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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Rigorous Definition of Scalars and Vectors?

What is the rigorous definition of "Vector" (& " Scalar")? Best I got was: https://www.youtube.com/watch?v=Ncx98PmXbZc
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What is the form of the radial gauge fixing condition in $p$-form fields?

In $1$-form fields (electrodynamics) the radial gauge fixing condition is expressed as $A_r=0$. How would that condition look like in higher-rank fields, for example a $2$-form field?
2 votes
1 answer
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Index position when varying an action with respect to the metric

I'm confused about where we should put tensor indices when we vary an action wrt the metric. For example, if I have in the Lagrangian a term such as $$ A_{\mu\nu}B^{\mu\nu}, $$ do I necessarily have ...
1 vote
1 answer
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Calculating Squared Angular Momentum Using Index Notation

I am trying to derive the form of the squared angular momentum using the index notation but I am stuck at the very last step. First, I defined the orbital angular momentum using the index notation: $\...
4 votes
1 answer
73 views

Wave equation in curved spacetime

Consider a tensor field $h_{\mu \nu}(t,\vec{x})$ which obeys the wave equation, given by (in units where $c = G = 1$): $$ \Box h_{\mu \nu} = a \text{ } \bar{T}_{\mu \nu} \tag{1}$$ where $a$ is some ...
6 votes
1 answer
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Is there are relationship between the Ricci scalar and the determinant?

On the one hand the determinant (technically the absolute value of the determinant) represents the "volume distortion" experienced by a region after being transformed. On the other hand the ...
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41 views

Frame-Dragging questions

I know the Kerr metric can be used to express Frame-Dragging for angular momentum around a Boyer-Lindquist coordinate, but relativistically according to the Lense-Thirring effect, spacetime in a ...
3 votes
2 answers
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What is a Tensor, intuitively?

Before reporting this a duplicate, this post explains the computations of tensors nicely, but I still have questions regarding why they are needed. Also, what makes a tensor actually tensor? The way ...
0 votes
2 answers
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For the Schwarzschild metric, are the values of the Ricci tensor and Ricci scalars always zero? [duplicate]

If we use the Schwarzschild metric to solve the Einstein field equations, would the values of the Ricci tensor and scalars always be zero?
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1 answer
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Is four-vector product always Lorentz invariant?

Let's say we have two four-vectors $a^{\mu}$ and $b^{\nu}$.Is it always true that any combination of those 4-vectors (1-rank tensors) multiplied together will yield an invariant quantity (0-rank ...
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1 answer
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Discrepancy in the transformation law for the Christoffel Symbols of the First Kind [duplicate]

I'm currently studying Tensor Analysis from Mathematical Methods: For Students of Physics and Related Fields by Sadri Hassani. In page 462, he introduces the "Affine Connection", i.e., the ...
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Operations of Tensors with Different Orders

I know that a 4th order tensor times a 2nd order tensor yields a 2nd order tensor; and a 2nd order tensor times a 2nd order tensor yields a 0th order tensor, or scalar. But from my linear algebra ...
7 votes
2 answers
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How to understand physical tensors?

The tensors used in physics, 2 rank tensors, have to be thinked as linear operators which send a vector into another vector or as multilinear maps which send two different vectors into a scalar? I'm ...
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1 answer
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Reshaping $U$ and $V^\dagger$ matrices resulting from an SVD into rank-3 tensors

Let's say, that we have a $6 \times 6$ matrix $M$. By conducting an SVD of $M$ we obtain $USV^\dagger$ matrices, where $U$ is of size $6 \times 6$ and is left-normalized, $S$ is diagonal with 6 ...
1 vote
2 answers
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Is the raised Levi-Civita symbol a tensor density of weight 1?

In Sean Carroll's GR book, pg 83, between eqs. (2.69-70), the Levi-Civita symbol with raised indices is defined as $$\tilde{\epsilon}^{\mu_1\mu_2...\mu_n}=\text{sgn}(g)\tilde{\epsilon}_{\mu_1 \mu_2...\...
3 votes
2 answers
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Why do we talk about inertia tensor?

When we talk about the inertia of a rigid body, in calculating the angular momentum as a function of the moment of inertia and angular velocity, the inertia tensor is introduced. But why is it a ...
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3 answers
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What can I substitute for tensors in special relativity? [closed]

Am in college (completed algebra, calc 2,trig) and willing to study special relativity this summer but really have problem with understanding tensors.
1 vote
1 answer
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Determinant metric tensor

I do not understand the relation between the determinant of the metric tensor $g$ and the non-tensorial symbol $\tilde{\epsilon}_{\mu_{0}..\mu_{n}}$. This is explained in Carrol's book as followed: \...
1 vote
2 answers
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Proof of the Piola Transform

Proof of the Piola Transform. As I understand it, the relationship between the second order tensor $\bf T$ over a reference configuration and the same tensor in a deformed configuration $\bf T^\prime$ ...
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1 answer
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Field of research treating tetrads (vierbeins) as fundamental objects?

After a lot of research on tetrads I think I found the subject I'd like to specialize in for postgrad/phd, as they seem to express many interesting and (perhaps) fundamental physical properties. So i ...
5 votes
2 answers
139 views

Proper conceptualization & notation for vectors, $n$-tuples, and matrices in physical space

This is a fairly basic question that I may be making longer than necessary. But it has plagued me for some time. It is essentially this: In what space do abstract physical vectors like a velocity ...
1 vote
0 answers
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Does anyone know a good paper about asymptotic symmetries at null infinity in $p$-form theories?

I've only found some about the analysis of asymptotic symmetries at spatial infinity (Afshar2018, Esmaeili2020). Is there a paper where they look at null infinity?
11 votes
7 answers
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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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1 answer
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Advection term for a matrix equation

How can I calculate a quantity like $(\vec{v} \cdot \nabla) M$ where $\vec{v}$ is the velocity vector, and $M$ is some 3x3 matrix? (if one wants, assume $M$ is a tensor) This would be the advective ...
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2 answers
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Finding equation of motion of Lagrangian density: What does the location of the indices mean?

We are given the following Lagrangian density: $$\mathcal{L}=F_{\mu \nu} A^{\mu} \mathcal{J}^{\nu}$$ where $F_{\mu \nu}$ is the electromagnetic field tensor, $ A^{\mu}$ the 4-vector of the vector ...
4 votes
1 answer
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Advanced atomic physics: From Liouville Equations to the Bloch equations

I'm trying to derive the Bloch equations from the Liouville equation. This should be possible according to this paper, which discusses higher order Bloch equations (second order spherical tensors). I'...
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1 answer
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Covariance of Euler-Lagrange equations under arbitrary change of coordinates

I'm trying to prove that the Euler-Lagrange equation $$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i})-\frac{\partial L}{ \partial q_i}=0$$ is invariant under an arbitrary change of coordinates $$...
3 votes
2 answers
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What is the physical meaning of the third invariant of the strain deviatoric?

In continuum mechanics of materials with zero volumetric change, the material condition can be expressed by the strain deviatoric tensor instead of the strain tensor itself. To express the plasticity ...
0 votes
1 answer
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Riemann curvature tensor components having 3 or 4 distinct components

When, if ever, will we see Riemann curvature tensor (RCT) components having 3 or 4 distinct indices?, like for $R_{txxy}$ or $R_{txyz}$ for ex. How this came about was I that I was reading that ...
1 vote
3 answers
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Question about Wald's example of a "derivative operator"

I am trying to study Wald's book on General Relativity. I thought the first two chapters were ok, but I'm completely stuck in chapter 3, on page 32 where he gives an example of a "derivative ...
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Index transposing in Einstein notation

When working with Einstein's summation convention, how do I have to transpose the indexes of the tensor? For example, supose I want to take the matrix product with its transpose. Which is the correct ...
0 votes
1 answer
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Why does contracting a term with a tensor means a portion of this term is a tensor?

I am looking at a problem in Guidry's Modern General Relativity, and the solution contains the following two sentences: In the scalar product expression $A\cdot B = g_{\mu \nu}A^{\mu} B^{\nu}$, the ...
-1 votes
2 answers
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Confusion about raising and lowering indices

Is it possible to take the following expression: $$U^\mu U^v\partial_\mu\partial_v$$ Where $U$ is the four-velocity, and simplify it the following way?: $$U^\mu U^v \eta_{\mu v}\partial^v\partial_v =c^...
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Can someone suggest me a book where whole Electrodynamics of master level is explained in tensor form

I am trying to study electrodynamics, can someone please suggest me a book or any other resource where I can get it in tensor form, Topics: waves in medium, resonators, etc.
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1 answer
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Divergence theorem in index notation

From Batchelor's book of fluid dynamics: I guess that's an easy question for anyone having more familiartiy than me in tensor calculus, anyways. First integral argument is the i-component of the ...
0 votes
1 answer
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Trace of second-order tensor and its invariance under coordinate transformation

Let's consider an arbitrary scalar field. If I act twice on the scalar field with a gradient operator, I will obtain second-order tensor. If I will take a trace of this tensor, I will obtain another ...
1 vote
1 answer
218 views

Levi-Civita symbol in 2-spinor notation

I'm reading An Introduction to Twistor Theory, by Huggett and Tod, and I don’t get the result we're being given page 17: the 2-spinor form of the 4 dimensional Levi-Civita symbol. \begin{equation} \...
1 vote
2 answers
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Lorentz transform of Levi-Civita Symbol

I was reading about Lorentz transformations and frequently I hear the notion of Lorentz transforming quantities like $\epsilon^{\mu \nu \rho \sigma}$. But I have never heard an explanation as to why ...
0 votes
1 answer
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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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1 answer
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Problem 2.1(b) in Peskin and Schroeder's Introduction to QFT

In this exercise the author claims that adding $\partial_\sigma K^{\sigma \mu \nu}$ does not affect the divergence of $T^{\mu\nu}$. In other words the author claims that $\partial_\mu \partial_\sigma ...
13 votes
2 answers
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Tensor decomposition under $\mathrm{SU(3)}$

In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning ...
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1 answer
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The displacement gradient tensor transformation rule

The transformation rule of a 2nd rank tensor expresssed in a given basis is often written as follow: $$F' = P^T FP $$ where $F$ is the matrix representation of the tensor in the old basis B, $F'$ its ...
2 votes
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Understanding the four-dimensional volume form in Action of Lagrangian

Into the following part below, I don't understand what is precisely a "four-dimensional volume form" implied in the integral below: For comparison, the ...
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1 answer
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How we find the contorsion tensor?

I know that the formula for contorsion tensor is $$K^{\mu\nu}_a=\frac12({T_a}^{\mu\nu}+T^{\nu\mu}_a-T^{\mu\nu}_a)$$ I want to know how I can find ${T_a}^{\mu\nu}$. What kind of contraction do I follow ...
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4 answers
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How do we make sense of $F^{\mu\nu}F_{\mu\nu}$? The book just assumes I understand it

Why are these upper and lower indices and what does that mean. I can't interpret the term with upper indices.
1 vote
3 answers
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Relation between divergence of unit normal and radius of curvature

I don't understand how does the divergence of a unit normal vector to a curve at a point gives the local radius of curvature. For simplicity consider a 2-D curve. $$\nabla.n=\frac{1}{R}$$ I want to ...
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1 answer
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Confusion on number of component of Cauchy stress tensor

The Cauchy stress tensor is often presented as a tensor having $(2,0)$ tensor having nine components in any given basis. However, I think it should actually be $6 \times 3 =18$ because a cube has six ...
4 votes
1 answer
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What does $\delta/\delta t$-derivative represent in tensor calculus?

Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the ...
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How to calculate the electric field of a polarization density?

By polarization density here I just assume I have a "blob" of free positive and negative charges, and instead of describing the system with a charge density $\rho(\pmb{r})$ I want to use the ...
1 vote
1 answer
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Stress energy tensor components for a perfect fluid

The stress energy tensor for a perfect fluid is given by $$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}-pg_{\mu\nu}$$ where U is the 4-velocity. The matrix components of the SEM are written as $$T_{\mu\nu}=diag(\...

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