1)how did muhammad's successors expand the arab empire after his death ?
2. why was there tension over who should rule the empire after the death of muhammad ?

1) They mostly spread Islam through conquering new countries. What they did also was unite all people under a single rule and ended religious uprisings.
2) There was tensions because one faction believed that Muhammad should be succeeded by someone chosen by people, while the other faction believed that his successor should be someone from his bloodline.

What war are you specifically talking about?