Questions tagged [tensor-calculus]

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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7
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1answer
281 views

Is the distinction between covariant and contravariant objects purely for the convenience of mathematical manipulation?

Two kinds of indices, covariant and contravariant, are introduced in special relativity. This, as far as I understand, is solely for mathematical luxury, i.e. write expressions in a concise, self-...
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Bracket Notation on Tensor Indices

I know about the () symmetrisation and [] anti-symmetrisation brackets on tensor indices so long as they appear on their own, such as : $$V_{[\alpha \beta ]}=\frac{1}{2}\left ( V_{\alpha \beta }-V_{\...
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Choice of Dirac gamma matrix representation and definition of adjoint spinor

Is the definition of the adjoint spinor $\bar{\psi}=\psi^\dagger \gamma^0$ forcing a particular choice of representation of the Dirac matrices (or a subset of the possible choices)? More precisely, I ...
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3answers
263 views

Is a metric tensor field the same thing as $ds² = -dt² + dx²+ dy² + dz²$?

I am having trouble understanding the nature of the metric tensor field on spacetime manifolds. In particular, a Riemannian manifold $(M,g)$ is defined as a real smooth manifold $M$ equipped with an ...
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2answers
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Variation of square root of determinant of metric, $\delta g$ [closed]

I am trying to calculate $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$ where $g = \text{det} g_{\mu \nu}$. We have $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 \...
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1answer
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Riemann tensor in 2d and 3d

Ok so I seem to be missing something here. I know that the number of independent coefficients of the Riemann tensor is $\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given ...
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3answers
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How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ \Lambda^{\mu_1'}{}_{\mu_1}...\Lambda^{\...
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766 views

Differential Forms and Densities

I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a ...
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1answer
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Covariant derivative and Leibniz rule

I read the Wikipedia page about the covariant derivative, my main problem is in this part: http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description Some of the formulas seem to lead ...
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1answer
199 views

Toroid moments tensor decomposition

I am currently working on my bachelor's thesis on the anapole / toroidal moment and it seems that I am stuck with a tensor decomposition problem. I have actually never had a course about tensors, so ...
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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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611 views

What does tensor mean in a physics context?

I am taking a fluid mechanics class and don't know very much physics. I was confused in class when the prof kept calling this derivative matrix of a fluid flow (A function from $\mathbb{R}^n\to\mathbb{...
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Derivation of $f(R)$ field equations, problem with integration by parts

I am following the derivation of the field equations on the the Wikipedia page for $f(R)$ gravity. But I do not understand the following step: $$ \delta S = \int \frac{1}{2\kappa} \sqrt{-g} \left(\...
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4answers
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Nature of Fields in QFT

I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere! There are specific types of fields used in physics: scalar fields (i.e. as ...
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2answers
845 views

Why are totally antisymmetric tensors more useful than totally symmetric tensors?

In an arbitrary number of dimensions, one can naturally define two tensors, Kronecker delta and Levi-Civita epsilon tensor. However, why isn't it advantageous to define some totally symmetric tensor ...
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3answers
321 views

Dimension of vector resulting from tensorial product

I'm quoting what I found in a book about quantum computation: Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then ...
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1answer
275 views

Can Yang-Mills field strength be defined as covariant derivative squared?

In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as $$ \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\...
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Can we generalize relativistic expressions found in specific frames, to arbitrary frames?

A claim is made in Sean Carroll's GR book multiple times that goes along the following lines: Given a problem in relativity, we can find a solution with ease if we choose a convenient reference ...
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How to define pseudovector mathematically?

The textbook describes pseudovector like this: Let $a,b$ be vectors and $c=a\times b$, $P$ be the parity operator. Then $P(a)=-a,P(b)=-b$ by definition. But $P(c)=c$ since both $a$ and $b$ reverse ...
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4answers
535 views

Why are stresses of continuum systems described via a tensor?

The tittle pretty much says enough. I have always been told so but no one really motivated it. So, I would like to know why do we use a tensor to describe the stresses in continuum mechanics.
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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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Significance of the Dual Electromagnetic Tensor $\tilde{\mathbf{F}}$/its derivation

In the context of Maxwell's equations, I was wondering whether there was any physical significance to the dual EM Field Tensor and/or its various derivations. It has components: $$\tilde{\textbf{F}} = ...
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2answers
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Tensor index notation with e.g. square brackets

I want to learn playing with indices and some notation in General relativity. But in every book just is used this notation. I know upper and lower but I don"t know the meaning of some combination of ...
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586 views

Positional probability density for combined spin and position states

In one dimension, given a particle in a quantum state $| \psi\rangle$, the probability density of position is given as $| \psi(x) |^2 = \psi^*(x) \psi(x) =\langle x | \psi \rangle\langle \psi | x \...
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848 views

Physical meaning of the Einstein tensor

The Einstein tensor is the tensor field $G$ on spacetime $M$ with components $$G_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R$$ so that Einstein's field equations can be written as: $$G_{\mu\nu}=\...
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3answers
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Interference between two photons, tensor product of individual wave functions?

I have learned that the wave function cannot be visualized as a real physical wave like for example the EM field, because for multi-particle systems, it is not a wave in $\mathbb{R}^3$ but in $\mathbb{...
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1answer
415 views

Physical visualisation of curvature

I was wondering-how do you visualise curvature in the context of general relativity. The gravity well and trampoline analogies are quite wrong, so I want a more realistic approach to it (say, the way ...
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1answer
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metric determinant and its partial and covariant derivative

question : $\nabla_a \nabla_b \sqrt{g} \phi =\partial_a \sqrt{g} \partial_b \phi$ is true ? because $\nabla_a \sqrt{g}=0$ so we can write $\sqrt{g} \nabla_a \nabla_b \phi$ , but because metric ...
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0answers
591 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
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1answer
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What does a colon mean in hydrodynamics equations?

In some hydrodynamics book I saw a notation like $e:e$ where $e$ is a matrix (shear stress tensor). This double dot product is in a scalar equation, so the result of this operation must be scalar. I ...
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2answers
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Mathematica package for supergravity and string theory [duplicate]

I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
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2answers
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Interference of two non-entangled photons, calculation using tensor product of Hilbert spaces

I'm trying to calculate the interference of two non-entangled photons, like in a double-slit experiment with two photon sources, one behind each slit (follow-up on this question). The individual ...
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1answer
164 views

Tensors and the Klein-Gordon Equation

Consider the Klein-Gordon equation: \begin{equation} \frac{\partial^2 \psi}{\partial t^2} = c^2 \Delta \psi - \frac{m^2 c^4}{\hbar^2} \psi, \end{equation} and define for each one of its solutions $\...
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1answer
110 views

How can you have $\frac{DA^\mu}{d\tau}$?

If a covariant derivative is given by: $$D_\nu A^\mu=\partial_\nu A^\mu +\Gamma^\mu_{\nu \lambda} A^{\lambda}$$ Then how does $\frac{DA^\mu}{d\tau}$ make any sense? Since there are no 'differentials' ...
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2answers
284 views

Tensor product postulate [duplicate]

Non relativistic quantum mechanics assumes that a composite system should be described with the tensor product of the component systems. This is the tensor product postulate of quantum mechanics. I ...
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1answer
221 views

Are there cases in which we should consider tensors as equivalence classes?

Usually in texts about Physics that uses tensors defines them as multilinear maps. So if $V$ is a vector space over the field $F$, a tensor is a multilinear mapping: $$T:V\times\cdots\times V\times V^...
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1answer
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Difference between $\partial$ and $\nabla$ in general relativity

I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\partial_\mu$ which would have the plain partial ...
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2answers
526 views

Higher dimensional relation between angular momentum, moment of intertia and angular velocity

In 3 dimensions we have the well known relation (summation convention is being used) $$ L_i = I_{ij} \omega_j $$ However, as is well known the angular momentum and angular velocity are not vectors ...
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2answers
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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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4answers
309 views

Is Lorentz transform a tensor?

I am confused whether Lorentz transform is a tensor or not, since it is a linear transform. If yes how can I verify that?
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1answer
481 views

How quantum field transforms in case of some particular spin

Except when a particle is spin-0, field of all particles transforms when frame of reference is changed, and this defines what spin is. The question is, specifically how does the quantum field ...
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2answers
768 views

How to show that the product of a Killing tensor and tangent vector is conserved along a geodesic?

I understand that a Killing vector $K^{\mu}$ satisfies, $$ K^{(\mu;\nu)} = 0 $$ I also know that along a geodesic, the quantity $$ p_{\mu} K^{\mu}$$ is conserved, where $p_{\mu}$ is the photon 4 ...
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2answers
543 views

What do the entries of the Einstein Tensor mean?

So if I understand correctly a tensor is something that transforms under certain laws and can be imagined as a combination of two vectors, e.g. the Stress tensor is a combination of surface normal ...
4
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1answer
248 views

Confusion about square bracket notation in the the Kronecker delta symbol

I am not sure I understand what the short-hand anti-symmetrization means. I.e. I know that $$\delta_{cd}^{[ab]} ~=~ \frac{1}{2}(\delta_{c}^{a}\delta_{d}^{b} - \delta_{c}^{b}\delta_{d}^{a})$$ but how ...
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2answers
591 views

Torsion tensor: definition

The definition of torsion tensor is the following: $$ \mathbf{T}(\mathbf{X},\mathbf{Y})=\nabla_{\mathbf{X}}\mathbf{Y}-\nabla_{\mathbf{Y}}\mathbf{X} -\left[\mathbf{X},\mathbf{Y}\right]. $$ In an ...
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2answers
984 views

Assumed symmetry of Christoffel Symbols

With reference to the discussion in an earlier question on the independence of metric and Christoffel symbols, it was discussed that the symmetry of the Christoffel symbols ($\Gamma_{\mu\nu}^{\alpha} =...
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1answer
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Metric signature explanation

Can anyone explain what metric signature is? I have a basic knowledge regarding tensors, btw. Also, how is it related to fundamental understanding of general relativity?
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2answers
404 views

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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3answers
2k views

What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
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0answers
914 views

Variation of the purely covariant Riemann tensor

I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$. I know that, $R_{\rho \sigma \mu \nu} = g_{\rho \...