Ciro Santilli @cirosantilli 35

Suppose that a rod has is length $L$ measured on a rest frame $S$ (or maybe even better: two identical rulers were manufactured, and one is taken on a spaceship, a bit like the twin paradox).

Question: what is the length $L^{\prime }$ than an observer in frame $S^{\prime }$ moving relative to $S$ as speed $v$ observe the rod to be?

The key idea is that there are two events to consider in each frame, which we call 1 and 2:Note that what you visually observe on a photograph is a different measurement to the more precise/easy to calculate two event measurement. On a photograph, it seems you might not even see the contraction in some cases as mentioned at en.wikipedia.org/wiki/Terrell_rotation

Measuring a length means to measure the $x_2-x_1$ difference for a single point in time in your frame ($t2=t1$).

So what we want to obtain is $x_2^{\prime }-x_1^{\prime }$ for any given time $t^{\prime }2=t^{\prime }1$.

In summary, we have:

By plugging those values into the Lorentz transformation, we can eliminate $t_{2}and{t}_1$, and conclude that for any $t_2^{\prime }=t_1^{\prime }$, the length contraction relation holds:

The key question that needs intuitive clarification then is: but how can this be symmetric? How can both observers see each other's rulers shrink?

And the key answer is: because to the second observer, the measurements made by the first observer are not simultaneous. Notably, the two measurement events are obviously spacelike-separated events by looking at the light cone, and therefore can be measured even in different orders by different observers.

Generalization of orthogonal group to preserve different bilinear forms. Important because the Lorentz group is $SO(1,3)$.

C plus plus is what you get when you want to have all of:

Java is good.

Its boilerplate requirement is a pain, but the design is otherwise very clean.

But its ecosystem sucks.

The development process is rather closed, the issue tracker obscure.

And above all, Google LLC v. Oracle America, Inc. killed everybody's trust in it once and for all. Thanks Oracle.

Has some of the best map data available for the United Kingdom, but their data appears to be proprietary?

Maybe the most famous one is Mycoplasma genitalium byt there are others, and notably with lower biosafety levels:

Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".

Furthermore, the non-compact part is always isomorphic to ${\mathbb{R}}^n$, only the non-compact part can have more interesting structure.

An overview of what you can do with a pre-shared key with tradeoffs can be found at: quantumcomputing.stackexchange.com/questions/142/advantage-of-quantum-key-distribution-over-post-quantum-cryptography/25727#25727 The options are: