##{dx}^2+{dy}^2=3^2+3^2=18##
##{dr}^2+r^2{d\theta}^2=0^2+3^2*(\theta/2)^2\neq18##
I have a feeling that what I'm doing wrong is just plugging numbers into the polar coordinate formula instead of treating it as a curve. For example, I naively plugged in 3 for r even though I know the radius...
I've been studying the Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds but have almost zero background in physics. The WRT of a 3-manifold is closely related to the Chern-Simons (CS) invariant via the volume conjecture. My question is, what does the CS invariant of a 3-manifold...
In SR, we know that ##\vec E \cdot \vec B## and ##E^{2}-B^{2}## are invariant.
Although I can prove those two invariant physical quantities mathematically, I do not know how to find at least
one example to demonstrate that ##\vec E \cdot \vec B## and ##E^{2}-B^{2}## are invariant.
Many thanks!
Hi everyone. Could you help me to find the way to prove some things?
1)Changing of body velocity or reference frame don't contribute to spacetime curvature
2)On the contrary the change of body mass causes the change of curvature in local spacetime
I use the assumption that if we have the same...
Here it is a simple problem which is giving me an headache,
Recall from class that in order to build an invariant out of spinors we had to introduce a somewhat
unexpected form for the dual spinor, i.e. ߰ψ = ψ†⋅γ0
Then showing that ߰ is invariant depends on the result that (ei/4⋅σμν⋅ωμν)† ⋅γ0...
Homework Statement
Let γ : I → ℝ2 be a smooth regular planar curve and assume 0 ∈ I. Take t ≠ 0 in I such that also −t ∈ I and consider the unique circle C(t) (which could also be a line) containing the 3 points γ(0), γ(−t), γ(t). Show that the curvature of C(t) converges to the curvature κ(0)...
Homework Statement
Show that the length of a curve γ in ℝn is invariant under euclidean motions. I.e., show that L[Aγ] = L[γ] for Ax = Rx + a
Homework Equations
The length of a curve is given by the arc-length formula: s(t) = ∫γ'(t)dt from t0 to t
The Attempt at a Solution
I would imagine I...
Hi,
The Tresca Critrion is given in the form of non continuous equations:
Max(½|σ1-σ2|,½|σ2-σ3|,½|σ3-σ1|) = k
How did they come up with the invarient equation
f(J2,θ) = 2√J2 * sin(θ+⅓π)-2k, θ from (0 to 60)
Hi.
I read that the Lorentz invariance Minkowski norm of the four-momentum
$$E^2-c^2\cdot \mathbf{p}^2=m^2\cdot c^4$$
has no analogue in Newtonian physics. But what about
$$E-\frac{\mathbf{p}^2}{2m}=0\quad ?$$
It might look trivial by the definition of kinetic energy, but it's still a relation...
Hi there.
How would I show that the eigenvalues of a matrix are an invariant, that is, that they depend only on the linear function the matrix represents and not on the choice of basis vectors. Show also that the eigenvectors of a matrix are not an invariant.
Explain why the dependence of the...
Homework Statement
Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves...
This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other?
Thanks!