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1answer
121 views

S. Weinberg, “The Quantum theory of fields: Foundations” (1995), Eq. 2.4.8

Unfortunately I'm struggling to understand how do we get eq. (2.4.8) from eq. (2.4.7), p. 60; namely how $(\Lambda \omega \Lambda^{-1} a)_\mu P^\mu$ is transformed into ...
3
votes
1answer
139 views

Type/Valence of the stress tensor

In classical continuum mechanics, the stress tensor is said to be of type/valence (1,1) and I do not see why. If I am correct, its maps a vector $n$ defined in $\mathbb{R}^3$ (which is the normal to ...
12
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2answers
700 views

Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
1
vote
1answer
435 views

Interpretation of the off-diagonal terms of the conductivity tensor

Say we have the electrical conductivity tensor expressed as a 3x3 matrix. I've seen that if it's cubic material then the conductivity tensor reduces to just the diagonal terms and these are equal, ...
-1
votes
1answer
262 views

Metric tensor in General Relativity or otherwise [closed]

What is the metric tensor? How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices? How it relates to distance function (metric) and angles? How ...
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7answers
1k views

Tensor Operators

Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ...
2
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2answers
152 views

Tensors in general relativity

This is a question on the nitty-gritty bits of general relativity. Would anybody mind teaching me how to work these indices? Definitions: Throughout the following, repeated indices are to be summed ...
5
votes
2answers
814 views

Irreducible Representations of SO(n) tensors

My interest is purely in $\text{SO}(n)$ tensors and how one works out their irrep decomposition. For instance, for rank 2 tensors we simply split into an antisymmetric part, a traceless symmetric part ...
4
votes
1answer
558 views

Irreducible decomposition of higher order tensors

I am familiar with the notion of irreps. My question refers simply to tensor representations (not tensor products of representations) and how can we decompose them into irreducible parts? For example, ...
3
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0answers
493 views

The connection between classical and quantum spins

I have two questions, which are connected with each other. The first question. In a classical relativistic (SRT) case for one particle can be defined (in a reason of "antisymmetric" nature of ...
3
votes
2answers
459 views

Tensor Product of Hilbert spaces

This question is regarding a definition of Tensor product of Hilbert spaces that I found in Wald's book on QFT in curved space time. Let's first get some notation straight. Let $(V,+,*)$ denote a set ...
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2answers
5k views

Center of a mass of a hemisphere [closed]

How can I show that position vector of the center of a mass of a hemisphere is $(0,0,\frac{3a}{8})$ where $a$ is radius of a hemisphere, $x$ and $y$ axis are laying on the base and $z$-axis is ...
0
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1answer
68 views

Why is it suffice to show Tensorial identity on a tensor composed of two vectors?

I've encounter many proves of Tensorail identity that begin with assuming our tensor can be written in form of: $T^{\alpha\beta}=u^{\alpha}v^{\beta}$ . As helpful is it might be, I'm not sure if its ...
1
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1answer
421 views

Symmetry of stress-energy tensor

Why in the general case of classical field theory canonical stress-energy tensor doesn't have symmetry of the permutation of the indices? For explanation, let's have a "derivation" of an expression ...
13
votes
1answer
504 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
2
votes
1answer
102 views

General expression of the redshift: explanation?

In some papers, authors put the following formula for the cosmological redshift $z$ : $1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$ where : $S$ ...
2
votes
1answer
407 views

Electromagnetic Tensor in Cylindrical Coordinates

I understand that the Electromagnetic Tensor is given by $$F^{\mu\nu}\mapsto\begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & ...
5
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5answers
285 views

Tensors and rotations

All the tensors that I have studied so far have always appeared with some kind of rotation. For example, spherical tensors rotate as spherical harmonics, tensors in the context of special relativity ...
7
votes
2answers
381 views

Tensor equations in General Relativity

In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent. Question: why is this true? I'm looking for a ...
1
vote
1answer
416 views

Arbitrary tensor covariant derivative

what are the rules for performing covariant derivatives on tensors of arbitrary rank? I found a few examples of Tensor derivatives: $$\nabla_{c} T^a {}_{b} = \partial_{c}T^a {}_{b}+ \Gamma^a{}_{cd} ...
10
votes
2answers
889 views

Symmetry of the $3\times 3$ Cauchy Stress Tensor

When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a ...
3
votes
0answers
144 views

Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
1
vote
1answer
124 views

Why elastic materials are discribed by tensors?

I am starting to read about elasticity of thin surfaces and I don't understand why tensors play such a major part? What are the tensors describing about the material? And just to clarify - Is there ...
1
vote
1answer
497 views

What is the Lorentz tensor with a superscript and subscript index?

I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar ...
5
votes
1answer
405 views

When a variation of a tensor is not a tensor?

In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a ...
2
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0answers
182 views

Solving the equation of relativistic motion

How does one solve the tensor differential equation for the relativistic motion of a partilcle of charge $e$ and mass $m$, with 4-momentum $p^a$ and electromagnetic field tensor $F_{ab}$ of a constant ...
2
votes
1answer
2k views

Stress energy tensor of a perfect fluid and four-velocity

In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units). We consider a metric $g_{\mu\nu}$ with a signature $(-, +, +, +)$ : ...
3
votes
1answer
249 views

Sign crazyness on the stress energy tensor?

I would like to know on what depends the sign of the stress energy tensor in the following formula : $T_{\mu\nu}=\pm(\rho c^2+P)u_{\mu}u_{\nu} \pm P g_{\mu\nu}$ In my case the metric is equal to ...
2
votes
0answers
138 views

Lecture Notes confusion: Constructing the Einstein Equation

This question is on the construction of the Einstein Field Equation. In my notes, it is said that The most general form of the Ricci tensor $R_{ab}$ is $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ ...
0
votes
1answer
50 views

Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?

More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:- $$ \Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left( \frac{\partial ...
2
votes
1answer
778 views

Ricci identity/Riemann curvature tensor and covectors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
314 views

Contracting the Riemann tensor issues, p540 hobson

I am stuck trying to work through something on p540 in Hobson (General Relativity: An Introduction for Physicists), one is supposed to use the variation of the full Riemann tensor and then contract it ...
2
votes
1answer
140 views

Non-diagonal elements when switching metric signature?

Considering a metric tensor with the signature $(-,+,+,+)$: $g_{\mu\nu}= \begin{pmatrix} -c^2 & g_{01} & g_{02} & g_{03}\\ g_{10} & a^2 & g_{12} & g_{13}\\ g_{20} & g_{21} ...
1
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1answer
232 views

Derivative of covariant EM tensor

I cannot seem to prove that the derivative of the duel tensor = 0. $$ \frac{1}{2}\partial_{\alpha}\epsilon^{\alpha \beta \gamma \delta} F_{\gamma \delta} = 0. $$ Writing this out I get (for some ...
0
votes
1answer
161 views

Tensor manipulation

Having a bit of trouble applying what I know about tensor manipulation, given, $T^{\mu \nu} = \left( g^{\mu \nu} - \frac{p^\mu n^\nu + p^\nu n^\mu}{p \cdot n} \right)$, I need to compute quantities ...
2
votes
2answers
145 views

When does $T^{ij} = T_{ij}$?

Suppose we have some tensor with components $T^{ij}$. Then suppose that we also have $T_{ij}$. When would $T^{ij}T_{ij} = (T^{ij})^2 = (T_{ij})^2$?
2
votes
1answer
94 views

Tensor perturbation inflation

During inflation the metric is de-Sitter so $dt^2-d\underline{X}^2 $. I know that the eqn.motion governing GW's from inflation (tensor perturbations) is $$2H\dot{h}+\ddot{h}-\nabla^{2}_{i}h~=~0,$$ ...
1
vote
0answers
128 views

Einstein +Maxwell 's tensor

Why is it true that we can deduce that Einstein's GR equations coupled with Maxwell's EM equations may be written in the form $$R_{ij}=C(F_{ik}F_j^{\,\,k}-{1\over 4}g_{ij}F_{mn}F^{mn})$$ without ...
2
votes
3answers
1k views

Maxwell Stress Tensor in the absence of a magnetic field

I'm having some trouble calculating the stress tensor in the case of a static electric field without a magnetic field. Following the derivation on Wikipedia, Start with Lorentz force: $$\mathbf{F} = ...
1
vote
1answer
229 views

Relationship between a formal vector derivative and time evolution of an operator

I'm an undergraduate in physics, with all the lack of knowledge inherent in that. In two of my classes, my professors introduced two equations which look eerily similar. The first, from general ...
0
votes
1answer
89 views

Two General Relativity questions

Hi When contracting $T^{\mu \nu}$ with $ g_{\mu \nu}$ does one get $T^{\mu \nu}_{\mu \nu} = T$? is the metric tensor already a sum over its component, so it is effectively a trace of a matrix with ...
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3answers
340 views

Is “entanglement” unique to quantum systems?

My text shows (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with $k$ and $l$ "bits of information", respectively, requires $kl$ bits to fully describe it. ...
1
vote
0answers
78 views

Equivalence of simple formulations of qubit entanglement

I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement. One definition states that (1) the bits of a ...
5
votes
1answer
312 views

Do partial derivatives commute on tensors?

Do partial derivatives commute on tensors? For example, is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
1
vote
1answer
398 views

Calculating electromagnetic invariant in matrix form

I'm kind of confused. I want to calculate the electromagnetic invariant $I := F^{\mu\nu}F_{\mu\nu} $, but I'm not sure what is the easiest way to do so. So, I was trying to do it in matrix form, i.e. ...
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3answers
7k views

What does this quote about the four dimensional divergence of an antisymmetric tensor mean?

In the beginning, God said that the four dimensional divergence of an antisymmetric second rank tensor equals zero and there was light. Can someone explain what is the meaning of this quote by ...
2
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1answer
197 views

Write $\epsilon_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta}$ as a total divergence $\partial_\mu G^\mu$

I have the following homework problem in theoretical electrodynamics: Show that the gauge invariant Lagrange density $\epsilon_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta}$ can be written as a ...
3
votes
1answer
675 views

Difference between $\partial$ and $\nabla$ in general relativity

I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\partial_\mu$ which would have the plain partial ...
2
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0answers
74 views

Expectation of 2-form field $B_{MN}$ in string theory

In the context of string theory, in particular when we are dealing with a low energy effective action, if we have an effective action of the form, $$S_{\mathrm{eff}} \sim S^{(0)} + \alpha S^{(1)} + ...
2
votes
1answer
194 views

Construction of the supersymmetric Faraday tensor

When I first learned gauge theories in my introductory quantum field theory course, I was taught that the Faraday (field-strength) tensor can be constructed by computing the commutator of the ...