Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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Lorentz Transformations in Minkowski space

If $\Lambda$ represents the Lorentz transformation matrix, then the transformation of contravariant components $x^\mu$ is given by $$x'^\mu=\Lambda^{\mu}{}_{\nu} x^\nu$$ and that of the covariant ...
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Are Fock spaces just a special type of tensor algebra?

Are Fock spaces just a special type of tensor algebra? The definitions I am using: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Tensor_algebra From what I can tell, the ...
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Projection of a tensor

Consider the following tensor (abstract index notation, e.g. Wald's) $B_{ab}$ and timelike vector field $X^{a}$ such that $X^aX_a=-1$ and \begin{equation} B_{ab}=\nabla_bX_a \end{equation} Then one ...
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Does $\partial_\mu =\frac{\partial }{\partial x^\mu}$ or $\partial_\mu =\frac{\partial }{\partial x_\mu}$? [migrated]

I am looking at the chain rule with covariant and contravariant vectors. I understand why we have: $$df=\frac{\partial f}{\partial x^\mu} dx^\mu$$ (Please correct me if I am wrong) since even though ...
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Relation between differentiation of one-form basis and Christoffel Symbols

If I want to covariantly differentiate a one form then I can write: $\nabla_\beta \tilde p = \dfrac{\partial p_\alpha}{\partial x^\beta} \tilde \omega^\alpha + p_\alpha \dfrac{\partial \tilde \omega^...
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Derivation of Christoffel Symbols

So I am reading a book on relativity & differential geometry and in the text, they gave the Christoffel symbols in terms of the metric and its derivatives, but I wanted to derive it myself. ...
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Give a strain state which describes a purely spatial change of volume [closed]

In a problem, I am asked to provide a strain state (strain tensor) that describes a "purely spatial change of volume". I am unable to understand what is meant by this. Does it mean that there is ...
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General expression of the redshift: explanation?

In some papers, authors put the following formula for the cosmological redshift $z$ : $1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$ where : $S$ ...
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How to prove the raising/lowering indices operation?

I've read this related question, though it didn't satisfy me; I hope this complements it. I know that if I contract a covariant tensor ${A_{\alpha\beta}}$ with a vector ${B^\beta}$, I get some other ...
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Defining electromagnetic stress tensor for non-linear media

In textbooks, the electromagnetic stress tensor (in vacuum also called Maxwell stress tensor) is usually derived for linear media, implying that $$ \vec D = \epsilon_0 \epsilon_r \vec E$$ My question ...
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Einstein Summation Convention: One as Upper, One as Lower?

My question refers to the often specified rule defining Einstein Summation Notation in that summation is implied when an index is repeated twice in a single term, once as upper index and once as lower ...
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What is the spin of the Kalb-Ramond field?

In bosonic string theory the massless states of the closed string are given by a rank 2 tensor, which is divided into its three irreducible spherical tensors: symmetric traceless, antisymmetric and ...
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Uncommon tensor notation $\partial_{(\mu}\xi_{\nu)}$

I came across this expression for the change in a metric under an infinitesimal gauge transformation $\epsilon\xi^\mu$. $$h_{\mu\nu}' = h_{\mu\nu}+2\epsilon\partial_{(\mu}\xi_{\nu)}$$ What does the $...
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Dirac notation - trace of product of (bipartite) density matrices

I'm getting confused by the Dirac notation. Suppose I have the following two objects. $$\rho = \sum_k p_k (\rho_A \otimes \rho_B) = \sum_k p_k |k \rangle \langle k | \otimes |k\rangle \langle k | ,$$...
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$SU(N)$ Yang-Mills Theory

Yang-Mills theory is based on the gauge group $G$ which we take to be $SU(N)$. Consider an example; $$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-\sum_{j=1}^N\bar{\psi}_j(i\gamma^\mu D_\mu-m)\...
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Covariant derivative of a covariant tensor wrt superscript

Is it true that when you take the covariant derivative of a covariant tensor, do you always have to do with a subscript? What if you do it wrt a superscript?Does the first term (with the partial ...
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What kind of product is $\prod^n_{j=1}\sigma^{(j)}_x$?

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. The adiabatic Hamiltonian is defined as $$...
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What is the connection between the coordinate transformation properties and graphical representation of covariant and contravariant components?

So right now I am studying General Relativity (in particular tensor analysis), and I have a question regarding covariant and contravariant components of a vector. I was taught how to transform ...
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Killing tensor in Minkowski space

I'm trying to solve the Killing tensor equation $\nabla_{(a}K_{bc)} = 0$ in Minkowski space. I'd like to generalise the method we use to find Killing tensors in Minkowski space. We can take $\...
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What expression of Ricci tensor should we choose in order to obatin a correct field equations?

I have doubt regarding the choice of the Ricci tensor $R_{ij}$. I have seen many books and papers use the expression $R_{ij}=\Gamma^i_{jp,i}-\Gamma^i_{ji,p}+\Gamma^i_{in}\Gamma^n_{jp}-\Gamma^i_{pn}\...
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Usage of tensors in physics [closed]

As I understand it, tensors are multi-linear maps that map vectors (and dual vectors) to real (or complex) numbers, but I'm hoping to gain some intuition as to why they are useful in physics. Is it ...
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Commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set $$\frac{\...
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Covariant Derivative of Kronecker Delta

I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
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If $T_{kl}=\epsilon_{kil}m_i$, how to show $m_i=0.5\epsilon_{ilk}T_{lk}$? [closed]

In a book I am reading (about magnetic dipole), it is given that $T_{kl}=\epsilon_{kil}m_i$. Then, it says since $T_{kl}=-T_{lk}$, it can be shown that $m_i=0.5\epsilon_{ilk}T_{lk}$. I understand that ...
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Rewriting Maxwell's equation in tensor form [closed]

Suppose $F_{ij}=\epsilon_{ijk}B_k $, how to prove the following: $\partial_iB_i=0$ becomes $\partial_iF_{jk}+\partial_jF_{ki}+\partial_kF_{ij}=0$ $B_iB_i$ becomes $F_{ij}F_{ij}/2$ I can see that ...
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Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
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260 views

Killing tensor and Riemann tensor identity

I know that if we have a Killing vector then it's straightforward to show the identity: $$\nabla_a \nabla_b K_c = R_{cba}^k K_d$$ I'm now trying to show the following identity for a $(0,2)$ Killing ...
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What is the difference between a tensor, vector, and a matrix? [duplicate]

I'm currently going through notes on a physics course and I'm having trouble understanding the difference between a tensor, a vector, and a matrix. I know that a vector is a kind of tensor and that a ...
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Reciprocal Dyadics: Menzel's Mathematical Physics

I am having difficulty justifying a step in Menzel's development of sets of reciprocal spanning vector sets. Here's an abbreviated development. All Fraktur letters represent vectors. Introduce the ...
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Covariant derivative of a covariant derivative

I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$. This is something I've taken for granted a lot in calculations, namely I though that by the ...
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Induced metric is a scalar for transformation from $x\to x'$? (Poisson E.A p.62)

I have a (simple) question about the induced metric $h_{ab}$. In Poisson E.A. (a relativist toolkit) it says in p. 62 that the induced metric $$h_{ab}=g_{{\alpha}{\beta}} \frac{\partial x^{\alpha}}{\...
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Flat space Solution of Einstein Field Equation

Does a trace-free energy-momentum tensor $T_{\mu}^{\mu} = 0$ ensure that the Einstein's field equations have a flat space solution?
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Tensor notation of Maxwell's equations

Tensor notation of Maxwell's equation read So when we explicitly try to find the Maxwell's equation from the above tensor equation we only get gauss law and curl of B. The div.B=0 and curl of E are ...
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How to show invariance using the Maxwell tensor?

I want to show the invariance of $E^2-c^2B^2$ under the Lorentz transformations. The obvious way to do this is to show that $$E^2-c^2B^2=E'^2-c^2B'^2,$$ where $E'$ and $B'$ are the Lorentz ...
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Peculiarity about a system of three electrons

Consider three (or any number bigger than 2) electrons without spatial degrees of freedom, thus the only degree of freedom is the spins. The Hilbert space is then formed by the tensor product of the ...
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Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
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How does the Einstein summation convention apply to the following equation?

This is the equation is in the "mathematical form" section of the following wikipedia article: http://en.wikipedia.org/wiki/Geodesics_in_general_relativity More specifically, the "Full geodesic ...
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An expression for stress power

I have seen it written that for a continuum undergoing deformation, if we ignore body forces and heat transfer, the work done is equal to stress power: $\cfrac{dW}{dt}=\sigma_{ij}D_{ij}$, where $D_{...
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Understanding basics of tensors

I am trying to understand tensors to learn General relativity. In the book that I am reading they claim that if the basis of a vector space undergoes a linear transformation $T$ then the components of ...
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Staggered Indices ($\Lambda^\mu{}_\nu$ vs. $\Lambda_\mu{}^\nu$) on Lorentz Transformations

I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes. What are the respective meanings of $\Lambda^\mu{}_\nu$ as ...
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Dielectric permittivity tensor of rank 2

In special relativity, one knows the Euclidean and Minkowski metric to raise or lower the index of a covector or vector. The Euclidean metric is a (1,1) tensor and is represented by a 3x3 identity ...
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Indicating that indices are equal in Einstein notation

tl;dr: I have an expression like this: (dramatization) $$ R_{\mu\nu} = \begin{pmatrix} B^{00}C_{00} & 0 & 0 & 0 \\ 0 & B^{11}C_{10} & 0 & 0 \\ 0 & 0 & B^{22}C_{20} &...
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Deriving $A^{\mu}_{;\nu}$ from $A_{\mu ; \nu}$

We have a covariant derivative of a covariant tensor: $$ A_{\mu ; \nu} = A_{\mu , \nu} - \Gamma^{\alpha}_{\mu \nu} A_{\alpha} $$ The covariant derivative of a contravariant tensor is: $$ A^{\mu}_{;\nu}...
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What is basic tensor algebra in teleparallel equivalent of general relativity?

Teleparallel gravity represents a viable alternative to general relativity where gravitation comes from torsion rather that curvature. The theory is based on a new modified connection, and the ...
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Finding the components of the tensor for potential and kinetic energy

I have a rather poor understanding of what a tensor is, but enough to apply it to the biggest part of the classical mechanics I'm studying. However, I've run into a small problem while studying "Free ...
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$SU(3)$ Tensor Methods in a Tetraquark

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
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Why define four-vectors to be quantities that transform only like the position vector transforms?

A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation ...
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Is this a Lorentz-scalar? How do I tell?

I'm struggling to identify whether a scalar is a Lorentz-scalar. E.g: $$\partial_i A^i \quad i \in {1,2,3}.$$ How do I determine if this is a Lorentz-scalar or not? If got the same problem with ...
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Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \...