The tensor-calculus tag has no wiki summary.
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0answers
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Superfields and the Inconsistency of regularization by dimensional reduction
Question:
How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)?
Background and some references:
...
15
votes
3answers
2k views
What is the physical meaning of the connection and the curvature tensor?
Regarding general relativity:
What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)?
What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
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 ...
13
votes
3answers
1k views
Irreducible tensors concept
This might be a little naive question, but I am having difficulty grasping the concept of irreducible tensors. Particularly, why do we decompose tensors into symmetric and anti-symmetric parts? I have ...
13
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4answers
1k views
History of Electromagnetic Field Tensor
I'm curious to learn how people discovered that electric and magnetic fields could be nicely put into one simple tensor.
It's clear that the tensor provides many beautiful simplifications to the ...
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?
8
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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 ...
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 ...
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 ...
6
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2answers
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What is the covariant derivative in mathematician's language?
In mathematics, we talk about tangent vectors and cotangent vectors on a manifold at each point, and vector fields and cotangent vector fields (also known as differential one-forms). When we talk ...
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 ...
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.
...
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. ...
4
votes
1answer
311 views
How do we know the geodesic is a minimum?
The geodesic equation is derived from the Euler-Lagrange equation, which (as I understand it) is a necessary but not sufficient condition to ensure that the geodesic is a minimum.
The introductory GR ...
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 ...
4
votes
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 ...
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 ...
3
votes
2answers
363 views
What is the mathematical formulation for buckling?
Argument: Buckling is an engineering concept that can only be applied to thin columns with compressive loading.
(Is it possible to) Prove the above sentence right or wrong with mathematical ...
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
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 ...
3
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0answers
276 views
I lost a factor of two in the electromagnetic field tensor
I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-)
...
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
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5answers
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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 ...
2
votes
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
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 ...
2
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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 ...
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 ...
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} ...
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 $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
174 views
Is there symmetry in 2d stress tensor in linear elastic fracture mechanics?
Assumptions:
Cross terms in strain tensor are defined as equal $\varepsilon_{xy} = \varepsilon_{yx}$.
pure mode I crack.
Far from crack tip, material is purely elastic and we are way below
yield ...
2
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0answers
43 views
Special case of the hodge decomposition theorem [migrated]
I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$.
any vector can be written as the ...
2
votes
0answers
128 views
Using the area element in derivation of geodesic
In the derivation of the geodesic, one starts with the integral of the line element (arclength):
$$L(C)=\int_{\tau_1}^{\tau_2}d\tau\sqrt{g_{\mu \nu}\dot{x}^{\mu} \dot{x}^{\nu}}$$
The integrand is ...
1
vote
2answers
250 views
Question with Einstein notation
Let’s consider this equation for a scalar quantity $f$ as a function of a 3D vector $a$ as:
$$ 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 ...
1
vote
2answers
430 views
What kind of invariants are proper time and proper length?
Under the Lorentz transformations, quantities are classed as four-vectors, Lorentz scalars etc depending upon how their measurement in one coordinate system transforms as a measurement in another ...
1
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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
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}$.
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.
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 ...
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 ...
1
vote
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
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 ...
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 ...
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.$$
The metric is given by $ds^2=h_1^2 dx_1^2+h_2^2 ...
1
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1answer
175 views
Tensor Introduction
I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, ...
1
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1answer
203 views
Tensors: relations between physics and linear algebra
In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as
$$
...
1
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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
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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 $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$
...
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
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 ...
