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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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Lorentz covariant derivative of the vielbein determinant

Denote by $\mathcal{D}_m$ the Lorentz covariant derivative, $$\mathcal{D}_m=\partial_m-\frac{1}{2}\omega_m{}^{ab}M_{ab} \tag{1}$$ where indices $m,n,p,\dots$ are world indices, indices $a,b,c,\dots$ ...
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Calculating surface gravity

I have some trouble with understanding how surface gravity/Killing horizon equation works, for example in following form: $$κ^2=-\frac 12 (\nabla^aK^b)(\nabla_aK_b)$$ with Killing vector $K$. I'm ...
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Electromagnetic Tensor - Magnetic Elements

Given the electromagnetic stress tensor $F^{\mu\nu}$ such that $F^{ij}=\epsilon_{ijk}B_k$, I attempted to write the 'inverse' and got the following. $$B_i=\frac{1}{2}\epsilon_{ijk}F^{jk}$$ This ...
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What is the physical meaning of a vector space, basis and tangent bundle? [on hold]

I've been studying General Relativity, mostly from Spacetime and Geometry by Sean Carroll and I don't really understand the physical meaning of all the mathematical terms used. I know what the ...
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1answer
38 views

What to do with an extra index in the definition of a tensor?

I came across this definition of a tensor while reading some vector calculus literature This definition contains the index $\ell$ in the last term, however the tensor itself only depends on $j$ and $...
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Why does the lowered Riemann tensor only have 20 independent components for the Schwarzschild metric?

I have seen quite a bit online about how there are only 20 independent components for the (lowered) Riemann tensor $R_{abcd}$ for the Schwarzschild metric. I've been told this follows from the ...
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1answer
29 views

4-Vector Upstairs vs Bottom Indexing

In placing the index top or bottom, I know that it's irrelevant for spacial components, but flips the sign temporally in the 4-position. Does this temporal flip extend to any 4-vector? I.e. for the ...
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106 views

4-Vector Definition

In most places I've looked, I see that 4-vectors are defined as 4-element vectors that transform like the 4-position under lorentz transformation. This is typically accompanied by generally, $$\...
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1answer
71 views

Formula for Christoffel Symbols in Terms of Derivatives of Riemannian Metric with Contravariant Indices? [on hold]

The following formula represents the usual textbook way of computing the Christoffel symbols of the second kind ${\Gamma^i}_{jk}$ via the Riemannian metric with covariant indices $g_{mn}$ and spatial ...
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Vanishing antisymmetrisation of bitwistors

Following the conventions in https://arxiv.org/abs/1406.7090 (mostly section 3.3). Let $T_{\hat{\alpha}}\,^{\mu}$ denote two linearly independent twistors forming a basis for a two plane in $\mathbb{...
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eigenvectors of two deformation measures described in the same basis system

I have two deformation gradient tensors, corresponding to the simple shear of a cube; one at time t (say $\textbf{F0}$) and another at time $t+\Delta t$ (say $\textbf{F1}$). The basis of $\textbf{F1}$ ...
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Definition of Alternating $(k,0)$- and $(k,l)$-tensors

I know that one can define the alternating subspace of $(0,l)$-tensors in a straightforward way. These are the renowned $l$-forms. However, I have been searching in the literature for a definition of ...
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1answer
72 views

Lorentz covariance of Pauli-Lubanski pseudo-vector

The Pauli-Lubanski pseudo-vector is defined as: $$W_{\mu}=\frac{1}{2}\epsilon_{\mu \nu \lambda \rho}J^{\nu \lambda}P^{\rho}$$ Where the rotation and translation operators transform as: \begin{align}...
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Computing the metric in the barred frame using a 2D coordinate transformation

I would like to apply the coordinate transformation $x^{\bar{1}} = 2x^1$, $x^{\bar{2}} = x^2$ in the 2D Cartesian plane. The metric in the barred frame is $g_{\overline{ij}} = \Lambda^i_{\bar{i}} \...
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1answer
42 views

Variation of scalar field - this equality is true?

I was using a computer program to do tensor computations, and found a mismatch between my result and the computer's result. The results would match if $$g^{cd} \nabla_c \phi \nabla_d \phi = g_{cd} \...
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32 views

Deducting from a special case that the electric field transforms as a 2nd rank tensor: Vanishing divergence concludes vanishing vector field?

I'm (as in the previous question) still working with Kobe's Paper. I try to understand his reasoning (given in Appendix A) why the electric field components should be components of a 2nd rank tensor. ...
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Stress Energy Tensor in language of differential forms

The motivation for this is that quantities like the electric current $J$ in maxwell's equations of motion can be expressed as a differential 3-form, so that the continuity equation can be written just ...
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What's the variation of a product of two metrics? [closed]

I was trying variate an action in General Relativity, and I come to the next calculus: $\delta(g^{\alpha\beta}g^{\mu\nu})$ And I did: $\delta(g^{\alpha\beta}g^{\mu\nu})=g^{\alpha\beta}\delta g^{\mu\...
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How can one define ordinary derivative of a vector field along a curve? [migrated]

Definition 1: A curve $C$ on a manifold $M$ is a smooth function $C:(a,b) \rightarrow M$. Definition 2: A vector field $v^a=v^a(t)$ along a curve $C$ is an assignment of vectors on the tangent space ...
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35 views

Index Notation Difficulty (Mixed index, i.e Torsion 3-form)

My question regards an index "notation" difficulty I've faced regarding the Torsion 3-form tensor (regarding properties of the Kalb-Ramond 2-form) and its Kronecker. I start with the co-variant: $$H_{\...
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1answer
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Intuition behind covariant and contravariant vectors

sorry is there any good intuition behind the following definitions. I am having trouble understanding these. Or is it recommended to just continue reading and accept these definitions for now? Update:...
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1answer
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Value of the invariant $R_{\mu \nu}F^{\mu \nu}$

Is there a simple way to find the value of $R_{\mu \nu}F^{\mu \nu}$ (where $R_{\mu \nu}$ is the Ricci tensor and $F^{\mu \nu}$ is the electromagnetic tensor), knowing that it is an invariant? ...
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definition of dual basis using gradient

I am trying to read up on tensor calculus, but am stuck at a definition I find unintuitive. I understand the normal basis definition with partial derivative, and can visualize what it represents ("...
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1answer
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Exterior and Covariant Derivatives

Is the following guaranteed to be true for any covariant vector $f_\mu$ (1-form $\boldsymbol{f}$) in the absence of torsion? $$\nabla_{[\alpha}\nabla_{\beta}f_{\mu]}=\partial_{[\alpha}\partial_{\beta}...
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2answers
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Trace of the Riemann Curvature Tensor

Referring to Wald's General Relativity, I have two questions. Let ${R_{abc}}^d$ be the Riemann curvature tensor. The author has never defined what it means by "trace of a tensor" before page 40 of ...
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1answer
50 views

Construct components of tensor operator [closed]

I'm reading Georgi's textbook on Lie Algebras and have been struggling with this question for quite awhile. The entirety of Chapter 4 (Tensor Operators) has been much more difficult than anything I've ...
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1answer
57 views

Confusion with Covariant Derivative

In Wald's General Relativity, the covariant derivative is an operator acting on $(k,l)$ type tensors. But referring to the second point in the yellow block in this post, the covariant derivative acts ...
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1answer
71 views

Tensor (anti)symmetrization for non-adjacent indices: how can I notate $T^{(ab)c}$ but symmetrizing over $a$ and $c$ instead?

Refer to the following passage from Robert Wald's General Relativity: More generally, for a tensor $T_{a_1\cdots a_l}$ of type $(0,l)$ we define \begin{align} T_{(a_1\cdots a_l)} & = \frac{1}...
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Are there alternative expressions for the d'Alembertian?

I know that the Christoffel symbols satisfy: $$ \Gamma_{ab}^a = \frac{1}{\sqrt{-g}} \partial_b (\sqrt{-g}),$$ where $g$ is the usual metric determinant. From this we can get an expression for the ...
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1answer
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Generalization of the Coulomb Force to the Lorentz-Force - Is it “guessing”?

it's me again, and I'm still stuck with the paper Generalization of Coulomb’s law to Maxwell’s equations using special relativity by Kobe, like in my previous question. My problem now lies in ...
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1answer
46 views

Finding the matrix form of Brinkmann's metric [closed]

I have the following problem: given Brinkmann's metric expressed as $$ds^2 = du dv - \delta_{i j} dx^i dx^j - K_{i j}(u) x^i x^j du^2$$ and $i,j=1,2, $ I have to find it's matricial form; my question ...
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How prove that the contraction of electromagnetic tensor and its dual, $F_{\mu\nu}\tilde F^{\mu\nu}$, is an total derivative? [closed]

I have tried to solve this exercise from Supergravity-Freedman and Van Proeyen (2012), Excercise 4.10 Show that the quantity $F_{\mu\nu}\tilde F^{\mu\nu}$ is a total derivative, i.e. $$F_{\mu\nu}\...
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General covariance and Maxwell equations

In General Relativity, I use the principle of general covariance such that \begin{equation} \eta_{\mu\nu}\to g_{\mu\nu}, \quad \partial_\mu\to \nabla_\mu \end{equation} so that I re-express physical ...
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Covariantly constant 2 Component Spinor

Note: For this question I am using the conventions of "Ideas and Methods of Supersymmetry and Supergravity" by Ioseph Buchbinder and Sergei Kuzenko (mostly p16 & p44). Let our space be equipped ...
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Contracted product of matrices with Lorentz indices

Let $\mathbf{A}_\mu$ and $\mathbf{B}_\mu$ be two matrix-valued spacetime vectors, i.e. $(A_\mu)_{ab}$ and $(B_\mu)_{ab}$ and let $\mathbf{C}$ be a matrix in the same space (external to spacetime), i.e....
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Massive Real Vector Field Identity

Let's consider the classical Lagrangian density for a real vector field $V^{\mu}$ of mass $M$: $$L_{V} = -\frac{1}{4} V_{\mu\nu}V^{\mu\nu} + \frac{1}{2} M^{2} V_{\mu}V^{\mu} $$ The Eulero-Lagrange ...
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Change in the metric due to a change in co-ordinates

I'm having a problem calculating the first order change in metric due to a transformation of co-ordinates. The transformation is as follows: $ \sigma'^{c} = \sigma^{c} + \epsilon^{c}$ Then $$ \...
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1answer
30 views

Scalar Flow Across a Small Area Element

I've just started reading the text "Vectors, Tensors, and the Basic Equations of Fluid Mechanics" by Rutherford Aris and I came across the following problem. If $\rho$ is any scalar property per unit ...
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1answer
76 views

Help with Chrstoffel symbols for geometric mechanics problem?

I am working through the book Geometric Control of Mechanical Systems by Bullo and Lewis https://www.amazon.com/gp/product/0387221956/ and I am stuck on a problem, E4-18. The problem was evidently at ...
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Re-expressing the Lorenz gauge condition in terms of the Faraday tensor (in curved spacetime)

I was wondering if the equation $\nabla_\mu A^\mu = 0$ could be written as a constraint equation solely on the $F_{\mu \nu}$ components. It seemed like the bulk of the problem was isolating terms such ...
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1answer
53 views

Is there a convention for the ordering of terms in the dot product of a vector with the gradient of a vector field? My left is another man's right

Note: I use $\left[\dots\right]$ instead of $\left(\dots\right)$ to indicated function arguments. I use $\left[\![\_,\_\right]\!]$ to signify a commutator. And I tend to use bracketing where others ...
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Decomposition of the symmetric part of a tensor

The rate of strain tensor is given as $$e_{ij} = \frac{1}{2}\Big[\frac{\partial v_i}{\partial x_j}+ \frac{\partial v_j}{\partial x_i}\Big]$$ where $v_i$ is the $i$th component of the velocity field ...
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1answer
48 views

Coordinate transformation in Tensor Calculus

I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field newtonian metric tensor $$ds^...
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1answer
84 views

Momentum transport equation

In the derivation of the momentum transport equation in Kirkwood's paper (https://aip.scitation.org/doi/10.1063/1.1747782), I am stuck at a particular point in the derivation. The rate of change of ...
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37 views

The foundations of geometric formulation of Newton's axioms

On Professor Frederic P. Schuller's Lecture about General relativity, where you can access it through this link: https://www.youtube.com/watch?v=IBlCu1zgD4Y, he clamed Newton's axioms can be converted ...
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1answer
62 views

Derivation of $j$ being a 4-vector in Landau-Lifschitz: Formulation with rigorous mathematical treatment?

Here on Stack exchange, there appeared the question on how to derive the 4 current actually being a Lorentz-tensor. One of the answers (How do we prove that the 4-current $j^\mu$ transforms like $x^\...
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2answers
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Is $\partial_\alpha f^\alpha$ coordinate-independent?

At this point in Schuller's 9th lecture on GR, he claims that Poisson's equation for the Newtonian gravitational field strength is $$-\partial_\alpha f^\alpha=4\pi G \rho,$$ where $\alpha=1,2,3$. But ...
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3answers
166 views

Are contravariant basis vectors and basis 1-forms identical?

The reason I'm asking this is because I am trying to develop a set of notes from my reading of MTW (and Wrede, Menzel, Bergman, etc.). I represent covariant basis vectors with $\mathfrak{e}_{i}$, ...
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1answer
63 views

What's the role of the tensor product in quantum mechanics?

There are several ways to define the tensor product. So there are multiple ways to look at it. I've seen it being used as an object to calculate metric, and I also seen in ring module theories. I'm ...
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Codazzi-Mainardi Equation & Field Equations

So the Codazzi-Mainardi Equation implies $$ P^{\alpha \mu} n^\nu G_{\mu\nu}=D^\alpha K - D_\mu K^{\alpha \mu} \tag{1} $$ where $G_{\mu\nu}$ is the Einstein-tensor, $D_{\mu}=P_{\mu}^\alpha \nabla_\...