AFTER DETECTIVE A NOSE GEAR DOOR PROB AFTER DEP ZZZ1; AN MD83 FLT CREW DIVERTS TO ZZZ.

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AFTER TKOF FROM ZZZ1; CLBING OUT OF 6000 FT; FO NOTICED ALL 3 GEAR LIGHTS WERE RED WITH GEAR HANDLE UP. I ASKED ATC TO STOP OUR CLB AT 10000 FT AND ADVISED ATC THAT WE HAD A PROB AND WOULD LIKE VECTORS TO WORK IT OUT. THE FO BEGAN WORKING THE RED GEAR LIGHT INFLT WITH GEAR HANDLE UP CHKLIST. WE CHKED THE MANUAL FOR ANY WRITE-UPS FOR GEAR PROBS. NONE WERE FOUND. THEN WE CYCLED THE GEAR DOWN. ONCE GEAR WAS DOWN; ALL INDICATIONS WERE NORMAL: 3 DOWN; GEAR DOOR OPEN LIGHT OUT. WE THEN CYCLED THE GEAR UP AND THE RED LIGHT STAYED ON; GEAR DOOR OPEN LIGHT WAS OUT; BUT IT SOUNDED LIKE ONE NOSE GEAR DOOR STAYED OPEN. AFTER ABOUT 5 MINS THE DOOR SLAMMED SHUT. DURING THE TIME THAT THE DOOR WAS OPEN I WAS IN CONTACT WITH ZZZ1 MAINT. THEY COULD NOT COME UP WITH ANY IDEAS TO FIX THE SIT. SO WE BOTH AGREED ON BRING THE AIRPLANE TO ZZZ. I THEN CALLED DISPATCH AND LET THEM KNOW WHAT WAS GOING ON. (THAT'S WHEN THE GEAR DOOR SLAMMED SHUT.) AT THAT POINT I DECIDED WE WERE GOING TO ZZZ. I THEN CALLED DISPATCH BACK AND ADVISED DISPATCH THAT I WAS DECLARING AN EMER AND GOING TO ZZZ. SOME THOUGHT WAS GIVEN TO GOING ON TO DEST; BUT WITH A SPD RESTR; BAD WX ENRTE AND AT DEST; THE BEST DECISION WAS TO GO TO ZZZ. AT THAT POINT I DECLARED AN EMER AND ATC VECTORED US TO RWY 4R AT ZZZ. WE THEN SET UP FOR AN OVERWT LNDG AT 144000 LBS ON A WET RWY WITH A HEADWIND. ON FINAL; CYCLED THE GEAR DOWN -- ALL INDICATIONS WERE NORMAL AND THE LNDG WAS SMOOTH AND NORMAL. AFTER THE LNDG; ALL INDICATIONS WITH THE GEAR WERE NORMAL. WE THEN TAXIED TO THE GATE WITH CFR EQUIP IN TOW. SUPPLEMENTAL INFO FROM ACN 639389: DEPARTED RWY 13 ZZZ1 WITH CLRNC TO 15000 FT. WE TOOK ALL THE TIME WE NEEDED AND WORKED THE PROB THOROUGHLY. OUR EXCELLENT TRAINING REALLY PAYS OFF. WE TAXIED TO GATE. POSTFLT INSPECTION WITH MAINT SHOWED NO OUTRIGHT EXPLANATION FOR PROB. PAX XFERRED TO FLT OUT OF ZZZ.