For parents and babies

we went to the registry office earlier today and as we're not married I was told that the mother chose the surname... Fair enough, but when I asked what difference this made I was told that if I did a runner then mother would be left holding the baby, quit literally, but surely whatever the name of the child and in the same circumstance then mother would be left holding the baby? I understand that it may add insult to injury, but surely in today’s world were I imagine quite a few couples are unmarried then the father has some rights here. The only reason for this I can think of is that it saves the government some paperwork later should I do a runner. As a matter of fact I would imagine that it would be considerably easier if the child had the fathers name and he did a runner to organise maintenance payments. Maybe I’m missing something here, but hasn’t the child always taken the fathers name? I’m very interested to hear your responses here