### Lightcone Structure of Spacetime

I want to start talking about some of the more philosophical aspects about physics but it won't be possible without going over important stuff. For this post I want to explain the lightcones that physicists use in relativistic physics. If you're already familiar with the concept of natural units you can go ahead and skip to the part labeled "lightcone structure of spacetime" and my feelings won't be hurt. But it may also help to watch Sabine Hossenfelder's video first. These are highly important to pin down in order to understand any branch of theoretical physics and what I discuss here is no exception.

Lightcone Structure of Spacetime

In mathematics and physics the metric of spacetime allows us to calculate the shortest distance between two points, the famous metric equation for Euclidean geometry and space is is the Pythagorean theorem for a flat two dimensional triangle

The theorem can be extended in three dimensions by adding a third dimension of space for a 3-dimensional universe but in special relativity this is still not enough. Space and time are combined into a single entity called "spacetime" so we need to modify the metric to include a variable for time as we have one dimension of time and three dimensions of space. We can easily do this but time is distinct from space. So our operation should be a negative rather than a positive sign

$(ds)^2 = (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2$

For a geodesic in a spacetime described by this metric the proper time between two points is taken to be the integral

In special relativity the metric only describes flat spacetimes and doesn't add the scale factor or take into account any gravitational effects. For particles moving much slower than the speed of light their motion is well approximated by the physics of Galileo and Newton. But according to general relativity, however special relativity is wrong globally but correct in local regions of spacetime. In infinitesimal regions our spacetime is described as "Minkowskian" and described by the metric given above. This metric allows for three possible solutions

$\Delta s > 0$

In this case we interpret the two events as being separated through a distance just like ordinary Euclidean geometry and we say they are "space like" separated.

In this case our solution is less than zero and what's under the square root is an imaginary number. So we flip the sign and interpret the answer as a time value and we say these events are "time like" separated from one another.

In this case the events are not separated through time or space and we say they are "light like" separated. These different possible values are connected as different regions of a lightcone, the future and the past lightcone are regions which are time like separated from us whereas the region outside of the lightcone are regions which are space like separated from us. Only events which you can send or receive a light ray from are included in your lightcone. For practical purposes we ignore the present.

Any three dimensional space like surfaces in which the elements of the collection is labeled by real numbers and increases steadily from one surface to the next can be used to define what is "present" what is "past" and what is "future" relative to your reference frame. Each event on one surface is simultaneous to the events on that same surface. This kind of structure might lead one to believe that time is a block rather than just the present but that's a rather controversial topic for another time.

### 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…