Friedmann's Acceleration Equation

The second Friedmann equation describes the universe's rate of expansion, how quickly the expansion is speeding up or slowing down. In short the equation tells us that if there is matter present in the universe then either its expansion rate is decreasing or its contraction rate is increasing. Friedmann originally solved Einstein's field equations with the Robertson-Walker metric and found two solutions, one was the Friedmann equation and the second was an equation which when subtracted by the first Friedmann equation gave the second Friedmann equation. However I want to explain how I first learned about the acceleration which I believe is he most easy and accessible way to derive it. We still need to use the first Friedmann equation that I spoke about previously.

Now multiply each term of the equation by the scale factor squared and you get

If you differentiate this equation with respect to time

Then divide by

Then the equation becomes

Finally substitute in the following term from the fluid equation

This gives us the equation in its final form

Although there is no force associated with pressure in a universe described by the Roberston-Walker metric, i.e., one that's isotropic, as you can see from the equation if we were to increase pressure we would further decelerate the expansion. On occasion you will see cosmologists who have mass density replaced with energy density. If you use natural units where c = 1 these become interchangeable. Though this need not concern us here.

The interesting thing about this equation is that if pressure p is positive then I can make the right hand side of the equation positive (if I also make the left hand side negative), then the equation tells me that if there is any matter in the universe the left hand side cannot be zero. So in standard cosmology with the cosmological constant omitted the expansion of the universe must be slowing down (or we equivalently say the contraction is spreading up). Perhaps, though we might speculate that I n the early universe the material content is such that it made the pressure negative in which the expansion would rapidly accelerate. This is known as "inflation" and is accepted by most cosmologists, with still a few reasonable sceptics but we'll discuss that in another post. For now I'll leave it at that.

1. P.S. I've talked about the first Friedmann equation and the fluid equation in previous posts. For anyone wondering what these are.

2. Under the first equation, it should read "scale factor squared."

Margaret Thatcher's Legacy for Britain

The following is an adaptation of my thoughts at UCL's Conservative Society some months ago concerning the issue of the Conservative Parties vote of no confidence that lead to the resignation of Margaret Thatcher, her legacy for Britain and why she's so undeserving hated by the hard left.

When one enters parliament through members lobby there are four prime ministers commemorated and immortalized in statue form. The first of these figures, David Lloyd George seeded the beginnings of the welfare state, the second Winston Churchill served his tenure protecting us from physical annihilation during the Second World War, the third, Clement Attlee nationalized the health service and sought to drive Britain down the road of socialism and the fourth, the late Baroness Thatcher brought great economic revolution at the end of the Cold War.

It's been said of British politics that these last two figures though diametrically opposed were the only elections that ever really mattered. B…

Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…

'Don't boo Labour, vote Conservative!' #ImWithHer

"My pitch is very simple, I'm Theresa May and I believe I'm the best person to be Prime Minister"

In an election one doesn't always get the option of voting for their primary candidate, for me that's been the case here. Originally I had supported Michael Gove and then Andrea Leadsom for leadership of the Conservative party but on June 8th we're expected to choose between Theresa May and Jeremy Corbyn. Whatever you think of the two personally, the choice of who offers better governance couldn't be clearer.

The most notable part of Jeremy Corbyn's "leadership" has been his support for the outrageous and policies of the far left. He supports the unilateral disarmament of British nuclear weapons, while supporting the right of Iran to have its own unrestricted nuclear program. He's had an industrial policy to nationalize the mining of coal but not to burn coal, and supports self-determination for the people of Palestine but not for the p…