General Relativity and Expansion
![[url=https://simple.wikipedia.org/wiki/Hercules–Corona_Borealis_Great_Wall#/media/File:Hubble_image_of_MACS_J0717_with_mass_overlay.jpg]"MACS J0717.5+3745"[/url] by NASA/ESA is licensed under [url=http://creativecommons.org/licenses/by/3.0]CC BY 3.0[/url]
A Hubble composite image of one of the largest structures ever seen. It approached 10 billion light years across.](https://www.geogebra.org/resource/Z35vW3Jf/Mv8MaEZ2bEx5bAPJ/material-Z35vW3Jf.png)
Universal Expansion and a Projectile
The most widely-held model posits that universe has been expanding outward ever since it began with the big bang. I wish to discuss the nature of this expansion with you. As a starting point, consider a projectile (drag-free) being launched outward from earth's surface at a certain speed v. The total energy of the projectile is the combination of the kinetic and potential energies, which gives
where the first term is the kinetic energy written with velocity as a derivative (for a good reason which you'll see shortly) and the gravitational potential energy which you'll recall is negative for universal gravitation.
Even if the earth were to explode and if the pieces were to spread out and chase after the projectile this equation will still be true so long as no parts overtake the projectile. This is a consequence of Gauss's law which is used in electrostatics.
As in electrostatics, where we often replace charge by charge density times volume, or we will do the same here. Therefore,
The idea of the exploding planet is that it remains spherical, but spreads out in space with assumed uniform density . The outer edge of that exploding planet is at the radius r of the original projectile. I imagine you see where we're going with this. We are looking for a way to understand not an exploding planet, but the universal expansion. The equation above can be rearranged to look like the following with some simple algebra:
General Relativity
Among other things, Einstein's equations of general relativity make predictions about the structure of space and the effect that gravitation has on it. Einstein's equations of general relativity are enormously complex to solve, but if we assume that space is homogenous (uniformly distributed) and isotropic (same in all directions), then Einstein's field equations reduce to an expression very similar to the one we see above for our exploding earth scenario. We will address these assumptions later in this section.
This equation was first derived by Alexander Friedmann in 1922, and allow us to briefly discuss physical cosmology, which is the study of the origin, structure, evolution, and ultimate destiny of the universe. The topic has historically been in the arena of philosophy and theology. It should be noted that the study of the origin of the universe is still in that arena, while physics is really only, by definition, capable of describing events within our universe once it exists. Physics has no intention of, and is incapable of discussing fundamental origins. Once a universe has formed, then physics may speak intelligibly about its progression and about physical eschatology - or the study of the ultimate destiny of the universe.
The Friedmann equation may be written:
The symbol R here is called the scale factor and is a measure of the size of the universe. The term K (not kinetic energy) is like the total energy E from our projectile. In this context it's called the curvature of space. K>0 is positive curvature, K<0 is negative curvature and K=0 is a flat universe. Just as with the projectile, when E>0 the projectile never returns but rather keeps expanding forever. When E=0 it does as well, but becomes static and infinite, and when E<0 the ball would eventually fall back down.
In the universal analogy K<0 would mean falling back down, and would reverse the big bang and create what's referred to as a big crunch, while both K=0 and K>0 would lead to a universe never to contract - thus making the process a one-time event in history. All the cosmological data we have points to the latter conclusion.
The Cosmological Constant
When looking at the Friedmann equation above, there is a density term for the universe which we should expect to drop as the universe grows in size. Thus the density should really depend on R. We can define it based on empirical values measured today. Since R itself is just a scaling factor, it is convenient to set it equal to 1, so that when R=1, where is the current density of the universe. As you can see, if the universe were to continue expanding to twice its current size (R=2), it would imply the universal density drops by 2 cubed, or by a factor of 8 - which makes sense.
While the equation seems complete, Einstein, as mentioned in the last section, added a term to his field equations to force them to predict a static universe (it is not). This term is called the cosmological constant. While it was a bad idea at the time since it was one introduced really to satisfy a philosophical bias rather than a scientific one, it has come back into favor in recent years, but for a different reason entirely.
Careful measurements of the universal expansion done in recent years seems to indicate not just that the universe is expanding, but is accelerating as it goes. The cosmological constant and the associated topic of dark energy represents a factor in the Friedmann equation to promote such expansion. Thus we add another term that acts like an undiluted density term independent of the size of the universe. The equation then becomes
where the last term represents the cosmological constant. I wish to solve this equation numerically with you, and a common thing to do is to simplify the expression by assuming not just that but also Lastly, the constants all get grouped and the equation becomes
The idea of the cosmological constant, or an undiluted vacuum density in the universe might trouble you. One way to make sense at least of its possible existence is to realize that perhaps the measure of smallness is independent of the size of the universe or of space-time. If Planck's constant is really a constant then even as the volume of the universe grows we should expect that each unit of volume may contain a constant energy density associated not with the universe's dimensions but with some absolute measure of smallness, which is Planck's constant. While this sounds sensible, it is nonetheless speculative.
Some Additional Comments
The assumptions about isotropy and homogeneity together form the cosmological principle which states that viewed on a large enough scale that the universe would appear the same to any observer. It should be noted that while the equations may become tenable with the assumptions of homogeneity and isotropy, that we shouldn't use them if they are not true. So is the universe isotropic and homogenous? Homogenous means an equal and uniform distribution of matter on a scale much smaller than the universe itself. Isotropic means that there is no preferred direction for any phenomenon.
The universe is generally assumed homogenous on scales of around 250 million light years, but several structures have been seen that are nearly 40 times larger than that scale, casting doubt on the cosmological principle. For example, see the Hercules-Corona Borealis Great Wall which extends around 10 billion light years across! A picture of a similar structure is at the top of this page.
The universe is nearly isotropic, but according to recent measurements by the ESA Planck mission of the cosmic microwave background - which is the afterglow of the big bang - it has statistically significant anisotropies. So the principle isn't rock solid by any means.
While I am not a cosmologist by training, my opinion is two fold: On the one hand I side with Karl Popper who was one of the best philosophers of science in the 20th century who criticized the principle on the grounds that it makes "our lack of knowledge a principle of knowing something". Forming a principle out of lack of understanding is certainly not the best way forward. It is true that without the cosmological principle our level of understanding about physical cosmology would be in its relative infancy.
On the other hand, what the principle allows is a starting point to seek understanding about our universe, and studying this wondrous place is not a bad idea. I would just caution you on taking any results too seriously if they rely on symmetries that can't be demonstrated.
Lastly, the anisotropies measured by the ESA are admittedly small, and while the inhomogeneous structures are not, most likely the results obtained by using the assumption of homogeneity are still rather accurate for most purposes. And how boring would it be if we had all the answers already?