Calculating row difference and insert into new column in R (without using for loop) [duplicate]

I have a huge dataset that I want to loop over each user in the first column and calculate interval column based on the difference (subtract) between each row with the next row of start_time column. The last row of each user can be 0 or NA because there is no other record after that user.

    df=read.table(text="
    user   start_time   
    1       4
    1       6
    1       10
    1       11
    2       1
    2       3
    2       5
    3       4
    3       4",header=T)

result
user   start_time  interval 
1       4          2 <-- (this is the result of 6-4)
1       6          3 <-- (9-6)
1       9          2 <-- (11-9)    
1       11         NA <-- (or can be 0) because it is last row of the user
2       1          2
2       3          3
2       6          NA <-- (or 0)
3       4          0
3       4          NA <--(or 0)

I would prefer something fast like group_by function. How can I do it in R?