The tensors tag has no wiki summary.
8
votes
3answers
334 views
Why do Maxwell's equations contain each of a scalar, vector, pseudovector and pseudoscalar equation?
Maxwell's equations, in differential form, are
$$\vec\nabla\cdot\vec{E}=~\rho/\epsilon_0,$$
$$\vec\nabla\times\vec B~=~\mu_0\vec J+\epsilon_0\mu_0\partial\vec E/\partial t,$$
$$\vec\nabla\times\vec ...
7
votes
6answers
862 views
What is a tensor?
I have a pretty good knowledge of physics but couldn't understand what a tensor is. I just couldn't understand it, and the wiki page is very hard to understand as well. Can someone refer me to a good ...
7
votes
1answer
257 views
Diffeomorphisms, Isometries And General Relativity
Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while.
Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
6
votes
3answers
133 views
From Manifold to Manifold?
Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...
6
votes
3answers
183 views
Is “entanglement” unique to quantum systems?
My text shows (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with $k$ and $l$ "bits of information", respectively, requires $kl$ bits to fully describe it. ...
6
votes
3answers
566 views
What is the difference between a spinor and a vector or a tensor?
Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
5
votes
1answer
141 views
Confused about indices of the Ricci tensor
In an intro to GR book the Ricci tensor is given as:
$$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu ...
4
votes
1answer
129 views
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?
4
votes
0answers
67 views
Shape of the state space under different tensor products
I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this).
Recall: In a ...
3
votes
1answer
150 views
Can one raise indices on covariant derivative and products thereof?
Can the following be true?
$g^{\sigma\rho}\nabla_{\rho}\nabla_{\mu} = \nabla^{\sigma}\nabla_{\mu}$
$g^{\sigma\rho}\nabla_{\nu}\nabla_{\sigma} = \nabla_{\nu}\nabla^{\rho}$
...
3
votes
2answers
135 views
Element of area in 4-dimensional space-time
How would you proof that
$$ \mathrm {Tr} (\mathbf{S\cdot \bar S })=0$$
where $\mathbf S$ is an element of area delimited for the 4-vectors $\mathbf u$ and $\mathbf v$ given by
$$S^{\alpha \beta}\equiv ...
3
votes
1answer
91 views
When a variation of a tensor is not a tensor?
In a comment about variation of metric tensor it was shown that
$$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$
which is contrary to the usual rule of lowering indeces of a ...
3
votes
2answers
192 views
What are $\partial_t$ and $\partial^\mu$?
I'm reading the Wikipedia page for the Dirac equation:
$\rho=\phi^*\phi\,$
......
$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$
with the conservation of probability ...
3
votes
1answer
164 views
Symmetrical Spinors and Symmetrical Tensors
In Quantum Electrodynamics by Landau and Lifshiz there is the following:
The correspondence between the spinor $\zeta^{\alpha \dot{\beta}}$ and
the 4-vector is a particular case of a general ...
3
votes
0answers
71 views
Curvature and spacetime
Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
3
votes
0answers
55 views
Expectation of 2-form field $B_{MN}$ in string theory
In the context of string theory, in particular when we're dealing with a low energy effective action, if we have an effective action of the form:
$$S_{eff} \sim S^{(0)} + \alpha S^{(1)} + (\alpha)^2 ...
2
votes
3answers
812 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
votes
1answer
178 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
votes
2answers
136 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
92 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
63 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
1answer
41 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
110 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 ...
2
votes
1answer
118 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 ...
2
votes
1answer
376 views
Levi Civita Symbol and contravariance vs covariance
I have a question regarding the Levi-Civita symbol and contravariance vs covariance. Some of this was asked in a previous post, but I think I need more clarification.
Consider the magnetic field:
...
2
votes
1answer
123 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 $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
58 views
Tensor perturbation inflation
During inflation the metric is de-Sitter so $dt^2-d\underline{X}^2 $.
I know that the eqn.motion governing GW's from inflation (tensor perturbations) is
$$2H\dot{h}+\ddot{h}-\nabla^{2}_{i}h~=~0,$$ ...
2
votes
3answers
264 views
Maxwell Stress Tensor in the absence of a magnetic field
I'm having some trouble calculating the stress tensor in the case of a static electric field without a magnetic field. Following the derivation on Wikipedia,
Start with Lorentz force:
$$\mathbf{F} = ...
2
votes
1answer
140 views
How quantum field transforms in case of some particular spin
Except when a particle is spin-0, field of all particles transforms when frame of reference is changed, and this defines what spin is. The question is, specifically how does the quantum field ...
2
votes
0answers
96 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 ...
2
votes
0answers
109 views
Stiffness tensor
Let's have a stiffness tensor:
$$
a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}.
$$
It has a 21 independent components for an anisotropic body.
How does body symmetry (cubic, hexagonal ...
2
votes
0answers
117 views
How do I extend the Lorentz transformation metric to dimensions>4?
How do I extend the general Lorentz transformation matrix (not just a boost along an axis, but in directions where the dx1/dt, dx2/dt, dx3/dt, components are all not zero. For eg. as on the Wikipedia ...
1
vote
1answer
60 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:
$$\nabla_{c} T^a {}_{b} = \partial_{c}T^a {}_{b}+ \Gamma^a{}_{cd} ...
1
vote
1answer
92 views
Relationship between a formal vector derivative and time evolution of an operator
I'm an undergraduate in physics, with all the lack of knowledge inherent in that. In two of my classes, my professors introduced two equations which look eerily similar. The first, from general ...
1
vote
1answer
281 views
Levi-Civita symbol in Euclidean space
Suppose a component of tensor field is described by $B^k=\varepsilon^{kij} \phi_{ij}$. If we define $B^k$ in an Euclidean space then does the rising or lowering of the indices of the Levi-Civita ...
1
vote
1answer
39 views
What is the Lorentz tensor with a superscript and subscript index?
I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar ...
1
vote
2answers
367 views
Tensor product notation [closed]
In the image there is a tensor product:
$$F_{\mu\nu}F^{\mu\nu}=2(B^2-\frac{E^2}{c^2})$$
It's about how this operation on the co- and contravariant field strength tensors can give one of the ...
1
vote
3answers
308 views
Understanding Tensors
I don't seem to be able to visualize tensors. I am reading The Morgan Kauffman Game Physics Engine Development and he uses tensors to represent aerodynamics but he doesn't explain them so I am not ...
1
vote
1answer
105 views
Calculating electromagnetic invariant in matrix form
I'm kind of confused. I want to calculate the electromagnetic invariant $I := F^{\mu\nu}F_{\mu\nu} $, but I'm not sure what is the easiest way to do so. So, I was trying to do it in matrix form, i.e. ...
1
vote
1answer
170 views
Tensor Introduction
I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, ...
1
vote
1answer
193 views
Tensors: relations between physics and linear algebra
In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as
$$
...
1
vote
2answers
113 views
What should I call an n>4 dimensional Minkowski metric?
I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be ...
1
vote
0answers
43 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 ...
1
vote
0answers
56 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
vote
1answer
166 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 ...
1
vote
1answer
67 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 ...
1
vote
0answers
74 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 ...
1
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0answers
54 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 ...
1
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0answers
155 views
Representing a polarization vector for light as a 'manifold of two state'
Explain me these projections please
Context: I was reading a paper (Phys. Rev. A 68, 052307) which involved mesoscopic coherent states of light. There, in order to calculate the uncertainty of a ...
0
votes
2answers
161 views
When and how do you represent a two body state as a tensor product?
I have read that in quantum mechanics, compound systems are constructed as tensor products.
But on page 177 of Griffith, for example, a two body wavefunction is introduced as
Psi ...
