Up to the work of Riemann and Gauss, this definition would have made clear to me examples of geometrical objects: a line, square, cube, hypercube; each of these possessing geometrical properties such as the number of sides, angles between faces, the dimension of space that contains them etc. Hence a geometrical object was a set of measurements associated with an object using a ruler.

Mariyah Bell 2022-10-16 Answered
Up to the work of Riemann and Gauss, this definition would have made clear to me examples of geometrical objects: a line, square, cube, hypercube; each of these possessing geometrical properties such as the number of sides, angles between faces, the dimension of space that contains them etc. Hence a geometrical object was a set of measurements associated with an object using a ruler.
After the work of Einstein and Minkowski who showed that time and space were a part of one another, would it be correct to say:
1. a geometrical object is a set of measurements associated with an object of distance and time using a ruler and a clock?
2. Geometrical objects includes an interval of time, the invariant space-time interval?
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Alexandria Rubio
In SR, you can have geometrical figures in spacetime. You don't need both rulers and clocks to define them. All you need is a clock. There is a nice presentation of this kind of thing in Laurent 1994.
In GR, you have a spacetime whose curvature varies from point to point. In this kind of spacetime, you can't in general transport geometrical figures from one place to another (and you also can't scale them). There is no notion of congruence. For these reasons there isn't much interest in studying geometrical figures for their own sake.
The ancient Greeks conceived of geometry in terms of figures and congruence, but today the notion is much more general than that. For instance, you can have projective geometry, which has no system of measurement, or geometries with finite numbers of points.
Bertel Laurent, Introduction to spacetime: a first course on relativity.
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Aryanna Blake
In relativity, yes you are on to something in that time is pretty much just another dimension like space, and clocks are just a special type of ruler.
However, a geometrical object is a bit more than what is stated. Physicists use the term loosely for a collection of measurements that, when taken together and interpreted correctly, is in some sense invariant. That is, the numbers themselves might change, but their collective interpretation does not, when you change coordinates.
The archetypical example is a vector/"arrow" in ${\mathbb{R}}^{2}$. Say we have a vector $\stackrel{\to }{v}$ representing a change of $+1$ in $x$ and $+2$ in $y$. We might write
$\stackrel{\to }{v}\stackrel{\left(x,y\right)}{⟶}\left(1,2\right).$
However, we could also parametrize the plane with another coordinates, say ($\left(w,z\right)$,$\left(w,z\right)$), related to the original coordinates by $w=2x$$w=2x$$w=2x$, $z=-y/2$$z=-y/2$$z=-y/2$. In this basis we have
$\stackrel{\to }{v}\stackrel{\left(w,z\right)}{⟶}\left(2,-1\right).$
The numbers have changed, but each set of numbers, when interpreted with the proper basis in mind, refers to the same geometrical object, $\stackrel{\to }{v}$ .