02:21
Hmm, might need to think about what it is like to live like a tachyon...
Do the accelerated frames of tachyons which tachyons are at rest in, contains indeterministic regions of spacetime
Other thoughts: Tachyon classifier:
Consider a pair of spacelike separates events A, B indeterminate causally (because it is A-> B in some frames and B-> A in others) correlated by a tachyon.
Let A,B be receivers with some initial binary value recorded, say 0, and 1
Upon emitting or absorbing a tachyon, a bit flip occurred at the receivers
Let $v_t$ be the velocity of the tacyhon
We also configure that A,B flashes a green pulse of light if it is 0 and yellow if it is 1
Now, in inertial frames $v < v_t$, A flips from 0 to 1 and emits a tachyon that travels faster than light to B, thus flipping B from 1 to 0
In inertial frames $v > v_t$ B flips from 0 to 1, retroabsorbs a tachyon, which travels to A to be retroemitted, and flips A from 1 to 0
Now suppose we have an event C that is in the future light cone of both A and B (hence C is timelike separated from A,B) which emits a blue light if it receives a green pulse of light from A, and stops emitting a blue light if it receives a red pulse of light from A
Then we end up with an interesting situation where depending on what $v$ is, C can switch from emitting blue light to not emitting blue light
Thus, because of the tachyon, a huge region of spacetime became casually correlated, and these events are related by a Lorentz transformation
In particular, for any timeline event that has a cause that involves events causally correlated by tachyons, the order of causality of the timelike event can be reversed when you overtake said tachyon, creating a scenario where basically, the whole configuration of events in the spacetime diagrams between different observers are related by a Lorentz transformation, not just events themselves
In particular, under such Lorenz transformation, the direction of causality of said events flips sign once it passes $v_t$
leading to a discontinuity in the direction of causality between two classes of reference frames
Let $(M^m,g)$ be a compact Riemannian manifold with smooth nonempty boundary, and $N^n\subseteq \Bbb R^d$ a boundaryless isometrically embedded Riemannian manifold. For $1\le p<\infty$ we define as usual
$$W^{1,p}(M,N):=\{u\in W^{1,p}(M,\Bbb R^d):u(x)\in N\text{ a.e.}\}.$$
Using the Euclidean tr...
