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1answer
93 views

Square of a tensor

I think, $$\sigma_{ij}\sigma^{ij} = \sigma^2.$$ However, on the Wikipedia page on Raychaudhuri equation, It was mentioned: $$\sigma^2=\frac{1}{2}\sigma^{ij}\sigma_{ij}$$ I am confused, but I think ...
1
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2answers
221 views

Tensor algebra doubt

Is it possible to take a tensor to the other side of the equation, and the tensor becomes its inverse(i.e contravariant becomes covariant and vice versa)? It is a stupid question, but It confuses me. ...
0
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0answers
57 views

QM: Commutation relations between irreducible vectors and angular momentum $[J^2,T_q^k]$

reading about the irreducible tensors and its commutation relations with the angular momentum one can find relations for $J_{z}$, $J_{+}$, $J_{-}$, but I was wondering, what about $J^2$ ? from ...
3
votes
3answers
114 views

Linear independence of the Covariant Derivative

What's the easiest way to show that the covariant derivative $\nabla U^{\mu}$ is linearly independent to $U^{\mu}$, which is a vector? I mean I'm assuming they are since I'm proving the second ...
5
votes
4answers
418 views

Why are stresses of continuum systems described via a tensor?

The tittle pretty much says enough. I have always been told so but no one really motivated it. So, I would like to know why do we use a tensor to describe the stresses in continuum mechanics.
5
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2answers
294 views

Perturbation metric problem

I know this is an already answered question, but I couldn't make head or tail of it, and it's bugging me. I know I'm probably asking a silly question, but please bear with me as I'm 14 and this is my ...
2
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3answers
496 views

Why is $ \vec{S}^{(A)} \otimes \vec{S}^{(B)} = \frac{\hbar^2}{4}(\sigma_x \otimes \sigma_x + \sigma_y \otimes\sigma_y + \sigma_z \otimes \sigma_z)$?

I haven't been taught tensor product in class but they have taught us addition of spin. I looked up online in this link->http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf (pg ...
4
votes
1answer
72 views

What's the point of a cobasis?

I've been learning about tensor analysis, and things have been going well so far, but I'm a bit stuck when it comes to the idea of a cobasis (by which I mean the reciprocal basis; not sure which term ...
0
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0answers
55 views

“+” and “-” sign in Maxwell Stress tensor

I have trouble in determining the "+" and "-" sign of momentum per unit time, per unit area of the following question. Why in the second part, $d\vec{a}$ is pointing in the $ -\vec{z} $ direction? I ...
2
votes
1answer
200 views

The Spin Connection

Why do we need to introduce the spin connection coefficients $\omega_{\mu \space \space b}^{\space \space a} $ in General Relativity? To me, they just look (mathematically) like the Christoffel ...
2
votes
2answers
185 views

Viscosity tensor in Navier-Stokes equation?

In my Hydrodynamics notes the viscosity term in the Navier-Stokes equation is of the form: $$ \nabla\cdot(\underline{\underline{h}}\cdot\nabla)\mathbf{u} $$ where $\underline{\underline{h}}$ is the ...
0
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1answer
65 views

Covariant derivative of a vanishing tensor component [closed]

Is the covariant derivative of a vanishing tensor component necessarily zero?
3
votes
1answer
66 views

Stress tensor in product of 2D CFTs

I was struggling with a question, hoping someone could point me in the right direction. I'm interested in 2D CFTs on a cylinder. I want to take the tensor product of two CFTs. My questions are these: ...
6
votes
0answers
92 views

What is the intepretation of the electromagnetic tensor?

Let $A$ be the four-potential, then we know that we can form the electromagnetic tensor as $F=dA$. This is usually done as a way to have a better writing of Maxwell's equations. So, to simplify the ...
0
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0answers
72 views

Einstein frame vs. Matter frame

What is the difference between Einstein frame and Matter frame in General Relativity? -A brief comment on each could be useful too. These two frames were used in this manuscript ...
2
votes
1answer
223 views

Stress energy tensor and the covariant derivative of the 4-momentum

Another basic question. I have usually seen the stress energy tensor $T^{ij}$ described as the flow of the 4-momentum field $p^i$ along direction $x^j$ in spacetime with $p^0$ as energy and $x^0$ as ...
1
vote
1answer
110 views

Second Rank Tensors [duplicate]

I'm a little confused, for the twentieth time, on what tensors are. I thought they were a generalization of matrices-but then they have special transformation rules. I'm looking for a concise ...
6
votes
2answers
244 views

Tensor decomposition under $\mathrm{SU(3)}$

In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning ...
6
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2answers
347 views

Do Dirac Gamma Matrices act like Tensors?

Do Dirac Gamma Matrices act like Tensors? Is it true that $$ \gamma_\mu = \eta_{\mu\nu}\gamma^\nu~? $$ Also what about $\sigma_{\mu\nu}$, where $\sigma_{\mu\nu}$ is defined to be: \begin{align*} ...
0
votes
1answer
149 views

Stress-energy tensor explicitly in terms of the metric tensor

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $R_{\mu\nu}$ and scalar curvature ...
2
votes
1answer
130 views

Hypersurface Normal

Could anyone explain why $$n^{a}n_{a}=\pm1$$ where $n^{a}$ is the normal to the hypersurface
1
vote
1answer
38 views

Soft brehmsstrahlung classical computation

On page 177 in Peskin & Schroeder there is a derivation I have a hard time with. They write the current for a charge at rest as $$j^\mu = (1,0)^\mu e \delta(x). $$ I don't understand what the ...
4
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0answers
112 views

Lie derivative of a scalar and PDE

I am reading about differential geometry, and in particular the Lie derivative and its relation to (relativistic) hydrodynamics. In particular, I was wondering if, given two scalar functions ...
1
vote
1answer
55 views

Transformation of Electromagnetic Four-Tensor

I apologize if I am missing something obvious, but I am in my first class with tensors and I am still learning the notation. I am running into a problem with the transformation of the transformation ...
1
vote
1answer
104 views

What is the difference between a skew-symmetric and an antisymmetric tensor?

What is the difference between a skew-symmetric and an anti-symmetric tensor? If they represent the same tensor, then why use different labeling.
1
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2answers
114 views

Inner products in relativity

In physics, the definition of a dot (inner) product is often between a vector (“contravariant vector”) and a covector (“covariant vector”). However, in mathematics, a dot product is always defined ...
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2answers
88 views

Why are these specific stress invariants chosen?

I've seen these invariants of the Cauchy stress tensor $S$ defined in multiple places: $$J_1 = \lambda_1+\lambda_2+\lambda_3 = Tr(S)$$ $$J_2 = \lambda_1^2+\lambda_2^2+\lambda_3^2$$ $$J_3 = ...
12
votes
4answers
458 views

Is the concept of tensor rank useful in physics?

The term 'tensor rank' is sporadically used in the mathematical literature to denote the minimum number of simple terms (i.e. tensor products of vectors) needed to express the tensor. This is ...
1
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0answers
55 views

Contracting Indices in General relativity [duplicate]

I was reading a book about general relativity and I came across these two equations $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+ ...
9
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1answer
313 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ...
1
vote
1answer
529 views

Proving Lorentz invariance of Maxwell equations

I've read somewhere that one does not need to prove Lorentz invariance of the Maxwell equations $F_{\mu\nu,\sigma}+F_{\nu\sigma,\mu}+F_{\sigma\mu,\nu}=0$ because it is "manifestly Lorentz invariant" ...
2
votes
1answer
175 views

Rank $L$ spherical harmonic tensor as a $2L+1$ dimensional Cartesian vector?

Rank two Cartesian tensors can be decomposed into $L=0,1,2$ spin like things 3x3=1+3+5. But the second equation below does not transform like a "tensor", it looks more like a vector transform in ...
4
votes
1answer
256 views

Interpretation of rank 2 spinors

While inspecting the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group and defining a right-handed spinor with upper dotted index and a left-handed spinor with lower undotted index and ...
2
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2answers
170 views

Recovering 4-vector Lorentz transformation from spinor formalism

I'm trying to recover the 4-vector transformation laws using spinors. I have defined $$v^{\dot{a}b} = v^{\nu} \sigma_{\nu}^{\dot{a}b}$$ as usual, with $\sigma_0=1$. Now with the rules for dotted ...
0
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0answers
41 views

The inertia tensor is “global”, how do I make it local?

A solid cuboid has the following moment of inertia: $$I= \begin{pmatrix} \frac{1}{12}h^2+d^2 & 0 & 0 \\ 0 & \frac{1}{12}w^2+d^2 & 0 \\ 0 & 0 & \frac{1}{12}w^2+h^2 ...
0
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1answer
100 views

Physical interpretation of $Q^i = \partial _\nu T^{i \nu}$

I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry: If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum ...
0
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2answers
95 views

What does this summation mean in relativity?

Equation 1.2 of 't Hooft's Introduction to General Relativity gives the Lorentz transformations: $$ (x^\mu)' = \sum\limits_{\nu = 1}^4 {L^\mu}_\nu x^\nu $$ Is this the sum of four square matrices ...
4
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2answers
434 views

Understanding the difference between co- and contra-variant vectors

I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear. One ...
0
votes
1answer
114 views

Derivation of normal shear stress

I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about. Specifically, what I don't understand is ...
0
votes
1answer
99 views

Quantum Mechanics mistake in partial trace

I have a given a density matrix by $\rho:=\frac{1}{2} |\psi_1 \rangle \langle \psi_1|+\frac{1}{8} |\psi_2 \rangle \langle \psi_2|+\frac{3}{8} |\psi_3 \rangle \langle \psi_3|.$ Where $|\psi_1\rangle ...
1
vote
2answers
199 views

How to prove the raising/lowering indices operation?

I've read this related question, though it didn't satisfy me; I hope this complements it. I know that if I contract a covariant tensor ${A_{\alpha\beta}}$ with a vector ${B^\beta}$, I get some other ...
0
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2answers
74 views

Quark space tensor product Vs Angular momentum space tensor product

For two triplet angular momenta states, say $J=1$ and $I=1$, if we wanna look at it in the coupled basis $F=I+J$, we use the regular Angular Momentum rules: $$|I-J|\leq F\leq I+J,$$ and from that ...
6
votes
4answers
384 views

Is the covariance or contravariance of vectors/tensors something that can be “visualized”?

I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how ...
3
votes
2answers
258 views

The signature of the metric and the definition of the electromagnetic tensor

I've read the definition of the electromagnetic field tensor to be ...
0
votes
1answer
167 views

Electrooptic Tensor, Relating Tensors to Orientations

So I'm trying to gain a better understanding of electrooptic tensors: An example of a quartz electrooptic tensor is given. I know in order to best implement this crystal, in order to get the highest ...
9
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5answers
353 views

Motivation for tensor product in Physics

This question is about a mathematical object (the tensor product) but thinking about the motivation that comes from Physics. Algebraists motivate the tensor product like that: "given $k$ vector spaces ...
7
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1answer
139 views

Are there cases in which we should consider tensors as equivalence classes?

Usually in texts about Physics that uses tensors defines them as multilinear maps. So if $V$ is a vector space over the field $F$, a tensor is a multilinear mapping: $$T:V\times\cdots\times V\times ...
2
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1answer
495 views

Deriving an equation involving Killing vectors

I'm currently studying Carroll's GR book Spacetime & Geometry, and ran into some trouble understanding the text. When discussing Killing vectors, Carroll mentions that one can derive ...
3
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2answers
136 views

Tensor contra- and covariance concept

Imagine we have $p$-contravariant and $q$-covariant tensor, such that $$ t~=~t^{i_1, \dots, i_p} _{j_1, \dots, j_q} e_{i_1}\otimes \dots \otimes e_{i_p} \otimes e^{j_1} \otimes \dots \otimes ...
4
votes
2answers
753 views

What's the idea behind the Riemann curvature tensor?

The Riemann curvature tensor can be expressed using the Christoffel symbols like this: $R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj} - \partial_l\Gamma^m{}_{kj} + \Gamma^m{}_{ki}\Gamma^i{}_{lj} ...