HPS 0410 Einstein for Everyone Fall 2018

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Assignment 11: Big Bang Cosmology

For submission

1. The dynamics that drive standard relativistic cosmologies are somewhat hard to understand. It turns out that this relativistic dynamics is mimicked in several important aspects by some simple dynamical systems in Newtonian theory. Those systems consist of a quantity of matter concentrated into a point in an empty Newtonian universe. That point explodes violently throwing out fragments of matter in all directions, producing an expanding cloud of debris. In Newtonian gravitation theory, every fragment of matter exerts an attractive gravitational force on every other fragment. These attractive forces act to pull the fragments of the cloud back together, slowing the rate of expansion of the cloud of debris.

There are three different types of histories for the cloud, according to the energy of the initial explosion:

I. Low energy explosion. The energy of the explosion is not great enough to overcome the attractive forces of gravitation and the cloud collapses back onto itself under gravitational forces.

Newtonian cosmology 2

II. High energy explosion. The energy of the explosion is sufficient to overcome the attractive forces of gravitation. The fragments continue to move apart without limit. The cloud is spread more and more thinly over time and never collapses back to a point. Only a part of the total energy of the explosion is needed to overcome the attractive forces of gravitation. The remainder fuels a continuing rapid expansion.

Newtonian cosmology 1

III. Critical energy explosion. The energy of the explosion is the exact minimum needed to prevent recollapse. Over time all of the energy of the explosion is used up in counteracting the attractive forces of gravitation. The critical energy level lies exactly on the boundary between the energies of I. and II.

(a) Which Newtonian model is associated with which relativistic cosmology?

(b) While these Newtonian models are remarkably good in mimicking the relativistic dynamics, the Newtonian models differ from the relativistic cosmologies in several very important ways. What are they?

2. How does a non-zero, positive cosmological constant affect the dynamics of the Newtonian models?

3. Stephen Hawking's singularity theorem assures us that there was a big bang singularity in our past. The theory makes assumptions that have to be true for the result to hold. What are they?

For discussion in the recitation.

A. In In FLRW spacetimes, there is unique way slicing up of spacetime into hypersurfaces of simultaneity. How can we reconcile this with the relativity of simultaneity of special relativity?

B. What is the difference between a Doppler shift in the frequency of light and the reddening of light by the cosmological red shift?

C. In the Lambda-CDM model of cosmology, the so called "standard model," 70% of matter is the mysterious dark energy and 25% is the dark matter in galaxies. We have no independent evidence of the existence of these two forms of matter other than through their gravitational effects. Should this worry us?

D. How sure can we be that the assumptions of Hawking's theorem are correct?

E. Some theorists find a singularity, such as the big bang, an affront to science and feel a strong need to find reformulated theories that will eliminate them. Are singularities to be avoided or eliminated from theories if possible? Why?