[projective-cone] (Figure)
Can be equivalently understood as positive-cone & positive quotient
Since the metric is zero on the light cone, many analysis cannot be done. Also, the quotient of rays on the light cone corresponds to metric-cannot-distinguish-colinear-light-like
induce bijective of
Proof , induce bijective of set of 1d linear subspace
identity induce
[complex-struct-of-4d-projective-lightcone] Complex structure of 4d projective-lightcone (Figure)
- Elliptic
- Hyperbolic
The hyperbolic case has two separate branches. There is a singularity region between the future light cone section and the past light cone section.
Is there a analogue? But is a Euclidean manifold, which is not suitable for the signature of split complex numbers , and stereographic projective hyperbolic projection seems quite complicated
Since the light cone can intercept , it is reasonable to lose the symmetry of corresponding to
Proof
Using to intercept the lightcone , we get
can be replaced with other non-zero real numbers, and the result is equivalent
Using to intercept the lightcone, we get space-like section . Divided into future and past two branches
โs projection cannot be intercepted by
Stereographic projection transition-function is quadratic inversion
and its coordinate
coordinate 1 , coordinate map
coordinate 2 , coordinate map
transition-function , or , i.e. the multiplicative inverse of . is a complex manifold
vs stereographic projection transition-function
A more direct mapping between the coordinates of , cf. Hopf-bundle
[linear-fractional]
acts on , , use the multiplicative inverse of to restrict it to , in coordinate 1
in coordinate 2
has the same
needs to be handled separately, the composition of cannot be expressed as ordinary matrix multiplication
Scaling gives the same linear-fractional, so can quotient to or
Prop (ref-13, p.172โ174)
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acts on in coordinate can be expressed as linear-fractional
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[Lorentz-group-spinor-representation]
Proof
in , 3 rotation , 3 boost , where is rotation in direction, is boost in direction
[rotation-boost-spinor-representation]
3 rotation 3 boost acts on the intercepted by of the projective light cone, calculate its representation in (one of) the stereographic projection coordinates
- rotation in
- act on
- act on , generator (with eigenvalue and eigenstate as base of )
- boost in
- act on
- act on , generator
Because the direction was chosen to construct the stereographic projection, the cases in the directions will be more complicated (I have not done the calculation and verification below)
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rotation in
act on , generator
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rotation in
act on , generator
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boost in
act on , generator
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boost in
act on , generator
It can be prove that , It can be prove that
Comparing of and of , at least locally isomorphic
-
for
where
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have form where (ref-2, Vol.1, p.180)
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from to . Solve it from the following to . Or use Polar decomposition to rotation boost + Euler angle
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from to , where
Use spacetime-momentum-spinor-representation to directly calculate
use Euclidean type topology, because metric is inherited from space-like inherited from metric
is the conformal transformation group of , represented as linear-fractional in stereographic projection coordinates
To calculate the conformal transformation factor of the metric, use coordinate and 3 rotation, 3 boost โฆ
[isotropy-on-projective-lightcone] Prop acts on projective-lightcone , isotropy is similar to
are surjective action, orbit number , so calculate isotropy only need to consider one point
Use the points , in coordinates , , corresponding to the point on the light cone projection
is isotropy ==>
So Isotropy
is similar to because
the group multiplication is
use the correspondence i.e. will give the usual semi-direct product , i.e.