(This should not be relied upon for practical halacha. When a question arises a Rabbi should be consulted.)
1. Many people have the practice to wear two pairs of tefillin each day, Rashi Tefillin and Rabbeinu Tam Tefillin. Rashi and Rabbenu Tam disagree with regard to the sequence of the Scriptural text contained in the parchment of the tefillin. While the accepted view is that of Rashi, some wear the Rabbeinu Tam Tefillin as well (for a full discussion see Shu”t Yabia Omer 1:3. See also a fascinating teshuva in Shu”t Min Hashamayim 3).
2. There is no blessing recited while wearing Rabbeinu Tam Tefillin. Yet, one may still not speak in between placing the tefillin shel yad and the tefillin shel rosh of the Rabbeinu Tam Tefillin (Yabia Omer 3:3).
3. The Shulchan Aruch (34:3) rules that Rabbeinu Tam Tefillin should be worn only by one who is known to be a very pious person. The Mishnah Berurah explains that it is a sign of haughtiness for anyone else to do this because the accepted practice is to wear only Rashi Tefillin. This led to the custom among many ashkenazim not to wear Rabbeinu Tam Tefillin.
4. Many chassidim and sefardim have the custom to wear Rabbeinu Tam Tefillin. Some explain that since today many people wear these tefillin, it is no longer considered a sign of haughtiness (see Yalkut Yosef vol. 1 page 46, shu”t Chaim Shaul 1, Birkei Yosef 34, shu”t Minchas Yitzchak 6:7, Yabia Omer 9:108).
In addition, the Teshuva Me’ahava (cited in the sefer Os Chaim V’Shalom 34:9) explains that one does not need to be pious to wear Rabbeinu Tam Tefillin. He feels that the restriction of the Shulchan Aruch was only for wearing both Rashi Tefillin and Rabbeinu Tam Tefillin at the same time. Wearing Rabbeinu Tam Tefillin after wearing Rashi Tefillin would be permitted according to all authorities.
5. Many have the custom that the chosson begins wearing Rabbeinu Tam Tefillin the morning following his wedding. The Sefer Shulchan Malachim writes that this is the common custom and that “one should not change or deviate from this practice.” The reason given for this is that the average single man is unfortunately prone to inappropriate thoughts. Therefore, they do not wear Rabbeinu Tam Tefillin so as not to have such thoughts while wearing the tefillin. Married men are not as prone to such thoughts and are permitted to wear Rabbeinu Tam Tefillin. This is also the view of Harav Ben Tzion Abba Shaul zt”l (shu”t Ohr Litzion 2:3:11) and Harav Yekusiel Yehuda Halberstam zt”l (Divrei Yetziv Orach Chaim 42).
[Although the aforementioned concern of inappropriate thoughts should apply when any single person wears Rashi Tefillin, we allow them to wear Rashi Tefillin so as to allow them to fulfill the great mitzvah of wearing tefillin each day. However, once they have fulfilled their obligation, they are not allowed to wear Rabbeinu Tam Tefillin, out of concern of impure thoughts (see Os Chaim V’Shalom 34:10, Shulchan Malachim page 348, shu”t Beis Shearim 29, and shu”t Mishneh Halachos 6:12).]
6. The custom among Chabad, Komarna, Nadvorna, and Breslov Chassidim is to begin wearing Rabbeinu Tam Tefillin at the bar mitzvah (see Netai Gavriel Nisuin vol. 2 page 74 and sefer Shaarey Halacha U’Minhag 29).
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