Appearance of The Night Sky
This isomorphism has a very interesting physical interpretation. We can identify the complex number
with a null vector in Minkowski space
or the Hermitian matrix
The set of real scalar multiples of this null vector, which we can call a null line through the origin, represents a line of sight from an observer at a particular place and time (an arbitrary event which we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars.
But by stereographic projection, we can identify with a point on the Riemann sphere. Putting it all together, we have identified the points of the celestial sphere with certain Hermitian matrices, and also with lines of sight. This implies that the Möbius transformations of the Riemann sphere precisely represent the way that Lorentz transformations change the appearance of the celestial sphere.
For our purposes here, we can pretend that the "fixed stars" live in Minkowski spacetime. Then, the Earth is moving at a nonrelativistic velocity with respect to a typical astronomical object which might be visible at night. But, an observer who is moving at relativistic velocity with respect to the Earth would see the appearance of the night sky (as modeled by points on the celestial sphere) transformed by a Möbius transformation.
Read more about this topic: Lorentz Group
Famous quotes containing the words appearance of, appearance, night and/or sky:
“This mesa plain had an appearance of great antiquity, and of incompleteness; as if, with all the materials for world-making assembled, the Creator had desisted, gone away and left everything on the point of being brought together, on the eve of being arranged into mountain, plain, plateau. The country was still waiting to be made into a landscape.”
—Willa Cather (1873–1947)
“Look into the world—how often do you behold a sordid wretch, whose straight heart is open to no man’s affliction, taking shelter behind an appearance of piety, and putting on the garb of religion, which none but the merciful and compassionate have a title to wear.”
—Laurence Sterne (1713–1768)
“They threw off their clothes, and he gathered her to him, and found her, found the pure lambent reality of her for ever invisible flesh. Quenched, inhuman, his fingers upon her unrevealed nudity were the fingers of silence upon silence, the body of mysterious night upon the body of mysterious night, the night masculine and feminine, never to be seen with the eye, or known with the mind, only known as a palpable revelation of living otherness.”
—D.H. (David Herbert)
“Two wooden tubs of blue hydrangeas stand at the foot of the stone steps.
The sky is a blue gum streaked with rose. The trees are black.
The grackles crack their throats of bone in the smooth air.
Moisture and heat have swollen the garden into a slum of bloom.
Pardie! Summer is like a fat beast, sleepy in mildew....”
—Wallace Stevens (1879–1955)