Can I calculate a tensor of a system( lots of particles) shaped like a sphere, then get exactly the radius of the system?
(I want to get lengths of three axes of ellipsoid, and I'm trying to examine the way with a sphere. )
I omitted ##\frac{1}{2}## only here. I actually calculated with it so it changes nothing, sorry for the mistake!
But I've done considering ##x^3## term and got ##\frac{2}{3}## factor. Thank you!
I'm reading Lambourne's <Relativity, Gravitation and Cosmology>, and I cannot get a result the book describes. It's on equation (6.7) in 173p.
When a person free-falls into a non-rotating black hole from ##r=r_0## to some position ##r=r'##, the proper time becomes...
Debye assumed sound wave dispersion relation for phonons(##ω=vK##) and this corresponds to acoustic modes in low frequency limits. That's why it explains low temperature heat capacity fairly well.
But how could this also explain high temperature limit(##C=3k_B## per atom)? I know Debye...
Imagining that an object spining around a spherical mass M has angular momentum that has z-component(θ=0) only, then
$$g_{μν}\frac{dx^μ}{dτ}\frac{dx^ν}{dτ}=(1-\frac{r_s}{r})c^2(\frac{dt}{dτ})^2-\frac{1}{1-\frac{r_s}{r}}(\frac{dr}{dτ})^2-r^2(\frac{dθ}{dτ})^2-r^2\sin^2θ(\frac{dφ}{dτ})^2=c^2$$...
It's really confusing if the frictional force IS an external force..
My guess is the frictional force isn't an external force
and therefore I can observe the momentum conservation even with friction if I carefully measure the velocity right before and after the collision.
But I'm not sure about it..
okay
I want to get how many phonon states(dN) are in [K, K+dK] in 1d K-space. For a state occupies small length ##\frac{2π}{a}## by periodic boundary condition,
##dN=\frac{dK}{\frac{2π}{a}}##.
?
I meant two electrons with two different spin types..up and down. That's why we should multiply 2 in electron case. What I calculated by ##\frac{dk}{\frac{2π}{a}}## is spatial part, not including spin part.
I'm to get the density of states of 1-dim linear phonons, with N atoms. I think it's a lot similar to that of 1-dim electrons, except that two electrons are allowed to be in one state by Pauli exclusion principle. To elaborate,
##dN=\frac{dk}{\frac{2π}{a}}=\frac{a}{2π}dk## for...
Riemann tensor is defined mathematically like this:
##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l##
Using covariant derivative formula for covariant tensors and covariant vectors. which are
##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c##
##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##,
I got these...