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 Exercise [18.04] 
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Joined: 23 Aug 2010, 13:12
Posts: 33
Post Exercise [18.04]
I wish somebody would help me understand this stuff!

Based on the way Penrose uses "transitive" to explain both the bundle of future timelike directions and also the homogeneity of space (preceding paragraph), I think what it means is "homogeneity of velocities" so that spacetime looks the same for every state of motion allowed by a vector in the interior of a future light cone. This is pretty fuzzy.

Presumably the base space of the bundle is the 3-dimensional space of allowed velocities. What, then, are the fibers - the 3-dimensional space orthogonal to the implied time direction ?


Last edited by Smith on 05 Dec 2010, 19:12, edited 1 time in total.

06 Sep 2010, 12:22

Joined: 03 Jun 2010, 15:18
Posts: 136
Post Re: Exercise[18.04]
I try to explain how I understood that (that does not mean my understanding is correct!).
Quote:
Based on the way Penrose uses "transitive" to explain both the bundle of future timelike directions and also the homogeneity of space (preceding paragraph), I think what it means is "homogeneity of velocities" so that spacetime looks the same for every state of motion allowed by a vector in the interior of a future light cone. This is pretty fuzzy. 

The "homogenety of space" means there is no special point (event) of Minkowski space. The same happens in an Euclidean space: there is no special point.
Then "homogeneity of velocity" means that there is no special velocity, i.e. that there is no special inertial frame, i.e. the special relativity principle. Every velocity can be obtained from any other one (transitivity) by an (active) Poincarè transformation, equivalent to move from one inertial frame to another.
The same happens in an Euclidean space: there is no a special direction at a given point, i.e. Euclidean space is isotropic.

Quote:
Presumably the base space of the bundle is the 3-dimensional space of allowed velocities. What, then, are the fibers - the 3-dimensional space orthogonal to the implied time direction ?


I think the "bundle of future timelike directions" is the (future timelike subspace of) tangent fiber bundle (chapter 15.5) of Minkowski space, which elements are the pairs (P,T) of an event P (t,x,y,z) (point of Minkowski space) and a tangent vector T at this point (i.e. a tangent to a world line, i.e. a 4-velocity vector). Hence the base space is the 4-dimensional Minkowski space itself, and a fiber is the space of 4-velocities allowed to a particle.
The 4-dimensional velocity space can be reduced to 3-dimensional (so making it very similar to classical velocity space) imposing that tangent vector length is unity, i.e. selecting an affine parameter (see chapter 14.5) for world lines: the affine parameter is the proper time for a particle being at that event and moving with that velocity. And then we can describe this unit vector with 3 parameters, that are the components of classical velocity.
This is the analogue of what we do in Euclidean space. We can describe a direction by a 3-dimensional vector; but its length is arbitrary, so we may decide to restrict ourselves to unit length vectors; and then we can describe this unit vector by 2 parameters, i.e. two direction angles.


07 Sep 2010, 15:37

Joined: 23 Aug 2010, 13:12
Posts: 33
Post Re: Exercise[18.04]
I believe the base space of the bundle is the 3-dim space of velocity vectors and each fiber is a 7-dimensional space of (possibly rotated) copy of the 4-dim Minkowski space. Each such 7-dim space defines a unque state of rest which constitutes the canonical projection from the fibers to the base (see p.329.)


08 Sep 2010, 18:12

Joined: 12 Jul 2010, 07:44
Posts: 154
Post Re: Exercise [18.04]
Smith, you have it backwards.

The fiber-bundle view has as the base space 4-dimensional Minkowski space.
The fiber at each point is the 3-dimensional space of normalized tangent vectors at that point.

(Even null vectors can be normalized, in the sense that multiplying by a positive constant doesn't change the "direction" of the 4-vector; there is thus only 2 degrees of freedom for null vectors, and 3 overall degrees of freedom for 4-vector direction. Just how you define a "normalized" null vector is up to you... you could simply scale so that the time component is 1 or -1, for example.)

A given point of the bundle therefore specifies both a particular point in space, and a normalized tangent 4-vector.

If we restrict our attention to only those tangent vectors that are future-pointing, then we get a new bundle contained in the one I just described. This new bundle is the one Penrose mentions.

When he says that the Poincare group acts transitively on this bundle, what he's saying is that for any two points in this bundle, there is some Poincare transformation that will map the first point to the second.

This is the same as saying we can find a coordinate transformation that will map the position and velocity of any particular object travelling through Minkowski space onto the position and velocity of any other such object.

Or, if these "objects" are the two observers you and I, then there's a Poincare transformation that maps my reference frame to yours!

(As Penrose mentions, the Poincare group also includes reflective symmetries, including reflection in time: You can map forward-pointing timelike vectors onto backward pointing ones. This reflects the fact that Minkowsi space, and special relativity, is symmetric whether time runs forwards or backwards - its mathematical structure is unchanged.)


10 Jul 2011, 07:53
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