Invariant Spacetime Interval for Classical Spacetime

[AcidRainLiTE]

In special relativity we have the invariant spacetime interval ds² = dx² - c²dt². If we think about classical (non-relativistic) space and time as one spacetime in which the transformation between reference frames is given by the Galilean transformation, is there a corresponding spacetime interval that is invariant (the same when computed in any reference frame)?

 

[PeterDonis]

If we think about classical (non-relativistic) space and time as one spacetime in which the transformation between reference frames is given by the Galilean transformation, is there a corresponding spacetime interval that is invariant (the same when computed in any reference frame)?

No. There can't be, because the Galilean transformation leaves the time coordinate unchanged but changes the space coordinates. In other words, a Galilean transformation can change $dx^2$ between two given events, but always leaves $dt^2$ unchanged.

 

[AcidRainLiTE]

So then is it not possible to think of classical space and time as a single spacetime since we cannot assign a definite meaning to the distance between spacetime points (independent of reference frames)?

 

[WannabeNewton]

So then is it not possible to think of classical space and time as a single spacetime since we cannot assign a definite meaning to the distance between spacetime points (independent of reference frames)?

Actually it is possible and there is a lot of literature on the geometrization of Newtonian gravity and Galilean relativity. The difference is that instead of a single space-time metric, one has a time metric and a spatial metric which are separate from one another.

EDIT: And the reason for this is that Galilean relativity offers an absolute global time function that foliates space-time by the same simultaneity planes for all inertial observers whereas in SR the simultaneity planes foliating space-time are relativized to inertial observers.

 

[AcidRainLiTE]

The difference is that instead of a single space-time metric, one has a time metric and a spatial metric which are separate from one another.

So while there is some notion of distance in classical spacetime, you cannot speak of a distance between two arbitrary spacetime points, right? You can only speak of the distance between points on the same time slice (this would be the spatial metric) or points at the same position slice (this would be the time-metric). You cannot speak of the distance of points on different time slices at different positions. Is this correct?