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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Difference between vielbein and the Jacobian matrix

In math books, I saw the metric tensor is defined with the use of the Jacobian matrix as $$g_{\mu \nu}=J_{\mu}^a \: J_{\nu}^b \: \eta_{ab}\tag{1}$$ where $J_{\mu}^a=\frac{\partial \bar{x}^a}{\partial ...
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Show that modified energy-momentum tensor defines conserved currents

In David Tong's notes on quantum field theory, one of the problems asks to show that a tensor defined by $$\Theta^{\mu\nu}=T^{\mu\nu}-F^{\rho\mu}\partial_\rho A^\nu$$ where $F^{\rho\mu}=\partial^\rho ...
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Transformation law for 3-velocity

I am kind of stuck with a problem mentioned in my current reading about special relativity. Given the Lorentz transformation $$x^{\bar{i}} = L^i{}_k \, x^k \quad ,$$ one has to find the transformation ...
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Index-free tensor expressions and what makes the metric tensor different

There are two doubts, but all are from the same section and closely related so I thought I'll ask in one post. I'm studying a section that introduces Christoffel symbols [Core Principles of Special ...
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1answer
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Is it correct to sum over either index of the metric the same way?

I don't know if the following is correct, i want to compute the following derivative $$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} \...
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The exact nature of the spacetime interval

The book I will reference here is (specifically page XXXV of) The Geometry of Physics by Frankel. I asked this question a long time ago when I was a complete newcomer to general relativity. There was ...
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1answer
46 views

Explanation of an equation in special relativity

$$ {\partial (0.5 (\partial_{\mu} A^{\mu})^2) \over \partial(\partial_{\mu} A_{\nu})} = {(\partial_{\rho} A^{\rho}) g^{\mu \nu} } $$ Can somebody explain why this is true?
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Demonstration of Electromagnetic Tensor antisymmetry

I've already made a post about this topic here, but I realized that I didn't understand the explanation on that post. in Chapter 7 of Rindler's book on relativity, in section about electromagnetic ...
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When can we use normal coordinates for a “proof”?

So I'm trying to find the equations of motion of a field in a particular metric. I know what the equations of motion of the field in flat space look like and how they simplify. I think it's always ...
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1answer
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Anti-symmetric tensors

I am little bit confused about the term anti-symmetric tensor. $$p_{ijk\ell}$$ is an anti-symmetric tensor. I would like to know its value when any of the two indices are same. For example, Is $$p_{...
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Quantum gravity in affine picture

I am studying an article which is about quantum gravity (M. Martellini, "Quantum Gravity in the Eddington Purely Affine Picture," Phys. Rev. D 29 (1984) 2746). I have come to eq. 2.11a which ...
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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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Question about the the velocity and acceleration in tensor notation

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $$V^i = (d/dt)Z^i$$ and the components fo ...
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Derivative of Christoffel symbols in a local inertial frame

I have a doubt regarding the Riemann tensor in a LIF. The general expression of the Riemann tensor is: $R^{\alpha}_{\beta \mu \nu} = \Gamma ^{\alpha}_{\beta \nu, \mu} - \Gamma ^{\alpha}_{\beta \mu, \...
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Ricci Tensor component, index lowering doubt [closed]

$v^j$$v^k$$[R(e_i,e_j)]e_k$.$e_i$ gives $v^j$$v^k$$R^l$$_k$$_i$$_j$ $e_l$.$e_i$ Then $v^j$$v^k$$R^l$$_k$$_i$$_j$ $g$$_l$$_i$ Finally, $v^j$$v^k$$R$$_i$$_k$$_i$$_j$ Which upon summation on i gives $v^j$...
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1answer
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Non-orthogonal transformations of the inertia tensor

The inertia tensor for a point is given by $$ I_{ij} = m(\delta_{ij} ||x||^2-x_i x_j)$$ where $m$ is the mass of the point and $x_{i}$ are the (covariant) coordinates of the given point in an ...
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How to get the Riemann curvature tensor from the commutator operating on a basis vector

In the following the basis vectors are assumed to be varying functions of position. This means that when a vector appears under the differentiation operator, both components and basis vectors will, in ...
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Laplace operator and tensor calculus:

I'm studying Tensor calculus and I found this interesting problem: Show that: $$ \Delta F=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_kF\right)$$ Here's some ...
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Trying to understand a visualization of contravariant and covariant bases

I was trying to intuitively understand the covariant and contravariant bases for a coordinate system and I came across this image on Wikipedia: Edit: After reading the first two answers I think I may ...
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9answers
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Is it foolish to distinguish between covariant and contravariant vectors?

A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a ...
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What is a tensor?

I have a pretty good knowledge of physics, but couldn't deeply understand what a tensor is and why it is so fundamental.
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Different variations of covariant derivative product rule

This is a follow-up question to the accepted answer to this question: Leibniz Rule for Covariant derivatives The standard Leibniz rule for covariant derivatives is $$\nabla(T\otimes S)=\nabla T\otimes ...
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1answer
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Problem with extending a vector field's action on another vector field

I'm following Schuller's lectures on gravity and light (this question specifically concerns this part of the video). Let $M$ be a smooth manifold with a smooth vector field $X$, and let $f\in C^{\...
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2answers
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Proving $ \vec{V} \cdot (\vec{\nabla}\vec{V}) = (\vec{\nabla}\cdot\vec{V})\vec{V} $ using index notation

In my fluid mechanics course we encounter a lot of vector calculus problems, one of which I have been struggling with for a while now. We must prove that $$ \vec{V} \cdot \left(\vec{\nabla}\vec{V}\...
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1answer
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Special Relativity: Interpretation of the partial derivate of Stress-Energy Tensor

This question is based on Carroll's book Spacetime and Geometry, specifically from page 33 to page 36. In the upper mentioned section we define the Stress-Energy Tensor as: The flux of the four ...
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Is current a tensor or a scalar quantity? [duplicate]

Is current a tensor or a scalar quantity? The internet seems to be divided on this one. Tensors are too complicated. So I am unable to find an answer. Can someone please clearly state whether it is a ...
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5answers
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Inverse and Transpose of Lorentz Transformation

I've seen this question asked a few times on Stack Exchange, but I'm still quite confused why the following "contradiction" seems to arise. By definition: $(\Lambda^T)^{\mu}{}_{\nu} = \...
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Is notational compactness in tensors (compared to linear algebra) relevant? [migrated]

In this post you can read: A matrix is a special case of a second rank tensor with 1 index up and 1 index down. It takes vectors to vectors, (by contracting the upper index of the vector with the ...
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3answers
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Proving that the Minkowski metric tensor is invariant under Lorentz transformations

I'm studying special relativity. A general Lorentz transformation is defined by $\Lambda^T\eta\Lambda=\eta$. Now, \begin{align} \eta'^{\mu\nu} &= \Lambda^\mu_{\;\;\alpha}\Lambda^\nu_{\;\;\beta}\...
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7answers
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Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
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Which basis 2-form elements represent positive traversals in Minkowski 4-space?

I'm certain some of this relies on arbitrary choice, for even in Euclidean 3-space, there is no a priori preferred choice of left versus right hand coordinates. In fact according to Einstein: There ...
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1answer
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Euler's Equation applied to Perfect Fluid

Consider this form of Euler's Equation: $$\rho \vec{a}=\nabla \cdot T+\rho \vec{f}$$ Where: $\rho$ is the density, $\vec{a}$ is the acceleration, $T$ is Cauchy's Stress Tensor and $\vec{f}$ is the ...
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A problem with differential forms in relativistic electromagnetism [duplicate]

We can write Maxwell's Laws as follow: $$\partial _\mu F^{\mu\nu}=\mu _0 J^\nu$$ $$dF=0$$ where $F^{\mu\nu}$ is the Faraday Tensor and $J^\nu$ is the Four-Current Density. You can see this for context....
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Parallel transport: Lie derivative vs covariant derivative

Given a manifold, we can generalize the idea of derivatives in multiple ways: two of them being the Lie derivative and the covariant derivative. Whereas Lie derivatives do not require any additional ...
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1answer
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Trying to intuitively understand Christoffel symbols

I recently watched Sean Carroll's YouTube series on "The Biggest Ideas in the Universe". In his Geometry and Topology video, he says that the connection in Riemannian geometry describes how ...
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1answer
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Relation between 4-divergence and 3-divergence of vector on the hypersurface

How the relation $\nabla_{a} u^{a}=D_{a} u^{a}-\epsilon v^{i} \nabla_{i} v_{j} u^{j}$ where $D_{a}$ is covariant derivate on the hypersurface between 4-divergence and 3-divergence of the vector on the ...
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2answers
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Commutator of derivatives with torsion

I am currently looking at "Physical aspect of space-time torsion" by IL Shapiro. There in eq. 2.10, it mentions that in space-time with torsion the commutator of covariant derivatives acting on the ...
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Derivative of Riemann tensor respect to Riemann tensor

I know that, for example we have $$\frac{\delta g^{jk}}{\delta g^{lm}}=\delta^{j}_{(l}\delta^{k}_{m)}.$$ This topic was discussed previously e.g. on Physicsforums.com and on Phys.SE. So I was ...
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1answer
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What is the convention for tensor indices for matrices?

The Lorenz-Transformation of the EM-Tensor F is given by the equation $$ F'^{\mu \nu} = \Lambda^{\mu}_{\ \ \rho} \Lambda^\nu_{\ \ \sigma} F^{\rho \sigma}$$ Then it says that this is equivalent to the ...
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Contravariant rank-2 tensor transformation in index notation

I'm slightly confused about the placement of upper and lower indices for the transformation of a rank-2 contravariant tensor. A contravariant rank-2 tensor transforms as $$M' = \Lambda M \Lambda^{T}$$....
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My understanding of General Relativity

My Background: In high school, I completed AP Physics C Mechanics and Electricity and Magnetism. In my first year of undergrad, I completed a course on Newtonian Mechanics and a course on Special ...
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Derivation of Maxwell's equation from Faraday tensor [closed]

If $F = F_{\alpha\beta}\text dx^\alpha \otimes \text dx^\beta$ Then how do I write the covariant derivative $\nabla F$ in component form. Here $F_{\alpha\beta}$ is a component of Faraday tensor and $\...
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3answers
77 views

Are $v^ie_{i}$ and $v^iv_{i}$ (where $v$ are the components and $e$ the basis vectors) both tensors? Or only the second one?

I am studying the math of tensors, I have an understanding of the concepts of covariance, contravariance, dual spaces, Einstein notation and so on. I am a bit confused about the notation though. My ...
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1answer
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Electromagnetic Potential in Relativity

Studying Special Relativity we discover that Maxwell's Equations can be also written in the following way: $$\partial _\mu F^{\mu\nu}=\mu_0J^\nu$$ $$dF=0$$ Where: $F$ is the Electromagnetic Tensor, $J$...
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1answer
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Contracted indices can be interchanged?

I am working on Lorentz transformations and I get a tensor of the form $$M_{abcd}=\epsilon_{ab\mu\nu}\Lambda^\mu\hspace{0.1cm}_c\Lambda^\nu\hspace{0.1cm}_d$$ Where $\epsilon_{abμν}$ is the totally ...
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3answers
348 views

Identifying Lorentz Covariant Equations

Statement: $\phi , A^{\mu}, T^{\mu \nu}$ are a Lorentz scalar, vector, and tensor. Which of the following equations are Lorentz covariant. a. $\phi = A_{0}$ b. $\phi = A^{\mu}A_{\mu}$ c. $\phi = ...
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2answers
218 views

Tensor and Matrices

Suppose that we are dealing with the following matrix: $$A=\begin{bmatrix}a_{00} & a_{01} \\ a_{10} & a_{11}\end{bmatrix}$$ but I don't want to use matrix notation, insted I want to use tensor ...
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How does Susskind get this fourth equation? [duplicate]

I'm studying Friedman and Susskind's Special Relativity and Classical Field Theory. In Lecture 6, they derive $$m{dU_\mu\over d\tau}=eF_{\mu\nu}U^\nu,\tag{6.33}$$ but only when $\mu=1, 2, 3$. ($F_{\mu\...
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1answer
56 views

Questions about Navier-Stokes equations, Einstein notation, tensor rank

I'm looking at Navier-Stokes equation in index notation and how to get them in vector notation: $$ {\partial u_i \over \partial t}+ u_j {\partial u_i \over \partial x_j}= -\frac{1}{\rho}{\partial p \...
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1answer
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Transformation Properties of Connection Coefficients

This question is about pages 95 and 96 of Carroll's book: Spacetime and Geometry. We have the formula for the covariant derivate: $$\nabla _\mu V^\nu=\partial _\mu V^\nu + \Gamma _{\mu\lambda}^\nu V^\...

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