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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Does completely antisymmetric tensor act on a tensor always produce a tensor or not?

So completely antisymmetric tensor $\epsilon$ act on a tensor can produce a new object. i.e. $G_{\alpha\beta}=\frac{1}{2}\epsilon_{\alpha\beta\mu\nu}F^{\mu \nu}$. However, According to Landau's ...
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Transforming covariant and contravariant derivatives

The covariant derivative of a contravector field is given by: \begin{equation} D_{k} A^{i} \equiv A^{i}_{\parallel k} = A^{i}_{\mid k} + \Gamma^{i}_{kp} A^{p} \end{equation} With $A^{i}_{\mid k} = \...
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The definition of quantities in special relativity as upper-index or lower-index

My question is for Minkowski metric $\eta_{\alpha\beta}=\mathrm{diag}(1,-1,-1,-1)$. While defining quantities like the four potential, four momentum or even space-time interval for that matter, why do ...
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Motivation for tensor theory of gravity

In class we were shown that $$\rho = \frac{dm}{dV}$$ has the transformation properties of the 00 component of a rank 2 tensor. So we'd like to turn the classical Poisson equation for gravity into a ...
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How does vectorization affect $\nabla$?

The homework and exercise was to prove $\nabla \times {A}$ transform as a vector, and I've solved it thorough hard algebra. However, something occurred to my mind and I have a hard time to resolve it....
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Co-spinors and contra-spinors

As i was reading my teacher's notes on $SU(2)$ and $SO(3)$, i have had this question. Why do co-spinors transform differently under a rotation than contra-spinors?
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Are rotation matrices tensors?

Are rotation matrices tensors? If no, why? I'm not sure about it, for example considering the $z$-axis rotation matrix when I rotate the coordinate system the rotation matrix around the old $z$-axis ...
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Why covariant derivative is partial derivative in SR?

I'm new in these topics and i've been confuse at some relations between the limit of SR for GR. In cartesian coordinates, basis do not change, so \begin{equation}\Gamma^{\mu}_{\alpha\beta}=0 \quad \...
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A form $F$ is simple if and only if $F\wedge F=0$?

Gravitation by Charles W. Misner, Kip Throne and John Wheeler page 93 Box 4.1 point 5 b. Applications: a. "In four dimensions, all 0 forms, 1- forms, 3-forms, and 4-forms are simple. A 2-form $F$ is ...
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How to find the mixed tensor, contravariant tensor and tensor trace of $F$

I have a question in particle physics that ask me to find the mixed tensor, contravariant tensor and tensor trace of $F$: Our professor didn't teach us that much about the math of tensor, which makes ...
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Metric (in)dependence of the electromagnetic field strength

In GR, the vector potential is defined as $A^\mu$ which is a contravariant vector. Then lowering the indices requires the metric $A_\mu =g_{\mu\nu} A^\nu$, using this vector one defines the field ...
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Given no strain and uniform angle of rotation, can the displacement field be expressed as a uniform rotation?

Consider displacement vector $\xi$ where $\mathbf{\xi}=\mathbf{\phi} \times \mathbf{x}$. $\phi$ is an angle vector along the axis of rotation and $\mathbf{x}$ is the position vector. Let $W_{ij}=\frac{...
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What is the physical difference between the mixed and fully contravariant Cauchy stress tensor components?

One traditional representation of the stress tensor among relativists is a rank-2 fully contravariant tensor, associating a contravariant force per unit area $t^i$ to a unit normal $n_j$ defined on a ...
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Action in Electromagnetism expressed in differential geometry and tensor notation

$$ S = -\frac{1}{4} \int F_{\mu\nu}F^{\mu\nu} = -\frac{1}{2} \int F \wedge *F$$ Trying to figure out why this identity holds true and getting stuck.
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Explicit form of the Christofffel Symbol used in Geodesic Equation

One way to motivate the Christoffel symbol is to consider the partial derivative of a tensor, $T_\alpha$ $\frac{\partial T_\alpha}{\partial x^\gamma}=\frac{\partial^2 x^\beta}{\partial x^\alpha \...
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Tensor analysis: confusion about notation, and contra/co-variance

I'm learning about tensors in the context of special relativity, and I'm a bit confused some notation. I understand a four-vector is a four dimensional vector, which is written in the form $(ct, x, y,...
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Tensor acting on another tensor

On page 22 of Sean Carroll's Spacetime and Geometry, he says that tensors can act on other tensors and gives the following example: $$ U^{\mu}_{\nu} = T^{\mu \rho}_{\sigma} S^{\sigma}_{\rho \nu}$$ ...
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Leibniz Rule for Covariant derivatives

I recently came across a video by prof Fredrick Schuller on general relativity where he defines the leibniz rule to be, $\nabla_X (T(\omega,Y))=\nabla_XT(\omega,Y)+T(\nabla_{X} \omega,Y)+T(\omega,\...
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How can I think of the flat space metric tensor as a multilinear function?

I'm pretty new to the idea of tensors, and I'm having a bit of confusion with how to think about the flat space metric tensor in special relativity. I understand that a good way to think about ...
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Uniqueness of affine connections

This is a problem from Carmelli book on general relativity. the conceptual problem is, given a spacetime, and hence a metric, can there exist more than one affine connection for which one can take ...
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Question about computing Christoffel symbols

I am trying to calculate the Christoffel symbols in polar coordinates, and I am confused on one step. Given that I am here, for example: $$\Gamma_{r \theta}^{\theta}=\frac{1}{2} g^{\alpha \theta}\...
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Differentiability of a Metric Tensor

As an introduction to Metric Tensors, I read the conditions to be met for a metric includes it being differentiability class $C^2$ i.e. all second-order partial derivatives of $g_{ij}$ exists and are ...
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How many different ways can Riemann-Christoffel Curvature Tensor can be derived? [closed]

In today's Relativity and Gravitation class, my prof was discussing about Riemann-Christoffel Tensor and he derived it. But in the end he told that there are many ways one can derive the Riemann ...
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Showing $\partial_{\mu}\tilde{F}^{\mu\nu}=0$ by the antisymmetric properties

The electromagnetic dual tensor is given by \begin{align} \tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\delta\rho}F_{\delta\rho} \end{align} Here, $\epsilon^{\mu\nu\delta\rho}$ is the Levi-Civita ...
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How to derive the angular velocity of circular orbits in Kerr geometry?

I am trying to derive the angular velocity of a circular orbit in Kerr geometry, eqn.(2.16) in Bardeen et al (1972) which reads $$\Omega=\dfrac{1}{r^{3/2}+a}$$ (Note that I am using the units in which ...
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What does diagonalizing inertia tensor do

In the normal case with moment of inertia, the angular momentum is parallel to the angular velocity $\vec{L} = I\,\vec{\omega}\tag{1}$ But when the object is rotating instantaneously about a point, ...
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Electromagnetic field tensor (and other tensors) with different sign conventions

In Wikipedia the components of the EM Field Tensor are listed as $$F^{\mu\nu}=\left( \begin{array}{cccc} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z &...
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How to translate this equation into physicist's notation? [closed]

I asked this in math stackexchange but no one has answered there so I ask here. How to translate this equation into physicist's notation, i.e. tensors with indices? $$\left\langle R_{N}\left(u,v\...
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Spacing in indices, and relation to matrices, in special relativity notation

I have some general confusion regarding notation on tensors in special relativity, and how indices correspond to the matrix representation of second-rank tensors. When one has a second-rank tensor $...
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How do you write $A A^T$ in Einstein notation?

In index notation it makes sense as $$\sum_j {A_{ij} A_{jk}^T} = \sum_j {A_{ij} A_{kj}}.\tag{1}$$ But this doesn't make sense for Einstein notation where in $$A^\mu_\sigma (A^\sigma_\nu)^T = A^\...
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Geodesic equation and spatial variation of time

I am trying understand the interpretation of geodesic equations. For simplicity, let us take a metric $$ds^2 = g_{00}(x)dt^2 + a(x,y,z)(dx^2 + dy^2 + dz^2).$$ I interpret the metric to be a spacetime,...
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Raising and lowering the indices of a perturbed metric

I am looking at a metric which is defined as (Eq 2.4 Glampedakis & Babak) $$ g_{\mu \nu} = g_{\mu \nu}^K + \epsilon h_{\mu \nu}$$ where $g_{\mu \nu}^K$ is the original unperturbed metric (Kerr) ...
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Electromagnetic tensor [closed]

How to prove the equality in this? $$F^{\mu\nu}F_{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}\left(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\right)\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\...
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Projection tensor in General Relativity

In MTW "Gravitation", the projection tensor is defined as $$\boldsymbol{P} = \boldsymbol{g} + \boldsymbol{u}\otimes\boldsymbol{u}$$ And one exercise asks to prove that a tangent vector $\boldsymbol{...
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What's the most common convention for torsion and contorsion tensor index position?

In Einstein-Cartan theory, the torsion tensor is usually defined as the antisymetric part of the connection: \begin{gather} \nabla_{\mu} \, A^{\lambda} = \partial_{\mu} \, A^{\lambda} + \Gamma_{\mu \...
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Raising and lowering indices and tensor contraction

I'm really confused by the notation of raising and lower indices in tensors when mixed with einstein summation notation and referencing the metric tensor. I need help separating several conflicting ...
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What exactly do raised indices mean in the context of 2-dimensional tensors?

I was reading Sean Carroll's Introduction to General Relativity On Page 12 there is an equation given for defining the Lorentz group as a collection of $4\times 4$ matrices that satisfy $$ \Lambda^...
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Transforming a tensor from Crystal to Laboratory frame of Reference

I want to transform the stiffness tensor of a rhombohedral crystal from crystallographic frame of reference to laboratory fame of reference, how to do it ? For crystal structures having orthogonal ...
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Is my thinking correct for partial derivatives and tensors?

So I was transforming the affine connection and I ended up with a term like this: $$ \frac{\partial^2 x'^a}{\partial x'^b \partial x^p} $$ where $x$ and $x'$ are two different coordinate systems ...
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Integral of the parallel transport equation

The parallel transport of a vector $v_0^\alpha$ along the curve $\gamma$ is given by a vector field $v^\alpha$ which satisfies the equation $$ \frac{\mathrm d x^\mu}{\mathrm d \lambda}\frac{\partial v^...
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Why are isotropic tensors not considered scalars?

In introductory textbooks (Griffiths, Shankar, Boas) a tensor is introduced as a mathematical objects which transform in a specific manner under changes of basis (i.e. changes of the coordinate system)...
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What's the covariant derivative of a normalized, timelike Killing vector?

I'm reading The large scale structure of spacetime and in page 72 the author says: A static metric admits a timelike killing vector $K$. We define the timelike unit vector $V$ as $V=K/f$, where $f^...
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The spinor metric, basic spinor calculations and spinor indices

I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24. Background Scharf introduces the index-raising symbol (spinor metric)...
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Coordinate-free proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic

I would like to know if the following proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic is correct. I already found a coordinate-...
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Moment of Inertia Tensor Terminology

I've learned about the moment of inertia tensor as a matrix that can be used to compute angular momentum, moment of inertia, etc. for a system. But why is it often described as a tensor instead of a ...
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Christoffel symbols in general coordinates

In order to understand the meaning of covariant derivative, I have seen the following argument. Let us consider a covariant vector $V_\mu$. We would like to understand whether $$T_{\mu\nu} = \frac{\...
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Silly doubt about relationship between Levi-Civita Connections and Koszul Form

In this paper [1] the author wrote and interresting relationship between differential geometry objects (the Levi-civita connection and Koszul form) by means of a musical isomorphism [2] (roughly ...
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Difference between position of indexes in tensor notation (SR) [duplicate]

I am learning SR, and don't understand the difference between the following notations of a Lorentz transformation $\Lambda$ $$\Lambda_{\mu\nu} , \Lambda_{\mu}\ ^\nu , \Lambda^{\mu\nu}$$ I know that ...
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162 views

Wedge product, tensor product, and Levi-Civita tensor/symbol

Source: Pages 89 and 90 of Sean Carroll's Spacetime and Geometry Quite a confusion in two steps of this quantity: $$ \begin{eqnarray} \sqrt{|g|}d^n x &=& \sqrt{|g|}dx^0 \land ... \land dx^{...
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Are there physical quantities constitute of magnitude, direction and rotation along that direction?

There are scalar quantities(magnitude) and vector quantities(magnitude and direction), but are there fundamental quantities that also depends on how it's oriented/rotated along the direction(magnitude,...