mitchell vitez

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Semi-Conserved Quantities

In physics, a conserved quantity is one that doesn’t change over time. A classic example is energy: you can transform it (say, from motion into heat), but can’t add to or subtract from the total energy of the universe. Energy isn’t alone, however. There are many other conserved quantities like mass, momentum, and charge. In fact, Noether’s Theorem tells us about many kinds of systems that necessarily have associated conservation laws, and it’s quite a nice result in physics.

This post is about a slightly different kind of quantity which I’ll call “semi-conserved”. These are quantities that do change over time, but change either so slowly or in such a balanced way that we can approximate them as conserved over short periods of time. These include many things that are relevant to humans: money, social status, or even moral values.

We know the value of money changes over time. In fact, we expect it to due to inflation. It’s important that it does grow, because it reflects a growth in the underlying economic value being generated. It’s also important that it doesn’t grow too fast, or we get all the negative effects of hyperinflation.

In general, an interesting thing to think about is what would happen if any of these semi-conserved quantities moved either towards being very close to truly conserved, or much farther away. Towards conservation, we see a reduced ability of the quantity to reflect major sweeping changes, but towards nonconservation we see volatility that again doesn’t accurately reflect the slowish near-constant growth of many of our major systems.

In some vaguer sense, many of our scientific models of the world are semi-conserved. For example, the shape of the universe could be considered to be something that’s semi-conserved. We know that spacetime curves around massive objects, but it’s often useful to use the approximation that spacetime is flat over large scales. Likewise, conservation is a useful approximation for any semi-conserved quantity at the right scale, but it’s also good to keep in mind the situations where these kinds of assumptions break down.

Of course, this leads us to the idea that some of our models of conservation fail to be themselves conserved. For example, the principle of conservation of mass isn’t strictly true. The true conservation (so far as I know) is of mass-energy, because according to \[ E = m c^2 \] they are equivalent and can be transformed into each other.

Is there anything we can be truly certain is conserved, and won’t turn out in a later model to be somewhat variable? This is sort of like asking how we can be sure of anything at all. Short of the most basic philosophical statements (e.g. “I think, therefore I am”), we’re probably out of luck. Despite this, we don’t need that kind of foundation to assign relative levels of correctness to our models based on the amount of evidence we acquire. It’s quite helpful to model mass as being conserved even if that’s not strictly true, because often we just don’t need insanely high levels of precision. I can’t be 100% sure the sun will rise tomorrow, but I’d still be willing to bet large amounts on it.

This notion of “conservation”, then, exists on a spectrum. We can list a bunch of quantities, from most to least conserved by some arbitrary metric.

Conservation isn’t necessarily a good or bad thing! While it’s probably good news for our everyday lives that mass isn’t popping in and out of existence all the time, if social standing were strictly conserved there’d be a lot less reason to try to improve.

I’ll end with a few questions that seem interesting.