Not knowing the exact cause of the problem we can only offer some general guidelines. It would be best to get back with the doctor by phone and ask this question for your son specifically.
Most musculoskeletal problems heal in about six weeks’ time. Rest is the key to avoid re-injury or microtrauma during the healing stage of recovery. If there’s a bone fracture such as occurs with spondylolysis then healing may take a little longer.
A recent study of soccer players with back pain from spondylolysis tried using three months as their time period. Everyone who took three months off got better but some got better sooner.
The authors say it’s not possible to say how long kids should stop sports activity. When they are pain free then exercise and activity can be added back gradually.