March 26, 2010

Big Bang for beginners-11: Relativity theory

(My latest book God vs. Darwin: The War Between Evolution and Creationism in the Classroom has just been released and is now available through the usual outlets. You can order it from Amazon, Barnes and Noble, the publishers Rowman & Littlefield, and also through your local bookstores. For more on the book, see here. You can also listen to the podcast of the interview on WCPN 90.3 about the book.)

For previous posts in this series, see here.

So far I have been simply describing what the Big Bang theory says without giving much of the theoretical background. But Einstein's General Theory of Relativity (like Darwin's theory of evolution by natural selection) has had such a profound effect on our relationship with the rest of the universe that I feel obliged to give readers, at least for cultural purposes, a glimpse of what the theory is and why it is so powerful, even if it remains obscure in its details. So for the sake of greater completeness and for the benefit of those who want to know more, in this post and the next I will give some of the theoretical background to what I have been saying so far, and hope that even those who are averse to algebra will stick with me through it and get some of the flavor of how the theory works.

A word of caution, though. This is not my field so I cannot guarantee that this is error-free or state-of-the-art knowledge. My goal here is to give a simplified understanding of how the important field of cosmology operates. In order to provide a narrative I will largely ignore the fact that this is a field in which there are spirited debates and disagreements over many of the details. I strongly recommend reading more authoritative works by real scholars in the field for a more complete understanding of all the alternative points of view.

The basic paradigm that the field of cosmology operates under is Einstein's General Theory of Relativity which generates the Einstein Field Equations:

Rij - (1/2)Rgij = (8πG/c4)Tij - Λgij

Without worrying too much about what each individual term means, the main idea is that the terms on the left of the equal sign (Rij and R) represent the curvature of space while the terms on the right (Tij and Λ) represent the mass and energy in the universe that causes this curvature. The quantity Tij is called the stress-energy tensor and in it is contained all the information about how all the mass and the 'normal' energy (i.e., excluding dark energy) is distributed throughout all space. Λ is what is called the cosmological constant and determining its value that has been the source of all the excitement within the last two decades. The quantity gij is called the 'space-time metric' and defines how space and time are related. So the above equation represents the fundamental relationship between the mass-energy of the universe and the curvature of space.

G is the universal gravitational constant and c is the speed of light and since these are such fundamental and important quantities, they have been measured with great precision and are found to have the values G=6.67x10-11Nm2/kg2 and c=3x108m/s. (For the most up-to-date and comprehensive compilation of data, see the work of the Particle Data Group at Lawrence Berkeley Laboratory, which has a section on astrophysics and cosmology that contains a very useful data table.)

If we treat the universe on a large enough scale as if all the mass and energy is homogeneously spread out (like a uniform gas or liquid) and ignore the clumping on small scales that make up the stars and planets, the equation above simplifies considerably by mathematics standards, although it is still difficult to solve. In that case, Λ is related to the density of the energy (ρΛ) of the 'vacuum' by Λ=(8πG/c2Λ, and it is this vacuum energy that is referred to as dark energy and is driving the accelerating expansion of the universe. The vacuum of space used to be considered as inert 'empty' space, but that is no longer the case.

The total energy density of the universe ρ is thus made up of what we might call matter density ρM (comprising regular matter such as protons, electrons and the like, plus electromagnetic energy and dark matter), and the energy density associated with dark energy. i.e., ρ=ρM&Lambda.

The critical density ρc that we encountered earlier and that determines the curvature and ultimate fate of the universe is something that we can calculate theoretically and is given by the expression ρc=3H2/8πG, where H is the Hubble constant (more about this and how it is measured in the next post). So &Omega=ρ/ρc, where Ω>1 gives us a positive curvature and a universe that will eventually stop expanding and start contracting, Ω<1 gives us an open universe that will expand forever, and Ω=1 gives us a flat universe that will also expand forever.

Hence &Omega = ρ/ρc = (ρM + ρ&Lambda)/ρc = ΩM + ΩΛ,

where ΩM = ρMc and ΩΛ = ρΛc.

The results obtained from the WMAP satellite say that the density of our universe is currently exactly equal to the critical density thus making Ω=1.0, and is made up of 4.6% 'ordinary' matter and energy, 23.3% dark matter, and 72.1% dark energy. This means that our current best estimates are that ΩM=0.28 and ΩΛ=0.72.

Note that since we know the values of G and H (more on this in the next post), the value of the critical density ρc=3H2/8πG can be calculated and it works out to be 1.0x10-26kg/m3. This is an extremely small number reflecting the fact that the universe is mostly empty space. This highly dilute distribution is one major reason why it is not easy to directly detect things like dark matter and dark energy.

When it comes to calculating the total energy density of the universe, the dark energy is added up with the other energies from ordinary matter and dark matter. But unlike those other forms of energy, its effect on cosmic expansion is to push outwards and increase the rate of expansion of the universe, and not pull on it and slow it down.

In those particular inflationary models that assert that Ω will always equal 1.0 for all time, since ΩM gets less as the universe expands and gets more dilute, the value of ΩΛ must increase with time to keep Ω=1, so that the outward pressure will ultimately win out over the gravitational attraction. In this model, we live in essentially a runaway expanding universe, with everything moving away from everything else with increasingly rapid speeds.

In fact, these theories suggest that the universe is expanding so rapidly that galaxies are disappearing from sight over the far horizon so we will see less and less of them as time goes by. So if we had happened to come along a hundred billion or so years later than we did, the only things we would see in the night sky would be the merged result of own Milky Way and the Andromeda galaxy, which are predicted to collide in the future. The sky would be really boring because the rest of the sky would be dark and people would have thought that there was nothing else in the universe. We would not have had the vast amounts of observational data that we have now that enable us to learn so much by making all these great inferences.

Lucky us!

Next: Measuring the universe.

POST SCRIPT: Mr. Deity has a better equation than Einstein's one


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"Rij - (1/2)Rgij = (8?G/c4)Tij - ?gij"

So much for Big Bang for beginners.

Posted by thoreau on March 26, 2010 10:26 AM

"The sky would be really boring because the rest of the sky would be dark and people would have thought that there was nothing else in the universe."

Poets and ancient mariners alike are happy this isn't the case :)

Posted by tim on March 27, 2010 02:19 PM