# What is the difference between a physical constant, a scalar, an invariant, and a conserved quantity?

I don't really know how to properly articulate this question. This question popped into my mind when pondering why the fact that a physical constant like the speed of light doesn't have an associated symmetry even though it's "conserved" in every frame of reference.

A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. Well-known examples include the speed of light (c), elementary charge (e), and Planck's constant (h). (More about physical constants here.)

A scalar (in the context of physics) is a physical quantity that can be described by a number and (generally) a unit of measurement. Examples include 41 lbs, 52 km/hr, and 96 degrees F. This is not to be confused with a vector, which includes both the quantity and direction. It should also be noted that a scalar is unchanged by rotations or reflections in Newtonian mechanics and unchanged by Lorentz transformations in relativity. (More about scalars in the context of physics here.)

A conserved quantity is a function of the dependent variables that is a constant (aka, conserved) along each trajectory of the system. Most laws of physics express some kind of conservation, meaning that conserved quantities are pretty common in mathematical models of real systems. As an example, any classical mechanics model will have energy as a conserved quantity as long as the forces involved are conservative. In the context of Noether's Theorem, the Noether charge and the Laplace-Runge-Lenz vector are both conserved quantities. If you look at this website about Noether's Theorem, conserved quantities are mentioned in more detail. (More about conserved quantities here.)

As for what you're pondering, I'm not sure I understand what you mean. If you could clarify what you mean by associated symmetry (sorry if that's a dumb question) that'd be helpful.

Hope this helps!

• Thank you for the answer! I meant associated symmetry in the sense of Noether's theorem. – Dargscisyhp Jul 8 '16 at 21:06
• @Dargscisyhp, my apologies! I'll update my answer. – heather Jul 8 '16 at 21:50
• @Dargscisyhp, I've updated my answer based on dmckee's and your comments. – heather Jul 8 '16 at 22:20
• A conserved quantity is a quantity whose value remains the same over time.
• An invariant, or scalar quantity is a quantity whose value is the same in all reference frames.

These two properties are completely independent. Energy is conserved but not invariant. Mass (i.e. $E^2 - c^2 \mathbf{p}^2$) is invariant but not conserved. Charge is both, and lots of things are neither. All physical constants are invariant, otherwise they wouldn't be worth calling constants.

You are wondering if the invariance of the speed of light leads to a conserved quantity. The answer is no, because Noether's theorem deals with symmetries, not invariant quantities.

However, there is a deeper sense in which you are right. The invariance of the speed of light is one feature of Lorentz symmetry, and this symmetry does produce conserved quantities! Since Lorentz transformations include rotations, three of them are just angular momentum. The three boosts produce conserved quantities that are, heuristically, the "speed of the center of mass". A more detailed derivation can be found here.

• Based on your definitions here, wouldn't a physical constant be both an invariant and a conserved quantity? For instance, the speed of light doesn't change over time (to the best of my knowledge), so why is it not also considered a conserved quantity? – Dargscisyhp Jul 8 '16 at 21:08
• @Dargscisyhp Conserved quantities are properties of the dynamics of a system, e.g. "this block and spring have $3 \mathrm{J}$ of energy". Physical constants are not associated with any system, they're part of the underlying theory. You build an invariant speed into spacetime itself, not into anything occupying it. – knzhou Jul 8 '16 at 21:17
• On a deeper level, though, the fact that a specific light beam will keep moving at speed $c$ forever is actually a consequence of the conserved quantities arising from Lorentz symmetry, as I hinted at in my answer. But this is completely different from the usage of $c$ to mean the invariant speed in SR. – knzhou Jul 8 '16 at 21:19