The term "New Moon" in astronomical terms means the position when the Sun, Moon and Earth are in conjunction (in one plane). At this time the dark side of the moon is towards the earth hence it is unobservable at this juncture. Whilst explaining the verse:

"By the sky, with its constellations" (Quran 85:1)

 many Commentators have mentioned the twelve prescribed paths on which the Sun and the Moon orbit throughout the days and nights. The sun completes its 12 paths (1 path per month) in a year due to its slower speed whereas the moon completes its paths in a month. The example of the sun is like that of the small hand in the clock and the moon that of the bigger hand.

Fig1 MoonPhases
Figure 1: Moon Phases

Every month, the moon orbits the earth until a stage where it confronts the sun. The sun rises in the east. Its rising place moves uniformly north-south during the year. As opposed to this, the visibility of the crescent moon is not uniform. Its visibility areas if plotted from the earth, looks like an ellipsoid (egg-shaped). This ellipsoid again is not uniform. It moves north-south throughout the year. Sometimes it covers northern areas (Alaska, Japan) and sometimes the southern areas (South Africa, New Zealand). The sighting of the crescent moon is completed all over the world in 24 hours. The reason for this becomes obvious when we learn that the moon moves in different orbits during every month and thus passes over different places of the globe.

When there is a sighting at a place that is within the visibility parabola, there will also be a positive sighting on the same day at all places that lie on the west. Places that are outside this visibility parabola (North, South and East of the place of sighting) will observe the crescent moon on the following day. This is the "law of nature" as defined by Allah (سبحانه و تعالى) regarding sighting of the crescent moon (Ru'yat-ul-Hilaal).

Fig2 VisibilityParabola

Figure 2: Visibility Parabola

Fig. 2 Schematically illustrates how along the latitude circles, the probability of visibility decreases to the east of ILDL (growing dark circles) and increase to the west of ILDL (growing white circles).

If the Islamic world follows this authentic sighting for Ramadan and Eidain, then most places will celebrate Ramadan & Eidain on the same day. Furthermore, with respect to their geographical locations (East or West), there will at the most be a difference of one lunar day (not solar day).  With respect to the lunar day, (there occurs a difference of two and sometimes three days between Middle Eastern countries and the rest of the Islamic world due to the unauthentic manner in which Ru'yah is made.