Tensor Definition and 1000 Threads

I am just wondering, is there a difference in meaning/definition between the indices of a tensor being right on top of each other A_{\mu }^{\nu } and being "spaced" as in A{^{\nu }}_{\mu } I seem to remember that I once read that there is indeed a difference, but I can't remember what it...

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Definitions like this one are common in books: For all ##k,l\in\mathbb N##, a multilinear map $$T:\underbrace{V^*\times\cdots\times V^*}_{k\text{ factors}}\times \underbrace{V\times\cdots\times V}_{l\text{ factors}}\to\mathbb R$$ is said to be a tensor of type ##(k,l)## on ##V##. Lee calls this...

Hi. I have a huge problem and without solving it I can't move forward. I will appreciate any help. Having the stress tensor S: 163.666557052527 -63.0272557558942 0.000000000000000E+000 -63.0272557558942 70.3802282767392 0.000000000000000E+000...

I was bored, so I tried to do something to occupy myself. I started going through withdrawal, so I finally just gave in and tried to do some math. Three months of no school is going to be painful. I think I have problems. MATH problems. :-p Atrocious comedy aside, Spivak provides a parametric...

I've PMd some of you with this question, but I got some conflicting replies or no replies at all lol, so I'm posting it here. I also did a Google search and found this which I'm almost sure answers my question, but I just want to confirm with you guys: ''In general, scalar fields are referred...

Homework Statement I'm confused about writing down the equation: \Lambda \eta \Lambda^{-1} = \eta in the Einstein convention. Homework Equations The answer is: \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho\sigma} However it's strange because there seems...

While reading this article I got stuck with Eq.(54). I've been trying to derive it but I can't get their result. I believe my problem is in understanding their hints. They say that they get the result from the Gauss embedding equation and the Ricci identities for the 4-velocity, u^a. Is the...

I have used GRTensorII and Cadabra for some time. And I think Cadabra have great potential for GR. But the current vision of Cadabra only deals with abstract tensor analysis, not with writing out of explicit components. So ,(eq :)when I try to check my final tensor expressions of solutions of...

Hello everyone, Say I have two irreducible representations \rho and \pi of a group G on vector spaces V and W. Then I construct a tensor product representation \rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right) by \left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v...

Hi, I am working through GR by myself and decided to derive the Friedmann equations from the RW metric w. ( +,-,-,-) signature. I succeeded except that I get right value but the opposite sign for each of the Ricci tensor components and the Ricci scalar e.g. For R00 I get +3R../R not -3R../R . I...

In a recent course on special relativity the lecturer derives the Lorentz transformation matrix for the four vector of position and time. Then, apparently without proof, the same matrix is used to transform the EM field tensor to the tensor for the new inertial frame. I am unclear whether it...

Hi All, I am reading the seminal paper by Eshelby on the elastic energy-momentum tensor, which I attach for your convenience. It is all beautiful but equation 4.4 at the beginning. He considers a surface S in the undeformed configuration of a body. The surface is translated by a vector u to a...

Hi everyone, I don't quite understand how tensor products of Hilbert spaces are formed. What I get so far is that from two Hilbert spaces \mathscr{H}_1 and \mathscr{H}_2 a tensor product H_1 \otimes H_2 is formed by considering the Hilbert spaces as just vector spaces H_1 and H_2...

Hello, some time ago I read that if we know the metric tensor g_{ij} associated with a change of coordinates \phi, it is possible to calculate the (Euclidean?) inner product in a way that is invariant to the parametrization. Essentially the inner product was defined in terms of the metric...

Is it the value of the metric tensor that determines the strength of a gravitational field at a specific point in spacetime?

Homework Statement What must the ratio of height to radius of a cylinder be so that every axis is a principal axis (with the CM as the origin)? Homework Equations Moment of inertia tensor. I need I_yy = \sum m *(x^2 + z^2) The Attempt at a Solution I calculated I_zz = MR^2 /12...

Can somebody please explain bilinear maps, tensor products and p-summing operators in an easy-to-understand way. As though explaining to an undergraduate student who just knows basic linear algebra and basic functional analysis. And please give some nice examples to make the explanations more...

I understand energy-momentum tensor with contravariant indices, where I think I get T^{αβ}, but how do I derive the same result for T_{αβ}? Why are the contravariant vectors simply changed to covariant ones, and why does it work in Einstein's equation?

hi, the delta symbol as a tensor (in the minkovski space, in case one has to be specific), what is it exactly? is it \delta^a_b = \frac{\partial{x^a}}{\partial{x^b}} is it \delta^a_b = g^{ac} g_{cb}or is there some other definition?thanks

Hi everyone, I'm reading through tensor product spaces and one question really bogs me. Why is it that the total Hilbert space of a system composed of two independent subsystems is the tensor product of the Hilbert spaces of the subsystems? It is always posed, but I've never seen a proof...

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...

Homework Statement Consider a stationary solution with stress-energy ##T_{ab}## in the context of linearized gravity. Choose a global inertial coordinate system for the flat metric ##\eta_{ab}## so that the "time direction" ##(\frac{\partial }{\partial t})^{a}## of this coordinate system agrees...

Hi everyone, I'm reading about the Wigner-Eckhart Theorem and the coupling of spherical tensor operators in Sakurai's Advanced Quantum Mechanics. I then came across this http://cc.oulu.fi/~tf/tiedostot/pub/kvmIII/english/2004/09_tensop.pdf, which states these concepts in Theorem 1 and 3...

Hi everyone. I'm working on deriving Friedmanns Equations from the Einstein Field Equations. I've got the '00' components worked out but I'm having some trouble with the spatial indices 'ii' of the stress energy tensor ## T_{ii} ##. I'm the FLRW metric with c=1 and signature (-,+,+,+) so that...

I'm trying to understand what kind of relation the metric can have with a general tensor B. d{{x}^{a}}d{{x}^{b}}{{g}_{ab}}=d{{s}^{2}} \frac{d{{x}^{a}}d{{x}^{b}}{{g}_{ab}}}{d{{s}^{2}}}=1 \frac{d{{x}^{a}}d{{x}^{b}}{{g}_{ab}}}{d{{s}^{2}}}=\frac{1}{D}g_{a}^{a}...

Greetings, I would like to find the commutator \left[Lx^2,Ly^2\right] and prove that \left[Lx^2,Ly^2\right]=\left[Ly^2,Lz^2\right]=\left[Lz^2,Lx^2\right] I infer from the cyclic appearance of the indices that using the index notation would be much more compact and insightful to solve the...

I was reading this page: http://en.wikipedia.org/wiki/Tensor which said the definition of a tensor was a relation between two vectors. I then went down to the examples section and it had some sort of (n,m) notation. I had some theories on what they meant but none of them made sense. What do n...

Hi, I am in the middle of revising for and a classical electromagnetism exam, and I've hit a wall when it comes to tensor equations. I know that for anisotropic materials: J=σE and E=ρJ And that in component form the first equation can be written as J_i = σ_{ij} E_j What I'm wondering...

(My question is simpler than it looks at first glance.) Here is Reynolds Transport Theorem: $$\frac{D}{Dt}\int \limits_{V(t)} \mathbf{F}(\vec{x}, t)\ dV = \int \limits_{V(t)} \left[ \frac{\partial \mathbf{F}}{\partial t} + \vec{\nabla} \cdot (\mathbf{F} \vec{u}) \right] \ dV$$ where boldface...

I have seen two definitions with oposite signs (for one of the pressure terms in the formula) all over the web and books. I suspect it is related to the chosen metric signature, but I found no references to that. General Relativity An Introduction for Physicists from M. P. HOBSON...

hello, Do I have the right to perform the following : gjo,i + g0i,j = (gj0 δij + g0i ),j = (2 g0i ),j Thank you, Clear skies,

Hi i have been following Hobson in their attempt to derive the einstein tensor, I have split the varied action into three terms and want to factor out \delta(g^{\mu\nu}) terms. The Riemann tensor R_{\mu \nu} must be expanded to R^{\rho}_{\mu \nu p} and then contracted back to the original...

I am new to tensor algebra. I have an expression involving a 2nd rank tensor (actually a dyadic) and a vector. I want to take divergence of the product i.e. ∇. (T.V) However, I am not sure if the simple product rule would work here. If I use that ∇. (T.V)= (∇.T).V + T. (∇.V)...

Suppose, I have the next metric: g = du^1 \otimes du^1 - du^2 \otimes du^2 And I want to calculate g(W,W), where for example W=\partial_1 + \partial_2 How would I calculate it? Thanks.

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 We have the Einstein tensor G_{αβ} = R_{αβ} - \frac{1}{2}g_{αβ}R where R_{\alpha \beta}, R are the Ricci tensor and scalar. Homework Equations We want the metric to be small perturbation of the flat space, so g_{\alpha \beta} = \eta_{\alpha \beta} + h_{\alpha...

I've seen it stated several times that a general covariant or contravariant tensor of rank n can be separated into it's symmetic and antisymmetric parts T^{\mu_1 \ldots \mu_n} = T^{[\mu_1 \ldots \mu_n]} + T^{(\mu_1 \ldots \mu_n)} and this is easy to prove for the case n=2, but I don't see how...

Hey everyone I was looking up the EM tensor on wikipedia, and they mention two invariants. There is the obvious one - The Lorentz invariant B^2- \frac{E^2}{c^2} And there is also the product of the EM tensor with its dual, which they call the pseudoscalar invariant: \frac{1}{2}...

How to calculate something relating to the determinant of metric tensor? for example, its derivative ∂_{λ}g. and how to calculate1/g* ∂_{λ}g, which is from (3.33) in the book Spacetime and Geometry, in which the author says that it can be related to the Christoffel connection.

I am trying to prove that C\otimesC (taken over R) is equal to C^2. The method I have seen is to show the following equivalences: C\otimesC = C\otimes(R[T]/<T^2+1>) = C[T]/<T^2+1> = C. (All tensor products taken over R). The only part I am having trouble with is showing that...

I'm having some trouble calculating the stress tensor in the case of a static electric field without a magnetic field. Following the derivation on Wikipedia, 1. Start with Lorentz force: \mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B}) 2. Get force density \mathbf{f} =...

Show that a symmetric tensor has n(n+1) \ 2 quantities. In a symmetric tensor we have that Aij = Aji which means that A12 = A21 A23 = A32 and so on. Thus these n quantites are similar. What do we do next?

Homework Statement Given a cylinder in the Ox1x2x3 coordinate system, such that x1 is in the Length direction and x2 and x3 are in the radial directions. The stress components are given by the tensor $$ [T_{ij}] = \begin{bmatrix}Ax_2 + Bx_3 & Cx_3 & -Cx_2 \\ Cx_3 & 0 & 0 \\ -C_2 & 0 &...

We are stating with equivalence principle that passing locally to non inertial frame would be analogous to the presence of gravitational field at that point, so g^'_{ij}=A g_{nm} A^{-1} where g' is the galilean metric and g is the metric in curved space, and A is the transformation which...

I'm trying to understand tensor product of vector spaces and how it is done,but looks like nothing that I read,helps! I need to know how can we achieve the tensor product of two vector spaces without getting specific by e.g. assuming finite dimensions or any specific underlying field. Another...

Hello, I know all my algebra, trig, and I'm still fine tuning calculus and I've solved ODEs using the Laplace transform. Now, my question is... what else must I know to study Tensor calculus/Tensor Analysis? I really want to know so that I have a true understanding of relativity(the famous...

my exploration of relativity followed by first reading various books which failed to explain to me how relativity worked but built a strong feel of how one can think about it. after which i decided to take the mathematical way of understanding it for which i am going on with the prof susskind's...

Use indicial notation to show that: $$ \mathcal{A}_{mi}\varepsilon_{mjk} + \mathcal{A}_{mj}\varepsilon_{imk} + \mathcal{A}_{mk}\varepsilon_{ijm} = \mathcal{A}_{mm}\varepsilon_{ijk} $$ I'm probably missing an easier way, but my approach is to rearrange and expand on the terms: $$...

\begin{alignat*}{3} A_{ij}B_{ij} & = & (A_{(ij)} + A_{[ij]})(B_{(ij)} + B_{[ij]})\\ & = & A_{(ij)}B_{(ij)} + A_{(ij)}B_{[ij]} + A_{[ij]}B_{(ij)} + A_{[ij]}B_{[ij]} \end{alignat*} $$ A_{(ij)}B_{[ij]} + A_{[ij]}B_{(ij)} = \frac{1}{2}(A_{ji}B_{ij} - A_{ij}B_{ji}) $$ Can I then say $A_{ji}B_{ij} =...