Newest Red Sox pitcher John Smoltz acknowledges applause during the Celtics-Raptors game in Boston Monday night.

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AP / Charles Krupa

BOSTON — John Smoltz had a problem.

The future Hall of Famer knew that the rehab on his shoulder, which was rebuilt during surgery last summer, was well ahead of schedule. But the reports that leaked out said he wouldn’t be ready to pitch until at least June, and possibly later.

So Smoltz grabbed a football, set up a video camera in a field and made a video of himself throwing passes. He sent this Heisman audition video to the teams he was interested in to prove the health of his shoulder.

To the Red Sox, he looked surprisingly good. Not next-starting-QB-for-USC good, but good enough that in early December they sent pitching coach John Farrell, VP of player personnel Ben Cherington and rehab coordinator Mike Reinold to Atlanta to see him throw –– this time, of course, with a baseball.

They liked what they saw that day and offered him a contract just before the holidays.

For Smoltz, whose long relationship with the Braves ended on a sour note, the attention and the trust meant the world.

“It did mean a lot to me for them to come down and come watch me throw,” Smoltz said. “All I ever said to my friends, privately, was that if somebody watches me throw, if they just put their arms on my shoulder and test me out, well, someone said miracle –– but well, it’s pretty incredible.”

The Red Sox’ early interest paid off. Smoltz, 41, passed his physical on Monday, attended a Boston Celtics game on Monday night and was officially introduced as the newest member of the Red Sox yesterday morning.

Smoltz signed a $5.5-million, one-year contract, with incentives that could pay him as much as $5 million more, mostly for time spent on the active roster. The Michigan native, who has a 15-4 postseason record, vowed that it will be money well spent because he hasn’t lost his desire to win.

“I’m as determined and as focused as I’ve ever been,” Smoltz said yesterday at Fenway Park. “I just want to win. If it meant running the resin bag out to the mound, I’d do that. … I don’t have anything to prove anymore. I don’t have any numbers to reach or anything — I just love to win.”

Smoltz spent the last 20 seasons with the Braves, for whom he won 210 games, saved 154 and won the 1996 N.L. Cy Young Award.

Although he has excelled as a reliever, Smoltz will be a starter in Boston, joining a rotation that includes Josh Beckett, Jon Lester, Daisuke Matsuzaka, Brad Penny and possibly Tim Wakefield.

Six months ago, this day would have seemed impossible. After Smoltz’s shoulder surgery last summer to repair multiple problem areas, including his rotator cuff and labrum, it was unclear whether he would ever pitch again.

But the right-hander recovered faster than expected, surprising even himself.

“I’m feeling great. I have, over the last five surgeries, been a quick healer. And I’m very determined to rehab,” Smoltz said.

General manager Theo Epstein said Smoltz could be ready to pitch by Opening Day, but that the team will hold him out for longer than that to make sure he is as strong as possible for the second half of the season.

“He passed our physical yesterday with flying colors,” Epstein said. “We feel like, if we absolutely had to have him ready in April, he could do that … Our doctors described him as a physical freak. He has already overcome more than most, and this is just another example.”

His health aside, however, it initially seemed ludicrous to think that Smoltz would ever leave Atlanta, where he and Chipper Jones were the faces of the franchise. That began to change when it became clear the Braves were skeptical that Smoltz’s shoulder would make a full recovery, and they weren’t prepared to offer him much guaranteed money, amid reports –– which Smoltz denied –– that there was no way Smoltz could return before June.

“They’ve thrown time frames out that don’t exist. They’re just not true. June 1, or whatever time frame that people said I was going to be ready, it just wasn’t the case,” Smoltz said.

The Braves reportedly offered only $2 million guaranteed, with roughly $7 million more in tough-to-reach incentives. Smoltz said it left him with little choice.

“I’m not bitter… they were taking a different direction and, for the most part, left me with really no options,” Smoltz said.

Once they assessed his health and realized he was serious about leaving Atlanta, the Red Sox became interested in his services.

The Rangers and Tigers were also said to be in the Smoltz bidding, and one team may have offered him as much as $8 million guaranteed.

But the Red Sox’ title chances, their personal attention and ability to stash him away and let him fully heal were enough to sway Smoltz to come to Boston.

“Words do not describe where I feel right now, to put on this uniform,” a smiling Smoltz said as he nodded toward his new No. 29 Red Sox jersey.

dbarbari@projo.com