<,HTML>, <,HEAD>, <,!-- This HTML file has been created by texi2html 1.38 from thisfigure.tex on 28 November 1997 -->, <,TITLE>,Seeing all Six Dimensions of the Lorentz Group of Special Relativity, in the Planetarium Sky<,/TITLE>, <,/HEAD>, <,BODY>, <,H1>,Seeing all Six Dimensions of the Lorentz Group of Special Relativity, in the Planetarium Sky<,/H1>, Look crosseyed directly at this figure, so that the right portion is viewed by the left eye, and the left portion by the right eye, until a fusion brings about a 3-dimensional image. (You might also try this crosseyed method to directly see any display usually intended for optically aided uncrossed viewing. Normal 3-d vision is crosseyed!) <,p ALIGN=center>, <,IMG SRC=cam1.jpeg ALT='<,Figure is a Sphere with Tangent Planes>,'>, <,p>, The resulting figure was created to make vivid the action of a Lorentz transformation ``on the planetarium sky''. The sphere is the sky. The transformation is the exponential of an infinitesimal Lorentz transformation, times a steadily increasing parameter, so the figure is actually a stage for a movie of a 1-parameter group of transformations. <,p>, The stars in the sky are points on the sphere for you to imagine---they are not shown in the figure. They move simply in relation to what I call ``latitudes'' and ``longitudes'': There are two fixed points, or ``poles'', where you see tangent planes touch the sky-sphere. The segment connecting these poles, I call the ``axis'' of the transformation. A steadily growing ``homeboost'' drives stars along the longitudes---this will be familiar to astronomers as Bradley's aberration of light (1729). The longitudes drawn in the figure, which as in geography run from pole to pole, are spaced equiangularly where their extended planes meet the tangent planes. Simultaneously, a steadily growing ``hometwist'' drives the stars along the latitudes, which like ``parallels'' of latitude in geography do not intersect on the sphere, though their planes here all pass through that line where the two tangent planes also meet. I call that line, the ``coaxis'' of the transformation. The axis is eccentrically placed, in the case presented. Taking the radius of the sky as 1, I call the eccentric displacement of the axis, the transformation's ``crossspeed''. <,p>, As in Mercator's projection in geography (1568), imagine the sphere spread out on a right circular cylinder or on a planar strip, so that the selected longitudes become equally spaced parallel lines. My selection of latitudes is such that on the Mercator map, they cut the longitudes in equal squares. That is why, unlike the convention in geography, my latitudes seen on the sphere get increasingly close as you approach either pole. Then, on that boring graph-paper-like ``generalized Mercator map'' image, not shown, the vector field for the flow of stars is uniform! <,p>, The mathematical sense of this 3-dimensional realization of what is more familiar to you as a 4-dimensional linear Lorentz transformation, comes about by collapsing the linear action on 4-velocities to a projective action on 3-velocities. (I prefer to think of my diagram's points as *reversed* 3-velocities, because when you point to a star, you point *oppositely* to the way the light goes.) Then the sky---the unit sphere---is all the 3-velocities at the speed of light (taken as 1). Hence the sky is preserved by Lorentz transformations---since Lorentz transformations are *defined* to leave the speed of light invariant. Points inside the sky are velocities slower than light, points outside are faster than light. How these off-sky points go---visually rather than algebraically---depends on how the stars go, by drawing any two lines through such a point which also punch through the sky, since lines go to lines, in any projective map. ...The homeboost acts alone on the axis, the hometwist acts alone on the coaxis, and one can instead build outwards from the action on this skew cross... ...The two poles can fuse, to give the exceptional or ``dipolar'' Lorentz transformations, where the crossspeed is 1, where the cross is no longer skew, and where Mercator projection yields to stereographic projection. <,p>, Four of the six dimensions of the Lorentz group are used up in telling where in the sky to place the two poles, as each place is worth two parameters. The homeboost and hometwist, finitely, or the fluxion 2-vector on the Mercator map, infinitesimally, use up the last two parameters. So you are indeed ``Seeing all Six Dimensions of the Lorentz Group of Special Relativity, in the Planetarium Sky'', four dimensions directly in the figure, and two more in your mind's eye by imagining a constant flow on the Mercator map suggested by the figure's latitudes and longitudes. --- eli@uwm.edu <,p>, <,A HREF='camnegative.gif'>,<,IMG SRC=camnegtiny.gif ALT=''>,another peek <,/A>, Source.


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