# What is metric (invariant) in Newton mechanics (equivalent to spacetime interval of Minkowski space)?

The answer might be obvious for those with much experience, but I could not get it via web search.

https://en.wikipedia.org/wiki/Minkowski_space

From the second postulate of special relativity, together with homogeneity of spacetime and isotropy of space, it follows that the spacetime interval between two arbitrary events called 1 and 2 is:

$${\displaystyle c^{2}\left(t_{1}-t_{2}\right)^{2}-\left(x_{1}-x_{2}\right)^{2}-\left(y_{1}-y_{2}\right)^{2}-\left(z_{1}-z_{2}\right)^{2}.}$$

The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of

$${\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}}$$

If we go back from second postulate (principle of the constancy of light speed) back to Newtonian, what is equivalent formula for space-time interval (metric) will be?

For context: I'm trying to assess correctness of physics in sci-fi https://en.wikipedia.org/wiki/Orthogonal_(series) where author tries to describe universe with (as I understood it) space-time interval of (changed Metric signature of "our" Universe of (1,3) to (4,0), also see http://www.gregegan.net/ORTHOGONAL/00/PM.html):

$${\displaystyle c^{2}\left(t_{1}-t_{2}\right)^{2}+\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}+y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}.}$$

Thinking about it I realized I could not even "guess" what the metric is in Newtonian mechanics.

• It is useful to remind oneself what the physical content of 'invariance' is. The surprising bit about special relativity, for the uninitiated, is exactly that things we naively consider invariant (because we move at such a low speed compared to $c$) are not. These things are time intervals and length measurements. Commented May 23, 2023 at 8:55
• The metric formulation of Newtonian mechanics can be found as en.wikipedia.org/wiki/Newton%E2%80%93Cartan_theory Commented May 23, 2023 at 8:59
• @Slereah, thanks. Your link says "defines two (degenerate) metrics", also I've found now physics.stackexchange.com/questions/368680/… where the accepted answer is "A single metric does not work, because the speed of light is infinite. ". I suspect my question is in essence duplicate of that. Commented May 23, 2023 at 9:05
• After some more reading (e.g. en.wikipedia.org/wiki/Galilean_invariance) I don't see Newtonian mechanics / Galilean relativity postulate infinite speeds (e.g of light), so I don't see how accepted answer to physics.stackexchange.com/questions/368680/… is correct. Commented May 23, 2023 at 10:48
• Commented May 23, 2023 at 12:39

In relativistic mechanics we have 4-dimensional space-time and the Minkowski metric $$(\Delta s)^2 = c^2(t_1-t_2)^2-(x_1-x_2)^2-(y_1-y_2)^2-(z_1-z_2)^2$$ which is invariant to Lorentz tranformations (containing boosts and space-rotations), or more generally to Poincaré transformations (containing boosts, space-rotations and space/time-translations).

On the other hand, in Newtonian mechanics we have 3-dimensional space and 1-dimensional time, which are two completeley separate concepts. So here we have two separate metrics for space and time. The space-metric is $$(\Delta\vec{r})^2 = (x_1-x_2)^2 + (y_1-y_2)^2+(z_1-z_2)^2$$ and the time-metric simply is $$(\Delta t)^2 = (t_1-t_2)^2.$$ These two metrics are invariant to Galilean transformations (including boosts, space-rotations, space-translations and time-translations).

A useful way to parametrize the cases is by writing $$\Delta \vec S \cdot \Delta \vec S= \Delta t^2 - E (\Delta x^2 +\Delta y^2+\Delta z^2),$$ where $$E=-1$$ for Euclidean space with signature $$(+,+,+,+)$$ and $$E=+1$$ for Minkowski spacetime $$(+,-,-,-)$$, and $$E=0$$ for a Galilean-Newtonian spacetime $$(+,0,0,0)$$.

For constant squared-interval, the $$E=-1$$ case is a 3-D hypersphere, the $$E=+1$$ case is a double-sheeted 3-D hyperboloid, and the $$E=0$$ case is a 3-D hyperplane.

• As noted in the comments to your question, the Galilean-Newtonian case has a degenerate metric (since the determinant is zero). It has two degenerate metrics with signatures $$(+,0,0,0)$$ [for the temporal metric] and $$(0,+,+,+)$$ [for the spatial metric]... and these are 4-dimensional metrics for a 4-dimensional vector space.

In the spacetime-cases, $$E$$ can be associated with a maximum signal speed, which is finite for special relativity and infinite for Galilean relativity.

(related: Euclidean space to Minkowski spacetime )

• While "infinite maximum-signal speed" is implicit in Galilean relativity,
historically, it was not featured like the "speed of light" postulate for special relativity. If it were, I would think that we would have found special relativity sooner than 1905.

(In https://www.physicsforums.com/threads/why-is-minkowski-spacetime-non-euclidean.1016402/post-6647528 , I argue that Felix Klein had the mathematical structure in place to propose Minkowski spacetime geometry... but not the physical intuition or motivation to do so.)

• Of course, the metric does more than assigning square-intervals to displacements. It also defines the notion of "orthogonality" between two vectors, via the dot-product (gotten from the polarization identities).