# Are there Galilean scalars?

In special relativity there are scalar quantities which are invariant under any Lorentz transformation, called Lorentz scalars. For example, the magnitude of the four-velocity is a Lorentz scalar. If we consider the usual Galilean transformations, however, magnitude of 3-velocity is not an invariant scalar anymore. Why is this? And are there some analogous "Galilean scalars" which are invariant under any Galilean transformation?

• and accelerations, forces. On the other hand, things like energy and work are generically frame dependent. – secavara Apr 28 '18 at 16:38
• @ZeroTheHero More precisely, the length of a displacement vector, but not the magnitude of a "higher-order" vector like velocity. Not sure whether that distinction was implicit in your use of the word "length" instead of "magnitude". – tparker Apr 28 '18 at 17:20
• @tparker good point. Yes I was thinking of $\Delta r$. – ZeroTheHero Apr 28 '18 at 17:36
• @ZeroTheHero: The length of a vector is not a scalar in Galilean relativity. What's invariant in Galilean relativity is the length of a vector connecting simultaneous events. – Ben Crowell Apr 30 '18 at 1:10
• @BenCrowell yes even more precise and correct statement. I better do away with mine as it will induce confusion, – ZeroTheHero Apr 30 '18 at 1:12

## 2 Answers

A key point in Galilean relativity is that time is a scalar: Everybody can agree on a single value of the time for an event. That, and the ideas of simultaneity that flows from it, seems to be so obvious that people don’t even think about it.

In special relativity, $p^2=m^2 c^2$ with $m$ the rest mass. In Galilean relativity, we can take the analogous invariant to be $m$. If you prefer a kinematic expression for it, we could choose $m = \frac{p^2}{2T}$, with $T$ the kinetic energy.

## protected by Qmechanic♦Apr 28 '18 at 17:54

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