לזכות ידיד נפשי האהוב והנאמן הר"ר אהרן ישראל בן ר' משה מרדכי לברכה והצלחה, הוא וכל בני ביתו!!!
וכן לזכות ר' משה מרדכי בן חנה לבריאות השלימה וכל טוב סלה!!
In Parshas Lech Lecha, Avraham Avinu tells Sarah to say that she is his sister and not his wife because if she says that she is his wife they will kill him in order to take her. The problem is that by saying that she is his sister she might be saving his life but she is causing the aveirah of gilui arayos because now they will be taking a married woman. So she should have said that she is his wife and even if it would have cost him his life it doesn't matter because regarding gilui arayos there is no heter of pikuach nefesh iand the law is yehareg vi-al yaavor??
The Maharal in Gur Aryeh answers that she thought that they might not care if she is married and take her anyway without bothering to kill Avraham, so it wouldn't help to say that she was married. Meaning, even though Avraham said that they would kill him it wasn't certain that they would do so. So saying that she was his wife might not have helped and thus saying that she was his sister was only a safek if it would be the cause of gilui arayos.
This would depend on the question of the Pri Megadim [Sefer Teivas Gomeh] as to whether we say יהרג ואל יעבור about gilui arayos even if it is only safek gilui arayos. According to the Maharal, Sarah held that for safek gilui arayos one need not sacrifice one's life.
The Sefer HaRokeach offers a different answer. He says that Sarah was being raped and had a din of קרקע עולם since she was completely passive. So there is no איסור of gilui arayos.
According to the Rokeach we learn a chiddush li-halacha: If a man is running after a women in order to violate her and if she escapes then he will instead kill innocent people - she shouldn't try to escape but because by giving in she will be saving lives.
But if a man is being forced to commit the sin of arayos and told that if he doesn't then innocent people will be killed, it is forbidden for him to commit the sin because he is a man and an active sinner in arayos which is not overriden by pikuach nefesh.
Great - but the Maharal didn't accept this answer. Why not?
Tosfos [Pesachim 25] say that this svara of קרקע עולם applies even to murder, meaning that if one is presented with the choice of being completely passive and being thrown on a baby [which would result in the baby's death] or being killed it is permitted to allow himself to be thrown on the baby. He is only קרקע עולם and is thus permitted to save his life.
But if he told to come to the place where they are going to throw him on the baby, it makes sense to say that it is forbidden. The Ritva in Pesachim  says that if a Goy says "let me take your weapon in order to kill or you will be killed", he may allow the Goy to take his weapon. But if the Goy says "bring me your weapon and let me take it" it is forbidden because he then becomes an active partner.
This law would apparently apply to speech as well and one is not allowed to aid and abate a murder through his words. So in our case by saying that she is Avaraham's sister, Sarah is in essence bringing about a situation where there will be gilui arayos, so the heter of קרקע עולם wouldn't apply.
But maybe we can make a distinction and say that in the case of the Ritva, the weapon is a direct cause of the murder, so bringing it and letting the Goy take it would be a direct cause of death. But in the case of Sarah, saying that she is Avraham's sister is not the direct cause of the gilui arayos.
Another difficulty with the answer of the Rokeach is that Rav Chaim Halevi says [Hil. Yesodei Hatorah] that the Rambam never mentions the heter of קרקע עולם so it is likely that he doesn't accept it.
The Klausenberger Rebbe [המדרש והמעשה פ' לך לך] suggests an answer based on the Rambam who says that when a non-Jewish couple agree that she is no longer his wife he sends her away and they are divorced. So by saying [with Avraham's consent] that she is his sister, this was essentially a desolution of their union and she is no longer an אשת איש [this assumes that the Avos had a din of Bnei Noach].
Others say simply that Goyim are not commanded to fulfill the mitzva of Kiddush Hashem [see Sanhedrin 74] so there is no יהרג ואל יעבור and she may save Avraham's life.
Others suggest that Avraham thought that they would never touch her even when she says that she is his sister because Hashem would guard her.
[בחלקו עפ"י הגר"א גנחובסקי ועי' גם בס' שירת משה על ז' מצוות בני נח סי' א' בס' פנינים מבי מדרשא פ' לך לך, מנחת אשר פ' וארא, קובץ מוריה שנה טו עמ' קיא והלאה, לקוטי שיחות ח"ה פ' חיי שרה בהערה 36]