Recent content by Sagittarius A-Star

This is not a full solution for problem (3b), but maybe a useful hint. When I substitute the RHS of (3b) into (3a), the two Levi-Civita-symbols share 2 indices, like in the given identity. Problem (3b) with renamed indices: ##F^{\gamma \delta}=(space/time-part)+\epsilon^{\gamma \delta}{...

Additional insight: The 4-vector ##E_{\vec U}^\alpha## from problem (3a), multiplied with charge ##q##, gives the Lorentz 4-force on a test charge with 4-velocity ##U##. ##f^\alpha=qE_{\vec U}^\alpha=qF^{\alpha\beta} U_\beta##.

The following refers to special relativity. It is sufficient to prove the validity of the equation of part (b) in only one basis. This would prove automatically it's validity in any basis. The headline of problem (3) is: “3+1 split of the electromagnetic field". A "3+1" split is...

According to Wikipedia, this is a non-relativistic equation. The relativistic equation is shown below in Wikipedia. https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force#Definition_and_description The same question regarding the relativistic equation is discussed in the Wikipedia article...

What may help is the following hint in the description of problem #3: ##\vec{e}_{\hat 0} = \vec U## In special relativity, in the restframe of the observer, his 4-velocity has the following components: ##U^\alpha = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix}## You may right-click ("Show...

As delimiters, you can use double $ or double # on each side. See in below LaTex Guide link under "Delimiting your LaTeX code".

For a rotating disk with only 3 rings (because of Wolfram Alpha calculation limitations): ##\lim_{v \rightarrow 1} {E/M} =## ## \lim_{v \rightarrow 1} {\sum_{i=1}^3 \frac{2i}{12\sqrt{1-v^2i^23^{-2} }}} = \infty## Calculation

I get from Wolfram Alpha 44.676683503771391589009225229630072178116008914652013623324856498 Calculation

I get from Wolfram Alpha 1.93277 Calculation 1 I get with else original values 4.23555 Calculation 2

I include in the summation the outer ring N and set it's speed slightly below the speed of light. ##E = \sum_{i=1}^N \frac{2\pi\lambda r_1i}{\sqrt{1-\omega^2r_1^2i^2}}## with ##\lambda = \frac{M}{\pi r_1 N(N+1)}##, ##v:=\omega R## and ##R=r_1N## ##\Rightarrow## ##E/M = \sum_{i=1}^N...

During his acceleration he can consider himself to be at rest in a pseudo-gravitational field, in which twin B ages more quickly than him because of gravitational time-dilation (equivalence principle). Source: https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

Close to the speed of light the proper area is ##2\pi R^2## instead of ##\pi R^2##. To keep proper mass density constant, you have at maximum to double the proper mass.

The proper area in the non-Euclidean geometry doubles close to the speed of light: ##A(R)= 2\pi \int_0^R \gamma r \, dr= {2 \pi c^2 \over \omega^2}(1-\sqrt{1-\omega^2R^2/c^2})## Source

The rest-mass depends on ##\omega## because of internal stress.

Purcell discussed also a scenario, in which, in the lab-frame, the object moves only vertically towards the horizontal conductor. Also in this scenario, the magnetic force in the lab frame turns out to be perpendicular to the velocity of the object. In the object's restframe, the neutral...