1. ### Trying to use polar coordinates to find the distance between two points

##{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...
2. ### A Chern-Simons Invariant

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...
3. ### I Examples of invariant quantities

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!
4. ### I Stress-energy tensor contribution to curvature

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...
5. B

### Dual spinor and gamma matrices

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...
6. ### Show curvature of circle converges to curvature of curve @ 0

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)...
7. ### Invariance of length of curve under Euclidean Motion

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...
8. ### Tresca criterion in terms of invariants

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)
9. ### I Newtonian analogue for Lorentz invariant four-momentum norm

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...
10. ### Eigenvalues are invariant but eigenvectors are not

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...
11. ### Find matrix representation for rotating/reflecting hexagon

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...
12. ### Spacetime interval and the metrric

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!