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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“Hence, the less government we have, the better,—the fewer laws, and the less confided power. The antidote to this abuse of formal Government, is, the influence of private character, the growth of the Individual; the appearance of the principal to supersede the proxy; the appearance of the wise man, of whom the existing government, is, it must be owned, but a shabby imitation.”
—Ralph Waldo Emerson (1803–1882)
“Hence, the less government we have, the better,—the fewer laws, and the less confided power. The antidote to this abuse of formal Government, is, the influence of private character, the growth of the Individual; the appearance of the principal to supersede the proxy; the appearance of the wise man, of whom the existing government, is, it must be owned, but a shabby imitation.”
—Ralph Waldo Emerson (1803–1882)
“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)
“Aunt Sally she was one of the mixed-upest looking persons I ever see; except one, and that was uncle Silas, when he come in, and they told it all to him. It kind of made him drunk, as you may say, and he didn’t know nothing at all the rest of the day, and preached a prayer meeting sermon that night that give him a rattling ruputation, because the oldest man in the world couldn’t a understood it.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Our roots are in the dark; the earth is our country. Why did we look up for blessing—instead of around, and down? What hope we have lies there. Not in the sky full of orbiting spy-eyes and weaponry, but in the earth we have looked down upon. Not from above, but from below. Not in the light that blinds, but in the dark that nourishes, where human beings grow human souls.”
—Ursula K. Le Guin (b. 1929)