Fig. 4. (Color online) Locally inertial frames in the Friedman-Lemaitre-Robertson-Walker space-time. Equal-time-line is curved toward direction of past. Here universal equal-time-line does not exist. A stationary observer O (observer O′) may conclude approximately, the α (β) event happens earlier than β (α) event. The gap between the coordinate scales become narrower as time goes, which means the universe is expanding. The red lines denotes paths of lights. The ratio of proper-time of another observer close to O (denoted by dotted line, not O′) to that of the observer O is greater than 1 (∼1.15). This corresponds to cosmological redshift. The lower two frames quite deviate from local inertial frame due to high curvature around big bang singularity.
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