# 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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### Transformation law for 3-velocity

I am kind of stuck with a problem mentioned in my current reading about special relativity. Given the Lorentz transformation $$x^{\bar{i}} = L^i{}_k \, x^k \quad ,$$ one has to find the transformation ...
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### Index-free tensor expressions and what makes the metric tensor different

There are two doubts, but all are from the same section and closely related so I thought I'll ask in one post. I'm studying a section that introduces Christoffel symbols [Core Principles of Special ...
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### Quantum gravity in affine picture

I am studying an article which is about quantum gravity (M. Martellini, "Quantum Gravity in the Eddington Purely Affine Picture," Phys. Rev. D 29 (1984) 2746). I have come to eq. 2.11a which ...
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### Derivation of Christoffel Symbols

So I am reading a book on relativity & differential geometry and in the text, they gave the Christoffel symbols in terms of the metric and its derivatives, but I wanted to derive it myself. ...
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### Question about the the velocity and acceleration in tensor notation

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $$V^i = (d/dt)Z^i$$ and the components fo ...
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### Is notational compactness in tensors (compared to linear algebra) relevant? [migrated]

In this post you can read: A matrix is a special case of a second rank tensor with 1 index up and 1 index down. It takes vectors to vectors, (by contracting the upper index of the vector with the ...
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### Proving that the Minkowski metric tensor is invariant under Lorentz transformations

I'm studying special relativity. A general Lorentz transformation is defined by $\Lambda^T\eta\Lambda=\eta$. Now, \begin{align} \eta'^{\mu\nu} &= \Lambda^\mu_{\;\;\alpha}\Lambda^\nu_{\;\;\beta}\...
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### Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
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### Which basis 2-form elements represent positive traversals in Minkowski 4-space?

I'm certain some of this relies on arbitrary choice, for even in Euclidean 3-space, there is no a priori preferred choice of left versus right hand coordinates. In fact according to Einstein: There ...
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### Euler's Equation applied to Perfect Fluid

Consider this form of Euler's Equation: $$\rho \vec{a}=\nabla \cdot T+\rho \vec{f}$$ Where: $\rho$ is the density, $\vec{a}$ is the acceleration, $T$ is Cauchy's Stress Tensor and $\vec{f}$ is the ...
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### A problem with differential forms in relativistic electromagnetism [duplicate]

We can write Maxwell's Laws as follow: $$\partial _\mu F^{\mu\nu}=\mu _0 J^\nu$$ $$dF=0$$ where $F^{\mu\nu}$ is the Faraday Tensor and $J^\nu$ is the Four-Current Density. You can see this for context....
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### Parallel transport: Lie derivative vs covariant derivative

Given a manifold, we can generalize the idea of derivatives in multiple ways: two of them being the Lie derivative and the covariant derivative. Whereas Lie derivatives do not require any additional ...
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### Trying to intuitively understand Christoffel symbols

I recently watched Sean Carroll's YouTube series on "The Biggest Ideas in the Universe". In his Geometry and Topology video, he says that the connection in Riemannian geometry describes how ...
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### Relation between 4-divergence and 3-divergence of vector on the hypersurface

How the relation $\nabla_{a} u^{a}=D_{a} u^{a}-\epsilon v^{i} \nabla_{i} v_{j} u^{j}$ where $D_{a}$ is covariant derivate on the hypersurface between 4-divergence and 3-divergence of the vector on the ...
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### Commutator of derivatives with torsion

I am currently looking at "Physical aspect of space-time torsion" by IL Shapiro. There in eq. 2.10, it mentions that in space-time with torsion the commutator of covariant derivatives acting on the ...
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### Derivative of Riemann tensor respect to Riemann tensor

I know that, for example we have $$\frac{\delta g^{jk}}{\delta g^{lm}}=\delta^{j}_{(l}\delta^{k}_{m)}.$$ This topic was discussed previously e.g. on Physicsforums.com and on Phys.SE. So I was ...
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### What is the convention for tensor indices for matrices?

The Lorenz-Transformation of the EM-Tensor F is given by the equation $$F'^{\mu \nu} = \Lambda^{\mu}_{\ \ \rho} \Lambda^\nu_{\ \ \sigma} F^{\rho \sigma}$$ Then it says that this is equivalent to the ...
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### Contravariant rank-2 tensor transformation in index notation

I'm slightly confused about the placement of upper and lower indices for the transformation of a rank-2 contravariant tensor. A contravariant rank-2 tensor transforms as $$M' = \Lambda M \Lambda^{T}$$....
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### My understanding of General Relativity

My Background: In high school, I completed AP Physics C Mechanics and Electricity and Magnetism. In my first year of undergrad, I completed a course on Newtonian Mechanics and a course on Special ...
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### Tensor and Matrices

Suppose that we are dealing with the following matrix: $$A=\begin{bmatrix}a_{00} & a_{01} \\ a_{10} & a_{11}\end{bmatrix}$$ but I don't want to use matrix notation, insted I want to use tensor ...
I'm studying Friedman and Susskind's Special Relativity and Classical Field Theory. In Lecture 6, they derive $$m{dU_\mu\over d\tau}=eF_{\mu\nu}U^\nu,\tag{6.33}$$ but only when $\mu=1, 2, 3$. (\$F_{\mu\...