I also wanted to thank you for leaving this comment. I'm guessing that you simply weren't aware that the camera could be controlled.
I've just made some changes to make it more apparent to the user that the visualization is 3D and comes with camera controls. Previously the instructions for...
If you rotate the camera in the visualization, I think you'll see that it's correct (only looks like 2 from that angle). Maybe I should consider an option to display grid-lines!
A few updates:
Dark mode.
Sliders (originally suggested by @robphy), though the input-boxes are still there for "custom" values.
On desktop, the options and controls now go side-by-side with the visualization, so that it's easier to adjust the parameters while viewing the results...
7 days is a lot longer than I'd realized is feasible. I wonder how much time you'd have (in this scenario) before the tidal forces became uncomfortable; I have a summer vacation to plan...
And the ultrarelativistic approximation in terms of rapidity:
$$1 - \beta \approx \dfrac{1}{2} \textrm{sech}^2 \, \theta$$
or just as good:
$$1 - \beta \approx \textrm{sech} \, (2 \theta)$$
This "ultrarelativistic" approximation can even be done without a binomial expansion (in case OP or other lurkers aren't familiar with that technique). Using ##\beta = v/c##:
$$
\begin{align*}
\gamma^{-2} &= 1 - \beta^2 \\
& = 1 - \left( 1 - \left( 1 - \beta \right) \right)^{2} \\
& = 1 -...
There are still exceptions, though. Hobson/Efstathiou/Lasenby used it in their GR book (2006), and more recently d'Inverno/Vickers continued using it in the 2nd edition of theirs (2022). John Kogut (2018) uses it in deriving the relativistic momentum, but then tells the reader that he'll always...
I don't interpret Einstein's original paper as using relativistic mass. Or I'm at least unsure. The thought experiment involves a body that gives off equally energetic radiation in opposite directions, such that its velocity is unaffected by the emission. That's the context that precedes the key...
If I'm not mistaken, it's a little more subtle than that, since both the time and the energy transform. In general, the "coordinate power" is given by ##P = \gamma^3(\vec v \cdot \vec a)m + \gamma \dot{m}##, and by time dilation and the chain rule we have ##\gamma \dot{m} = dm/d\tau ## (i.e...
I'm wondering if ultimately it boils down to the chosen "canonical" ordering of the basis ##n##-form (for an ##n##-dimensional space).
For example, in ##4##-dimensional Euclidean space (so we don't get side-tracked by the "different" nature of time), half of the basis ##2##-forms must be true...
I think the approach taken in Chapter III.2 ("Einstein's Clock and Lorentz's Transformation") of Anthony Zee's Einstein Gravity in a Nutshell might be more or less what the OP is looking for. Zee derives the LT from the interval's invariance, and his derivation of the interval's invariance isn't...
I should add that in the case of Minkowski spacetime, I'd guess that the basis ##2##-forms involving ##dt## are all of the same "type" (either the pseudo-tensors or the true tensors). But again, my question is more general: how to tell in the general case which basis forms are pseudo-tensors and...
My understanding is that the Hodge dual of a pseudo-form is always a "true" pseudo-form, and vice versa. However, I'm a little confused about how this applies to basis-forms in general.
I believe I understand how it works for the ##0##-form case: the basis ##0##-form is the scalar ##1##...