Greg Egan
Nov4-06, 03:36 PM
John Baez wrote:
>So one thing you're saying
>is that a product of rotations in two different rest frames can be
>a boost??
One way to see why the products of some rotations will be boosts is as
follows.
Suppose that for a pair of linearly independent vectors u and v in R^3 we
can find an element of O(2,1) with the following properties:
* it preserves both u and v
* its determinant is -1
* its square is the identity
If we can find such an element, we'll call it ref(u,v).
Now, if we pick three linearly independent vectors u, v and w so that
ref(u,v), ref(u,w) and ref(v,w) all exist, we can construct the following
elements of SO(2,1):
g1 = ref(u,v) ref(v,w) which preserves v
g2 = ref(v,w) ref(u,w) which preserves w
g3 = ref(u,v) ref(u,w) which preserves u
and we have g3 = g1.g2 because ref(v,w)^2 = I.
So if we pick a spacelike u and a timelike v and w, we'll have a boost
that's equal to a product of rotations.
The one question that remains is, when can we find ref(u,v)? This is
trivial in O(3), but O(2,1) is trickier. Generically we can construct a
normal n to the plane spanned by u and v:
n^a = g^{ab} eps_{bcd} u^c v^d
and if it's not a null vector we can construct ref(u,v) as the projector
into the plane minus the projector onto n. But if n is null, I don't
think ref(u,v) exists.
>So one thing you're saying
>is that a product of rotations in two different rest frames can be
>a boost??
One way to see why the products of some rotations will be boosts is as
follows.
Suppose that for a pair of linearly independent vectors u and v in R^3 we
can find an element of O(2,1) with the following properties:
* it preserves both u and v
* its determinant is -1
* its square is the identity
If we can find such an element, we'll call it ref(u,v).
Now, if we pick three linearly independent vectors u, v and w so that
ref(u,v), ref(u,w) and ref(v,w) all exist, we can construct the following
elements of SO(2,1):
g1 = ref(u,v) ref(v,w) which preserves v
g2 = ref(v,w) ref(u,w) which preserves w
g3 = ref(u,v) ref(u,w) which preserves u
and we have g3 = g1.g2 because ref(v,w)^2 = I.
So if we pick a spacelike u and a timelike v and w, we'll have a boost
that's equal to a product of rotations.
The one question that remains is, when can we find ref(u,v)? This is
trivial in O(3), but O(2,1) is trickier. Generically we can construct a
normal n to the plane spanned by u and v:
n^a = g^{ab} eps_{bcd} u^c v^d
and if it's not a null vector we can construct ref(u,v) as the projector
into the plane minus the projector onto n. But if n is null, I don't
think ref(u,v) exists.