Absolute time and space
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In physics, the concept of absolute time and absolute space are hypothetical models in which time either runs at the same rate for all the observers in the universe or the rate of time of each observer can be scaled to the "absolute time" by multiplying the rate by a constant. Similarly the notion of absolute space says that space is an inert arena on which other physical phenomena take place. The space affects the other phenomena, but the space itself is not affected by those phenomena. Absolute space underlies the laws of classical physics of Isaac Newton. Newton's absolute space implies that there is one inertial system out of an infinite number that is unique.
These separate concepts were common in physical theory prior to the advent of special relativity theory, which united the two and showed both to be relativistic. In his theories, the idea of the absolute time and space was superseded by the notion of spacetime in special relativity and especially dynamically curved spacetime in general relativity.
Absolute simultaneity refers to the hypothetical coincidence of two or more events in different points in space for all observers in the universe. It is shown in the theory of relativity that there may be always observers for whom simultaneity won't correspond to the same moments in time and therefore simultaneity is always relative. The theory of relativity doesn't allow the existence of such time because of non existence of absolute simultaneity.
[edit] Common moment
Common moment is a hypothetical moment in time that is measured as the same time for two or more events at different points in space by all observers in the universe. It assumes existence of a time that runs at the same rate for all observers in the universe or at least that can be "scaled" to such a common rate, in which this "common moment" can be determined. Such hypothetical time is called absolute time. In cosmology it is usually called "cosmic time".
According to relativity theory there can't be such time and each observer (whether stationary or non-stationary relative to other observers) has its own time running at different rate than the times of at least some other observers in the universe. Therefore, strictly speaking, there are no "common moments" in nature since it is not possible to establish uniquely the simultaneity of two events in two different points in space for some observers. Nature doesn't need "simultaneity" for anything since nature doesn't operate at a distance, but only on contact between interacting agents, so the simultaneity, as not existing in nature, is a human rather than a physical idea, following from imprecise measurements of time. So is the "common moment".
However the differences between measurements of time may be smaller than the ability to detect them and so we may postulate an "approximate simultaneity" and "approximately common moment" for most practical purposes.