The tensor-calculus tag has no wiki summary.
2
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3answers
107 views
On Einstein notation with multiple indices
On Einstein notation with multiple indices: For example, consider the expression:
$$a^{ij} b_{ij}.$$
Does the notation signify,
$$a^{00} b_{00} + a^{01} b_{01} + a^{02} b_{02} + ... $$
i.e. you ...
2
votes
0answers
43 views
Special case of the hodge decomposition theorem [migrated]
I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$.
any vector can be written as the ...
2
votes
5answers
98 views
Why define four-vectors to be quantities that transform only like the position vector transforms?
A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation ...
1
vote
1answer
33 views
Vector analysis in curvilinear coordinates using the metric tensor
In Weinberg's Gravitation,
the formula for the volume element in curviliniar coordinates is given by
$$dV=h_1 h_2 h_3 dx^1 dx^2 dx^3.$$
The metric is given by $ds^2=h_1^2 dx_1^2+h_2^2 ...
3
votes
0answers
72 views
Uniqueness of the vector in $\mathbb{R}^n$ specified by the curl, divergence and the normal component [migrated]
If I know the curl, and divergence of a n-component vector in a region, and its normal component around its boundary, is the vector uniquely specified? If yes, how do I prove it? Also, is there a ...
1
vote
2answers
56 views
Kronecker delta confusion
I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same):
...
1
vote
1answer
51 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
82 views
Derivation of the volume element (which uses the metric tensor)?
I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be ...
3
votes
1answer
100 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
1answer
126 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
70 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
62 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
Contraction of the metric tensor
This is perhaps a simple tensor calculus problem -- but I just can't see why...
I have notes (in GR) that contains a proof of the statement
In space of constant sectional curvature, $K$ is ...
0
votes
1answer
36 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 $\Gamma_{00}^\lambda$:-
$$
\Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left(
\frac{\partial ...
2
votes
1answer
138 views
Ricci identity/Riemann curvature tensor and covectors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
4
votes
2answers
117 views
Difference between slanted indices on a tensor
In my class, there is no distinction made between,
$$
C_{ab}{}^{b}
$$
and
$$
C^{b}{}_{ab}.
$$
All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. ...
0
votes
0answers
110 views
covarient derivative of electromagnetic field tensor
I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is Divergence-less directly(Without using general lie derivative method which can prove any energy momentum ...
1
vote
1answer
99 views
Pauli matrices and the Levi-Civita symbol
This is just a quick question. I would figure this out myself if I wouldn't have an exam about this tomorrow.
I am working on the non-relativistic approximation of the Dirac equation for an electron ...
2
votes
1answer
44 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} ...
0
votes
2answers
118 views
Weinberg's proof that Levi-Civita Symbol is a tensor
In Gravitation and Cosmology, S.Weinberg states the following:
$$\Lambda_{\epsilon}^{\alpha}\Lambda_{\zeta}^{\beta}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa ...
0
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0answers
47 views
Vector identities equivalence under different coordinates
I've learned to represent curl, rot and Laplacian in the general form using scaling factors, Levi Civita symbol and delta.
I was asked to prove some general identities in vector calculus.
I was ...
2
votes
2answers
120 views
Differential Forms and Densities
I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a ...
1
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1answer
95 views
Question about contraction with metric tensor
I just starting to study GR and I could not prove the following: if I have to tensors $T_{\mu\nu}$ and $Q_{\mu\nu}$ such that $T_{\mu\nu}=Q_{\mu\nu}$, why can I multiply both sides of the equation by ...
1
vote
2answers
173 views
What is the Riemann curvature tensor contracted with the metric tensor?
Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example
$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
0
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3answers
222 views
Relativistic basic question - four vector, Lorentz matrix
I have heard relativistics only very compressed during my student time. Now I looked up the definitions again and a question comes into my mind:
A contravariant vector is transformed like this: ...
1
vote
1answer
113 views
Covariant derivative with upper index
I just need clarification, that is, to see that I'm doing the right thing.
When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have ...
7
votes
3answers
714 views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu ...
1
vote
1answer
175 views
Tensor Introduction
I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, ...
9
votes
2answers
351 views
Can any rank tensor be decomposed into symmetric and anti-symmetric parts?
I know that rank 2 tensors can be decomposed as such. But I would like to know if this is possible for any rank tensors?
1
vote
1answer
50 views
Testing covariance of an expression?
This is something I've been unsure of for a while but still don't quite get.
How does one tell whether an expression (e.g. the Dirac equation) is covariant or not? I get it for a single tensor, but ...
0
votes
1answer
57 views
Writing a tensor with respect to a particular basis
When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way:
$$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu ...
1
vote
1answer
203 views
Tensors: relations between physics and linear algebra
In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as
$$
...
5
votes
4answers
298 views
Are covariant vectors representable as row vectors and contravariant as column vectors
I would like to know what are the range of validity of the following statement:
Covariant vectors are representable as row vectors. Contravariant
vectors are representable as column vectors.
...
7
votes
3answers
621 views
What does the dual of a tensor mean (e.g. dual stress tensor in relativistic ED)?
I know what the dual of a vector means (as a map to its field), and I am also
aware of of the definition a dual of a tensor as,
$$F^{*ij} = \frac{1}{2} \epsilon^{ijkl} F_{kl}\tag{1}$$
I just don't ...
4
votes
1answer
564 views
Covariant derivative and Leibniz rule
I read the Wikipedia page about the covariant derivative, my main problem is in this part:
http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description
Some of the formulas seem to lead ...
8
votes
6answers
892 views
What is a tensor?
I have a pretty good knowledge of physics but couldn't understand what a tensor is. I just couldn't understand it, and the wiki page is very hard to understand as well. Can someone refer me to a good ...
1
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0answers
104 views
How to integrate twice of this viscous term?
I am reading a paper, and I do not understand why the author said the following term when integrated twice will become,
$\int\limits_\Omega {{\rm{d}}\Omega {{\bf{\psi }}^{\bf{u}}}\cdot\nabla ...
0
votes
1answer
98 views
Positive Permutation Tensor
I have seen that one can make an operator with
$$
L^i=\epsilon^{ijk}x_{j}\partial_{k}
$$
But what if I want to make instead items that are sums, instead of differences. For instance
...
1
vote
1answer
63 views
true three index tensors
is such a tensor, $T_{\alpha\beta\, \gamma}$, possible such that
$$T_{\alpha\beta\, \gamma}=T_{\beta\alpha\, \gamma}=-T_{\alpha\gamma\, \beta}=-T_{\gamma\beta\, \alpha}$$
That is, symmetric under two ...
1
vote
1answer
242 views
metric signature explanation
Can anyone explain what metric signature is?
I have a basic knowledge regarding tensors, btw.
Also, how is it related to fundamental understanding of general relativity?
Thanks.
0
votes
0answers
72 views
How do I write the energy of a constant, uniform 2D charge distribution?
Let's consider a 2D electromagnetic field defined in a square domain $[0,\Lambda]^2$, with periodic boundary conditions, with a constant charge distribution, uniform all over the aforementioned ...
1
vote
2answers
250 views
Question with Einstein notation
Let’s consider this equation for a scalar quantity $f$ as a function of a 3D vector $a$ as:
$$ f(\vec a) = S_{ijkk} a_i a_j $$
where $S$ is a tensor of rank 4. Now, I’m not sure what to make of the ...
0
votes
1answer
1k views
Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?
I have earlier posted the same question here on math stackexchange but without any answer. As the question concerns tensors, I guess that I have come to the right ...
2
votes
1answer
96 views
Question from Schutz's
In q. 22 in page 141, I am asked to show that if $U^{\alpha}\nabla_{\alpha} V^{\beta} = W^{\beta}$, then $U^{\alpha}\nabla_{\alpha}V_{\beta}=W_{\beta}$.
Here's what I have done:
$V_{\beta}=g_{\beta ...
2
votes
1answer
127 views
Are there any clear and expressive plainword sense of metric tensor components?
Can the following groups of components of metric tensor can assigned a clear sense?
https://docs.google.com/drawings/pub?id=1kVqkN1gT-a2fDy2S851l9iQKaMfaatCDo517OSZBHEo&w=467&h=228
I have ...
4
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1answer
59 views
Spatial and polarizing beam splitters in a graphical calculus
Suppose I have four wires, and I tensor product them together
$A \otimes B \otimes C \otimes D$
I pass $A \otimes B$ through a spatial beam splitter
$Spl: A \otimes B \rightarrow A^\prime \otimes ...
14
votes
4answers
2k views
Are matrices and second rank tensors the same thing?
Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:
1-Are matrices and second rank tensors ...
2
votes
1answer
174 views
Is there symmetry in 2d stress tensor in linear elastic fracture mechanics?
Assumptions:
Cross terms in strain tensor are defined as equal $\varepsilon_{xy} = \varepsilon_{yx}$.
pure mode I crack.
Far from crack tip, material is purely elastic and we are way below
yield ...
5
votes
2answers
87 views
Torsion and gauge invariant EM kinetic term
Everytime I hear about adding torsion to GR, something struggles me: how do you create a kinetic term for the electromagnetic field that is still gauge-invariant? One of the consequences of torsion is ...
0
votes
3answers
282 views
Need some basic help with notation and the Christoffel symbols
Apologies in advance if some of the questions below seem overly simple.
In an introductory GR book, I find the following expression for the autoparallel of the affine connection (the upper bound of ...
