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:
“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)
“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)
“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)
“Now days are dragon-ridden, the nightmare
Rides upon sleep: a drunken soldiery
Can leave the mother, murdered at her door,
To crawl in her own blood, and go scot-free;
The night can sweat with terror as before
We pieced our thoughts into philosophy....”
—William Butler Yeats (1865–1939)
“The sky it seems would pour down stinking pitch,
But that the sea, mounting to the welkin’s cheek,
Dashes the fire out.”
—William Shakespeare (1564–1616)