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?
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.