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This page contains links to some outreach activities I've done, a few random notes on some interesting bits of physics, and even more random links to useful resources.

Random posts about science

The below are notes on interesting things in astronomy and cosmology. No other rationale than they may be interesting and/or useful.

Why distances and times in Cosmology are hard

It is a truth universally acknowledged that ~~a single man in possession of a good~~ distances and times are hard to conceptualize when it comes to the Universe. There are different concepts of distance and time, and it is not always obvious which one to use. Below is a small, tale that illustrates some of the challenges: Back in 2005, a singer by the name of Katie Melua released a song called "Nine Million Bicycles". It did rather well. You can listen to it here, if you must. The second verse of this song goes like this:

We are 12 billion light-years from the edge,
That's a guess,
No one can ever say it's true,
But I know that I will always be with you.

If you read that and you are a normal person you probably get vaguely warm fuzzies. If you read that and you're a physicist then you are probably fluffing up in righteous indignation. But bear with me in either case. Shortly thereafter, an article appeared in The Guardian which expressed no small degree of frustration with this verse of the song for its scientific inaccuracy, commenting that we are not twelve billion light years from the edge of the observable universe, but 13.7 billion light years, and that the age of the Universe is a well measured number, not a guess. The article proposes a replacement verse, which reads:

We are 13.7 billion light-years from
the edge of the observable universe,
That's a good estimate with
well-defined error bars,
Scientists say it's true, but
acknowledge that it may be refined,
And with the available information, I predict that I will always be with you

Fewer warm fuzzies, but scientifically accurate? Well, no. The article contains at least one, and possibly two, misconceptions about distances and times in cosmology

The first is that the article (and Katie's song, to be fair) seems to treat a light year as a measure of time. This is wrong. A light year is a measure of distance, not time. One light year is, and here I'll be careful with my definition, the speed of light travelling in vacuum in (say) meters per second, times the number of seconds in a year. In other words, one light year is how far light will travel in a year, assuming that the fabric of spacetime is'nt doing anything wierd, like expanding. One light year is equal to 9.461 trillion kilometers, or 5.879 trillion miles, or 5.108 trillion nautical miles, or... well, you get the idea. Astronomers use "light years" as a measure of distance to get rid of all those trillions. They're distracting. So, when the revised lyric say "We are 13.7 billion light-years from the edge of the observable universe", with the intent that a "light year" means a duration of time, then this is at best deeply confusing. It would be like saying that a soccer pitch is three weeks across, with the justification that it would take an especially slothful snail about three weeks to get across one.

But... perhaps the article did mean a light year as a measure of distance? Aftr all, if the Universe is 13.7 billion years (not light years) old, and nothing can travel faster than light, then surely the distance to the edge of the observable universe is indeed 13.7 billion light years? In other words, the number 13.7 billion is both the age of the universe in years, and the distance to the edge of the Universe in light years?

Well, no, this is also wrong. The distance to the edge of the observable Universe is quite a bit more than 13.7 billion light years, due to the expansion of the Universe itself. As a ray of light scampers merrily across the Universe (I'm a physicist, trust me, light does scamper merrily), the underlying fabric of spacetime is expanding. This means that the ray of light travels a total distance equal to how far it would have gone if space were not expanding, plus the amount that space expanded by (I'm glossing over some subtleties there but they're not that important). A good analogy would be a car driving down a road that is itself expanding by (say) one mile in length every minute. If the car dries down this road for an hour, with its speedometer reading 60 miles perhour, then the total distance that car will travel after an hour is not 60 miles, but 120 miles. It's the same deal with light in the Universe. The Universe may be 13.7 billion years old, so a ray of light starting from the "edge" will have travelled for 13.7 billion years (well, very slightly less than this, but that's another subtlety we'll gloss over), but the total distane light travels in that time is much greater, due to the expansion of space, making the edge of the observable Universe much further away.

How much further? Well, this gets complicated as it depends on how you build the model that tells you exactly how space expanded since the Big Bang. Using recent estimates for this model, then a good number for the distance to the edge of the observable Universe is about 46.5 billion light years. So, a scientifically correct first line of the second verse might read: "We're nearly fifty billion light years from the edge"

A nice way to understand the Christoffel Symbol

When first learning General Relativity, a common stumbling block is when the Christoffel symbol is first encountered. The Cristoffel symbol, $\Gamma^{\alpha}_{\beta\gamma}$, is explained as "how do the basis vectors change as you move around in spacetime", or "To what degree is spacetime not flat", or "How does the coordinate system adopted relate to the underlying geometry". Slightly more directly, $\Gamma^{\alpha}_{\beta\gamma}$ is "how does the $\alpha$ basis vector change as a function of the $\beta$ basis vector as you move along the $\gamma$ basis vector". But this is not, necessarily, all that enlightening.

There is, however, a cnceptualy striaghtforward(ish) way to understand the Christoffel symbol, and in particular what those three indices mean in a physical sense. To do so, we start with the Hubble Constant, otherwise known as $H_{0}$, which tells us how rapidly the Universe around us is expanding. Or, if you prefer, how rapidly is the Universe currently expanding. Observationally, it is defined as how fast galaxies move away from us, divided by how far away those galaxies are from us. So, to measure it, we take a whole bunch of galaxies (or anything bright enough to see far away), measure their velocities away from us (this is basically redshift), independently measure their distances away from us, plot the two against each other, fit a linear relation, measure the slope of that relation. Ok I'm glossing over some subleties there, but that's the basic idea. So, the numerical value of the Hubble Constant is:

$H_{0} = \frac{v}{d}$

in which v is the velocity of the galaxies (or whatever object is being used), and d is their distances. The units of the Hubble Constant are thus "distance per unit time, per unit distance". Which just means "For every unit distance you move away from Earth, by how much does the velocity away from us change?" $70$ km s$^{-1}$ Mpc$^{-1}$, which means "for every Mpc you move away from Earth, the rate at which galaxies get further away from Earth increases by 70 km s$^{-1}$". The further away from Earth you go, the faster galaxies move away from Earth, in other words.

What has all this got to do with Christoffels? To make the connection (pun intended, if you are wondering "what pun" then don't worry about it, very nerdy joke, if you're thinking "these physicists and their in jokes" then okay fair) we first remind ourselves that, when interpreted through the lens of General Relativity, the positions and velocities of galaxies that we measure should not be interpreted in the usual "day to day" sense, but instead as measurements of what the underlying geometry of the Universe is doing. In other words, the movement of galaxies away from us arises because the geometry of the Universe itself is changing. The galaxies themselves are not moving relative to that geometry. When we measure v and d for a galaxy far, far away we are atually measuring what the geometry of the Universe is doing "there" (the quotes are to avoid spending several paragraphsto make that bit formally correct) and not how that galaxy itself is moving "there".

To progress further, some math is unavoidable. I'm not just saying that ecause I've been itching to throw equations around, it really is. Anyway, To describe the geometry of a spacetime within General Relativity, we often try to start with a function called a metric (often also called the absolute interval). The most commonly used metric for the Universe is called the Friedmann-Lemaitre-Robertson-Walker metric. Written as an absolute interval, it looks like this

$ds^{2} = -dt^{2} + a^{2}(t)\left(\frac{1}{1-kr^{2}}dr^{2} + r^{2}d\theta^{2} + r^{2} \sin^{2}\theta d\phi^{2} \right)$

and written as a metric, as a matrix, it looks like this:

$g_{\mu\nu} = \begin {pmatrix} -1 & 0 & 0 & 0 \\ 0 & \frac{a^2}{1-kr^{2}} & 0 & 0 \\ 0 & 0 & a^2r^2 & 0 \\ 0 & 0 & 0 & a^2r^2\sin^2\theta \end {pmatrix}$

For our purposes, the key part of this metric is the thing out from of the spatial part of the interval: $a(t)$ (it's square in the interval). This function tells us how "big" things are at a given time (more subtelty glossing there, but I don't have the paragraphs). Looked at this way, then our measurements of galaxy velocities and distances can be thought of as measurements of $a(t)$ and its first time derivative. Specifically, we can write:

$H_{0} = \frac{v}{d} = \frac{\dot{a}}{a}$

in which $\dot{a} = da/dt$

To see the relevance of this to Christoffels, there is no option but to look at how the Christoffel symbol itself is calculated. This is done directly from the metric. The general expression for doing so is

$\Gamma^{\beta}_{\mu\nu} = \frac{1}{2}g^{\beta\alpha}(\partial_{\mu}g_{\nu\alpha} + \partial_{\nu}g_{\alpha\mu} - \partial_{\alpha}g_{\mu\nu})$

which looks non-trivially unfriendly, but it's not quite as bad as it looks when you get stuck into it. Honestly! Decide what components of the Christoffel symbol you want to calculate, plug in the (components of the) metric you want to calculateit for, slog away fror a little bit, all done.

If we do all this fto calculate a specific component of the Christoffel symbol for the FRLW matric, we get this:

$\Gamma^{i}_{0i} = \frac{\dot{a}}{a}$

This tells us that this particular component of the Christoffel symbol is the Hubble Constant! Or, if I'm being preise, the Hubble parameter. In essence, the Hubble parameter can be thought of as $\Gamma^{km}_{s, Mpc}$. What's more, it is equally straightforward to show that

$\Gamma^{i}_{i0} = \frac{\dot{a}}{a}$

which neatly explains another sometimes tricky idea - the torison tensor. This tensor is defined:

$T^{\alpha}_{\mu\beta} = \Gamma^{\alpha}_{\mu\beta} - \Gamma^{\alpha}_{\beta\mu}$

In many instances, the torsion tensor is asserted to be zero, but looking at it through the lens of the Hubble Constant helps us see why, from a physical perspective. Zero torsion says, for the FLRW metric, that "for every Mpc further away, galaxies increase in recession velocity by 70 km per second" is the same as "Every seond, one Mpc gets bigger by 70 kilometers". Non-zero torsion would mean the first statement does not imply the second. Which would be, to put it mildly, odd.

Useful Links

Random posts about science

Why distances and times in Cosmology are hard

A nice way to understand the Christoffel Symbol