If my wife is out of the country for an extended trip, am I required to light Shabbat candles in the house if I am home alone? Does it matter how long she is going to be away? Or that the time difference between her and me is five hours?
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Of course, as the comment notes, consult your local Orthodox rabbi.
But see here for a halacha sheet shown before and approved by Rav Shternbuch:
To proffer a reason: The purpose of lighting candles is kavod shabbat, shalom bayit, and oneg shabbat. (See here.) To cite a definition of the last item:
If so, if a man's wife is away, he still has this obligation of lighting the candles.
See also this, from Rabbi Doniel Neustadt:
Footnote 8 reads:
No distinction was made between the wife lighting or not lighting elsewhere.