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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Non-zero Riemann tensors for the BTZ solution without charge or angular momentum

Can someone tell me which are the non-zero components of the Riemann tensor for the BTZ solution with the absence of charge and angular momentum? I tried searching it up but couldn't find any ...
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Why can integrals be written as $I=\int \phi(x) \epsilon$?

Carroll's book Spacetime and Geometry gives this new way to think about integrals: $$I=\int \phi (x) \epsilon\tag{2.98}$$ $\epsilon$ is the Levi - Civita tensor (2.96). I don't see how the RHS equals ...
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Electromagnetic tensor in a FRW metric

In some papers [like https://arxiv.org/pdf/2204.06883.pdf, eq. (31) ], I see that the Electromagnetic tensor field, for a FRW metric (written in a conformal way) \begin{equation} ds^{2} = a^{2}(\tau) \...
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Thinking of a linear operator as a (1,1) tensor

I am reading that a linear operator $A$ can be thought of as a (1,1) tensor [where $(r,s)$ corresponds to $r$ vectors and $s$ dual vectors]. This can be done by saying $$A(v,f) \equiv f(Av)$$ where $v$...
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Index position of spherical harmonic function and tensorial properties

The spherical harmonics functions are denoted as $Y_l^m$ or $Y_{lm} $in https://en.wikipedia.org/wiki/Spherical_harmonics e.g., \begin{align} Y_{lm} &= \dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^...
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How to sum with $\epsilon_{ijk}$?

How does one find the sum with $\epsilon_{ijk} $? For example, $\sum\limits_{i,j,k=1}^3 \epsilon_{ijk} \frac{a_i b_j}{c_k} T_k$? Here $a_i, b_j$ and $c_k$ are scalars and $T_k$ is an operator.
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How to tell if the covariant derivative of something is timelike or spacelike

How can I tell if the covariant derivative of something is timelike or spacelike?
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Question on reciprocal metric tensor

First of all, to simplify we will assume 2 dimensional space with a symmetrical metric tensor $g_{\mu\nu}$ It's known that d'Alembert operator (we will use it for example) is defined as $\partial_\nu \...
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What is the divergence $S^{k}{}_{i,k}$ of a force component of a stress tensor called?

The following is from Hermann Weyl's Space-Time-Matter. Notice that $$\mathfrak{p}=-\left\{ p_{i}=S^{k}{}_{i,k}\right\}$$ is equivalent to a volume force. But, from the infinitesimal view, it is the ...
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Twistor equation and Killing equation

Yesterday was the birthday of Roger Penrose. And reading again about twistors I realized that twistor equation is strikingly similar to a Killing equation. My question is, are they "equivalent&...
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Proof that a Lorentz-invariant scalar function can only depend on scalar products

If we have a Lorentz-invariant scalar function $f$ of a single four-vector $x^{\mu}$ we can show that $f$ can only depend on $x^2$ (see Argument of a scalar function to be invariant under Lorentz ...
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Lie derivative acting on a function

I'm a little confused about Lie derivatives. In fact all that definitions of pull-back and push-forward and one-parameter family of diffeomorphisms and integral curves and so on seems strongly ...
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Confused about tensor notations of how vector and covectors act on each other

I'm learning/playing around with tensors and somehow got this contradiction, suppose $\{v_i\}$ and $\{w_i\}$ are basis for a vector space $V$ and $\{ v^i \}$ and $\{w^i\}$ are basis for the dual space ...
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Tensor that depends on first-order derivatives of the metric

The question is pretty straightforward. Just as all tensor fields depending on the metric's second derivative can be derived from the Riemann curvature tensor, is there any well-known tensor, such ...
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Valid tensor expression?

The following given tensor expression is:$$A^i=B_i+C_i\tag{1}$$ which is invalid expression since the free index on L.H.S is in upper part and on R.H.S it is in lower part so to make valid tensor ...
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Classical electromagnetism field strength with index up and down

In Classical electromagnetism, we know the Lagrangian density read $$ \mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu} $$ where $$ F_{\mu \nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu} $$ However, I ...
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Is there only one convention to define the electromagnetic field tensor?

I know that the electromagnetic field tensor depends on which metric is used. For example wikipedia uses the $(+---)$ sign convention, but in the Griffiths we have the $(-+++)$ sign convention. That's ...
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Riemann curvature tensor

I am little bit confused on Riemann curvature tensor, Riemann curvature tensor written in component form as; $$R^d_{cab}=\partial_a\Gamma^d_{bc}-\partial_b\Gamma^d_{ac}+\Gamma^i_{bc}\Gamma^d_{ai}-\...
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How do you use the Riemann tensor to show curvature in time?

I know that the Riemann tensor shows curvature in space. However, in the case of general relativity, where space AND time is curved, how would one use the Riemann tensor to show curvature in a ...
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Schwarzschild Christoffel Symbol Problem

Schwarzchild metric is static, none of its components change over time. Curiously, basis vector $e_r$ does change over time. If we differentiate $e_r$ with respect to time and take the time component ...
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Index-free notation indulgence!

I have been try to find as an exercise for myself the most suitable coordinate-free form of the following equation found in Misner, Thorne, Wheeler's Gravitation (p. 84) \begin{equation} v^\ell = (F_{...
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Navier-Stokes Equations in Einstein Notation and its relation to Poisson's Equation

The Navier-Stokes equations are: $$\partial_t v + v \cdot \nabla v = - \nabla p + \nu\nabla^2 v, \quad v \in \mathbb{R}^3\\ \nabla \cdot v = 0$$ I have seen that, using Einstein notation (which I am ...
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Transverse component of distorsion tensor in GR

On pages 164-165 of Eric Gourgoulhon's lecture notes on Numerical Relativity, the author introduces the decomposition (9.49) for the distorsion tensor related to a foliation $(\Sigma_t)_{t\in \mathbb{...
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Divergence of tensor fields

I have found numerous definitions for the divergence of a tensor which makes me confused as to trust which one to use. In Itskov's Tensor Algebra and Tensor Analysis for Engineers, he begins with ...
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Are the linear Lie groups matrices, tensors, or both?

In some ways, this is a question about notation. In my experience, I have only seen the classical Lie groups — such as $\operatorname{GL}(n,\mathbb{R})$, $\operatorname{SL}(n,\mathbb{R})$, $\...
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What is the intuition behind the ${\nabla u^T}$ term of the stress/strain tensor for a Newtonian fluid?

$$\epsilon=\frac{\nabla u+ \nabla u^T}{2},$$ $u$ is vector displacement, and $\nabla u$ is the gradient matrix of $u$. Now for a Newtonian, incompressible fluid, this describes the shear stress forces ...
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Calculating conjugate momenta for a spin-2 field

Consider a symmetric spin-2 field $h_{\mu \nu}$. I have the following Lagrangian for this field: $$\mathcal{L} = - \frac{1}{4}\left(\partial_{\lambda}h_{\mu \nu} \text{ } \partial_{\phi}h_{\alpha \...
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Kalb-Ramond current fall-offs at future null infinity

I can couple the electromagnetic field to a current generated by the complex scalar field for example: $S=- \int d^4x \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + A_\mu J^\mu$ with $J_\mu = i(\partial_\mu \phi^...
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Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory

1) Introduction From standard tensor calculus, here restricted to Minkowski spacetime, we learned that: A scalar field is a object that transforms as: $$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$ A ...
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Tensor symmetry: A symmetric mixed $(1,1)$ tensor is necessarily a multiple of the identity tensor

I found this observation in a book. "It is not possible to have an invariant definition of symmetry in one contravariant and one covariant index". That's all right, my problem is that to ...
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Determinant of the inverse of a Lorentz transformation

In many text book (Ashok Das Quantum Field Theory) $$(\Lambda^T)_\nu{}^\mu=\Lambda^\mu{}_\nu$$ that gives $\Lambda^T$ = $\Lambda^{-1}$, where $\Lambda$ is Lorentz Transformation matrix. However, this ...
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How to contract spinor indices?

In normal vector representation, vectors can be contracted as follows: $$v^\mu v_\mu$$ with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of ...
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What unit of measurement are used for 𝑀 and 𝑟 in the Schwarzschild metric?

So I understand that 𝑀 in any metric solution stands for mass. But what unit of measurement do you use to define 𝑀?
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Can you convert the Christoffel Symbol to the form of a scalar?

Given some tensor $T_{\mu v}$, you can use the metric tensor to contract its indices, converting it into the form of a scalar: $$g^{\mu v}T_{\mu v}=T$$ Even though the Christoffel Symbol is not a ...
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Taking the covariant derivative of the derivative of the metric tensor

Does the covariant derivative of the derivative of the metric tensor exist? if so, how do you evaluate it? $$\nabla_a (\partial_b g^{\mu v})=?$$ It would seem natural to assume that this cannot exist, ...
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What are some good textbooks to learn EM with differential forms? (With focus on condensed matter)

As the title suggests, does anyone have recommendations for learning EM with differential forms? Both undergraduate and graduate level textbook suggestions are welcomed. I know there's Lindell's ...
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Tensor manipulations in Landau & Lifschitz "Classical theory of fields"

Landau & Lifshitz "Classical theory of fields" section 6 p. 19 define: $$ df^{ik} = dx^i dx'^k - dx^k dx'^i $$ and $$ df^{*ik}=\frac{1}{2} \; \epsilon^{iklm}df_{lm} \tag{6.11} $$ and ...
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Reading "A Convex Darboux Theorem" by Chiappori & Ekeland

This is cross-posted from https://math.stackexchange.com/q/4452087/ but got no reply I am reading "A Convex Darboux Theorem" by Chiappori and Ekeland [1]. I do not pretend to understand the ...
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Is it possible to derive the Weyl tensor from the Ricci tensor or Ricci scalar? If so, how?

I understand that the Ricci tensor is derived from the Riemann tensor, but I know that the Weyl tensor is also derived from the Riemann tensor. So, is it possible to calculate the Weyl tensor from the ...
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Rigorous Definition of Scalars and Vectors? [closed]

What is the rigorous definition of "Vector" (& " Scalar")? Best I got was: https://www.youtube.com/watch?v=Ncx98PmXbZc
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Calculating Squared Angular Momentum Using Index Notation [closed]

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: $\...
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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 ...
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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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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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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 ...
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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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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 ...
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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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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 ...
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