After blowing the shofar on Rosh Hashanah we say hayom harat olam. The word harat is connected to pregnancy and birth. Herayon means pregnancy in modern Hebrew, and horeh is the name for a parent. The medieval commentator Rashbam, on Genesis 49:26, further connects the word to har meaning mountains - parents and ancestors being the ancient mountains from which we are hewn.
Following these associations, English machzorim tend to render our phrase as "Today is the birthday of the world", which may conjure up images of a big cake with 5771 candles.
Rabbi Matis Weinberg, however, translates it : "today is the conception of the universe". In his striking phrase, Rosh Hashanah is the "womb of the year". On these days, all of the possibilities of the coming year exist in embryo. "The developing foetus is exquisitely vulnerable to minute changes in the uterine environment... in the same way, the nascent year is sensitive to small variations in Rosh Hashanah the womb, which defines the year's potential."
This understanding helps us to makes sense of numerous details of Rosh Hashanah. To note just one, it is remarkable that the Torah readings and haftarot of Rosh Hashanah all feature (or in the case of the Akedah reading, midrashically imply) images of women crying for their children.
On Rosh Hashanah we pray for the safe birth and growth of the potential in the New Year.