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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38 views

Thinking of the Faraday field-strength tensor as a 2-form

Background I'm familiar enough with the Faraday tensor $F_{\alpha\beta}$ to know that it's is a 2-form. Hence, at each point $P$ in spacetime $V$, it's a multilinear map $$F: T_PV\times T_pV\to\mathbb{...
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Are Lagrange's equations physical laws?

Well, I studied that a physical law is an equation between tensors that are function of position and time because when the frame is changed tensors change in order to leave the equation true: $$T_1(\...
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Still confuse about tensor

In special relativity, a four-vector $\mathbf{x}$ in an inertial frame is related to $\mathbf{\overline{x}}$ through a Lorentz transformation $\mathbf{\Lambda}$: \begin{align} \overline{\mathbf{x}}...
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Timelike and spacelike projections in General Relativity and associated conservation laws

For any timelike curve $p_\mu$ in General Relativity (section 3 of this review), we can project this into its timelike and spacelike components. Further, these projections are associated with ...
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Scalar coupled to Gauss-Bonnet invariant vs Horndeski theory

So here it is a somewhat tormenting question. The first statement will be a little specific but then I will make clear what the jargon indicates. How can we show that a Lagrangian made of a scalar ...
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Indices misprint in Sean Carroll's Spacetime and Geometry?

To my knowledge, 3 or more indices may not appear in a given term, as I've found in a video produced by "Faculty of Khan": However, on page 30, Sean Carroll writes: As obvious, the indices 0 and 1 ...
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Is $X\otimes X$ not the simultaneous position operator?

I had thought that $X\otimes X$ would be the operator on $H_1\otimes H_2$ to simultaneously measure the x-positions of two particles. But there seems to be something wrong with this -- for a given ...
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Orientation and sign convention in 2D electrodynamics using differential forms

I've been following this paper for a treatment of electrodynamics using differential forms. In particular, they demonstrate that Maxwell's equations expressed using differential forms are form-...
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Compact expression of Maxwell's equations: missed minus sign

With much courtesy I ask a simple explanation to be able to obtain a minus sign missing from the compact form of Maxwell's equations: $$\boxed{\square \overleftrightarrow F=\mu_0 \boldsymbol{\mathcal{...
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Einstein summation in tensor calculus

I am looking at the Schaum's Outlines "Tensor Calculus" by David C. Kay, and on page 3, the following non-identity and identity are presented: $$ \begin{align} a_{ij}(x_i + y_j) &\neq a_{ij} x_i +...
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Electromagnetism on 3 torus

We all know Maxwell equations in 3+1 spacetime, where the "space" is $\mathbb{R}^3$ and time is $\mathbb{R}$. Moreover, it is easy to construct (using differential forms) the corresponding theory in a,...
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Numerical examples of covariant and contravariant tensor transformations [closed]

I've examined dozens of textbooks and searched the internet for numerical examples of a tensor transformation. I remember only seeing symbolic explanations and examples in the now standard symbolism ...
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Geodesic equation for spherical symmetric gravitational field

Let $h_{\mu\nu}= g_{\mu\nu}-\eta_{\mu\nu}$ be of the form: $$\begin{pmatrix} -2\phi(r) & 0&0&0 \\ 0& -2\psi(r)&0&0\\ 0& 0&-2\psi(r)&0\\ 0& 0&0&-2\psi(...
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Pseudotensors for describing physical quantities

I have been reading about tensors from Mathematical methods for Physics and Engineering, by K.F. Riley, M.P. Hobson and S.J. Bence. And there are a couple of things i am not getting. On page 949 (...
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Confused about the gauge transformation of the amplitude tensor for gravitational waves

Far away from the field sources, where the energy-momentum tensor $$T_{mn}=0 \tag{m,n=0,1,2,3}$$ The linearized EFE becomes $$\Box \bar h_{mn}=0 \tag{1}$$ where $\bar h_{mn}$ is the trace-reverse ...
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Transformation of dielctric constant tensor

I have a dielectric tensor $$K = \begin{pmatrix} 2000 & 0 & 0 \\ 0 & 2000 & 0 \\ 0 & 0 & 50 \end{pmatrix};$$ which I want to transform to a new coordinate system given by ...
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Is there any meaning of tensor contraction?

Is there any meaning behind tensor contraction. Or is it just randomly getting rid of some components by only selecting those with same index and sum them up? For example, I know tensor is ...
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Torsion form and exterior covariant derivative

The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
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Shear stress sign convention and Rotation

If one uses the Rotation matrix to do calculate the component of a Tensor (Tensor A) gets something like this: Now,one can get the same results for the stress tensor by means of equilibrium My ...
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Lorentz boost tensor notation confusion

I have been given this$$ \delta X^{\mu}=\omega_{\mu \nu}\left(M^{\mu \mu}\right)_{\sigma}^{\rho} X^{\sigma} $$ and I think it should be equal to this but I'm confused if I'm doing it correctly $$ \...
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Torsion tensor in Relativity

While reading Sean Carroll's book on general relativity, I came across something called as a 'Torsion Tensor' which is defined as, $$\Gamma{^\lambda}{_{\mu\nu}} - \Gamma{^\lambda}{_{\nu\mu}} = T{^\...
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Relativity and components of a 1-form

I have a question regarding Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973), Gravitation ISBN 978-0-7167-0344-0. It is a book about Einstein's theory of gravitation. At page 313, ...
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Is the interval $ds^2$ NOT invariant under translation in an inhomogenous space?

In the Chapter 9 Symmetries, Section 9.1 The Killing vectors (page 101) are Killing vectors defined such that an infinitesimal translation along the vector keep the line element invariant. It means ...
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Moment of inertia- a tensor quantity [duplicate]

The moment of inertia is a tensor and the matrix contains nine elements. The off-diagonal elements are like Ixy, Ixz, Iyx and so on. Ixy = mxy. But M.I. = mass × (perpendicular distance of the ...
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Dual of an antisymmetric tensor

Consider the construction of the dual of $F_{ik}$, which is an antisymmetric tensor. The dual is given by the expression $$F^{*lm} = \frac{1}{2} \epsilon^{iklm} F_{ik}\tag{1}$$ The question I'm ...
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Ricci Curvature Tensor in a static gravitational field (non-relativistic)

Pg 171 of "Tensors, Relativity and Cosmology" The non-relativistic limit of the metric in a static gravitational field is defined as $$ds^2=\left(1+\frac{2 \phi}{c^2}\right)(dx^0)^2+g_{\alpha \beta}...
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Using symmetry of Riemann tensor to vanish components

The Riemann tensor is skew-symmetric in its first and last pair of indices, i.e., \begin{align} R_{abcd} = -R_{abdc} = -R_{bacd} \end{align} Can I simply use this to say that, for example, the ...
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Dirac Notation Tensor product

We can write a Singlet state of two $\frac{1}{2}$ spin particles like this: $$|S\rangle = \frac{1}{\sqrt{2}}\left( |+ \rangle ⊗ |-\rangle - |-\rangle ⊗|+\rangle \right) $$ is this the same as ...
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How electromagnetic energy-momentum looks like for arbitrary 4-velocity vector?

I need to expresse the electromagnetic energy-momentum tensor in a vacuum $$T^\nu_{\ \ \ \mu} = F_{\mu\alpha}F^{\nu\alpha} - \frac14 F_{\alpha\beta}F^{\alpha\beta}\delta^\nu_{\ \ \mu}$$ in terms of ...
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Action & Energy-Momentum Tensor for Matter Fields

Pg 163 of "Tensors, Relativity and Cosmology" The action integral of a given matter distribution can be written in the form $$I_K=-c\int_\Omega\frac{\rho}{\sqrt{-g}}\frac{ds}{dt}\sqrt{-g} d\Omega ...
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Raising & lowering indices of 3-pseudovectors?

Now, let space tmie metric is $$\eta_{\mu\nu}=\text{diag}(+,-,-,-)$$ now $$x_{\mu}=(x^0,-\mathbf{x})$$ and $$x^{\mu}=(x^0,\mathbf{x})$$ and $$x^{\mu}=\eta^{\mu\nu}x_{\nu}$$ also $$\partial_\mu=(\...
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Timelike, spacelike etc. for higher-order tensors

Vectors $V^\mu$ in relativity can be classified into those which are timelike, spacelike and null. A similar classification is available for tensors: A tensor $$T^{\mu_1\mu_2...\mu_p}_{\phantom{\mu_1\...
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Renaming tensor indices in summation

I am doing some tensor product calculations, which involve terms like this: $F^{\mu\lambda}\partial^\nu A_\lambda$ I am trying to write it as a total differential of some quantity. To achieve this I ...
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longitudinal and transverse components in higher dimensions

I am familiar with the Helmholz decomposition of a vector field in three dimensions: $$\vec{V}=\vec{\nabla}\wedge\vec{A}+\vec{\nabla}\phi$$ But I am interested to show that something similar can be ...
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Raychaudhuri scalar

In Carroll's 'Space-time and Geometry', appendix F on congruences, the Raychaudhuri equation is derived. However, in the process, I seem to miss a calculation step that changes the sign of the ...
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Stokes's theorem in tensor field

On pg 73 of "Tensors, Relativity and Cosmology" The generalized Stokes's theorem in arbitrary $N$-dimensional space is given by: $$\int_c A_mdx^m=\frac{1}{2}\int_S F_{mn}dS^{mn} \tag{1}$$ where $...
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Lowering index of Riemann tensor

I'm trying to undertand the lowering of index of Riemann curvature tensor, but I'm not sure what I have to do. I know that $R_{ebcd} = g_{ea}{R^a}_{bcd}$. But let's say I have the coordinates ($t,r,\...
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Divergence of a tensor

On pg.70 of Dalarsson's "Tensors, Relativity and Cosmology" For a mixed tensor of contravariant order 2 and covariant order 1 $(T^{mn}_{p,m})$, the divergence with respect to m is defined as:$$T^{...
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Rewriting a lagrangian in terms of Hodge duals?

Spinors have been found to have some interesting applications in general relativity (such as Wittens positive energy proof). Recently I'd come across a series of papers 1 2 3 (there are many more ...
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Proof of Schur's Theorem

On Pg. 123 of Schaum's Tensor Calculus: At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$ for any ...
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How to consider a moment of a force as a 2-form?

My major is mechanical engineering. Recently, I'm reading "The Geometry of Physics An Introduction (3ed)" by Theodore Frankel. On page lix in the section O.r, the author discussed the concept of ...
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Understanding Riemann Curvature Tensor in Misner, Thorne and Wheeler's Gravitation

I'm trying to understand section 11.4 of Misner, Thorne and Wheeler's Gravitation textbook, which explains how the output of the Riemann Curvature Tensor $Riemann(...,A,u,v)$ is a vector describing ...
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Proof of first Bianchi identity

The proof is often simplified by using the following theorem: "If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g_{...
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Why do we need invariants to represent real life quantities?

Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical ...
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How to prove a 4D vector is a 4-Vector?

This is a fairly open ended question. Given a set of 4 Components, that is, a 4D Vector, what is the process for determining rather or not it is a "4-Vector" as defined in special relativity? I want ...
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Indices of the Riemann Tensor of the first kind

When establishing the identity $V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . ...
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Why Electrical conductivity tensor is symmetric? Or is it not always symmetric?

How to show that the electrical conductivity tensor is symmetric? (or it's not always symmetric?)
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Einstein notation: can a free index be upper in one term and lower in another term?

Consider a linear combination of terms written using Einstein notation. Consider one free index in the linear combination: is it necessary that the index is upper in all terms or lower in all terms, ...
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Doubt about the history of the tensor object in physics

The word "tensor" (or, maybe the whole physical idea of this object) could be traced date to 1898 on the work on Crystals due to Voigt [1]. It seems that the references [2] and [3], introduce the idea ...
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What physical quantity remains equal in different frames? [closed]

I recently came across a problem which involved going into rotating frames. And it was pretty tiring. (And difficult..). So I wondered if there was some quantity that you could measure from the ground ...