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2
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2answers
38 views

How do we determine if a certain physical quantity is a vector?

For instance in Newtonian physics we treat position of objects, displacements, velocities, forces, momenta, angular velocities etc all as vector quantities (little arrows in space which have a certain ...
3
votes
1answer
84 views

How to write the Clebsch-Gordan decomposition in tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is known that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ may be decomposed through the ...
1
vote
0answers
39 views

Indicating that indices are equal in Einstein notation

tl;dr: I have an expression like this: (dramatization) $$ R_{\mu\nu} = \begin{pmatrix} B^{00}C_{00} & 0 & 0 & 0 \\ 0 & B^{11}C_{10} & 0 & 0 \\ 0 & 0 & B^{22}C_{20} ...
1
vote
2answers
75 views

Using $\sqrt{-g}$ in integrals of proper volume

I am a little confused over integration using proper volume element. When do we use $\sqrt{-g}$ in calculations? For example, in many calculations involving stars, say when using TOV equation, this ...
1
vote
1answer
18 views

Exponential decay of correlation in PEPS

PEPS (Projected Entangled Pair State) is a tensor network that plays the same role in two dimensional lattice as MPS (Matrix Product State) plays in one dimensional spin chain. A good introduction can ...
7
votes
3answers
254 views

Converting between matrix multiplication and tensor contraction

If I have $A^{\alpha \beta} B_{\beta \gamma}$ then this should be the equivalent to the following matrix multiplication: $AB$ since we're summing over the columns of $A$ and the rows of $B$. By the ...
0
votes
1answer
29 views

Tensor products of Hilberts spaces: definition of outer products and commutators

Suppose one has two single-particle Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ and consider the tensor product of these such that $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is a two-particle ...
0
votes
1answer
18 views

Placement of indices in canonical commutation relations of coordinates and conjugate momenta as well as fields and conjugate momenta

The canonical commutation relations between generalised coordinates $q_a$ and their conjugate momenta $p^a$ are given by $[q_a,q_b]=[p^a,p^b]=0$ $[q_a,p^b]=i\delta^b_a$. Furthermore, the canonical ...
0
votes
0answers
7 views

Warping function for torsion of non-circular prism

I have a few questions regarding the case of torsion of a prism, as encountered in continuum mechanics. Specifically, a prism (which can be a cylinder, a rectangular prism, elliptical prism, etc.) has ...
0
votes
0answers
48 views

Covariant and contravariant tensors and divergence - notation [closed]

This is in the context of 4-tensors and special relativity. My question is this. I want to carry out this operation: $$A_{0,0}+A_{1,1}+A_{2,2}+A_{3,3}.$$ How do I notate this in summation form?
0
votes
1answer
40 views

Raising and lowering operators for a composite isospin $SU(2)$ system

Consider pion states composed of $q \bar q$ pairs where $q \in \left\{u,d \right\}$ transforms under an $SU(2)$ isospin flavour symmetry. These bound states transform in the tensor product $R_1 ...
3
votes
1answer
53 views

Is time a vector in Minkowski space? [duplicate]

I am arguing about this topic with my school teacher in so long time, I want to finish this debate. My teacher's opinion is "Yes, Time is vector" because four-vector has $t$ component, and mine is ...
1
vote
1answer
60 views

On the proof of the existence of geodesics coordinates

From "Introducing Einstein’s Relativity" by Ray D’Inverno page 77-78 In my calculation, the process is $$\frac{\partial{x^{'a}}}{\partial{x^d}}=\frac{\partial{x^{a}}}{\partial{x^d}}+\frac{1}{2} ...
2
votes
2answers
258 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}} = ...
10
votes
1answer
311 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
5
votes
1answer
424 views

Fierz identity for Weyl spinors in tensor currents

Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish $$\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 = -\frac{3}{2} ...
0
votes
2answers
46 views

What's meant by: “All observers agree on the combination of basis vectors and components for a tensor”?

There's a Youtube video given by Dr Dan Fleisch on Tensors, where he states at 11:22: You may be wondering, what is it about the combination of components and basis vectors that makes tensors so ...
0
votes
1answer
66 views

Dirac Notation With Comma

Does $\langle A,B\rvert$ mean $\langle A\rvert\langle B\rvert$? If so how is an operator applied to this in $\langle A,B\rvert \hat O $? For an example say the annihilation operator acting on ...
0
votes
1answer
51 views

When dealing with spinor indices, how exactly do we obtain the barred Pauli operator?

In the set of SUSY notes I'm following, the Pauli operator is given as: ${(\sigma^\mu)}_{\alpha\dot{\alpha}} = (I_2, \sigma^1, \sigma^2, \sigma^3)$. The antisymmetric tensor that lowers and raises ...
11
votes
2answers
507 views

Tensor product in quantum mechanics?

I often see many-body systems in QM represented in terms of a tensor products of the individual wave functions. Like, given two wave functions with basis vectors $|A\rangle$ and $|B\rangle$, belonging ...
0
votes
4answers
115 views

Tensor calculus in special relativity

We say that any function $f(t,x,y,z)$ is a tensor of rank {0 0} because it takes no vectors or one-forms in order to give a real number. But couldn't we have just written the same function as ...
5
votes
1answer
91 views

Proportionality Constant in Einstein Field Equations

The Einstein Field Equations: $$G_{ab}~=~8\pi T_{ab}.$$ I am familiar with how to obtain the $8\pi$ proportionality factor through correspondence with Newtonian gravity, but am wondering if this ...
8
votes
7answers
809 views

Is there a physical interpretation of a tensor as a vector with additional qualities?

What is a tensor? has been asked before, with the most highly up-voted answer defining a tensor of rank $k$ as a vector of a tensor of rank $k-1$. But if a scalar is defined as a physical quantity ...
0
votes
0answers
16 views

An expression for stress power

I have seen it written that for a continuum undergoing deformation, if we ignore body forces and heat transfer, the work done is equal to stress power: $\cfrac{dW}{dt}=\sigma_{ij}D_{ij}$, where ...
5
votes
1answer
249 views

Finding the components of the tensor for potential and kinetic energy

I have a rather poor understanding of what a tensor is, but enough to apply it to the biggest part of the classical mechanics I'm studying. However, I've run into a small problem while studying "Free ...
1
vote
0answers
30 views

Ordering of basis elements of a Lie-group representations tensor product [migrated]

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \oplus_{J}\textbf{r}_J $$ where $\textbf{r}_J$ are ...
-1
votes
0answers
17 views

relation between stress and strain tensor in Navier Stokes equations

Yesterday I had posted a question requesting elucidation of a part of Navier Stokes equation but I removed it as it(and I myself) was not so specific/clear. After going through few resources online I ...
3
votes
2answers
506 views

Is Newton second law covariant or invariant?

Is Newton second law covariant or invariant between two inertial frames, moving with uniform traslational motion with respect to each other? If it is invariant then, indipendently from the frame, ...
3
votes
2answers
222 views

Tensor index notation with e.g. square brackets

I want to learn playing with indices and some notation in General relativity. But in every book just is used this notation. I know upper and lower but I don"t know the meaning of some combination of ...
1
vote
1answer
44 views

Coordinate form of divergence of anti-symmetric tensor field [closed]

just a quick question on something that might save me a little bit of time and effort. In a general curved metric, the divergence of a vector field, $A^\mu$, can be written as: $ \nabla_\mu A^\mu = ...
2
votes
2answers
188 views

What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
1
vote
1answer
65 views

Derivation of the relativistic equation of energy conservation for a perfect fluid

I'm currently attempting to struggle through the first chapter of Sean M. Carrol's spacetime and geometry. I'm a bit stuck, most likely because of not understanding the mathematical operation. ...
0
votes
1answer
26 views

What is the short time limit of Maxwell viscoelastic fluids?

The Maxwell model for viscoelastic fluids writes: $$ \tau\stackrel{\triangledown}{\sigma}+\sigma=2\eta D(v) $$ where $D(v) = \frac{1}{2}(\nabla v +\nabla v^T)$, $v$ velocity and $\sigma$ stress tensor ...
3
votes
1answer
136 views

Fermi-Propagated Jacobi equation in the book The Large scale structure of space-time

On page 81, equation (4.6), the author use the Fermi derivative to write the Jacobi equation \begin{equation} \tag{4.6} \frac{{D^2}_\text{F}}{\partial s^2} {}_{\bot}Z^a = ...
3
votes
1answer
62 views

Derivation of the Cartan Field equation

Please help me understand how, in this introduction to spacetime and fields, the Einstein Cartan equation: $$C^k_{\hspace{2mm} [ji]}-\delta_{[i}^{k}C^l_{\hspace{2mm} ...
0
votes
1answer
94 views

The Lie derivative of the metric $g_{ab}$ and index notation

I don't quite know where to start this question. I'm essentially not understanding how to compute the Lie derivative of a given metric and vector. So I have the following definition: $$ ...
1
vote
1answer
165 views

Jacobi equation in the book The Large scale structure of space-time

On pp. 79, it is obvious that equation (4.2) \begin{equation} \frac{D}{\partial s}Z^a = {V^a}_{;\ b}Z^b \end{equation} holds, where $Z$ is the deviation vector and $V$ is the unit tangent vector along ...
1
vote
1answer
73 views

A question regarding $f(R)$ Lagrangians

Consider the class of Lagrangian known as $f(R)$ Lagrangians where the Lagrangian is some function $f(R)$, \begin{equation} S=\int\sqrt{g}d^4x\ f(R) \end{equation} assuming there are no (or ...
1
vote
1answer
49 views

Why is the spatial term for contravariant 4-gradient negative, whereas for other 4-vectors it is the covariant part that is negative spatially?

The contravariant 4-displacement is: $${x}^{\alpha} = (ct,\mathbf{r})$$ And the contravariant 4-gradient is: $${\partial}^{\alpha} = (\frac{1}{c}\frac{\partial}{\partial{t}},-\nabla)$$ From what I ...
0
votes
1answer
40 views

Ordering of Contravariant and Covariant spinors. Understanding the spinor space

I've been referring to Pg.36-Pg.38 in Introduction to Supersymmetry by Wiedamann. For understanding the precise origin of dotted, undotted indices on Spinors. He starts off my saying that $M$ acts on ...
0
votes
1answer
34 views

Calculating motion of equation in tensor form

for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$ how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$ ...
1
vote
0answers
31 views

Turning a k-space integral into an energy integral for a conductivity tensor

Looking over a derivation of the conductivity tensor for magneto-resistance, I got stuck trying to go from (1.133) to (1.134), transforming the k-space integral into one over energy. In this ...
1
vote
0answers
55 views

Einstein's Equations [closed]

Can one please explain tensors, specifically stress energy tensors and its application in Einstein's Equations? I am a beginner cosmology learner and want to know the meaning and significance of ...
0
votes
2answers
45 views

Can you set all dummy indices equal to each other? [closed]

According to the Einstein summation convention, can you set all dummy indices within a same expression equal to each other? Example, if both α and β are dummy indices in a same expression can you set ...
1
vote
0answers
30 views

Relativity Coordinate transformation of Vector [closed]

I'm taking a first course in General Relativity but I've been struggling with coordinate system transformation. For example, if I have a Vector defined in Cartesian (x,y) coordinates as $V_x=x^2+3y$ ...
0
votes
0answers
47 views

Moment of inertia tensor of cube with objects inside

I'm currently trying to write a simulation that, essentially, calculates the moment of inertia of a hollow cube in 3D, where it's possible to attach additional cuboid shapes inside of it - to see how ...
0
votes
0answers
50 views

Splitting different aspects of a system in Quantum Mechanics with tensor products

My understanding from Classical Mechanics is that the degrees of freedom of a system are the generalized coordinates which we use to describe the system. In that case the number of degrees of freedom ...
3
votes
1answer
85 views

How is the electromagnetic tensor expanded?

The electromagnetic tensor is given by $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$, and it appears in the Lagrangian as $L = -\frac{1}{4}F_{\mu\nu}^2 - A_{\mu}J_{\mu}$. The text I'm ...
1
vote
0answers
33 views

Identity in continuum mechanics [closed]

For a problem in the textbook I am reading, I need to prove that $$\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS,$$ where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a ...
3
votes
2answers
82 views

Tensor product in many electron atoms

One of the quantum mechanics postulates states that a composite system can be described with the tensor product of the component systems. I've read some rationalization about this fact in some post ...