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 the, appearance of, appearance, night and/or sky:
“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)
“A bureaucracy is sure to think that its duty is to augment official power, official business, or official members, rather than to leave free the energies of mankind; it overdoes the quantity of government, as well as impairs its quality. The truth is, that a skilled bureaucracy ... is, though it boasts of an appearance of science, quite inconsistent with the true principles of the art of business.”
—Walter Bagehot (1826–1877)
“The true, prescriptive artist strives after artistic truth; the lawless artist, following blind instinct, after an appearance of naturalness. The one leads to the highest peaks of art, the other to its lowest depths.”
—Johann Wolfgang Von Goethe (1749–1832)
“A baby nurse is one that changes diapers and loves ‘em dearly. Get up at all hours of the night to give ‘em the bottle and change their pants. If the baby coughs or cries, you have to find out the need. I had my own room usually, but I slept in the same room with the baby. I would take full charge. It was twenty-four hours. I used to have one day a week off and I’d go home and see my own two little ones.”
—Ruth Lindstrom (c. 1892–?)
“The sky sinks in the morning, this fact has been insufficiently observed.”
—Samuel Beckett (1906–1989)