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
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“To educate the wise man, the State exists; and with the appearance of the wise man, the State expires. The appearance of character makes the state unnecessary. The wise man is the State.”
—Ralph Waldo Emerson (1803–1882)
“The fear of death is indeed the pretence of wisdom, and not real wisdom, being the appearance of knowing the unknown.”
—Plato (c. 427–347 B.C.)
“February is a suitable month for dying. Everything around is dead, the trees black and frozen so that the appearance of green shoots two months hence seems preposterous, the ground hard and cold, the snow dirty, the winter hateful, hanging on too long.”
—Anna Quindlen (b. 1952)
“As it grew darker, I was startled by the honking of geese flying low over the woods, like weary travellers getting in late from Southern lakes, and indulging at last in unrestrained complaint and mutual consolation. Standing at my door, I could hear the rush of their wings; when, driving toward my house, they suddenly spied my light, and with hushed clamor wheeled and settled in the pond. So I came in, and shut the door, and passed my first spring night in the woods.”
—Henry David Thoreau (1817–1862)
“The sky calls
To the deaf earth. The proverbial disarray
Of morning corrects itself as you stand up.
You are wearing a text.”
—John Ashbery (b. 1927)