I desperately need some help, if any kind sole out there may be able to help I would be very grateful

My grandfather was born in Beamish in Dec 1891. He married my grandmother in July 1913. His son was born Oct 1913. As I was growing up my grandfather was not around. I was told that he had died soon after my father had been born. I have searched death indexes from 1913 to 1920 looking for his death but can find nothing anywhere. My feelings now are that I have been told a little white lie to stop me asking further questions. Did he do a runner. After all he married my grandmother late in her pregnancy. If my feelings are correct has anyone got any ideas how I could continue to trace him?
Please can you help?

Mercel
Thank you to all who helped. Special thank you to DS. His death certificate showed the incorrect spelling of his surname. He did die in 1916, death cert should John Robert Foster.