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)
“Do not look on his appearance or on the height of his stature, because I have rejected him; for the LORD does not see as mortals see; they look on the outward appearance, but the LORD looks on the heart.”
—Bible: Hebrew, 1 Samuel 16:7.
“The moon shines bright. In such a night as this,
When the sweet wind did gently kiss the trees,
And they did make no noise, in such a night
Troilus methinks mounted the Troyan walls,
And sighed his soul toward the Grecian tents,
Where Cressid lay that night.”
—William Shakespeare (1564–1616)
“The sky sinks in the morning, this fact has been insufficiently observed.”
—Samuel Beckett (1906–1989)