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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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 ...
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32 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 ...
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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 ...
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1answer
40 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 ...
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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 ...
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1answer
67 views

On the proof of the existence of geodesics coordinates [closed]

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} {...
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1answer
78 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 \...
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51 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 ...
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1answer
84 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 $\...
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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 ...
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1answer
122 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 ...
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4answers
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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 $f(t,x,y,...
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1answer
106 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 ...
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1answer
38 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 $D_{...
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2answers
530 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, $\...
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1answer
49 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 = \...
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1answer
56 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 ...
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1answer
69 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. ...
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1answer
27 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
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1answer
157 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 = -{R^a}_{bcd}{}_{\bot}Z^cV^bV^...
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1answer
103 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: $$ \left(\...
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1answer
171 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 ...
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1answer
59 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 ...
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1answer
78 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 ignoring)...
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1answer
46 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 $...
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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 $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$ ...
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35 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 ...
3
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2answers
241 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 ...
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2answers
58 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 ...
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31 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 $V_x=x^2+3y$ ...
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71 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 ...
3
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1answer
64 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 "...
3
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1answer
65 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} j]l}=\frac{\kappa}{2}s_{ij}^{\...
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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 ...
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Need some help understanding Relativistic Notation

My question originates from what is done in the book on Quantum Field Theory book by Mark Srednicki on page 21 (if anyone has it). So say you have an inertial frame that is represented in the ...
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2answers
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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 ...
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53 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 ...
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3answers
90 views

Tensor manipulation and showing equality

Im taking GR for the first time and it is definitely throwing me for a loop. The question I am working on is this: Prove that if a contravariant tensor $A^{uv}$ is symmetric, then it remains ...
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1answer
100 views

Working with indices of tensors in special relativity

I'm trying to understand tensor notation and working with indices in special relativity. I use a book for this purpose in which $\eta_{\mu\nu}=\eta^{\mu\nu}$ is used for the metric tensor and a vector ...
2
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3answers
166 views

Why is the Ricci tensor defined as $R^\mu _{\nu \mu \sigma}$?

The Ricci tensor is defined as the contraction of the Riemann tensor in its upper and the second lower index. I was wondering why it is defined this way. What happens if the Ricci tensor is defined ...
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Writing Breit-Pauli spin-spin-coupling Hamiltonian as a sum of irreducible spin tensor operators

The spin-spin interaction part of the Breit-Pauli Hamiltonian for two electrons contains the term \begin{equation} 3(\mathbf S_1 \cdot \mathbf r)(\mathbf S_2 \cdot \mathbf r) - r^2 \mathbf S_1 \cdot \...
0
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1answer
65 views

How do I derive geodesic equation using variational principle? [duplicate]

I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got ...
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Can the permittivity tensor always be diagonalized?

For an anisotropic medium, permittivity is a symmetric tensor, with elements as $\epsilon_{ij}$ How can one be sure that it is diagonalizable, if some elements, say $\epsilon_{xy}$ and $\epsilon_{yx}$...
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Identifying Lorentz Covariant Equations

Statement: $\phi , A^{\mu}, T^{\mu \nu}$ are a Lorentz scalar, vector, and tensor. Which of the following equations are Lorentz covariant. a. $\phi = A_{0}$ b. $\phi = A^{\mu}A_{\mu}$ c. $\phi = ...
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1answer
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Transformation of self-dual and anti-self-dual tensors and irreducibility of representations

I am working out exercise 2.5 of Maggiore's book. Part of the exercise is the following: Verify that self-dual and anti-self-dual tensors are irreducible representations of (real) dimension three of ...
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0answers
45 views

Work by Gravity using Tensors [closed]

Now I'm familiar with the various methods for deriving work done by gravity, but I noticed a few things about the situation, and wanted to see if I could properly apply a tensor treatment to the ...
2
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1answer
85 views

Components of dual vectors

(This is a close retelling of Wald, problem 2.4b. Not for homework; just curiosity and an increasingly alarming suspicion that I've never actually understood anything.) Let $Y_1 ... Y_n$ be a ...
3
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2answers
41 views

Possible confusion, the inertia of something yields a tensor? (trying to understand an example)

I was reading the text by Dan Fleisch titled a A Student’s Guide to Vectors and on first pages he says: An example of a tensor is the inertia that relates the angular velocity of a > rotating ...
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0answers
62 views

Calculus formulas for buoyant force? [duplicate]

I am launching a high-altitude balloon as a part of a physics project I am working on. I know that the amount of helium I need corresponds to about 40 newtons of lift for the launch, which is all I ...
3
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1answer
67 views

Tensor indices and row and column labels of corresponding representation matrices

When reading undergraduate GR literature, I often see that the authors represent tensors ${\eta^\alpha}_{\beta}$, ${\eta^\beta}_{\alpha}$, $\eta_{\alpha \beta}$, $\eta^{\alpha \beta}$ as matrices. ...