lagrangian: $$\mathcal{L} = \bar\psi (\mathrm{i} {\not{\!\partial}} - m) \psi \; ,$$ where I used a matrix notation to supress all the colour indices (i.e., ##\psi## is understood to be a three-component vector in colour space whilst...

Hello all, I'm trying to calculate a commutator of two covariant derivatives, as it was done in Caroll, on page 122. The problem is, I don't get the terms he does :-/ If ##\nabla_{\mu}, \nabla_{\nu}## denote two covariant derivatives and ##V^{\rho}## is a vector field, i need to compute...

I'm having trouble understanding those concepts in the title. Can someone explain those concepts in an easy to understand manner? Please don't refer me to a wikipedia page. I know some linear algebra and multi-variable calculus. Thank you.

For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?

Homework Statement Hi I am reviewing the following document on tensor: https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Homework Equations In the middle of page 27, the author says: Now, using the covariant representation, the expression $$\vec V=\vec V^*$$...

Homework Statement So I've been trying to derive field strength tensor. What to do with the last 2 parts ? They obviously don't cancel (or do they?) Homework EquationsThe Attempt at a Solution [D_{\mu},D_{\nu}] = (\partial_{\mu} + A_{\mu})(\partial_{\nu} + A_{\nu}) - (\mu <-> \nu) =...

Hii! Newton's law of gravity is ∇.(∇Φ) = 4πGρ. A book on GR gives a suggestion to make it Lorentz covariant by using de' Alembertian operator on 'Φ' in the LHS of above equation instead of Laplacian. Then it explains that this won't work because we have to include in 'ρ' all the energy...

Hey all, I'm just starting into GR and learning about tensors. The idea of fully co/contravariant tensors makes sense to me, but I don't understand how a single tensor could have both covariant AND contravariant indices/components, since each component is represented by a number in each index...

I want to prove that differentiation with respec to covariant component gives a contravariant vector operator. I'm following Jackson's Classical Electrodynamics. In the first place he shows that differentiation with respecto to a contravariant component of the coordinate vector transforms as the...

Hi I am doing some exam revision for an EM class and I'm trying to understand a few things about this derivation. Specifically in equation 18.23 why do we not consider the derivative of the four velocity i.e ##\partial^{\alpha}U^\beta## Then going from 18.23 to 18.25 why is...

I was reading through hobson and my notes where the covariant acts on contravariant and covariant tensors as \nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma \nabla_\alpha V_\mu = \partial_\alpha V_\mu - \Gamma^\gamma_{\alpha \mu} V_\gamma Why is there a minus...

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...

Hello, I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference \vec{V} = \vec{V}(M') - \vec{V}(M) : In order to evaluate this difference, we do a parallel transport of \vec{V}(M') at point...

As we can not meaningfully compare a vector at 2 points acted upon by this operator , because it does not take into account the change due to the coordinate system constantly changing, I conclude that the elementary differential operator must describe a change with respect to space-time, How do...

Hey all, I've just started tensor analysis but do not understand why in contravarient uses 1 and covarient uses 2, could someone please explain these? Perhaps my understanding of the definitions is causing me to misunderstand why its written like this. Any help appreciated.

E.g - considering co variant differentiation, The issue with the normal differentiation is it varies with coordinate system change. Covariant differentiation fixes this as it is in tensor form and so is invariant under coordinate transformations.'If a tensor is zero in one coordinate system...

Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You

I am reading a notes about tensor when I came across this which the notes did not elaborate more on it. As a result I don't quite understand why. Here it is : " Note that we mark the covariant basis vectors with an upper index and the contravariant basis vectors with a lower index. This may...

Not sure where to post this thread. That being said, can someone explain to me simply what covariant and contravariant tensors are and how covariant and contravariant transformation works? My understanding of it from googling these two mathematical concepts is that when you change the basis of...

Homework Statement Given that ##x_\mu x^\mu = y_\mu y^\mu## under a Lorentz transform (##x^\mu \rightarrow y^\mu##, ##x_\mu \rightarrow y_\mu##), and that ##x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu##, show that ##x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu##. Homework Equations...

I am trying to reconcile the definition of contravariant and covariant components of a vector between Special Relativity and General Relativity. In GR I understand the difference is defined by the way that the vector components transform under a change in coordinate systems. In SR it seems...

Homework Statement Show U^a \nabla_a U^b = 0 Homework Equations U^a refers to 4-velocity so U^0 =\gamma and U^{1 - 3} = \gamma v^{1 - 3} The Attempt at a Solution I get as far as this: U^a \nabla_a U^b = U^a ( \partial_a U^b + \Gamma^b_{c a} U^c) And I think that the...

In Euclidean space, we may define covariant basis by the partial derivative of position vector with respect to each coordinates, i.e. ##∂R/(∂z^i )=z_i## But in curved space (such as, the two dimensional space on a sphere)[FONT=Arial] how can we define covariant basis 'intrinsicly'?(as we have no...

I was trying to see what is the covariant derivative of a covector. I started with $$ \nabla_\mu (U_\nu V^\nu) = \partial_\mu (U_\nu V^\nu) = (\partial_\mu U\nu) V^\nu + U_\nu (\partial_\mu V^\nu) $$ since the covariant derivative of a scalar is the partial derivative of the latter. Then I...

I wondered if anyone had a good online reference on the covariant formulation of Maxwell's macroscopic equations and the other equations of classical electromagnetism? The wikipedia article talks about constituitive equations in vacuum, which doesn't make a lot of sense to me since M and P...

Dear all, I was reading this https://sites.google.com/site/generalrelativity101/appendix-c-the-covariant-derivative-of-the-ricci-tensor, and it said that if you take the covariant derivative of a tensor with respect to a superscript, then the partial derivative term has a MINUS sign. How? The...

[SIZE="4"]Definition/Summary Covariant derivative, D, is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0 The adjustment is made by a linear operator known both as the...

Doing some problems in D'INVERNO GR textbook and I am stuck on taking the covariant derivation of a tensor twice. Please see the attached picture and please do inform me if something is not clear :smile:

Homework Statement I need to calculate \square A_\mu + R_{\mu \nu} A^\nu if \square = \nabla_\alpha \nabla^\alpha , and is the covariant derivate SEE THIS PDF arXiv:0807.2528v1 i want to get the equation (5) from (3) Homework Equations A^{i}_{{;}{\alpha}} =...

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.