Special Conformal Transformations of Conformal Geometry

[rloldershaw]

Hello,

In conformal geometry there is a 15-parameter symmetry group.

I have an rough conceptual understanding of the 3 spatial translations, the 1 temporal translation, the 3 rotations, the 3 Lorentz "boosts", and the 1 dilation transformation.

I am having trouble conceptualizing the remaining 4 "special conformal transformations", which apear to be combinations of translations, rotations and possibly something referred to as "inversion"

If there are any math aficionados out there who can give me a more intuitive, conceptual, visual understanding of what is going on physically with these special conformal transformations, I would be eternally grateful.

Maybe one approach might be to start with a homogeneous sphere and say: "Ok, when we subject the sphere to a SCT, these physical processes happen to the sphere.

Also, why are there specifically 4 SCTs?

Any help would be welcome and the more diferent perspectives, the better.

Yours in science
Knecht
www.amherst.edu/~rloldershaw

[Tglad]

I wonder if 6 years is a bit late for replying? ! 

You are talking about Minkowski space, there is also conformal geometry for Euclidean space, but I'll answer for Minkowski space.

The easiest way to think of them is as rotations around a circle (rather than around an infinite line). So space squashes and squeezes through the middle as you rotate it around the circle.
The circle (centred around 0) can be defined by a 3d vector (its length being the circle radius) so this gives 3 extra dimensions.

The last is a temporal inversion, which leaves the time = 1 (or -1) constant, ie an inverse around the double hyperboloid.

I haven't quite grasped what combinations of circle rotations and time inversions look like though.

[DarkBeam]

Uh, Lorentz boost is a very cool name!

(Whisper in my ears what it is anybody )

[Tglad]

It is like a velocity but more funky because Einstein is involved
If a rotation rotates points between two space dimensions, a boost 'rotates' between one space dimension (the velocity direction) and the time dimension.

It answers the question-
How can it be that if I travel at half the speed of light or even 99% the speed of light, a light ray still travels at the speed of light right past me? I can't catch up with light.

[kram1032]

Minkowski Space is a way to treat space and time on equal footing. Conformal Space is a way to treat n-planes and n-lines on equal footing.

So Cobformal Minkowski Space says thateverything you can do with a line (for instance, rotating around it), you can also do with a circle and everything yiu can do with a plane (reflecting in it) you can do with a sphere and you can use all those transformations equally on space and time.
With mild complications because space and time have different signs which basically trades geometry of circles with that of hyperbolas.

[Tglad]

If you Lorentz boost in a particular direction, say towards your front door, then you have a velocity towards the door, but also the view towards the door shrinks by n% and the view behind you grows by n%, and the world around you slows down.

Mathematically it is dead simple, just plot the distance along your boost direction on the x axis, and time on the y axis, so that the speed of light is 45 degrees. Then a Lorentz boost scales up by n% in the one 45 degree axis and scales down by n% in the other 45 degree axis. e.g. double lengths along one diagonal and halve them along the other.

This graphic shows a continuous Lorentz boost (an acceleration), x is space, y is time.

[kram1032]

that graphic also shows a crystal structure in minkowski space, which is pretty awesome. It's a space-time-crystal! (in one space and one time dimension)

[DarkBeam]

Mind bending theory, but my brain may burn if I try to understand