[polar-decomposition-of-Lorentz-group] polar decomposition to rotation and boost (ref-2, Vol.1, p.165)
let where
(Using and polar decomposition of positive definite symmetric matrix)
where
is boost, map to
, have diagonal form where
with
[Euler-angle-Lorentz-group] Question
- rotation
Use the rotation of the axis to generate
in ,
in
- boost
Use the boost of the axis
in
in
[Lorentz-group-Lie-bracket] with boost and rotation decomposition and Lie-bracket
Where mimics cross product. Example
Where mimics dot product. Example ==> or
Written to mimic cross product
have form where (ref-2, Vol.1, p.180)
[Lorentz-group-orbit-isotropy]
or act on
| orbit type | isotropy-group type |
| or | |
| or | |
| or |
[isotropy-on-lightcone] Prop acting on lightcone is similar to (exactly the Euclidean affine group)
Proof Using spinor technology
Prop acting on lightcone (not projective-lightcone), isotropy is similar to
is surjective action, orbit number , so calculate isotropy only need to consider one point
- is light cone isotropy
- is a scaling of
- is a spatial rotation of . It can give the entire light cone cross-section
==> isotropy where
Similar to the calculation of isotropy-on-projective-lightcone , here it will be similar to
[isotropy-on-lightcone-intuition] Intuition of the isotropy-group of orbit lightcone. According to
Discuss the two situations separately
-
. is rotation in
-
let with boost and rotation decomposition (not the in )
Linear isomorphism to a new basis
where
- is rotation in
- is boost in
- and are analogous to lightcone coordinate , keeping
Or written as
where will change , remains fixed
Two-dimensional Lie algebra is commutative, corresponding to or in
On the light cone (figure)
- Generally generates a hyperbolic orbit
- Generally generates an elliptical orbit
- can be generalized to general lightcone combinations e.g. , which will generate parabolic orbits
Specific calculation of the action of
let . metric will be
in