Is Mutah Marriage Allowed?
Question

Could you please tell if there is such a concept as temporary marriages (mut’ah marriage) in islam? I would like to know because a friend of mine has read a book by Professor Abui Qasim Gourgi and is under the impression that if they are already married it is okay for them to do mut’ah (the name for a temporary marriage according to Islamic shari’ah). 
His definition for a temporary marriage is that if you like someone it is okay for you to have your nikah read with them for a short period of time. 
Please could you tell me more about the issue of mut’ah and which schools of thought believe in such an idea? Could you support your answer using references from hadith and Quran?
Praise be to Allah.What is mut’ah marriage?
Mut’ah or temporary marriage refers to when a man marries a woman for a specific length of time in return for a particular amount of money. 
History of mut’ah marriage
The basic principle concerning marriage is that it should be ongoing and permanent. Temporary marriage – i.e., mut’ah marriage – was permitted at the beginning of Islam, then it was abrogated and became haram until the Day of Judgement. 
Ali (may Allah be pleased with him) narrated that the Messenger of Allah (peace and blessings of Allah be upon him) forbade mut’ah marriage and the meat of domestic donkeys at the time of Khaybar. According to another report, he forbade mut’ah marriage at the time of Khaybar and he forbade the meat of tame donkeys. (Narrated by al-Bukhari, 3979; Muslim, 1407) 
Al-Rabi’ ibn Sabrah al-Juhani narrated that his father told him that he was with the Messenger of Allah (peace and blessings of Allah be upon him) who said, “O people, I used to allow you to engage in mut’ah marriages, but now Allah has forbidden that until the Day of Resurrection, so whoever has any wives in a mut’ah marriage, he should let her go and do not take anything of the (money) you have given them.” (Narrated by Muslim, 1406)
Purpose of marriage in Islam
Allah has made marriage one of His signs which call us to think and ponder. He has created love and compassion between the spouses, and has made the wife a source of tranquility for the husband. He encouraged us to have children and decreed that a woman should wait out the ‘iddah period and may inherit. None of that exists in this haram form of marriage. 
A woman who is married in a mut’ah marriage, according to the Rafidis – i.e. the Shi’ah, who are the ones who say that this is permissible – is neither a wife nor a concubine. But Allah says (interpretation of the meaning): 
“And those who guard their chastity (i.e. private parts, from illegal sexual acts)
Except from their wives or (the slaves) that their right hands possess, for then, they are free from blame;
But whoever seeks beyond that, then those are the transgressors.” [al-Muminun 23:5-7]
Mut’ah marriage according the Rafidis
The Rafidis quote invalid evidence to support their argument that mut’ah is permissible. For example: 

They quote the verse in which Allah says (interpretation of the meaning):

“…so with those of whom you have enjoyed sexual relations, give them their Mahr as prescribed…” [al-Nisa 4:24]
They say: this verse indicates that mut’ah is permissible, and the word ‘their mahr (ujurahunna – lit. their dues or their wages)’ is evidence that what is meant by the phrase ‘you have enjoyed sexual relations’ is mut’ah. 
The refutation of this is the fact that prior to this Allah mentions the women whom a man is forbidden to marry, then he mentions what is permissible for him, and He commands the man to give to the woman he marries her mahr. 
The joy of marriage is expressed here by the word enjoyment (‘of whom you have enjoyed sexual relations’). A similar instance occurs in the Sunnah, in the hadith of Abu Hurayrah according to which the Messenger of Allah (peace and blessings of Allah be upon him) said: “A woman is like a bent rib, if you try to straighten her you will break her. If you want to enjoy her, then enjoy her while she still has some crookedness in her.” (Narrated by al-Bukhari, 4889; Muslim, 1468) 
The mahr is referred to here as ajr (lit. dues or wages), but this does not refer to the money which is paid to the woman with whom he engages in mut’ah in the contract of mut’ah. The mahr is referred to as ajr elsewhere in the Book of Allah, where Allah says (interpretation of the meaning): 
“O Prophet (Muhammad)! Verily, We have made lawful to you your wives, to whom you have paid their Mahr (bridal-money given by the husband to his wife at the time of marriage)…” [al-Ahzab 33:50]
Thus it becomes clear that there is no evidence in this verse to suggest that mut’ah is permissible. 
Even if we were to say for argument’s sake that this verse indicates that mut’ah is permitted, we would still say that it is abrogated by the reports in the sahih Sunnah which prove that mut’ah is forbidden until the Day of Resurrection. 

The reports that some of the Companions regarded it as being permissible, especially Ibn 'Abbas.

The refutation here is the fact that the Rafidis are following their own whims and desires, because they regard the Companions of the Prophet (may Allah be pleased with them) as disbelievers, then you see them quoting their actions as permissible in this instance and in others. 
With regard to those who said that it is permissible, they are among those who did not hear that it had been forbidden. The Companions (may Allah be pleased with them) – including ‘Ali ibn Abi Talib and ‘Abd-Allah ibn al-Zubayr – refuted Ibn ‘Abbas’s view that mut’ah was permitted. 
‘Ali narrated that he heard Ibn ‘Abbas permitting mut’ah marriage, and he said, “Wait a minute, O Ibn 'Abbas, for the Messenger of Allah (peace and blessings of Allah be upon him) forbade it on the day of Khaybar and (he also forbade) the meat of tame donkeys.” (Narrated by Muslim, 1407)
For more, please see this category: Rulings on Marriage.
And Allah knows best.