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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
18 views

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
38 views

Work by Gravity using Tensors [on hold]

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 ...
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1answer
77 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 ...
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2answers
34 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
21 views

Calculus formulas for buoyant force?

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 ...
2
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1answer
49 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. ...
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1answer
61 views

To prove uniqueness of Rotation Tensor [closed]

How can you prove that a rotation tensor which rotates some given vector is a unique tensor? Let's say we have a vector 'a' and we take a tensor product of that vector with some tensor 'Z' such that: ...
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1answer
77 views

How should Christoffel symbols be written (in LaTeX)? [closed]

I'm writing a summary of a lecture on relativity, and we've recently introduced the Christoffel symbols. It seems that the upstairs indices are the "leftmost" and the downstairs indices are somewhat ...
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2answers
76 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 ...
3
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1answer
129 views

Repeated index in covariant derivative using abstract index notation

The same index showing up twice in the charge conservation law $\nabla_a j^a = 0$, as stated using abstract index notation, highly confuses me. If we chose a coordinate basis ...
2
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1answer
136 views

Invariant tensors in a general representation and their physical meaning

I'm trying to use tensor methods to find invariant elements of representations. Specifically I'm looking at representations of $SU(5)$. I can show that the invariant element in $5\otimes\bar{5}$ (or ...
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1answer
61 views

Is an event formally a 4-vector? [duplicate]

An event is a 4D point in spacetime. At every point in spacetime there is a tangent space. 4-vectors live in the tangent space. One can contract two events using a metric tensor. Is there a process ...
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1answer
64 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
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3answers
126 views

Covariant and contravariant 4-vector in special relativity

I've just learned about contra- and covariant vector in the context of special relativity (in electrodynamic) and I'm struggling with some concept. From what I found, an intuitive definition of ...
3
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1answer
64 views

Cauchy stress tensor in different coordinate system

The general form of the cauchy stress tensor is given by the dyadic decomposition $$\boldsymbol \sigma = \sigma_{ij}\,\,\mathbf{e}_i \otimes \mathbf{e}_j$$ I want to know how this can be expanded in ...
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0answers
11 views

Spin coefficient transformation for null rotation with $l$ fixed [closed]

In Newmann-Penrose formalism, a Null rotation with $l$ fixed is $$l^a->l^a\\ n^a-> n^a + \bar{c}m^a + c\bar{m}^a+c\bar{c}l^a \\ m^a-> m^a+cl^a \\ \bar{m}^a-> \bar{m}^a+\bar{c}l^a $$ Using ...
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2answers
81 views

Understanding Tensor-operations, covariance, contravariance, … in the context of Special Relativity

I'm currently learning about special relativity but I'm having a really hard time grasping the Tensor-operations. Let's take the Minkowski scalar product of 2 four-vectors: $$\pmb U . \pmb V = ...
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57 views

Cauchy stress tensor for a spherically symmetric problem [closed]

Given a sperically symmetric problem, I am asked to show that its Cauchy stress tensor, in spherical coordinates will assume the form: ...
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1answer
67 views

What exactly does the Kretschmann scalar implies and how does it work?

From the General Relativity class lectures I understood that this particular invariant, the Kretschmann scalar namely $$R_{\mu\nu\lambda\rho} R^{\mu\nu\lambda\rho}$$ is really important because, ...
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1answer
59 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$? [closed]

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where $R_ab$ ...
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0answers
37 views

What is the significance of 'energy ellipsoid'?

Well, today I was reading Tensors by Feynman in his lectures, where he introduced the concept of 'energy ellipsoid'. This is the following excerpt: [...] the polarization tensor can be measured ...
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2answers
77 views

Gradient of vector dot tensor dot vector [closed]

Im new to tensor notation. How would one take the gradient of the expression below? $$\nabla (\vec{r}\dot{}A\dot{}\vec{r})$$ $A$ is a 3 by 3 symmetric tensor independent of $\vec{r}$.
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1answer
62 views

Why is not the D'Alembert operator a scalar?

I am taking a course on classical electrodynamics and my professor has defined the D'Alembert operator to me as: $$\square=\eta^{\mu \nu} \partial_{\mu} \partial_{\nu}$$ I have been operating using ...
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0answers
56 views

How do I decide when to use raised/lowered indices when calculating the amplitude of a Feynman diagram?

I am learning the Feynman rules for QCD. The book I am reading tells me that gluon propagators contribute a factor of $$\frac{-ig_{\mu\nu}\delta^{\alpha\beta}}{q^2}$$ However, in one of the ...
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42 views

How is the Von Mises stress used in 1D or 2D?

The Von Mises Stress is given by: $\sigma_{VM} = \sqrt{\frac{3}{2}\boldsymbol{\sigma}^{'}:\boldsymbol{\sigma}^{'}}$ My understanding is that the $\frac{3}{2}$ is included to ensure that the von mises ...
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1answer
77 views

What is the difference between scalar and vector mesons?

My understanding is that vectors and pseudooscalars change sign under parity operation and pseudovectors and scalars do not. However, I don't understand what the difference between a vector and ...
0
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1answer
55 views

Possible inconsistency of mixed index tensor notation

I am posting this here, because in my experience, this sort of thing exists in physics-related works only. Given a local frame $\{e_{(i)}\}$ on some $n$-dimensional manifold $M$, and given a local ...
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1answer
40 views

physical meaning of major symmetry of the stiffness tensor

What happens if a stiffness tensor does not have the "major symmetry" $C_{ijkl}=C_{klij}$? Background: In linear elasticity (generalising Hooke's law from a spring to a continuous medium), the ...
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1answer
44 views

Notation: tetrad indices

I am trying to understand the meaning of upper and lower indices as used in the Newman-Penrose formalism. The tetrad is $\lbrace l^{a},n^{a},m^{a},\overline{m}^{a}\rbrace$, where the upper index ...
2
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1answer
210 views

Riemann tensor with 2nd and last indice the same will vanish?

I calculated that Riemann tensors are antisymmetric with respect to 2nd and last indice,as the symmetry properities of $R_{\rho\nu\sigma\mu}$ goes. $$R^{\omega}_{\ \ \ ...
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1answer
56 views

Is it right to write $\varepsilon_{ijk} \delta_{jl}=\varepsilon_{ilk}$? (indices notation)

Consider the $l$ component of vector position $\vec{r}$, $r_l$, and the $i$ component of angular momentum $\vec{L}$, $L_i$. We have that $$L_i=[r\times p]_{i}=\varepsilon_{ijk}r_jp_k$$ ...
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1answer
47 views

Show that $R_{\mu\nu}=C g_{\mu\nu}$ from the vacuum Einstein equation with a nonzero $\Lambda$ [closed]

If I begin with the vacuum field equation with a nonzero cosmological constant: $$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=0$$ How can I show that $$R_{\mu\nu}= ...
2
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1answer
70 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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0answers
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 ...
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1answer
111 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 ...
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0answers
53 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
30 views

Topics required before studying Tensor Calculus [closed]

Which topics do I need to know before going through tensor calculus?
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0answers
53 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 ...
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1answer
104 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
99 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
108 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
63 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
43 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}} ...
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2answers
64 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
83 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 ...
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2answers
58 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 ...
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3answers
116 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
24 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, ...
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1answer
136 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. ...