We’re rewarding the question askers & reputations are being recalculated! Read more.

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.

Filter by
Sorted by
Tagged with
1
vote
2answers
245 views

Compact expression of Maxwell's equations: is there a missing minus sign?

The compact form of Maxwell's equations: $$\boxed{\square\, \boldsymbol{\mathsf{F}}=\mu_0 \boldsymbol{\mathcal{J}}} \tag{1}$$ where the current density quadrivector is given by the relation $\...
1
vote
1answer
64 views

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 +...
0
votes
0answers
51 views

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,...
3
votes
0answers
43 views

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 ...
0
votes
1answer
36 views

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 (...
3
votes
0answers
72 views

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 ...
0
votes
0answers
28 views

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 ...
1
vote
1answer
64 views

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 ...
0
votes
0answers
37 views

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 $\mathfrak g$ ...
0
votes
0answers
74 views

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 ...
1
vote
2answers
43 views

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 $$ \...
2
votes
2answers
97 views

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{^\...
1
vote
1answer
89 views

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, ...
3
votes
2answers
117 views

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 ...
0
votes
0answers
32 views

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 ...
0
votes
0answers
27 views

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 ...
0
votes
0answers
54 views

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}...
2
votes
1answer
86 views

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 ...
1
vote
1answer
41 views

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 ...
4
votes
1answer
107 views

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 ...
2
votes
0answers
123 views

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 ...
0
votes
1answer
72 views

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=(\...
3
votes
0answers
55 views

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\...
0
votes
0answers
44 views

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 ...
1
vote
0answers
50 views

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 ...
2
votes
1answer
35 views

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 ...
0
votes
1answer
85 views

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 $...
1
vote
1answer
62 views

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,\...
2
votes
1answer
52 views

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^{...
2
votes
1answer
66 views

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 ...
1
vote
1answer
67 views

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 ...
1
vote
0answers
87 views

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 ...
2
votes
0answers
90 views

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 ...
0
votes
1answer
70 views

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_{...
1
vote
2answers
65 views

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 ...
2
votes
3answers
539 views

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 ...
1
vote
2answers
65 views

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 . ...
2
votes
1answer
182 views

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?)
1
vote
2answers
96 views

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, ...
2
votes
0answers
41 views

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 ...
1
vote
1answer
82 views

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 ...
1
vote
0answers
155 views

Hodge dual with Levi-Civita symbol or tensor?

In a curved spacetime with a general Minkowskian metric $g_{\mu\nu}$, I understand the difference between the Levi-Civita symbol $\pmb{\in}^{\mu\nu\rho\sigma}$ (which is a tensor-density of weight $W=+...
2
votes
3answers
66 views

Can I contract index in this expression?

I'm reading Carrol text on general relativity, on page 96 they arrive to the term \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\...
0
votes
0answers
34 views

Order of positions of tensor/vector components in an inner/outer product

Show that if $T_i$ are the components of covariant vector T, then $S_{ij}=T_iT_j-T_jT_i$ are the components of a skew-symmetric covariant tensor S. The question is whenever working with equations of ...
0
votes
1answer
64 views

Covariant derivative in a basis

Reading through this paper, I saw that the energy momentum conservation: $$\nabla_\mu T^{\mu\nu}=0$$ can be evaluated as: $$\partial_t(\sqrt{-g}T^{t}_\nu)=-\partial_i(\sqrt{-g}T^{i}_\nu)+\sqrt{-g}T^...
0
votes
0answers
63 views

A specific derivation of Yang-Mills equations of motion

I am not happy about the derivation of Yang-Mills equations of motion (YM eom) given here @Prahar https://physics.stackexchange.com/a/312681/42982: @Prahar said: Yang-Mills action is $$ S = \int ...
3
votes
0answers
59 views

Variations of tensors are tensors?

Recently I posted a question about variation of metric. I thought I understood it and talked with my friend about it. After that he said he's not convinced because he can't prove variation of metric ...
1
vote
3answers
118 views

Functional derivative of metric

To do functional derivative of some actions, we need to know a functional differential of metrics $g_{\mu \nu}(x)$. One of the formulae about that is: $$g_{\mu\nu}\delta g^{\mu\nu} = - g^{\mu\nu} \...
0
votes
0answers
35 views

Derivative with respect to a coordiante differential (geodesic equation)

If the arc length is chosen to be the action integral, that is $$ S=\int \sqrt {g_{kn}\frac{dx^k}{ds} \frac{dx^n}{ds}} dx \tag{11.13} $$ Then Lagrangian is given by $$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
1
vote
0answers
42 views

Lorentz transformation rules using the field strength tensor

A rank 2-tensor transforms under Lorentz transformation in the following way: $$T^{\mu \nu} \rightarrow T^{' \mu \nu} = \Lambda^\mu _\sigma\Lambda^\nu _\rho T^{\sigma \rho}$$ The field strength ...