PeterDonis said:
But how do you define a "circle" in Galilean spacetime? There is no metric, so the usual definition of a circle as the set of points that are all at the same distance from the center doesn't work.
Also, if we are talking about boosts, we are talking about vectors, not geometric figures, and the whole idea of a "circle" doesn't really make sense for vectors.
From my draft at
https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-TrilogyArticle.pdf
(as part of
https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-GRposter.pdf
https://www.aapt.org/doorway/Posters/SalgadoPoster/SalgadoPoster.htm
http://www.aapt.org/doorway/TGRU/ )
- as Euclid defines a circle in the plane as
"the locus of points (say) t=1 km from a point [along radial paths from the point, using odometers measuring t]"
- Minkowski might define the Minkowski-circle (a hyperbola) on a spacetime diagram as
"the locus of events (say) t=1 s from an event [along inertial worldlines, using wristwatches]".
- in hindsight, let's pretend that Galileo might define the Galilean-circle (a t=1 line) on a position-vs-time diagram as
"the locus of events (say) t=1 s from an event [along inertial worldlines, using wristwatches]"... just like Minkowski.
as seen in
https://www.desmos.com/calculator/kv8szi3ic8
- Vary the ##E##-slider: E=-1 for Euclid, E=+1 for Minkowski, E=0 for Galilean.
(Observe the unit "circles", the radial lines, and the tangent lines.)
- Then, for fixed-E, vary the ##v_2##-slider to trace out the "circles".
- Open the "Boost" folder, vary the ##v_{LAB}##-slider to see the "circle" get preserved by the boost.
(That is, the boosted radial unit-vectors remain on the "circle".)
The metric for these affine Cayley-Klein geometries is
$$ds^2=dt^2-Edy^2.$$
(Yes, the Galilean metric is degenerate... one needs another quadratic form for the spacelike vectors,
as is well known. See the papers from
Trautman,
Ehlers, etc...
See also
https://www.physicsforums.com/threads/why-is-minkowski-spacetime-non-euclidean.1016402/post-6647305 and links therein.
While "degeneracy" sounds bad,
note that from a projective-geometry point of view, Euclidean geometry has degenerate features,
but there they manage to get along and develop Euclidean geometry.)
The idea of my approach is to carry out all constructions and calculations keeping track of the ##E##,
and to do so in a unified way as much as possible.
A construction which seems difficult to be unified is demoted in favor of one that can be unified.
GOAL: Same storyline, different ##E##.
(So, I want to define "timelike", "spacelike", and "null" in a unified way from these structures.)
The boosts for the above are ( from
https://www.physicsforums.com/threads/why-is-minkowski-spacetime-non-euclidean.1016402/post-6648907 )
$$\left( \begin{array}{c} t' \\ y' \end{array} \right)
=
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1-E\beta^2}} & \frac{E\beta}{\sqrt{1-E\beta^2}}
\\
\frac{\beta}{\sqrt{1-E\beta^2}} &
\frac{1}{\sqrt{1-E\beta^2}} & \end{array}
\right)\\
\left( \begin{array}{c} t \\ y \end{array} \right)$$
I learned of these 2-D
Cayley-Klein geometries (which include these 9 geometries:
Euclidean, Hyperbolic, Elliptical/Spherical, Minkowski, Galilean, deSitter and anti-deSitter and their Galilean limits) from
"A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity" by I.M. Yaglom
https://www.amazon.com/dp/0387903321/?tag=pfamazon01-20
and, more recently,
Richter-Gebert's "Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry"
https://www.amazon.com/dp/3642172857/?tag=pfamazon01-20 .
To see the unified metric for all 9 geometries, look at page 3 of my poster.)
So, I am carrying out the constructions and calculations,
especially as it applies to physics (PHY101, Special Relativity, and hopefully the deSitter spaces).