Question: Am I allowed to message my family in America after Shabbos goes out here even though it’s still Shabbos there?
Answer: The Shulchan Aruch (OC 263:17) writes that one who has accepted Shabbos early may ask another Jew who hasn’t yet accepted Shabbos to do a melacha on their behalf. Following this, R’ Moshe Sternbuch (Teshuvos Vehanhagos 5:92) allows one in Eretz Yisrael to send a fax after Shabbos to someone in America even though it is still Shabbos there providing that they won’t come to break Shabbos by reading business communication.
Likewise, R’ Ephraim Greenblatt (Rivevos Ephraim 6:115:2; 8:152:3; 8:158:2) quotes R' Yisroel Yaakov Fisher, R’ Yosef Shalom Elyashiv and R’ Yisrael Belsky who allowed one to send a fax before or after Shabbos to one abroad even though it was Shabbos for the recipient. R’ Yisroel Pesach Feinhandler (Avnei Yashpei 5:47:1) explains that we only look at where the melacha is being performed. As it isn’t Shabbos for the one performing the melacha, they may send the fax then (See Shevet Halevi 8:40:3; Cheshev Haefod 3:86).
R’ Moshe Sternbuch adds, however, that one may not ask another explicitly to send them a fax while it is still Shabbos for them.In conclusion, one may message people before or after Shabbos even if they are in a different time zone and it is still Shabbos there, providing that they won’t come to break Shabbos by doing so. One mustn’t specifically ask another to message them during Shabbos.