# Formal Term for an invariant constant to all observers

I was thinking of the speed of light and realized I don't know how to quickly name the concept of "physical quantity that is measured to be the same in all reference frames". Are there examples of other such constants and how generally do we refer to these?

• I think it's just an "invariant". Other examples are the invariant mass and the charge of a particle. – knzhou Apr 16 '18 at 20:07
• In any sensible system of units for relativity, the speed of light is 1, and 1 is the same for all observers because it's just a number defined as part of the real number system. – Ben Crowell Apr 16 '18 at 22:34

They are generically called invariant quantities. Not to be confused with conserved quantities or scalars.

Particularly for the case of observers or reference frames, it is called Lorentz invariant or sometimes Poincaré invariant (which also includes Lorentz) quantities since the symmetry of the theory we are talking about is the Lorentz or Poincaré symmetry, respectively.

Actually, all kinds of symmetries have invariants but speed of light is a little bit different. It is a universal constant. It does not depend on a particle field or a region, on the contrary, charge or mass is a field-dependent quantity. Planck constant and Newton's gravitational constant are other examples similar to speed of light.

Coupling constants (i.e., $e$ the unit electric charge, $g$ the weak or $g_S$ the strong coupling constants, etc.) are also other examples.

Conserved quantities, on the other hand, are quantities that are not necessarily constant but particularly conserved with respect to time (or proper time). They can vary for different observers but not with time. For example energy is a conserved quantity but is not invariant from an observer to another. Alsı, charge of a particle field is a conserved quantity, that is, it does not change locally (time or space), but differs from field to field.

Invariant mass of multiple particles also is a conserved quantity, rather than an invariant quantity. Because it is the conserved quantity for the translation symmetry (a part of the Poincaré symmetry) which is similar to the electric charge. Also both electric charge and invariant mass depend on the object.

Scalars are not necessarily invariant for a different observer. For example a scalar field would change by a phase factor when we switch to a translated observer.

ANNEX: (edit)

As a matter of fact, there are also non-scalar Lorentz invariants. For example, the Kronecker delta, $\delta_\mu^\nu$, is a constant tensor (not a scalar) and it is invariant under Lorentz (or Poincaré) transformations.

Another non-scalar Lorentz invariant example is the Levi-Civita symbol (not the tensor), $\epsilon_{\mu\nu\alpha\beta}$. It does not change under any transformation because it is just made of 0's and ±1's. For that matter, note that the determinant of a metric is transformed as a scalar density because only the components of metric will transform while the symbol would not (otherwise it would be a scalar).

• Scalars are not necessarily invariant for a different observer. For example a scalar field would change by a phase factor when we switch to a translated observer. Here you seem to be making a case that invariants are a proper subset of scalars. However, there are also examples of Lorentz invariants that are not scalars. An example is the determinant of the metric, $\operatorname{det} g$. To see that it's not a scalar, note that under a coordinate dilation $(t,x,y,z)\rightarrow(kt,kx,ky,kz)$, we have $\operatorname{det} g\rightarrow k^{-2}\operatorname{det} g$. – Ben Crowell Apr 16 '18 at 22:42
• I am not saying that scalars are a subset of invariants or anything like that. I am saying that they are two different things and tried to make the distinction. By the way, indeed the determinant of metric is a scalar density but not a Lorentz invariant. Please read the edit I made to my answer. Thanks for the contribution. – Oktay Doğangün Apr 21 '18 at 20:19

A scalar is by definition the same in every reference frame.

• Is speed not a scalar? – WillO Apr 16 '18 at 22:53
• @WillO: If you click through to the WP link, you'll see that FrodCube means a relativistic scalar, which is defined differently than a scalar in the context of freshman physics. – Ben Crowell Apr 16 '18 at 23:44