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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$SU(3)$ Tensor Methods in a Tetraquark

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
2
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1answer
80 views

relativistic addition of velocities using tensor notation? [closed]

I know the way of deriving the formula using usual lorentz transformation formulas,,but is there a way out of deriving it using 4-vector notation??please help
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22 views

Differentiation of functions defined of invertible matrices

Let $f:GL(n,\mathbb{R}) \rightarrow \mathbb{R}$. I would like to compute the directional derivative of this function. An example of one such function would be $\det(A)$, $A \in GL(n,R)$. Can I use the ...
3
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1answer
120 views

Tensor decomposition

I came across what a Physicist called "decomposing a tensor with respect to a congruence", something I simply cannot grasp. I searched a lot and I couldn't find any reference on that. I know that ...
2
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0answers
67 views

Tensor product of spin states

I just wanted to check that I carried out this problem correctly. I got the correct answer, but I'm not sure if what I did to get it is completely correct. This is from the second part of problem 3.4 ...
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0answers
73 views

Irreducible representations of $SU(2)$, Tensor-operators under rotations

First of all: this is my first question so please give feedback to the way I'm formulating the question! The question is about an exercise I have to solve, but I simply get nowhere. It is given the ...
0
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1answer
252 views

Commutation of vector operators

I'm supposed to show that $\left[\mathbf A,\mathbf B\right]=0$ (for two vector operators $\mathbf A$ and $\mathbf B$) if and only if all components of $\mathbf A$ commute with all components of ...
1
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1answer
114 views

Does $g_{\mu\mu}$ in an expression follow the Einstein summation convention?

Assume that I have the expression for a Christoffel symbol: $$ \Gamma^\mu_{\alpha \beta}=\frac{1}{2}g^{\mu \lambda}(\partial_\alpha g_{\beta \lambda}+\partial_\beta g_{\alpha \lambda} - ...
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2answers
241 views

I have problem understanding the direction of the shear stress [duplicate]

Is it in the direction of the velocity ? If it is, can anybody describe it somehow that matches my physical feeling? How the shear stress affects another layer of fluid ? I faced a question ...
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0answers
107 views

Constructing Killing tensors from Killing vectors

Background: After reading about Carter constant and symmetries in GR, I became interested in Killing tensors. I tried reading this paper by Alan Barnes, Brian Edgar and Raffaele Rani, discussing ...
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0answers
50 views

Explicit calculation using metric tensor

I'm trying to calculate the following quantity: $$S_{\bar{\alpha} \bar{\beta}} \equiv \left( \Lambda^{-1} \right)^\alpha_{ \ \bar{\alpha}} \left( \Lambda^{-1} \right)^\beta_{ \ \bar{\beta}} ...
2
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2answers
71 views

Does the definition of tensor include basis?

From my understanding, a vector is a geometric object, which can be expressed as $$ v = v_i e^i $$ where $v_i$ and $e^i$ are components and basis, respectively. It seems to me that many people, e.g. ...
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2answers
111 views

Tensor product postulate [duplicate]

Non relativistic quantum mechanics assumes that a composite system should be described with the tensor product of the component systems. This is the tensor product postulate of quantum mechanics. I ...
1
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2answers
70 views

Interference of two non-entangled photons, calculation using tensor product of Hilbert spaces

I'm trying to calculate the interference of two non-entangled photons, like in a double-slit experiment with two photon sources, one behind each slit (follow-up on this question). The individual ...
4
votes
3answers
131 views

Interference between two photons, tensor product of individual wave functions?

I have learned that the wave function cannot be visualized as a real physical wave like for example the EM field, because for multi-particle systems, it is not a wave in $\mathbb{R}^3$ but in ...
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0answers
25 views

LinAlg based physics textbooks [duplicate]

I'm in my second year studying physics, and ever since I took LinAlg, I've been noticing LinAlg-related concepts pop up all over the place, but it has never been presented directly as matrices, bases, ...
1
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1answer
225 views

Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
0
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1answer
76 views

Two Point Correlator

I have a problem to reproduce the following identity: \begin{equation} \Pi_{\mu\nu}(q^2) = i \int d^Dx e^{iqx} \langle 0 | T \{j_\mu(x) j_\nu(0) \} | 0 \rangle = (q_\mu q_\nu - g_{\mu\nu} q^2 ) ...
2
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1answer
111 views

Lowering/raising metric indexes

So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing. Consider Minkowski spacetime. The trace of a matrix $A$ can be ...
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0answers
66 views

Transformation of Christoffel symbols [closed]

Friends I have little problem with transformations:) In General relativity is Christoffel symbol of second kind defined as: $$ \Gamma^{l}_{ij}=g^{lk}\left(\frac{\partial g_{ki}}{\partial ...
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2answers
52 views

From tensor seen as linear maps on vector spaces to index contraction

I am considering a second-order tensor $\mathbf T$ seen as a linear maps on vector spaces, acting on a vector $\mathbf v$ to create a new vector $\mathbf u$, thus, $\mathbf u= \mathbf{T}(\mathbf{v})$. ...
0
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2answers
276 views

What is a Christoffel symbol?

What is a Christoffel symbol? I often see that Christoffel symbols describe gravitational field and at other times that they describe gravitational accelerations. Then, on some blogs and forums, ...
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2answers
292 views

What exactly is $T_{\mu\nu}$?

Continuous matter is described in special relativity by the matter tensor which is the so-called stress-energy-momentum tensor. I am finding a difficulty understanding how a tensorial tool ...
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0answers
28 views

How do you solve for flux density when you are given the E(x,y,z) and relative permittivity as a tensor?

This given material is not isotropic therefore the relative permittivity is represented as a tensor given by the matrix: $$\left(\begin{matrix} 3 & 1 & 2 \\ 2 & 3 & 3 \\ 2 & 2 ...
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0answers
88 views

Minkowski metric and Null tetrad metric

I'm starting with the Newman-Penrose formalism and have a very basic question that I'm very confused about. The standard Minkoswki metric is $\eta_{ab}=\mathrm{diag}(-1,1,1,1)$. Is then the null ...
0
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1answer
74 views

The Ricci tensor and its relation to volume

From Wikipedia's entry on Ricci tensor, In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in ...
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1answer
51 views

Tensor as outer product

This is a problem I am trying to solve and need help with. Given a $ \left( \begin{array}{} 0 \\ 2 \end{array} \right)$ tensor h such that h$(\quad ;A)=\alpha $h$(\quad ;B)$ for any two vectors ...
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0answers
42 views

Show that $M^{\mu\nu}$ describes the angular momentum of the system

Define $M^{\mu\nu}$ = $\int d^3x(x^\mu T^{0 \nu}-x^{\nu}T^{0 \mu})$ describes the angular momentum of the system. I don't want you to solve it but I'm not really sure what kind of criterion it ...
4
votes
1answer
150 views

Gradient one-form [duplicate]

I am trying to understand what gradient one-form means actually. In the book that I'm following (A first course on General Relativity by Schutz) it's told that gradient is a one-form and it's ...
3
votes
0answers
55 views

What kind of math do I need got general relativity? [duplicate]

I'm 15 this year and have a passion in physics What kind of math do I need to tackle general relativity? Also what year in uni do we learn about general relativity?
3
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3answers
172 views

4-velocities in different frames

We have an observer in an inertial frame $S$ who measures a particle's 4-velocity as $U$. We then have another inertial frame $S'$ with $X'=\Lambda{X}$, where $\Lambda$ is a matrix representing a ...
1
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1answer
46 views

Metric Tensor and Strain Rate Tensor- Comparison of Units

Is there any way the metric tensor can have a dimension in general relativity? I ask because there is an equation where the strain rate tensor of continuum mechanics is expressed as a difference of ...
0
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1answer
107 views

Proving the invariance of the inner product

If we define the inner product as ${\textbf{u}\cdot\textbf{v}=g_{ij}u^{i}v^{j}}$, where ${g_{ij}}$ is the metric tensor, ${S}$ and ${T}$ are transformation matrices, ${S}$-for covariant indices and ...
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2answers
123 views

Is $X^{\mu\nu} \equiv A^{\mu}+B^{\nu}$ a tensor? [closed]

Consider $X^{\mu \nu}$ an "object" with two indices, defined as $X^{\mu\nu} = A^{\mu}+B^{\nu}$. Is $X^{\mu\nu}$ a tensor? Exists some transformation law to carry $X$ to a new coordinate system ? What ...
1
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1answer
78 views

Work out components $F^{01}$ and $F^{ij}$ of the antisymmetric tensor $F^{\mu\nu}$ under the Lorentz Transform [closed]

Work out explicitly how the components $F^{0i}$ and $F^{ij}$ of the antysymmetric tensor $F^{\mu\nu}$ introduced in chapter I.6 transform under a Lorentz transformation This problem is from Zee, ...
4
votes
1answer
159 views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) ...
5
votes
3answers
653 views

How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ ...
0
votes
1answer
90 views

How can you have $\frac{DA^\mu}{d\tau}$?

If a covariant derivative is given by: $$D_\nu A^\mu=\partial_\nu A^\mu +\Gamma^\mu_{\nu \lambda} A^{\lambda}$$ Then how does $\frac{DA^\mu}{d\tau}$ make any sense? Since there are no 'differentials' ...
2
votes
1answer
80 views

Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?

The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is: ...
4
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2answers
304 views

Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...
0
votes
1answer
86 views

In field theory, why are some symmetry transformations applied to the field values while other act on the space that the fields are defined on?

My basic understanding is that a field theory consists of symmetry groups, a space $S$ that the symmetry groups act on and of fields defined on that space $S$. In other words, the space $S$ is the ...
0
votes
2answers
527 views

Variation of square root of determinant of metric, $\delta g$ [closed]

I am trying to calculate $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$ where $g = \text{det} g_{\mu \nu}$. We have $$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 ...
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2answers
122 views

Transformation of four-velocity in special relativity

I am revising special relativity introducing more matrix form in the equation. Currently I am reading book in which transformation matrix is defined as $${\Lambda= \begin{bmatrix} \gamma & ...
-1
votes
1answer
89 views

The Riemannian Curvature in Deformations

Is there a direct correlation between the Riemannian Curvature tensor and the deformation gradient tensor in continuum mechanics?
2
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2answers
287 views

Jaumann deviatoric stress rate

Background about terms in this question: Hookes law and objective stress rates From my understading, the Jaumann rate of deviatoric stress is written as: $$dS/dt = \overset{\bigtriangleup}{{S}} = ...
1
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1answer
92 views

Covariant derivative commutator on spinors [closed]

What is this object $[\nabla_{\mu},\nabla_{\nu}]\epsilon$ in terms of curvature tensor $R_{\mu\nu}$? Where $\nabla_{\mu}$ is the covariant derivative on a four sphere and $\epsilon$ is spinor. PS: I ...
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0answers
63 views

Prove that $T_{00}$, $T_{10}$, $T_{01}$, and $T_{11}$ are all $L/(4\pi x^2)$ at $(ct, x, 0, 0)$ for star of constant luminosity $L$

We have a star of constant luminosity $L$. We want to prove that the components $T_{00}$, $T_{10}$, $T_{01}$ and $T_{11}$ are all the same for the event $(ct,x,0,0)$ and they are all $L/(4\pi x^2)$. ...
0
votes
1answer
89 views

When is Einstein summation implied by Lorentz indices?

I would like to ask if it is possible to find out whether Einstein summation is used in an equation. For example, $$A^{\mu \nu} = 1$$ can either mean $\sum_{\mu\nu} A^{\mu \nu}=1$ or $A^{\mu \nu}=1$ ...
2
votes
4answers
405 views

Is this a Lorentz-scalar? How do I tell?

I'm struggling to identify whether a scalar is a Lorentz-scalar. E.g: $$\partial_i A^i \quad i \in {1,2,3}.$$ How do I determine if this is a Lorentz-scalar or not? If got the same problem with ...
1
vote
0answers
39 views

What are the dimensions of the stress energy tensor in relativity? [duplicate]

Can anyone tell me what the dimension of the stress energy tensor is? Also, if it represents energy density, will calling it kinetic or potential energy be appropriate?