Exploring Epsilon Simultaneity: Advantages and Applications

[etotheipi]

What is the advantage of considering the generalised simultaneity criterion $t = (1-\epsilon)t_1 + \epsilon t_2$ for $\epsilon$ between $0$ and $1$? How does varying the parameter $\epsilon$ help to elucidate the structure of the special theory? I think the surfaces of simultaneity are no longer so intuitive. I wondered whether this is helpful to solve some problems or just a gimmick.

 

[Dale]

I would say that it is not helpful, at least I have never seen a place where it is helpful. It's only value is that it establishes that the one-way speed of light is a convention and that you can (if you are a masochist) adopt a convention where the one-way speed of light is not c and still be consistent with the data.

 

[Ibix]

It's analogous to adopting a coordinate system in Euclidean space where one of the axes isn't perpendicular to the others. That can be useful in crystallography, I seem to recall, because the natural directions in some crystals are non-orthogonal.

One thought - didn't we discuss recently that clocks on the surface of the Earth are usually synchronised in the Earth-centered frame, but their worldlines are not orthogonal to that? So (locally) we're all using an $\epsilon$ that isn't quite 0.5?

 

[etotheipi]

I was just revising this today! Body-centred lattices have a primitive basis $\{ \frac{a}{2}(\hat{\mathbf{y}} + \hat{\mathbf{z}} - \hat{\mathbf{x}}), \frac{a}{2}(\hat{\mathbf{z}} + \hat{\mathbf{x}} - \hat{\mathbf{y}}), \frac{a}{2}(\hat{\mathbf{x}} + \hat{\mathbf{y}} - \hat{\mathbf{z}})\}$ whilst face-centred lattices have a primitive basis $\{\frac{a}{2}(\hat{\mathbf{y}} + \hat{\mathbf{z}}), \frac{a}{2}(\hat{\mathbf{z}} + \hat{\mathbf{x}}), \frac{a}{2}(\hat{\mathbf{x}} + \hat{\mathbf{y}}) \}$. But we hardly ever used these in favour of the canonical basis. The silver lining for the primitive basis is that the Weiß zone law $hU + kV + lW = 0$ holds in any crystallographic system, but apart from that you are just stuck with annoying calculations with the metric

 

[Ibix]

You doubtless know more crystallography than I remember... I was just thinking of it as a physical circumstance where we might reasonably choose to use non-orthogonal coordinates. The Earth's rotation forces a vaguely analogous circumstance where the only sensible global simultaneity criterion is not orthogonal to the helical worldlines of clocks at rest on the surface. So Einstein-synchronised clocks on the east and west sides of a lab aren't quite synchronised per GMT, I think.

 

[Ibix]

The easy way to see it is to imagine the set of helical worldlines of clocks on the equator. The congruence forms a cylindrical worldsheet. Except in the special case of zero rotation the planes orthogonal to the worldlines are all "slanted" in the same sense as you go around the cylinder. You can't have a closed loop without slanting the loop in the opposite sense in at least one place.