The tensors tag has no wiki summary.
3
votes
2answers
110 views
What's the idea behind the Riemann curvature tensor?
The Riemann curvature tensor can be expressed using the Christoffel symbols like this:
$R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj}
- \partial_l\Gamma^m{}_{kj}
+ \Gamma^m{}_{ki}\Gamma^i{}_{lj}
...
2
votes
1answer
63 views
Spin tensor and Lorentz group operator in bispinor case
For infinisesimal bispinor transformations we have
$$
\delta \Psi = \frac{1}{2}\omega^{\mu \nu}\eta_{\mu \nu}\Psi , \quad \delta \bar {\Psi} = -\frac{1}{2}\omega^{\mu \nu}\bar {\Psi}\eta_{\mu \nu}, ...
0
votes
0answers
25 views
Density matrix and irreducible tensor operators
I'm reading those lecture notes on atomic physics. Yesterday I posed a question on reducible tensors, and today I have a question on their relation to the density matrix.
If there's any information ...
4
votes
1answer
70 views
Riemann tensor notation and Christoffel symbol notation
In paper by Barnich and Brandt Covariant theory of asymptotic symmetries,
conservation laws and central charges they defined the Riemann tensor like this:
$$R_{\rho\mu\nu}^{\quad \ \ ...
2
votes
3answers
80 views
What does “transform among themselves” mean?
I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context:
An arbitrary rotation of the ...
1
vote
1answer
57 views
Transformation rule of a partial derivative
We know the following transformation rule:
$$ \partial'_b = \frac{\partial}{\partial x'^b} = \frac{\partial x^c}{\partial x'^b} \, \frac{\partial}{\partial x^c} = \frac{\partial x^c}{\partial x'^b} ...
2
votes
2answers
114 views
Notation for anti-symmetric part of a tensor
I know that
$A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$
But how can write $E_{[a} F_{bc]}$ like the above?
Can you provide a reference where this notational matter is discussed?
2
votes
1answer
101 views
Weight of a tensor density
Is there any freedom in choosing the weight of a tensor density?
I have seen in some papers that they introduce a tensor density made from metric with a special weight.
There is a tensor density with ...
15
votes
2answers
288 views
In relativity, can/should every measurement be reduced to measuring a scalar?
Different authors seem to attach different levels of importance to keeping track of the exact tensor valences of various physical quantities. In the strict-Catholic-school-nun camp, we have Burke ...
1
vote
0answers
47 views
Direct sum of the spinors and EM field tensor
EM field tensor refer to the direct sum of $(1, 0), (0, 1)$ spinor representation of the Lorentz group. How to show it?
Each of these spinor representations corresponds to the symmetrical spinor ...
2
votes
1answer
115 views
Riemann tensor in 2d and 3d
Ok so I seem to be missing something here.
I know that the number of independent coefficients of the Riemann tensor is $\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given ...
2
votes
2answers
110 views
Coordinate Transformation of Scalar Fields in QFT
By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct?
...
1
vote
2answers
43 views
How would one show that a nonabelian field strength tensor transforms in a certain way under a local gauge transformation?
How would one show that the nonabelian ${F_{\mu\nu}}$ field strength tensor transforms as ${F_{\mu\nu}\to F_{\mu\nu}^{\prime}=UF_{\mu\nu}U^{-1}}$ under a local gauge transformation? Rather than going ...
1
vote
2answers
64 views
How do you show from the index notation that the change of frame formula for a metric must involve the transpose?
Let $x^\mu$ and $x^{'\mu}$ be two coordinate systems related by $$dx^{'\mu}~=~S^\mu{}_\nu~ dx^\mu.$$ In index notation the metric in both systems are related by: ...
1
vote
1answer
81 views
Weyl & Riemann curvature tensors and gravitational “physical” quantities in Einstein vacuum equations
If we look at the Einstein vacuum equations, that is without matter (there is the possibility or curvature without matter), for instance we may consider gravitational waves. The question is: Is there ...
1
vote
1answer
98 views
S. Weinberg, “The Quantum theory of fields: Foundations” (1995), Eq. 2.4.8
Unfortunately I'm struggling to understand how do we get eq. (2.4.8) from eq. (2.4.7), p. 60; namely how $(\Lambda \omega \Lambda^{-1} a)_\mu P^\mu$ is transformed into ...
3
votes
1answer
70 views
Type/Valence of the stress tensor
In classical continuum mechanics, the stress tensor is said to be of type/valence (1,1) and I do not see why.
If I am correct, its maps a vector $n$ defined in $\mathbb{R}^3$ (which is the normal to ...
6
votes
0answers
123 views
Introduction to spinors in physics, and their relation to representations
First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
1
vote
1answer
78 views
Interpretation of the off-diagonal terms of the conductivity tensor
Say we have the electrical conductivity tensor expressed as a 3x3 matrix. I've seen that if it's cubic material then the conductivity tensor reduces to just the diagonal terms and these are equal, ...
0
votes
0answers
14 views
How to calculate independent componets of tensor for crystal symmetry?
I have a question as title says.
For a formula $j_i = \beta_{ijl} e_j e_l^*$, where $i$, $j$, $k$ are coordinate and $\beta_{ijk}$ is the tensor, if I know the tensor should be satisfied for crystal ...
1
vote
0answers
97 views
Eigenvalues of the square of Pauli-Lubanski operator
Let's have Pauli-Lunanski 4-operator:
$$
\hat {W}^{\nu} = \frac{1}{2}\varepsilon^{\nu \alpha \beta \gamma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma},
$$
which easy transforms to
$$
\hat {W}^{\nu} = ...
-1
votes
1answer
112 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 ...
17
votes
5answers
471 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
votes
2answers
104 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 ...
4
votes
2answers
114 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
117 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, ...
2
votes
0answers
202 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
145 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 ...
-1
votes
2answers
277 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
votes
1answer
53 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 ...
0
votes
1answer
98 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 ...
10
votes
1answer
246 views
Uniqueness of Riemann curvature tensor
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
60 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 :
$S$ ...
1
vote
1answer
104 views
Electromagnetic Tensor in Cylindrical Coordinates
I understand that the Electromagnetic Tensor is given by
$$F^{\mu\nu}\mapsto\begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\
E_{x} & 0 & -B_{z} & B_{y}\\
E_{y} & B_{z} & ...
5
votes
5answers
159 views
Tensors and rotations
All the tensors that I have studied so far have always appeared with some kind of rotation. For example, spherical tensors rotate as spherical harmonics, tensors in the context of special relativity ...
5
votes
2answers
466 views
What is the physical significance of the off-diagonal moment of inertia matrix elements?
The tensor of moment of inertia contains six off-diagonal matrix elements, which vanishes if we choose the principle axis of the rotating rigid body and the components of the angular momentum vector ...
4
votes
0answers
181 views
Tensor equations in General Relativity
In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent.
Question: why is this true?
I'm looking for a ...
1
vote
1answer
111 views
Arbitrary tensor covariant derivative
what are the rules for performing covariant derivatives on tensors of arbitrary rank?
I found a few examples of Tensor derivatives:
$$\nabla_{c} T^a {}_{b} = \partial_{c}T^a {}_{b}+ \Gamma^a{}_{cd} ...
3
votes
0answers
96 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 ...
0
votes
1answer
78 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
115 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 ...
3
votes
1answer
143 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
votes
0answers
77 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
338 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 $(-, +, +, +)$ :
...
2
votes
1answer
111 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 ...
1
vote
0answers
80 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
39 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 $\Gamma_{00}^\lambda$:-
$$
\Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left(
\frac{\partial ...
2
votes
1answer
200 views
Ricci identity/Riemann curvature tensor and covectors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
219 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
58 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} ...
