The law if a woman died r”l during childbirth:
A woman which sat on the birthing stool and died, one is to bring a knife on Shabbos even through a public property, and tear her stomach and take out the baby as perhaps [the baby] will be found alive
The reason for why we suspect that he is alive: Now, although majority of times the infant dies prior to the mother, nevertheless by the danger of a life we do not follow the majority, and at times it is able to live after its mother’s death. Now, although this infant never had a living status, Shabbos is to be desecrated due to doubt just as one who did have a living status.
The reason for why today we are no longer accustomed to tear the mothers stomach: The reason that [cutting the mother’s stomach] is not in practice today even during the week is because we are no longer expert in the mother’s death to such close proximity that the child still can live, as [we suspect that] perhaps she fainted and if her stomach were to be torn she will die, and thus one needs to wait [to verify whether she has died or merely fainted] and until then the child has certainly died. [See Q&A for Halacha today]
Summary: The law if a woman died r”l during childbirth:
If a woman died during labor r”l, one is to bring a knife on Shabbos even through a public property, and tear her stomach and take out the baby, as perhaps [the baby] will be found alive. Today however this in no longer in practice being that by the time we determine her death the baby has certainly died.
Practically today is one to desecrate Shabbos to remove the baby if the mother died?
Being that today there are machines which determine the exact time of death for the mother as well as whether the baby is still alive one may obviously desecrate Shabbos to help save the baby in accordance to the readings of these machines.