Question: This year Rosh Hashanah is a 3-day holiday – Thursday and Friday are RH and then Shabbat. Is permissible to vacation in Washington DC on Rosh Hashanah and Shabbat where one can attend an Orthodox Shul and then visit the free museums and national monuments?
Answer: In general, it is forbidden to visit a museum on Shabbat in Israel – whether it is free or one for which you can purchase a ticket before Shabbat (like the Israel Museum) because of the Chilul Shabbat for Jews involved. In this case, however, outside of Eretz Yisrael, it is theoretically permissible to visit a museum since the majority of visitors are non-Jews, and there is no extra effort for the Jews. However, there is a problem of Uvdin De-Chol. While this is a difficult concept to define, it basically means being involved in weekday activities, the performance of which impinges on the holiness of Shabbat. Therefore, one should refrain from doing so, and this is indeed the opinion of the Rabbis of America.
And this is the rule for a regular Shabbat. It is certainly forbidden all the more so on Rosh Hashanah which is a serious day, a day of Teshuvah, a day of self-reflection and not a day for entertainment and vacation.