Metric Signatures, Choice, and Reflection

The point of this post is two-fold: First, I've just tried to implement MathJax in the site, and I'm gonna try to test it out. If you're seeing this, it probably worked. Secondly, I've been thinking a lot about conventions in physics, how they're useful, when they're not, and what we gain by thinking about them. In a more abstract sense, the line of thought leads (or at least has lead me) to the unsurprising but important realization that the order in which learn things can substantially effect our tastes, and not always benignly.

So I've been taking a course called "Quantum Field Theory for Cosmology" at Perimeter this semester. It's a spiritual successor to a course I took the previous semester, which focused on general relativity. In both of these courses, we've got this thing called a "metric", which is essentially a way of telling us how far away certain events are in spacetime. In Minkowski (flat) spacetime, we've got two ways of representing the metric. In general relativity and quantum gravity, the convention is: $$\eta^{\mu \nu}= \eta_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$$ and in quantum field theory and particle physics, the convention is $$\eta^{\mu \nu}= \eta_{\mu \nu}= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{pmatrix} $$

It's important to note for the non-specialist that these will give you the same physics, as long as you're consistent in your choice. I'll give an example: take the Lagrangian density of a real scalar field- $$\mathcal{L} = \frac{1}{2}\eta_{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi - \frac{1}{2}m^2\phi^2$$ This is going to look a little different depending on our choice of metric: in the first, it's going to be $$-\frac{1}{2}\dot{\phi}^2 + \frac{1}{2}(\nabla\phi)^2- \frac{1}{2}m^2\phi^2$$ and in the second, $$\frac{1}{2}\dot{\phi}^2 - \frac{1}{2}(\nabla\phi)^2- \frac{1}{2}m^2\phi^2$$ Solving the equations of motion, $$\frac{\partial\mathcal{L}}{\partial\phi} = -m^2\phi \quad , \quad \frac{\partial\mathcal{L}}{\partial( \partial_{\mu}\phi)} = (-\dot{\phi}, \nabla\phi)$$ Very similarly, in the second metric, $$\frac{\partial\mathcal{L}}{\partial\phi} = -m^2\phi \quad , \quad \frac{\partial\mathcal{L}}{\partial( \partial_{\mu}\phi)} = (\dot{\phi}, -\nabla\phi)$$ These both lead to very similar Euler-Lagrange equations: $$\ddot{\phi} - \nabla^2\phi+m^2\phi = 0$$ For the negative signature, or $$-\ddot{\phi} + \nabla^2\phi+m^2\phi = 0 $$ for the positive. These both lead to, in relativistic notation, identical reductions to what is termed the Klein-Gordon equation: $$(\partial^{\mu}\partial_{\mu} + m^2)\phi = 0$$ So they both work. I first learned the first one, and so in the tribal way that people tend towards, I decided it was the best way to do things and that the other way wasn't as good. But they both have their strengths and their weaknesses, but only really in the aspects that become more obvious depending on the sign. For the $(+,-,-,-)$ convention, you get timelike 4-vectors with positive magnitude. For $(-,+,+,+)$, you keep familiar 3-d formalism. You also don't have to worry about your determinant flipping in a generalized dimensionality case. It's an aesthetic choice, and (as far as I can tell) nothing deper. I still prefer the first one, but only because I'm used to it.

The point I'm trying to make, and I swear there is one, is that I've been thinking a lot these past couple of days about how much of what we learn is stylistic in nature rather than substantiative, and the interrelation of style and substance. No serious conclusions yet, but I'm going to use this example as a reminder to be more open-minded about ways to frame information, ways of knowing, and ways of understanding.