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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Physical intuition on $\mathbf{v}\otimes \mathbf{w}$

On Physics there's one very clear intuition on what a vector $\mathbf{v}$ is: they represent things with direction and magnitude (although when no metric is available there's no clear concept of ...
8
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1answer
146 views

What does Ricci tensor do with two vectors?

I have found it easier to understand the meaning of a particular tensor if I can find out what does it do if I cancel all its lower indices with vectors in short: $g_{ij} u^i v^j$: dot product of ...
0
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2answers
228 views

What does it mean to “contract” a tensor identity?

I'm taking a GR course at the moment, completely stumped on this step here: starting from the Bianchi identity: Then it says "Contracting the Bianchi identity..." How does this work and what ...
2
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1answer
506 views

Prove Christoffel Symbol Identity

In a book I am reading, the following identity is claimed and then "left to the reader to prove." $g_{ij}$ is the metric tensor, and $\Gamma$ is the Christoffel symbol of the second kind with the ...
0
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1answer
57 views

How do I construct the Maxwell tensor $\bf{^*F}$ from Fadaray one $\bf{F}$ in a non-flat spacetime?

In the book Gravitation (Misner, Throne and Wheeler), it's said that to consider the line element of the flat space on the derivation of Maxwell tensor $\bf{^*F}$ from the Fadaray tensor $\bf{F}$ ...
2
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0answers
146 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
3
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1answer
178 views

In continuum mechanics, why is the stress vector $T=\sigma\cdot n$ not a covector?

In continuum mechanics, the stress vector (see Cauchy stress tensor) $T=\sigma\cdot n$ is the surface density of a force. Forces are covectors, since they map a displacement vector to a scalar energy. ...
3
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1answer
250 views

Tensor components change under rotation-translation

I am currently working on a research project in a non-physics field, where I would like to work on a very constrained 2nd order tensor (3x3, symmetric, traceless). The tensor represents probability of ...
2
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1answer
371 views

Does the velocity vector always point in the same direction as the momentum vector?

I was told that the angular velocity vector does not always have to point in the same direction as the angular momentum vector. This is due to the fact that they are related by the equation $L=I ...
1
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1answer
68 views

Photon propagator inverse

If i have the operator $D^{\mu\nu}=\partial^{\mu}\partial^{\nu}+m\epsilon^{\mu\alpha\nu}\partial_{\alpha}$. What's your inverse $(D^{\mu\nu})^{-1}$?
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2answers
127 views

Index gymnastics and representing bra-kets as covariant and contravariant tensors

I am trying to figure out how to write, in Einstein notation as well as pick out elements of $$\langle A|[\mu]|B\rangle \langle X|[\nu]|Y\rangle$$ where $[\mu] = \begin{bmatrix} \mu_{11} & ...
0
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2answers
218 views

Off-diagonal terms in metric for 4D space-time [closed]

Consider a delta between two events in 4D space-time written as a 4-vector, $x^\mu=(dt, dR)$. The time $dt$ is a scalar difference in time. The 3-vector $dR$ points some direction in space. One ...
1
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2answers
89 views

How do I represent $A$ transpose $A$ in indicial notation?

I know this question sounds lame, but the book I am following doesn't use the answer I expect and it has been using a similar notations everywhere else which has confused me. I think Q[Any tensor] ...
0
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0answers
52 views

Eigenstates of operators on constituent systems in tensor product space

Suppose I have two entangled physical systems $\mathcal{A}$ and $\mathcal{B}$ with respective hilbert spaces $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$. If $A,B$ are operators on ...
1
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2answers
115 views

Can someone explain how Weinberg's definition of the affine connection for the geodesic equation matches the definition of an affine connection?

Consider the geodesic equation \begin{equation} 0=\frac{d^2 x^\lambda}{d\tau^2}+ \Gamma^\lambda_{\mu\nu} \frac{d x^\nu}{d\tau}\frac{d x^\mu}{d\tau} \end{equation} In Gravitation and Cosmology, on page ...
1
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1answer
131 views

Bianchi Identity using null tetrad

I'm currently looking at the Newman-Penrose Formalism, and trying to understand where there sets of equations come from. For that, I need to know how I can write the second Bianchi identity for the ...
-1
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1answer
104 views

Weyl scalar calculation

I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take ...
2
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1answer
149 views

Transpose of (1,1) tensor

When we transpose a (1,1) tensor, shall we simply switch the two indices while keeping their upper/lower positions or switch them and also switch their upper/lower positions? In general, would the ...
-1
votes
1answer
99 views

Decomposition of group representation using tensor method

I am dealing with the decomposition of the representation $5\otimes5$ of $SU(5)$: $$5\otimes5=15\oplus10 $$ demonstration: $$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$ ...
0
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0answers
24 views

I wasn't able to find a good resource for Bipartite state and Bell's theorem

Our professor used tensor product to explain bipartite operator and states and then he used the new operator and state to explain Bell theorem. I wasn't able to find a good resource or reference for ...
2
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0answers
388 views

Covariant versus “ordinary” divergence theorem

Let $M$ be an oriented $m$-dimensional manifold with boundary. As stated in Harvey Reall's general relativity notes (here) or Sean Carroll's book, the "covariant" divergence theorem (i.e. with ...
3
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1answer
55 views

Product on Tensor Products

I'm trying to understand how products on tensor products work. For instance, in quantum mechanics, you have ($x$ tensor $y$) times ($z$ tensor $a$), where $x$, $y$, $z$, $a$ are all operators acting ...
0
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2answers
98 views

Proving a relation with Four-velocity tensor [duplicate]

I'm trying to show that: $U^a_{\space\space;b}U^bU_a = 0$ (Where U is four-velocity) and I'm stuck on how to go about it. I tried expanding it out into the Christoffel symbols, but that didn't seem ...
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7answers
335 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
1
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3answers
281 views

Tensor product in quantum mechanics

In Cohen-Tannoudji's Quantum Mechanics book the tensor product of two two Hilbert spaces $(\mathcal H = \mathcal H_1 \otimes \mathcal H_2)$ was introduced in (2.312) by saying that to every pair of ...
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2answers
227 views

Relation between Vector space $V$ and its dual $V^{*}$ [closed]

I asked the same question in Math.SE, but I was suggested to ask it here as well. I am studying relativity, and as you know the theory extensively uses the notion of covariant and contravariant ...
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2answers
125 views

Suggested operatonal definition for a tensor [duplicate]

The two tensor definitions I'm (newly) familiar with, by transformation rules, and as a map from a tensor product space to the reals, don't tell me what a tensor does, and to the best of my knowledge ...
2
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1answer
334 views

$SU(3)$ irreducible representations with tensor method

I am dealing with the tensor product representation of $SU(3)$ and I have some problems in understanding some decomposition. 1) Let's find the irreducible representation of $3\otimes\bar{3}$ we have ...
1
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1answer
143 views

What does a colon mean in hydrodynamics equations?

In some hydrodynamics book I saw a notation like $e:e$ where $e$ is a matrix (shear stress tensor). This double dot product is in a scalar equation, so the result of this operation must be scalar. I ...
0
votes
1answer
121 views

Why tensor product? [duplicate]

Let $A$ an $B$ be two discrete observables (like spins). When exactly and why we have to consider their tensor product when talking about the mutual observation of the corresponding phenomena?
1
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1answer
232 views

What is the dyad corresponding to a stress tensor?

(As I understand it ... qualifies every sentence in what follows).. a stress tensor is a rank 2 tensor that maps a unit vector normal to a surface to the stress (or traction) vector corresponding to ...
0
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1answer
79 views

Energy-Momentum Tensor with mixed indices

I know that $T_{\mu\nu}$ is the Energy-Momentum Tensor and $T=g^{\mu\nu}T_{\mu\nu}$, but does anyone know what $T^{\nu}_{\mu}$ is and how its calculated?
2
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1answer
483 views

Inverse Metric Tensor

First the setup: Let $\mathcal M$ be a $2$-dimensional manifold. Let $U_P$ be some open neighbourhood of a point $P \in \mathcal M$. Let $\mathcal F : U_P \rightarrow \mathbb R \times \mathbb R$ be ...
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votes
1answer
70 views

Simple question about the electromagnetic tensor written as a 2-form

I noticed that the 2 form (Electromagnetic tensor) is written as: $$F= F^{ab}e^a \wedge e^b$$ while we know that $$F= F_{\mu\nu}dx^\mu \wedge dx^\nu$$ Is there something wrong with the indices ...
0
votes
1answer
83 views

What is the relation between $\eta^{ab}x^2$ and $x^ax^b$?

What is the relation between $\eta^{ab}x^2$ and $x^ax^b$? Here $\eta$ is the Minkowski metric in $d=4$ and $x$ is a 4-vector. In particular, a tensor like $$x^a x^b x^c (\eta_{ab}x^2 - x_ax_b) ...
1
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1answer
127 views

Rotation matrix in yo-yo problem?

I need to solve the yo-yo problem not in the normal sense. Instead, I need to include the position vector $r$ and rotation matrix $R$. Assume the yo-yo is rotating in the plane. In the problem yo-yo ...
1
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2answers
73 views

Compute the inertial tensor and then solve the equation? [closed]

If the $J_{\Omega}$ is the following matrix, which is solved by ja72 in How to compute the inertia tensor ${\bf{J}} _{\Omega}$ of a body of revolution: $${\bf J} = \rho\, \begin{bmatrix} ...
1
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2answers
179 views

Is a vector field not a vector quantity?

I'm trying to make sense of Poisson bracket relation $$\{L_i,A_k\}_{PB}~=~\epsilon_{ikl}A_l,\tag1$$ where $L_i$ is $i$th component of angular momentum, $A_k$ is $k$th component of an arbitrary ...
1
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1answer
127 views

How to compute the inertia tensor ${\bf{J}} _{\Omega}$ of a body of revolution

Suppose that $\Omega$ is a body of revolution of the function $y=f(x), a\le x \le b$ around the $x$-axis, where $f(x)>0$ is continuous. How to compute the inertia tensor ${\bf{J}} _{\Omega}$? ...
0
votes
2answers
46 views

Is it inevitable to compute the quadruople tensor in components? Why? [closed]

I was trying to determine the quadrupole tensor for a given charge distribution in one go from this equation: $$\overleftrightarrow{D}=\int d^3r \varrho(\vec{r})\left(3\vec{r} \circ ...
1
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0answers
37 views

Is energy-momentum of curvature a boundary/holographic density?

Since the beginnings of General Relativity, we have had this awkward, unholy separation of the universe in marble versus wood. divergence of the stress-energy momentum holds at all points of ...
0
votes
1answer
51 views

Electromagnetic tensor notation

How do you transform between the electromagnetic tensors $F_{\mu\nu}$ and $F^{\mu\nu}$? $$ F_{\mu \nu}= \begin{pmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & -B_z & B_y \\ -E_y ...
0
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0answers
105 views

Quasi-primary fields and usual fields

How do i see that the way quasi-primary/primary fields transform contain the transformation rule for fields as we know it (scalar, vector fields) in QFT?
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3answers
864 views

Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
2
votes
1answer
134 views

components of mixed tensor with same indices

If my tensor $a^{\mu\nu}=$ matrix of 4*4 size (let's say, in 1+3 dimensions with mostly negative convention for the metric), what is $a^{\mu}_{\mu}$ ? Is it the trace or the vector of diagonal ...
5
votes
2answers
186 views

Metric tensor in SRT

I just read on this webpage that we have (click me) $g_{\alpha \beta} = g_{\alpha}^{\beta} = g^{\alpha \beta}.$ Now, although I understand that the first and the last one are equal, I don't think ...
0
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1answer
63 views

Trouble getting the matrix representation of a 4-state Hamiltonian

$\newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\ket}[1]{\left| #1 \right\rangle}$I have a simple 4-state Hamiltonian and am trying to find the matrix representation (in order to determine ...
3
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1answer
98 views

Naturalness of tensor fields in general relativity?

In the third chapter of the book The Large Scale Structure of Space-Time, the authors say regarding the matter fields in general relativity: These fields will obey equations which can be expressed ...
0
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1answer
82 views

Little problem with indexes

Suppose I have a diagonal matrix metric, like $$b_{\mu\nu} = \mbox{diag}(1, -1, -1, -1)$$ namely there are nonzero values only for $\mu = \nu$. My problem is this (please be quiet to explain me ...
2
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2answers
94 views

Why can tensors be broken up into parts?

I have found these notes: http://www.physics.usu.edu/Wheeler/QuantumMechanics/QMWignerEckartTheorem.pdf Which state on page two that a matrix (M) can be broken up into rotationally independent pieces ...