ASTRONOMY 9: HISTORY OF COSMOLOGY

Assignment #15--Solutions

2000 April 5

1.

Hawley & Holcomb, p. 229, #2

Spherical (k=+1) Flat (k=0) Hyperbolic (k=-1)

Size Finite Infinite Infinite

Circle Circum. $<2\pi r$ $2\pi r$ $>2\pi r$

Sum of triangle $>180^\circ$ $180^\circ$ $<180^\circ$

# of parallels 0 1 $\infty$

2.

Hawley & Holcomb, p. 229, #3 Two observers are initially at rest with respect to each other, and the first then accelerates away from the second. Newton will say that there is an absolute space which defines the non-accelerating frames, so if the pair were initially not accelerating, then the first will feel a force, as he is the one who is accelerating against the backdrop of absolute space.

Mach will say that it is the overall distribution of mass throughout the universe which defines the non-accelerating frames, and so if the pair were initially not accelerating relative to the distant galaxies, then the first will again feel a force.

Newton doesn't care if the two are the only objects in the universe; the unchanging background of absolute space is still there. Mach, on the other hand, would say that neither of them will feel a force in this case, because there is no other mass to define the non-accelerating frames.

3.

If a tree falls in a forest with no one around, what happens? Discuss in terms of the ideas presented in the article on Schrodinger's cat in the reader. In the standard Copenhagen interpretation, Schrodinger's cat gets into this superposition state of being both alive and dead at the same time, and the tree might similarly get into a state of being both fallen and not-fallen. (Einstein famously asked another quantum physicist if he really believed the moon wasn't there when he wasn't looking.) Then when an observation is made, the tree picks a definite state to be in. So it hinges on what an ``observation'' actually is; the reader article discusses evidence that this could simply be any interaction with the environment, and a conscious observer is not required. Since the tree is interacting with lots of things around it, it remains in a boring state of classical normalcy, either fallen or not, but not both. In the many-worlds interpretation, the universe forks when an observation is made: one universe with a dead tree, and one with an alive one.

4.

Which is your favorite interpretation of quantum mechanics? Why? The results of this survey are that many-worlds is the winner with 53% of the vote, standard Copenhagen comes in a close second with 41%, and hidden variables a distant third with 6%. Interestingly, I have heard anecdotally that a poll of leading quantum physicists and cosmologists found that 58% of them chose many-worlds, but among scientists in general, more chose Copenhagen. So you may be closer to Stephen Hawking than you think!

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	Spherical (k=+1)	Flat (k=0)	Hyperbolic (k=-1)
Size	Finite	Infinite	Infinite
Circle Circum.	$<2\pi r$	$2\pi r$	$>2\pi r$
Sum of triangle	$>180^\circ$	$180^\circ$	$<180^\circ$
# of parallels	0	1	$\infty$