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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If $T_{kl}=\epsilon_{kil}m_i$, how to show $m_i=0.5\epsilon_{ilk}T_{lk}$?

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 [on hold]

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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Usage of tensors in physics

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 ...
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1answer
52 views

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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31 views

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 ...
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Writing Tensor in LaTeX [closed]

How can I write this in LaTeX? Each time I am trying to write it down the compilers is showing me error for multiple subscription.
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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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Manipulating tensors in relativistic quantum mechanics

I was doing a problem that involved showing a Heisenberg equation of motion was consistent with the Dirac equation. The question involved a lot of algebra which was generally fine but something done ...
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55 views

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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245 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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98 views

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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73 views

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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74 views

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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15 views

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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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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92 views

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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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: $$ ...
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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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Question about tensor operators [closed]

I have learned there is a so called tensor product in linear algebra when one is trying to pair vectors (or operators) from two different vector spaces, they use the symbol $\otimes$. But then in QM I ...
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Tensors Contracting Indices

I'm pretty confused regarding the components of a tensor once you take its trace (or contraction). I'll use $B\in T_2^1(V)$ to be specific. Let $V$ be an $n$-dimensional vector space with basis ...
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Tensor product - Addition of angular momenta

In the book Quantum Mechanics - Cohen-Tannoudji, in chapter X, equation (B-5) says $$ \vec{S^2} = (\vec{S_1} + \vec{S_2})^2 = \vec{S_1^2} + \vec{S_2^2} + 2\vec{S_1}\cdot\vec{S_2} $$ and $$ ...
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How do we determine if a certain physical quantity is a vector?

For instance in Newtonian physics we treat position of objects, displacements, velocities, forces, momenta, angular velocities etc all as vector quantities (little arrows in space which have a certain ...
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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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Exponential decay of correlation in PEPS

PEPS (Projected Entangled Pair State) is a tensor network that plays the same role in two dimensional lattice as MPS (Matrix Product State) plays in one dimensional spin chain. A good introduction can ...
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Converting between matrix multiplication and tensor contraction

If I have $A^{\alpha \beta} B_{\beta \gamma}$ then this should be the equivalent to the following matrix multiplication: $AB$ since we're summing over the columns of $A$ and the rows of $B$. By the ...
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Tensor products of Hilberts spaces: definition of outer products and commutators

Suppose one has two single-particle Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ and consider the tensor product of these such that $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is a two-particle ...
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Using $\sqrt{-g}$ in integrals of proper volume

I am a little confused over integration using proper volume element. When do we use $\sqrt{-g}$ in calculations? For example, in many calculations involving stars, say when using TOV equation, this ...
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Placement of indices in canonical commutation relations of coordinates and conjugate momenta as well as fields and conjugate momenta

The canonical commutation relations between generalised coordinates $q_a$ and their conjugate momenta $p^a$ are given by $[q_a,q_b]=[p^a,p^b]=0$ $[q_a,p^b]=i\delta^b_a$. Furthermore, the canonical ...
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Warping function for torsion of non-circular prism

I have a few questions regarding the case of torsion of a prism, as encountered in continuum mechanics. Specifically, a prism (which can be a cylinder, a rectangular prism, elliptical prism, etc.) has ...
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64 views

On the proof of the existence of geodesics coordinates [closed]

From "Introducing Einstein’s Relativity" by Ray D’Inverno page 77-78 In my calculation, the process is $$\frac{\partial{x^{'a}}}{\partial{x^d}}=\frac{\partial{x^{a}}}{\partial{x^d}}+\frac{1}{2} ...
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Raising and lowering operators for a composite isospin $SU(2)$ system

Consider pion states composed of $q \bar q$ pairs where $q \in \left\{u,d \right\}$ transforms under an $SU(2)$ isospin flavour symmetry. These bound states transform in the tensor product $R_1 ...
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What's meant by: “All observers agree on the combination of basis vectors and components for a tensor”?

There's a Youtube video given by Dr Dan Fleisch on Tensors, where he states at 11:22: You may be wondering, what is it about the combination of components and basis vectors that makes tensors so ...
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Dirac Notation With Comma

Does $\langle A,B\rvert$ mean $\langle A\rvert\langle B\rvert$? If so how is an operator applied to this in $\langle A,B\rvert \hat O $? For an example say the annihilation operator acting on ...
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Tensor product in quantum mechanics?

I often see many-body systems in QM represented in terms of a tensor products of the individual wave functions. Like, given two wave functions with basis vectors $|A\rangle$ and $|B\rangle$, belonging ...
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104 views

How to write the Clebsch-Gordan decomposition in tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is known that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ may be decomposed through the ...
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Tensor calculus in special relativity

We say that any function $f(t,x,y,z)$ is a tensor of rank {0 0} because it takes no vectors or one-forms in order to give a real number. But couldn't we have just written the same function as ...
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1answer
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Proportionality Constant in Einstein Field Equations

The Einstein Field Equations: $$G_{ab}~=~8\pi T_{ab}.$$ I am familiar with how to obtain the $8\pi$ proportionality factor through correspondence with Newtonian gravity, but am wondering if this ...
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25 views

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 ...
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520 views

Is Newton second law covariant or invariant?

Is Newton second law covariant or invariant between two inertial frames, moving with uniform traslational motion with respect to each other? If it is invariant then, indipendently from the frame, ...
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Coordinate form of divergence of anti-symmetric tensor field [closed]

just a quick question on something that might save me a little bit of time and effort. In a general curved metric, the divergence of a vector field, $A^\mu$, can be written as: $ \nabla_\mu A^\mu = ...
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When dealing with spinor indices, how exactly do we obtain the barred Pauli operator?

In the set of SUSY notes I'm following, the Pauli operator is given as: ${(\sigma^\mu)}_{\alpha\dot{\alpha}} = (I_2, \sigma^1, \sigma^2, \sigma^3)$. The antisymmetric tensor that lowers and raises ...
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Derivation of the relativistic equation of energy conservation for a perfect fluid

I'm currently attempting to struggle through the first chapter of Sean M. Carrol's spacetime and geometry. I'm a bit stuck, most likely because of not understanding the mathematical operation. ...
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What is the short time limit of Maxwell viscoelastic fluids?

The Maxwell model for viscoelastic fluids writes: $$ \tau\stackrel{\triangledown}{\sigma}+\sigma=2\eta D(v) $$ where $D(v) = \frac{1}{2}(\nabla v +\nabla v^T)$, $v$ velocity and $\sigma$ stress tensor ...
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Fermi-Propagated Jacobi equation in the book The Large scale structure of space-time

On page 81, equation (4.6), the author use the Fermi derivative to write the Jacobi equation \begin{equation} \tag{4.6} \frac{{D^2}_\text{F}}{\partial s^2} {}_{\bot}Z^a = ...
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The Lie derivative of the metric $g_{ab}$ and index notation

I don't quite know where to start this question. I'm essentially not understanding how to compute the Lie derivative of a given metric and vector. So I have the following definition: $$ ...
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166 views

Jacobi equation in the book The Large scale structure of space-time

On pp. 79, it is obvious that equation (4.2) \begin{equation} \frac{D}{\partial s}Z^a = {V^a}_{;\ b}Z^b \end{equation} holds, where $Z$ is the deviation vector and $V$ is the unit tangent vector along ...