I wrote:(adsbygoogle = window.adsbygoogle || []).push({});

>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.

I think the only way the normal to the plane can be null is if the plane

is tangent to the light cone, in which case it will contain a single null

ray (which is itself normal to the plane), and all the other vectors in

it will be spacelike.

So if (but not only if) u and v are timelike, ref(u,v) will exist.

What's the significance of this construction failing if two of the

vectors lie on a tangent plane to the light cone? Well, if u and v are

*not* on such a plane, then you can choose w to be almost anywhere:

anywhere such that (u,w) and (v,w) also don't lie on tangent planes to

the light cone. In other words, given such generic u and v, you can find

SO(2,1) elements g_u and g_v that preserve them, *and* such that the

eigenvector of g_u g_v lies *almost* anywhere in R^3.

However, if u and v lie on a tangent plane to the light cone, then I

think the eigenvector of (g_u g_v) will always lie on that same tangent

plane, for all g_u and g_v that preserve u and v respectively.

The upshot of this for collisions in 2+1 gravity would be that certain

tachyon-tachyon and tachyon-luxon collisions (which yield either tachyons

or luxons as the outgoing particle, never tardyons) involve *coplanar*

vector-valued momenta.

What I'm claiming amounts to the statement that for each null ray n there

is a subgroup H(n) of SO(2,1), consisting of those elements whose

eigenvectors are normal to n.

So the simplest way to check this would be to look for a subalgebra of

so(2,1) consisting of elements whose null space lies in the plane normal

to n.

If we take the example of n being the null ray generated by (1,1,0), then

so(2,1) elements of the form:

| 0 a b |

| a 0 b |

| b -b 0 |

have their null space generated by (b,b,-a), which is always normal to

(1,1,0). And this set turns out to be closed under the Lie bracket, with

[e_a,e_b]=e_b (where by e_a I mean the element above with a=1, b=0, and

by e_b vice versa).

So it looks like those subgroups of H(n) do exist. But they won't be

Abelian! So those weird coplanar collisions in 2+1 gravity still won't

obey the old vector addition law.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Re: This Week's Finds in Mathematical Physics (Week 232)

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**