Why are theoretical physics equations so simple?

 

I…beg your pardon?

I'd like to present to you:

That monstrosity there is known as the “Standard Model Lagrangian”. It's an equation that encapsulates everything we currently know about particle physics.

It's, um, not exactly simple is it?

Even when equations do look simple, it's often because they're hiding away layers of abstraction and difficulty.

The standard model is often written as:

This is somewhat shorter and nicer! However, all that's been done is package some terms up into some other terms.

Another example: the governing equation of Einstein's General Relativity is the following:

That's a beautifully elegant equation. So concise. You have spacetime on the left, and the stuff residing in spacetime on the right.

However, this only looks simple because I've packaged up the awfulness into several objects. I've got , the Ricci tensor, and I haven't told you what is….

If I were to write this equation in terms of , the spacetime metric that is the fundamental object of study you get the following:

Unable to parse this math expression.\tiny \partial_\rho \left(\frac{1}{2} g^{\rho \sigma} \left( \partial_b g_{\sigma a} + \partial_a g_{\sigma b} - \partial_\sigma g_{ab} \right)\right) - \partial_b \left(\frac{1}{2} g^{\rho \sigma} \left( \partial_a g_{\sigma \rho} + \partial_\rho g_{\sigma a} - \partial_\sigma g_{\rho a} \right) \right) + \left( \frac{1}{2} g^{\rho \sigma} \left( \partial_\lambda g_{\sigma \rho} + \partial_\rho g_{\sigma \lambda} - \partial_\sigma g_{\rho \lambda} \right)\right) \left( \frac{1}{2} g^{\lambda \sigma} \left( \partial_a g_{\sigma b} + \partial_b g_{\sigma a} - \partial_\sigma g_{ba} \right) \right) - \left( \frac{1}{2} g^{\rho \sigma} \left( \partial_\lambda g_{\sigma b} + \partial_b g_{\sigma \lambda} - \partial_\sigma g_{b \lambda} \right) \right) \left(\frac{1}{2} g^{\lambda \sigma} \left( \partial_a g_{\sigma \rho} + \partial_\rho g_{\sigma a} - \partial_\sigma g_{\rho a} \right) \right) - \frac{1}{2} g_{a b} \Big[ \partial_\rho \left(\frac{1}{2} g^{\rho \sigma} \left( \partial_c g_{\sigma c} + \partial^c g_{\sigma c} - \partial_\sigma g^c_{c} \right)\right) - \partial_c \left(\frac{1}{2} g^{\rho \sigma} \left( \partial^c g_{\sigma \rho} + \partial_\rho g_{\sigma }^c - \partial_\sigma g_{\rho }^c \right) \right) + \left( \frac{1}{2} g^{\rho \sigma} \left( \partial_\lambda g_{\sigma \rho} + \partial_\rho g_{\sigma \lambda} - \partial_\sigma g_{\rho \lambda} \right)\right) \left( \frac{1}{2} g^{\lambda \sigma} \left( \partial^c g_{\sigma c} + \partial_c g_{\sigma }^c - \partial_\sigma g_{c}^c \right) \right) - \left( \frac{1}{2} g^{\rho \sigma} \left( \partial_\lambda g_{\sigma c} + \partial_c g_{\sigma \lambda} - \partial_\sigma g_{c\lambda} \right) \right) \left(\frac{1}{2} g^{\lambda \sigma} \left( \partial^c g_{\sigma \rho} + \partial_\rho g_{\sigma }^c - \partial_\sigma g_{\rho }^c \right) \right) \Big] + \Lambda g_{a b} = \frac{8 \pi G}{c^4} T_{a b}\tag*{}

But even this grotesque monstrosity isn't everything, I have still made several simplifications to my notation:

1. I've used the Einstein summation convention, so , if I were to write all the sums in explicitly, this would be awful.
2. I've used where is the coordinate in my system
3. Again, I still haven't told you how to calculate

As physicists, we don't want to be dealing with nonsense like that all the time. We want to simplify our notation such that we can actually get stuff done.

Therefore, we define intermediary objects, which allow us to simplify things. In GR, we define the Christoffel symbols , using the metric and then the Ricci tensor using the Christoffel symbols. Each layer of abstraction hides a bunch of nasty mathematics away, so we don't have to think about it explicitly.

An example from my own theoretical work, is this equation, which deals with the self-gravity of two components of a model galactic disk:

Where , and are rotated coordinates.

Now — when I'm number crunching, I don't care about the specifics of how this quantity is calculated.

All I care about is that, in principle, it can be calculated. I can press a button on my computer and can get the value of this function at any point.

Once this is true, why would I ever bother with the full equation?

In every subsequent piece of mathematics, I'm going to call this quantity and get on with it.

Which of the following would you rather work with:

Or:

One of these looks simple. But they're exactly the same thing. And this isn't even conceptually a difficult integral — it's just the gravitational field of an oddly shaped object.

That's why theoretical physics equations sometimes look really simple and neat — it's because all of the grotesque mathematics is hidden away, tucked inside definitions in definitions in definitions.

Crack open one of these “simple” things, and you'd very quickly find yourself changing your mind!