relativity intuition

Here is a space-time grid.

What would happen to it if we were at the same place, but at different speeds ?

Now let's look at constant acceleration.

A parametrization of such trajectory in flat space is given by $$ \ca{ t(τ) = \frac{c}{α} \sinh \frac{ατ}{c} \\ x(τ) = \frac{c²}{α} \left( \cosh \frac{ατ}{c} - 1 \right) } $$