why is there a time lag from south to north?

There is no time lag from south to North rather it is from East to West. This is so because time is determined by longitudes and not latitudes.

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since the Earth turns from East to West (or West to East) timezones running East to West would not show how far the sun has gone. Think about it: how could it possibly be the very same time at a given moment all along the equator? The equator runs through the entire sunlit and night time parts of the earth.

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