I 3 clocks thought experiment - Absolute vs. relative aspects?

[Dale]

Does that mean you are talking about an absolute spacetime geometry, independent of an observer?

I thought it might be worthwhile to go a little further into what the spacetime geometry entails.

Imagine a sheet of paper. This paper represents spacetime (1D space and 1D time).

Absolute spacetime says that there is one and only one unique way to draw a grid where one set of grid lines is space and the other is time. Relative spacetime says you can take any straight line and draw parallel and perpendicular lines to form a valid grid.

The geometric approach says that the grid lines are entirely optional. You can use them if you like, for convenience, but the geometric approach looks for things that are independent of the grid lines altogether.

Think of Euclidean geometry. Points, lines, angles, and distances were all concepts developed without coordinate grids. Straight, parallel, perpendicular, all of these concepts do not require coordinates in Euclidean geometry. Furthermore, if you choose to use a coordinate grid, doing so does not change any of those geometrical concepts.

The difference between spacetime geometry and Euclidean geometry is the metric. In Euclidean geometry to find the distance from $A$ to $B$ we draw a set of circles centered on $A$ and then the distance to $B$ is the radius of the circle that contains $B$. In spacetime geometry we draw a set of hyperbolas centered on $A$ and the interval is the semi major axis of the hyperbola that contains $B$.

This change in geometry contains all of the rules of special relativity. The most important can be stated in terms of the geometry, without reference to grids. But if one desires to use coordinate grids for convenience, the underlying geometry establishes the rules for that too.