Isotropic Definition and 124 Threads

I would like to ask some questions about an interesting paper that was written back in the late 90's (https://arxiv.org/abs/astro-ph/9701131) There, the authors propose how the universe may evolve from the near future to extremely far time scales Near the end of it (Section VI, D.), they...

Hi, reading "Mechanics" book by Landau-Lifshitz, they derive from spatial homogeneity that the Lagrangian ##L## of a free particle cannot explicitly depend on spatial coordinates ##q## in an inertial frame. However my point is as follows: suppose to consider the Lagrangian ##L= \frac 1 2...

Consider an expansion for the density ##\rho(t,\mathbf{x})## of the form$$\rho(t,\mathbf{x}) = \sum_{l=0}^{\infty} a_{i_1 i_2 \dots i_{\mathscr{l}}}(t,r) \hat{x}_{i_1} \hat{x}_{i_2} \dots \hat{x}_{i_{\mathscr{l}}}$$where ##r = |\mathbf{x}|## and ##\hat{x}_i = x_i/r##. Also, ##a_{i_1 i_2 \dots...

Hi, I was thinking about the following. From a mathematical point of view, SR assumes the following postulate: spacetime is a flat Lorentzian smooth manifold. From the above and a minimal interpretation (i.e. a minimal set of "rules" to define the correspondence between mathematical objects...

Question: Solution first part: Have I done it right? I don't know how to begin with second part since the dielectric is non-lineair, and most formulas like $$ D=\epsilon E$$ and $$P= \epsilon_0 \xhi_e E$$, only apply for lineair dielectrics. What to do?

The question is in the title. I believe the answer is yes.

In an isotropic universe, every observer sees themself as being at the center. But consider 3 observers, A, B, and C who are 5 billion light years apart and all lined-up in a straight line with B at the center. B knows this to be true because A is in one direction and C is in exactly the...

I don't understand how pressure must be constant in all directions to balance out the force? Arent the forces in each direction independent, so that pressure forces in the x direction and y direction and z direction can all be different to each other, as long as they are balanced in that...

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...

At 100m: (a) 0.03315 W/m (b)4166 W Since E is inversely proportional to 1/r^2, then E at 150m is 2.22 V/m. (a) 2.22/377= 0.00654 W/m (b) 4*pi*r^2*Wrad= 1665 W Is this reasoning correct?

Hello, My question is simple. I have read that isotropic biaxial strain does not lower C2 symmetry, but no proof whatsoever was provided. I would like to know if it is actually true and have a solid proof. If someone can provide it, that would be wonderful. But also explaining me how to start...

Hi I am reading Landau's mechanics So in the first chapter page 5 It reads : since space is isotropic, the lagrangian must also be independent of the direction of v , and is therefore a function only of it's magnitude ... I can't understand why , I think Landau's book has many fans in this...

I am currently having trouble deriving the volume element for the first octant of an isotropic 3D harmonic oscillator. I know the answer I should get is $$dV=\frac{1}{2}k^{2}dk$$. What I currently have is $$dxdydz=dV$$ and $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference...

Homework Statement When a point charge is positioned at the origin = 0 in an isotropic material, a separation of charge occurs around it, the Coulomb field of the point charge is screened, and the electrostatic potential takes the form \phi(r) = \frac{A}{r} \exp\left( -\frac{r}{\lambda}...

Homework Statement Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates. Homework Equations $$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...

Hi there, I am currently trying to understand the theoretical frame work of diffusive shock acceleration. I am having trouble understanding a step in the derivation given by drury 1983 (http://www.oa.uj.edu.pl/user/mio/Ast-Wys-En/Literatura/drury.pdf). In the derivation of eq. 2.47 it is stated...

I've been reading up on Killing vectors, and have got on to the topics of homogeneous, isotropic and maximally symmetric space-times. I've read that for an isotropic spacetime, one can construct a set of Killing vector fields ##K^{(i)}##, such that, at some point ##p\in M## (where ##M## is the...

Hello guys, I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here...

I read the Special Theory of Relativity in Jackson's textbook, Classical Electrodynamics 3rd edition. Consider the wave front reaches a point ##(x,y,z)## in the frame ##K## at a time t given by the equation, $$c^{2}t^{2}-(x^{2}+y^{2}+z^{2})=0 --- (1)$$ Similarly, in the frame ##K^{'}## the wave...

I don't understand the reasoning for any of the three constraints imposed. why would ##dtdx^i## terms indicate a preferred direction? what if there was identical terms for each ##x^i## would there still be a specified or preferred direction? (or is it that in this case we could rename ##t## to...

If space-time is isotropic does this imply it is spherically symmetric? why doesn't it need to be both isotropic and homogeneous?

I'm studying fluid dynamics and we just had a lecture about the momentum equation. We started the lecture by talking about pressure in terms of molecules moving across a hypothetical surface element and carrying their momentum with them (in both directions). There are 2 things confusing me about...

In order to use Solid65 in Ansys workbench for simulating concrete, we shall define the multi-linear isotropic stress-strain curve as well. I have the concrete compression stress-strain data in excel. I would like to ask that how could I get the multi-linear isotropic stress-strain curve in...

In Weinberg's book it is said that a Static, Isotropic metric should depend on ##x## and ##dx## only through the "rotational invariants" ##dx^2, x \cdot dx, x^2## and functions of ##r \equiv (x \cdot x)^{1/2}##. It's clear from the definition of ##r## that ##x \cdot dx## and ##x^2## don't...

In the lecture notes http://top.electricalandcomputerengineering.dal.ca/PDFs/Web%20Page%20PDFs/ECED6400%20Lecture%20Notes.pdf at page 15 eq. (2.46) it says that the dielectric tensor in an isotropic media can be represented by: δi j A(k,ω) + ki kj B(k,ω) I understood that in the case of I. M...

Homework Statement Attached Homework EquationsThe Attempt at a Solution So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold? I don't quite understand by what is meant by 'these vectors give preferred directions'. Can...

I have known what Ornstein-Zernike equation is. I try to plug in the form as follow to the isotropic materials: Still, I cannot show the pair correlation function as follow. Can anyone know what I have missed?

Suppose I apply a pair of equal and opposite harmonically varying forces perpendicular to an infinite drum membrane. Consider the following forcing functions at two nearby points,(x=0,y=a) and (x=0,y=-a), separated by a distance 2a, F(t,0,a) = Acos(ωt), F(t,0,-a) = -Acos(ωt) Let the forcing...

What is the process? And can/cannot it filter out?

Hello dear friends, today's question is: In a non static and spherically simetric solution for Einstein field equation, will i get a non diagonal term on Ricci tensor ? A R[r][/t] term ? I'm getting it, but not sure if it is right. Thanks.

Schwarzschild coordinates for the Schwarzschild black hole solution become very weird near the event horizon because the radial coordinate is based on the proper circumference of a sphere but that has a minimum at the event horizon. This is easy to see in isotropic coordinates, where the...

Homework Statement I am told that the gravitational force of a mass m located inside an isotropic distribution of spherical radius R and total mass M is given by Fg = -GmM(r)/r^2 where r is the distance between m and the center of distribution and M (r) is the mass contained below the distance...

It is common is cosmology to study density fluctuations in the early universe. However, it is also common to assume that the background space is homogeneous and isotropic and use the FRW metric. I do not see how density fluctuations can be possible in a homogeneous and isotropic space. Can you...

In Woodhouse's 'General Relativity' he finds an expression for the energy-momentum tensor of an isotropic fluid. If W^a is the rest-velocity of the fluid and \rho is the rest density then the tensor can be written as T^{ab} = \rho W^aW^b - p(g^{ab} -W^aW^b) for a scalar field p. The...

I'm doing some back of the envelope calculations for the potential of a turbine thermal generator using ammonia as a working fluid. I've never done thermodynamics before so I'm looking for a reality check. Isotropic turbine work done by a unit mass is given as h2 - h1 or simply dh between the...

Homework Statement Hi everybody! I'm a bit stuck in this problem, hopefully someone can help me to make progress there: A mass point ##m## is under the influence of a central force ##\vec{F} = - k \cdot \vec{x}## with ##x > 0##. a) Determine the equation of motion ##r = r(\varphi)## for the...

The Schwarzschild radial coordinate ##r## is defined in such a way that the proper circumference of a sphere at radial coordinate ##r## is ##2\pi r##. This simplifies some maths but creates some rather odd side-effects, so to get a more physical picture I like to use isotropic coordinates...

Hi PF, As you may know, is the the elasticity/stiffness tensor for isotropic and homogeneous materials characterized by two independant material parameters (λ and μ) and is given by the bellow representation. C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} +...

1. Does anyone know why for an isotropic distribution function, pressure tensor reduces to a scalar pressure? For instance, for a Maxwellian distribution P=A ∫ vx vy exp-(vx2 + vy2 + vz2) dvx dvy dvz is not zero. I think everybody should realize how bogus some of the authors are. Google...

The metric $$ds^2=-R_1(r)dt^2+R_2(r)dr^2+R_3(r)r^2(d\theta^2+sin^2d\phi^2)$$ when changed to $$ds^2=-R_1(r)dt^2+R_2(r)(dr^2+r^2d\Omega^2)$$ upon setting ##R_2(r)=R_3(r)##, the later metric holds the name of isotropic metric. My question what is the difference between the first and the second...

Homework Statement Hello everyone, I am having a problem whether or not a turbulence at a specific location (let's say A) is isotropic or not. I have calculated the two root mean square values of velocity fluctuations measured at the point A in a fully developed turbulent pipe flow. the first...

Hi，I was wondering if someone could tell me the name of this equation, where does the equation come from? “If light is isotropically generated in a medium then the fraction transmitted to the outside world is given by: F = (1/4)(n2/n1)2[1-{(n1-n2)/(n1+n2)}2]” Thank you so much :)

Homework Statement The Hamiltonian is given by: H = \frac{1}{2} \sum_{i=1,2}[p_i^2 + q_i^2] We define the following operators: J = \frac{1}{2} (a_1^+ a_1 + a_2^+ a_2) J_1 = \frac{1}{2} (a_2^+ a_1 + a_1^+ a_2) J = \frac{i}{2} (a_2^+ a_1 - a_1^+ a_2) J = \frac{1}{2} (a_1^+ a_1 - a_2^+...

Hey all, I realize a question on this topic has been asked elsewhere, but the links to references they use seem to be dead, so I'll press on! I'm reading some introduction to antenna theory and I've often puzzled on the equation: A_{eff} = \frac{\lambda^2}{4\pi} which relates the effective...

Can anybody give me a simple and easy example to understand it

I come cross one proof the Landau-Yang Theorem, which states that a ##J^P=1^+## particle cannot decay into two photons, in this paper (page 4). The basic idea is, the photon's wavefunction should be symmetric under exchange, however the spin part is anti-symmetric and the space part is...

The energy of photon is $$E=\frac{hc}{\lambda}$$ Now if we have an isotropic point light source of power P, Number of photons $$N=\frac{P}{E} = \frac{P \lambda}{hc}$$ Hence one can find the change in momentum and hence the force exerted by a beam or light sources. But let's say we keep an...

Hi, I have some trouble understanding if linear momentum and angular momentum (and their conservation laws) are completely independent or not. For example, one can calculate the angular momentum of a uniformly moving body with respect to a fixed point in space and show that it is indeed...

It is pretty straight forward to prove that the Kronecker delta \delta_{ij} is an isotropic tensor, i.e. rotationally invariant. But how can I show that it is indeed the only isotropic second order tensor? I.e., such that for any isotropic second order tensor T_{ij} we can write T_{ij} =...

Homework Statement A two-dimensional isotropic harmonic oscillator of mass μ has an energy of 2hω. It experiments a perturbation V = xy. What are its energies and eigenkets to first order? Homework Equations The energy operator / Hamiltonian: H = -h²/2μ(Px² + Py²) + μω(x² + y²) The...