- #1
andrea.dapor
- 4
- 0
We call a group G "simply connected" if every curve C(t) in G which is closed (that is, C(0) = C(1) = I) can be continuously deformed into the trivial curve C'(t) = I (where I is the unit element in G). This is formalised saying that, for each closed C(t), there exists a continuous function F: [0, 1]x[0, 1] -> G such that
1) F(0, t) = C(t), for all t
2) F(1, t) = I, for all t
3) F(s, 0) = F(s, 1) = I, for all s
Now, Wald (General Relativity, 1984) says that the Poincare group is not simply connected, beacuse in particular for a rotation of [tex]2\pi[/tex] about an axis - say z - such a function F does not exist.
My question follows.
Consider the function
F(s, t) := sI + (1 - s)C(t),
where C(t) is the closed curve in Poincare group G associated to a rotation of [tex]2\pi[/tex] about z, that is,
C(t) =
(1 0 0 0)
(0 cos2\pi t -sin2\pi t 0)
(0 sin2\pi t cos2\pi t 0)
(0 0 0 1)
with t in [0, 1].
This F seems to verify (1)-(3)... where is my mistake?
I thank you for your help, and apologize for the "matrix" above...
1) F(0, t) = C(t), for all t
2) F(1, t) = I, for all t
3) F(s, 0) = F(s, 1) = I, for all s
Now, Wald (General Relativity, 1984) says that the Poincare group is not simply connected, beacuse in particular for a rotation of [tex]2\pi[/tex] about an axis - say z - such a function F does not exist.
My question follows.
Consider the function
F(s, t) := sI + (1 - s)C(t),
where C(t) is the closed curve in Poincare group G associated to a rotation of [tex]2\pi[/tex] about z, that is,
C(t) =
(1 0 0 0)
(0 cos2\pi t -sin2\pi t 0)
(0 sin2\pi t cos2\pi t 0)
(0 0 0 1)
with t in [0, 1].
This F seems to verify (1)-(3)... where is my mistake?
I thank you for your help, and apologize for the "matrix" above...