I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##.
##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}##
##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...
This proof was in my book.
Tensor product definition according to my book: $$V⊗W=\{f: V^*\times W^*\rightarrow k | \textrm {f is bilinear}\}$$ wher ##V^*## and ##W^*## are the dual spaces for V and W respectively.
I don't understand the step where they say ##(e_i⊗f_j)(φ,ψ) = φ(e_i)ψ(f_j)##...
Can we consider the E and B fields as being irreducible representations under the rotations group SO(3) even though they are part of the same (0,2) tensor? Of course under boosts they transform into each other are not irreducible under this action. I would like to know if there is in some...
In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and
the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector.
i) Calculate the angle between the...
I know that a tensor can be seen as a linear function.
I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.
Hello,
I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it.
∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1)
L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2)
Here is Eq (1) the...
I have this statement:
Find the most general form of the fourth rank isotropic tensor. In order to do so:
- Perform rotations in ## \pi ## around any of the axes. Note that to maintain isotropy conditions some elements must necessarily be null.
- Using rotations in ## \pi / 2 ## analyze the...
There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
Hi,
I've been watching lectures from XylyXylyX on YouTube. I believe they are really great !
One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...
I am now reading this paperhttps://arxiv.org/pdf/gr-qc/0405103.pdf, which is related to the energy condition in wormhole. Nevertheless, I got a problem in Eq.(6), which derives from so-called ANEC in Eq.(2): $$\int^{\lambda2}_{\lambda1}T_{ij}k^{i}k^{j}d\lambda$$
And I apply the worm hole space...
I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors.
What people usually do is
take the covariant derivative of the covector acting on a vector, the result being a scalar
Invoke a product rule to...
Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged.
Hello!
I am struggling with understanding the meaning of "each member is a unit vector":
I can see that N would represent the number of samples, and the pointy bracket represents an...
On pages 42-43 of the book "Tensors: Mathematics of Differential Geometry and Relativity" by Zafar Ahsan (Delhi, 2018), the calculation for the angle between Ai=(1,0,0,0) (the superscript being tensor, not exponent, notation) and Bi=(√2,0,0,(√3)/c), where c is the speed of light, in the...
My attempt at ##g_{\mu \nu}## for (2) was
\begin{pmatrix}
-(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta)
\end{pmatrix}
and the inverse is the reciprocal of the diagonal elements.
For (1) however, I can't even think of how to write the...
Hi, I'm worried I've got a grave misunderstanding. Also, throughout this post, a prime mark (') will indicate the transformed versions of my tensor, coordinates, etc.
I'm going to define a tensor.
$$T^\mu_\nu \partial_\mu \otimes dx^\nu$$
Now I'd like to investigate how the tensor transforms...
Hi,
I'm currently working through a tensor product example for a two qubit system.
For the expression:
$$
\rho_A = \sum_{J=0}^{1}\langle J | \Psi \rangle \langle \Psi | J \rangle
$$
Which has been defined as from going to a global state to a local state.
Here
$$ |\Psi \rangle = |\Psi^+...
Hello
I have been going through the cosmology chapter in Choquet Bruhats GR and Einstein equations and in definition 3.1 of chapter 5 she defines the sectional curvature with a Riemann( X, Y;X, Y) (X and Y two vectors)
I don't understand this notation, regarding the use of the semi colon, is it...
Homework Statement
Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$
where q is a constant vector.
Homework Equations
The Attempt at a Solution
$$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial...
Homework Statement
This isn't exactly a problem but rather a problem in understanding the derivation of the phenomenon, or more precisely, one step in the derivation.
In the following we will consider the EPR pair of two spin ##1/2## particles, where the state can be written as
$$ \vert...
Im trying to calculate the principals moments of inertia (Ixx Iyy Izz) for the inertia tensor by triple integration using cylindrical coordinates in MATLAB.
% Symbolic variables
syms r z theta R h M; % R (Radius) h(height) M(Mass)
% Ixx
unox = int((z^2+(r*sin(theta))^2)*r,z,r,h); % First...
I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates.
A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...
I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are:
\Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y}
knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...
I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...
I know that a vector is a tool to help with quantities that have both a magnitude a direction. At a given point in space, a vector has a particular magnitude and direction and if we take any other direction at the same point we can get a projection of this vector in that direction.
Tensor is a...
Hello!
Does anyone have an idea of how can I obtain information from a band diagram about the directions along which the system conducts best and worst ?
Thank you in advanced! :)
Hello guys!
I have to solve a problem about crystal symmetry, but I am very lost, so I wonder if anyone could guide me.
The problem is the following:
Using semiclassical transport theory the conductivity tensor can be defined as:
σ(k)=e^2·t·v_a(k)·v_b(k)
Where e is the electron charge, t...
Homework Statement
Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds:
## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ##
Homework Equations
## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ##
The...
It is known that vectors change them sing under the influence of parity when ##(x,z,y)## change into ##(-x,-z,-y)##
$$P: y_{i} \rightarrow -y_{i}$$
where ##i=1,2,3##
But what about vectors in Minkowski space? Is it true that
$$P: y_{\mu} \rightarrow -y_{\mu}$$
where ##\mu=0,1,2,3##.
If yes how...