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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How to obtain the Newtonian limit of specific energy $\mathcal{E}=-u_t$ of general relativity?

In general relativity, the conserved specific energy is expressed by the equation $$\mathcal{E}=-u_t$$, where $u_t$ is the time-component of the four-velocity. This is the total energy of the system ...
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Variation of the Gauss-Bonnet action and Palatini identity for the purely covariant Riemann tensor

I'm taking the variation of the Gauss-Bonnet action $$\mathcal{L}_{GB} = \frac{1}{2}\left(R^{2} - 4R^{\mu\nu}R_{\mu\nu} + R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}\right)$$ to obtain the equations of ...
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Definition of stress-energy tensor for the EFEs and other arbitrary cases

Suppose that there exists manifold $M$ s.t $T_{\mu\nu} \in M$, where $T_{\mu\nu} $ is the stress-energy tensor. The einstein field equations are given by $$R_{\mu\nu}-\frac12Rg_{\mu\nu}+\Lambda g_{\...
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Christoffel symbol metric connection

Suppose that you are given an arbitrary metric $g_{\mu\nu}$ such that you want to calculate all of the Christoffel symbol $\Gamma^{\lambda}_{\mu\nu}$. The equation for a Christoffel symbols are $$\...
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Christoffel symbols for certain metric using different formulas

Let M be differentiable manifold that represents spacetime, s.t. $\Gamma^{\lambda}_{\mu\nu}$ is the Christoffel symbol/connection coefficient. The general formula for the christoffel symbol is defined ...
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Variation of tensor $\sqrt{1-A_{MP} B^{P}{}_{N}}$ in terms of $A$ and $B$

I want to vary the tensor $X_{MN}=\sqrt{\delta_{MN}-A_{MP} B^{P}{}_{N}}$ in terms of $A$ and $B$. Perturbatively I can do this since \begin{equation} X_{MN} = \delta_{MN} - \frac{1}{2} (AB)_{MN} - \...
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A little clarification on Cartesian tensor notation

Goldstein pg 192, 2 ed In a Cartesian three-dimensional space, a tensor $\mathrm{T}$ of the $N$th rank may be defined for our purposes as a quantity having $3^{N}$ components $T_{i j k}$.. (with $N$ ...
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Confusion on two tensors constructed from Riemann curvature tensor and its dual

Assuming the metric signature is $(-+++)$ and solves vacuum Einstein equation, we start from Riemann curvature tensor $R_{\mu \nu \rho \sigma}$ and its dual ${}^*\!R_{\mu \nu \rho \sigma}$ and ...
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Question about indices and matrix

This is essentially a trivial question, which can be answer probably immediately, but i have this doubt anyway. If, say, $$\Lambda^{a}_{b} = \begin{pmatrix} f & -fc\\ -fc & f \end{pmatrix}$$ $...
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Differentials in index notation

I'm trying to show the following $$ \frac12 F_{\sigma\beta}\frac{\partial F_{\sigma\beta}}{\partial(\partial_\mu A_\alpha)} = F_{\mu\alpha} $$ where the EM field tensor $F_{\sigma\alpha} = \partial_\...
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Christoffel symbols calculation

Suppose that you want to take a covariant derivative $$\nabla$$ of some arbitrary (2,0)-tensor $$T^{\mu\nu}$$such that $$\nabla_{\alpha}T^{\mu\nu}=\partial_{\alpha}T^{\mu\nu}+\Gamma^{\mu}_{\alpha\beta}...
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Different Lagrangian formulas for geodesics [duplicate]

Suppose that you want to find the geodesics of some metric $ds^2$. In order to do so, one must solve the geodesic equation $$\frac{d^2x^\mu}{d \tau^2}+\Gamma^\mu_{\rho\sigma}\frac{dx^\rho}{d \tau}\...
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"Proof" that zero curvature implies $\partial_a \Gamma^b_{cd}$ is symmetric in $a$ and $c$

I know the claim is wrong. I just want to know where this "proof" goes haywire: Assume curvature is 0 implies Parallel transport is path independent implies Path integration of Christoffel ...
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About covariant derivative on scalar functions

My question is . When we discuss covariant derivative on any tensorial object we say we apply an operator which reduces to ordinary partial derivative while acting on scalar function. Why it reduces ...
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Tensor contractions in the transformation coefficients of a primed tensor

Consider some arbitrary primed tensor of rank (2,2) $${T'}^{\mu\nu}_{\rho\sigma},$$ where $${T'}^{\mu\nu}_{\rho\sigma}\equiv T^{\mu'\nu'}_{\rho'\sigma'}.$$ The tensor $T$ transforms as $${T'}^{\mu\nu}...
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Derivative of the christoffel symbol

Consider $\partial _dC_{ab}^c$, where $C_{ab}^c$ is a field of Christoffel symbols. Is it not true that the tensor field $\partial _dC_{ab}^c$, anti-symmetrized over $d$ and $a$, is $0$ if and only if ...
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Tensor Manipulation in Wald's General Relativity by Robert M. Wald at page 334

I don't understand the example, just after the "i.e.", at the end of the paragraph in the image. Why is it zero when the condition is fulfilled?
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Space-time metric in tensor form

In space time metric in tensor form: The distance is given by $$ds^2=c^2dt-dx^2-dy^2-dz^2$$ Which in tensor form is: $$ds^2=\sum_{\alpha \beta}g_{\alpha \beta}dx^\alpha dx^\beta$$ Using Einstein ...
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Why are non-central nuclear forces also called tensor forces?

Experiments suggest that nuclear forces are non-central. Sometimes this is called tensor forces. Why?
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Derivative of the logarithm of a tensor

Consider a generic tensor, for example a rank-2 tensor $R_{\mu \nu}$ (but we can take whatever object with indices). I can of course take the logarithm of it, $\log R_{\mu \nu}$. However, what happens ...
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Self-dual condition implies dimension 2 modulo 4

Reading the article of Henneaux and Teitelboim Dynamics of Chiral (Self-Dual) p-Forms they state that "In order for self-dual fields to exist it is necessary that $F$ and $^*F$ should have the ...
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Matrix representation of Hamiltonian - Issues with the tensor product

I am struggling with the following problem: Consider the Hamiltonian: $$ \hat H = \sum_{ij} \Gamma_{ij} (\hat S_i \times \hat S_j) + \sum_i B^z_i S^z_i + \sum_{ij} J_{ij} (\hat S_i \cdot \hat S_j)$$ ...
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Trace of gravitational Weyl spinor

If two of the indices of the gravitational Weyl spinor $\Psi^{ABCD}$ is contracted, does it vanish? I mean does it follow from the traceless nature of the Weyl tensor that $\Psi^{A}_{~~BAD}=0$?
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About general covariance

\begin{equation} u^{\mu}=\frac{d}{d\tau}x^{\mu} \end{equation} \begin{equation} \partial_{\lambda}(u_{\nu} u^{\nu}) = (\partial_{\lambda}u_{\nu}) u^{\nu} + u_{\nu}(\partial_{\lambda}u^{\nu}) = 0 \end{...
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Decomposition of product of two antisymmetric Lorentz tensors

Suppose I have a tensor $A_{\mu\nu}$ in the $(3,1)\oplus (1,3)$ representation of the Lorentz group where $(a,b) =(2s_a+1,2s_b+1)$. I was wondering on how to decompose explictly in terms of tensors ...
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Torsionfree condition of the covariant derivative operator

In R. Wald's book, this condition is written as: $$\nabla_a \nabla_b f=\nabla_b \nabla_a f.$$ But aren't $\nabla_a$ and $\nabla_b$, like, the exact same derivative operator with a re-naming of index? ...
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Squared Summation of Terms using Einstein's summation convention

In working with QFT and Maxwell's equations, terms such as:$$\left(\partial_\mu\,A^\mu\right)^{2}$$ often appear. Since I am new to this, I am not sure of the expansion. That is, is it 4 terms ...
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How can I calculate Christoffel symbols from this metric? [closed]

I have a problem, I am given the following equation $ds^2=g_{\mu\nu}dx^\mu dx^\nu=R_o^2(d\theta^2+\sin^2\theta\cdot\bar{g}_{ab}(\varphi)d\varphi^ad\varphi^b)$ and I am asked to calculate the ...
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The volume element is a tensor density

I'm following Carrol's book on general relativity and in Chap. 2 he talks about tensor densities, differential forms and integration. He says we can identify the integration measure $dx^{n}$ as $dx^{0}...
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Definition of a rank-3 spin tensor operator?

For a spin $F$ it is common to define a rank-2 tensor, also called spin alignment tensor, which takes the form $$ T_{ij} = 3 \dfrac{F_i F_j + F_j F_i}{2} - \mathbf{F}^2 \delta_{ij} \;. $$ Note that ...
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Geodesic deviation equation in a non-coordinate basis

I've been trying to derive the geodesic deviation equation in an anholonomic basis and something is quite off. I'm pretty sure the equation should be the same no matter what set of vector fields we ...
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Doubt about the coordinates of a moving Schwarzschild black hole [duplicate]

My friend asked me about the geometry of a moving black hole with constant velocity. My first attempt to cast its metric tensor was: $1.$ Start with Schwarzschild metric in spherical coordinates $(t,...
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Schwarzchild Metric ODE for θ solution

Consider the schwarzchild metric and one of its geodesic differential equation $$r^2\sin(\theta)\cos(\theta)\phi'^2-r^2 \theta''=0$$ where $f'\equiv \frac{d}{d \tau}:\forall f \in \mathbb{R}$. How ...
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How often is a non-coordinate and non-orthonormal basis used in GR?

I wrote a program that takes as input the basis vectors if electing to use an orthonormal basis, or metric components if using the coordinate basis, and outputs non-zero Christoffel symbols and ...
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Writing the columns or Rows of a Matrix in Suffix Notation

I have the following matrix: $$M=\begin{bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end{bmatrix} = \begin{bmatrix} \vec{a}\\ \vec{b}\\ \vec{c} \end{bmatrix}$$ ...
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Inner product of tensor-product

If I have states $\psi_{a,b,c,d}$ ,then is the following relation true: $$\langle \psi_a \otimes \psi_b| \psi_c \otimes \psi_d\rangle = \langle \psi_a |\psi_c \rangle \cdot \langle \psi_a |\psi_d \...
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Tensorial direct product

The direct product of two tensors is also a tensor. I would like to know if we can write a tensor as a product of only two other tensors. For example, how to find $A^{\mu}$ and $ B^{\nu}$ so that $\...
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Deriving relativistic equations of motion using scalar field stress-energy tensor

Question: Stress energy tensor of a minimally coupled scalar field is $T_{\mu\nu} = \partial_\mu\phi\partial_\nu\phi - \left[\frac{1}{2}(\nabla\phi)^2+V(\phi)\right]g_{\mu\nu}$. Derive the scalar ...
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Differentiation of tensors in lagrangian formalism: two questions

I think I have this right, but I have no way to check it and would appreciate a second opinion. I want to calculate the following: $$ \frac{\partial}{\partial\rho_\nu}\left[\left(\partial_\mu\rho_\nu-\...
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Geodesic equation fastest solution

Suppose that you want to solve the geodesic equation for the Schwarzschild metric. The geodesic equation is $$\frac{d^2 x^{\mu}}{d \tau^2}+\Gamma^{\mu}_{\rho\sigma}\frac{d x^{\rho}}{d \tau}\frac{d x^{\...
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How compute the expression of electromagnetic tensor explicitly as given here?

I am trying to understand how the second line arrives at the last line of this expression. For $F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu$ And $F^{\mu\nu} = \partial^\mu A^\nu -\partial^\nu ...
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Why is it natural to make the electromagnetic field an antisymmetric, rank 2 tensor?

As far as I know, the electromagnetic field strength tensor is defined to be the simplest object involving the electric and magnetic fields that transforms properly under Lorentz transformations. ...
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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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3 answers
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Covariant and contravariant for a beginner

I saw that people were representing matrices in two ways. $$\sum_{j=1}^n a_{ij}$$ It is representing a column matrix (vector actually) if we assume $i=1$. $$\begin{bmatrix}a_{11} & a_{12} & ...
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How we will find the contorsion tensor?

How we can swap indices while finding contorsion tensor \begin{equation} {K^{\rho}}_{\mu\nu} = \frac{1}{2} \left( {T^{\rho}}_{\mu\nu} - {{T_{\mu}}^{\rho}}_{\nu} - {{T_{\nu}}^{\rho}}_{\mu} \right) \end{...
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Christoffel symbol identity?

In multiple questions (e.g. A helpful proof in contracting the Christoffel symbol? or https://physics.stackexchange.com/a/101677/290999), I have now seen the following identity being used: $${\Gamma^\...
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Symmetries of Dual Riemann tensor

In four dimensions, the dual of the Riemann tensor is defined as \begin{align} ^*R^{\mu}{}_{\nu}{}^{\gamma\delta}=\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}R^{\mu}{}_{\nu\alpha\beta}\,. \end{align}...
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Measuring the spacetime curvature of an object

Suppose that I want to measure the amount of curvature of spacetime that an object in space creates, like the star Rigel? How would I proceed to do so? Would I use the Ricci tensor $R_{\mu\nu}$ or the ...
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Proof that the maximal slicing condition maximizes the volume

I am struggling with proving that the maximum slicing condition $$ K = 0 $$ not only extremizes, but maximizes the volume $\mathcal{V}$ enclosed within a closed two-dimensional surface $S$. The ...
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Maxwell equation in tensor form

I have seen many times on different sources that the equation: $$ \epsilon^{\mu\nu\rho\sigma}\partial_{\nu}F_{\rho\sigma}=0$$ Is true identically following from it being a product of an anti-symmetric ...
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