The observance of Jewish law ( halakhah) in the polar regions of Earth presents unique problems. Many mitzvot, such as Jewish prayer and the Jewish sabbath, rely on the consistent cycle of day and night in 24-hour periods that is commonplace in most of the world. However, north of the Arctic Circle (and south of the Antarctic Circle) a single period of daylight can last for a month or more during the summer, and the night lasts for a similar length of time in the winter. The question for religious Jews that live in or visit these regions is how to reconcile the observed length of days in the polar regions with common practice elsewhere in the world. Should a "day" be defined solely based on sunrise and sunset, even if these events do not occur for long stretches of time; or should the definition of a polar "day" be consistent with the length of a day in the rest of the world?
The problem was first identified in the 18th century, when Jewish émigrés began to move in greater numbers to the northern parts of Scandinavia. A number of different opinions on the question have been presented in responsa and are reviewed in a 2005 essay by rabbi J. David Bleich.  And, in a more recent STAR-K Kosher Certification article published in Kashrus Kurrents, Summer, 2007, "When Does One Pray When There is No Day", by Rabbi Dovid Heber: https://www.star-k.org/articles/articles/travel/515/when-does-one-pray-when-there-is-no-day/.
The definition of a "day" in polar regions affects mitzvot that must be performed during the day, or at a particular time of day. It also affects the passage of time in the Jewish calendar for the purpose of observing Shabbat and other Jewish holidays.
A long list of mitzvot must be performed at any time during the day but not at night, or vice versa. These are listed in a mishnah in Tractate Megillah.  (Many of these mitzvot, as part of the sacrificial service in the Temple of Jerusalem, could not be performed in the polar regions.) In addition, a lender is required to return clothing used as collateral to a poor borrower if he needs it to sleep at night,  and an employer must pay a day laborer his wages on the same day that the work is done. 
The most familiar mitzvah that depends on the time of day is Jewish prayer. The morning Shema must be read between dawn  and three variable hours after sunrise. ("Variable hours" are each one-twelfth of the time between sunrise and sunset, or according to another opinion between dawn and the appearance of stars at twilight. Variable hours are longer than 60 minutes in the summer, and shorter than 60 minutes in the winter.) The prayers of Shacharit, Mussaf and Minchah are also limited to certain hours of the day. The evening Shema and Ma'ariv, though acceptable at any time of the night, should preferably be done in the first half of the night. It is possible that during very long days of the polar summer, evening prayers are not recited, and during very long nights of the polar winter, daytime prayers are not recited.
The passage of days from one to the next most prominently drives the observance of Shabbat on every seventh day. During the polar summer, hundreds of hours can pass without sunset, and it is possible that this entire period is just one day of a week. However, since Shabbat is observed on the same day throughout the world (allowing for differences in time zones), it stands to reason that Shabbat should be observed simultaneously even in polar regions.
The Bible, Talmud, and individual pre-modern Jewish writers do not address this issue because Jews did not visit the polar regions and were not aware of its distinctive nature before modern times. However, the section in Talmud regarding the "desert wanderer" has been used by modern authorities to analyze this issue. The Talmud contains the following discourse:
- Rav Huna says, if a man is wandering in the desert and he does not know when is the Sabbath, he should count six days [as weekdays] and keep one day as the Sabbath. Hiyya bar Rav says he should keep one day as Sabbath, then count six days [as weekdays]. ...
- Rava says, on each day he may do whatever he needs in order to survive, except for his Sabbath. But should he die on the Sabbath? He could prepare extra food the day before his Sabbath, but that might be the real Sabbath. So every day he may do whatever he needs in order to survive, even on the Sabbath. How is the Sabbath recognizable to him? By kiddush and havdalah [which he performs on his Sabbath but not on other days].
- Rava says, if he knows which day he departed on the journey, he may do work on the same day of the week [i.e. 7 or 14 days after he departed, because he certainly would not have departed on a Sabbath]. 
The law is in accordance with the first opinion, that a confused desert wanderer keeps six "weekdays" followed by one "Shabbat", but he may not perform activities forbidden on Shabbat on any day except to aid his own survival.  The law is based on a principle that a person who is unaware of reality should create his own Sabbath while acting out of concern that the real Sabbath may be on a different day. 
Rabbi Israel Lipschutz, in his commentary Tiferet Yisrael, writes that in polar regions there is a 24-hour day, as evidenced by the fact that the sun rotates in the sky from a high point at noon to a low point near the horizon at midnight. He does not offer a means of measuring the passage of a 24-hour day during the polar winter when the sun is invisible.  He advises that a Jewish traveler observe the beginning and end of the Sabbath based on the clock of the location whence he came. It is unclear whether this refers to his residence or his port of embarkation. 
A result of this view is that two Jews who leave from different cities will always observe Shabbat on Saturday, but at different times. A Jew who leaves from America will observe the Sabbath according to the clock of his hometown, while a Jew from Europe will use the clock of his European hometown, which begins and ends Sabbath about five hours earlier than in America. Thus, there is no uniquely identifiable beginning and end of the day in the polar regions. 
- "Mizvot in the Polar Regions and in Earth Orbit." J. David Bleich. Contemporary Halakhic Problems, volume 5, chapter 3, pages 75-128. Targum Press, 2005. ISBN 1-56871-353-3
- Chapter 2; page 20b.
- Deuteronomy 24:12-13
- Deuteronomy 24:14-15
- Specifically, this is the time between dawn and sunrise that a person may first distinguish between light and blue, according to the Mishnah in Berakhot.
- Babylonian Talmud, Tractate Shabbat, 69b (translated from Aramaic)
- Joseph Caro. Shulhan Arukh, Orakh Hayyim, chapter 344.
- The above strictly applies when one is truly unsure what day it has become. In a case where one knows exactly what day s/he left civilization, and has kept careful records since, s/he would still continue to keep the calendar normally.
- In practice, weather permitting, this could be accomplished by watching the constellations rotate about Polaris.
- And at that, it is unclear whether the "port of embarkation" is the initial port of embarkation or the final point of the journey where day and night were well established before crossing the Arctic/Antarctic Circle–such as, for example, Anchorage, if one is traveling toward Prudhoe Bay.
- Bleich pp. 85-92