Einstein's theory of relativity grew out from a question of simultaneity in time of events at different locations in space, which Einstein could not answer in a non-ambiguous way and then jumped to the conclusion that a fundamental revision of our concepts of space and time was necessary. Einstein took so on the responsibility in the service of science and humanity to make the revision and thereby open the door to a modern physics of "curved space-time" with all its wondrous new effects of time dilation and space contraction, albeit too small to be detected.
It is clear that simultaneity plays an important role in our society, to set schedules and allow people to meet at the same place and for these purposes we all have clocks synchronized to a reference clock. And to decide which scientist first submitted an article reporting a certain new scientific break-through and to navigate...
But what role does simultaneity play in physics? In what sense do distant physical objects care about simultaneity? Do they all have synchronised clocks? Of course not. What they do is to react to local forces acting locally in time, and no simultaneity with the action of distant objects is involved.
Or is it? What about gravitation, isn't it supposed to act instantly over distance and thus require a form of exact simultaneity? Yes, it so seems because in Newtonian gravitation the Earth is instantly acted upon by a gravitational force from the Sun directed towards the present position of the Sun, and not towards the position where we see the Sun because of the 8 minute time delay of the light from the Sun.
The standard view on gravitation, is thus that the presence of matter instantly generates a gravitational potential/force (Newton) or "curvature of space" (Einstein) at distance. This view comes with the following questions:
- What is the physics of the instant action at distance? Gravitons?
- What is the physics of the simultaneity associated with instant action?
- $\Delta\phi =\rho$. (*)
More precisely, simultaneity does not appear to be a matter of the physics of the world, since atoms are not equipped with a man-made system of synchronised clocks, and so it is not reasonable to make a complete revision of Newtonian mechanics starting from an ad hoc idea of probably little significance.