Actor John Shea takes on another 'case' - as nuclear gumshoe.

Right up there next to Bogie's photo on John Shea's dressing room mirror is a quotation from Einstein: ''We shall require a substantially new manner of thinking if mankind is to survive.''

Both are clues to his starring role in Arthur Kopit's new nuclear whodunit, ''End of the World (With Symposium to Follow).''

In the play, Shea appears as a playwright who turns private eye when a billionaire commissions him to write a play about thermonuclear warfare.

It was just another case for Shea, who played Charles Horman (the journalist-filmmaker in Costa-Gavras's film ''Missing'') and starred as Robert F. Kennedy in a British TV miniseries, ''Kennedy.''

So Shea gumshoes on stage in a trench coat with raised collar, a fedora with lowered brim, and the sang-froid of Bogart facing the Fat Man in ''The Maltese Falcon.''

Backstage in his dressing room before that night's Kennedy Center performance there is a different John Shea. He has the face of a gladiator who reads Plato: strong, fierce, and thoughtful, with a jaw so square it's almost clenched. The angles of his face are softened somewhat by a helmet of curly black hair and eyes so dark they might be onyx. It's a surprise then to find him dressed like a comparative-literature graduate student: mellowed-out brown corduroy pants, brown plaid shirt, brown argyle socks, and brown suede shoes. Tall and lean, he has a contemplative walk.

Shea sits on an oatmeal couch with his back to the mirror and murmurs a line from the play we have been talking about: ''A playwright is like a detective. A crime is at the heart of nearly every play, a crime against the spirit, crime against the flesh.''

The plot thickens when we learn that playright Arthur Kopit, who wrote ''The End of the World,'' was actually commissioned by wealthy insurance czar Leonard Davis to write a play about a possible nuclear Armageddon. And Kopit chose to do it sleuth style.

''Arthur gave me that particular picture'' - he nods up at Bogart - ''as a reminder of one of the heroes of the genre . . . the downtrodden detective, who is tough but honest, trying to make the best of a bad situation, hired by a man who is like a Sidney Greenstreet character. . . . '' He slips into Sidney Greenstreet for a second with a dry, sinister laugh and a ''Well, sir, you're the one for me, Mr. Spade.'' Clearly he enjoys the role of nuclear gumshoe he plays in ''End of the World,'' although only a part of it is private-eye high jinks.

The rest is doomsday serious, as detective Michael Trent is drawn deeper and deeper into the mystery of whether a mortal self-destructiveness will push the world into nuclear warfare. It is a demanding role in which he's on stage virtually every moment.

''I've never had a workout on stage like this before,'' says Shea, who's done Brecht, O'Neill, and Shakespeare.

Before the curtain goes up, says Shea, he doesn't have time to get nervous. ''I am cranking myself up to such a high pitch of (energy) that I can go out there and propel the play, project it for two hours out to 1,200 people.''

After devoting hunks of time to membership in Performing Artists for Nuclear Disarmament and numerous political appearance requests stemming from ''Missing, '' Shea decided to bow out of politics. ''To become engaged politically means you lose your artistic credibility, your artistic energy and strength . . . so I decided I would no longer affiliate myself with anyone or anybody, only try to tell stories (as an actor) and make ideas clear through my work.''

He has come to the stage via the drama department at Bates College and Yale Drama School under Robert Brustein. How does he surface the character he wants to play? Does he climb into the role like a pair of jeans?

With a slow smile he backs into it: ''Usually Americans work from the inside out and the English work from the outside in. I've been involved, with other actors, in a mid-Atlantic approach that's neither one nor the other, but both. I am concerned with how the character walks and talks, moves and speaks, because the external characteristics provide clues to character. But I'm also concerned about how a character thinks, believes, feels - his inner life.

''The idea is to create a total character, so that while you are on stage (or camera) you are thinking the actual thoughts that character is thinking . . . this is where the actor's true act of creation comes in. It's called the subtext. The writer provides the text and the actor provides the sub-text, all the thoughts the character is thinking and all the emotions he's feeling. . . . it's an organic process, so that the thoughts you're thinking are leading to the lines the writer has given you, the way your own thoughts lead to what you say. That's why you can do a long run. . . .''

Shea sauntered into acting and into one of the big roles of his career. He was a government major at Bates College in Lewiston, Maine, when he strolled into a Little Theater rehearsal with a fellow football team member who was in the cast. Theater director Lavinia Shaeffer asked him to read a part because they were one actor short; he strolled out of the reading with the lead in Shakespeare's ''Much Ado About Nothing'' and changed his major to acting. He never auditioned for the starring role in ''Missing'' either. The film's European director, Costa-Gavras, read three rave reviews in the New York dailies on his way in from the airport the morning after ''American Days'' opened with Shea starring as a British punk rocker. He drove from the airport to Universal Casting, asked for Shea, and signed him up for ''Missing'' after catching the play. ''It's like when the time is right, it just happens. Don't you feel that? Sometimes the best things in life happen most effortlessly. They're just BANG suddenly they're there,'' says Shea.

His other credits include the male lead in the Chelsea Theater production of ''Yentl,'' ''Romeo and Juliet'' on and off Broadway, a couple of TV series (''Eight Is Enough'' and ''The Man From Atlantis''), and then a much juicier role in the highly successful Playwrights Horizon Production of A. R. Gurney's ''The Dining Room.''

Next fall his new film, ''Windy City,'' will be released. ''It's about a writer who's down and out and living in Chicago,'' he says. ''He's worshipped movie heroes, action heroes like Errol Flynn, all his life, but finds himself incapable of taking action to save his own life.''

Shea grew up in Springfield, Mass., where his father is superintendent of schools and his mother is a painter. In his junior year at Bates he spent the summer as an apprentice at the Berkshire Playhouse in Stockbridge, Mass., and met a girl sitting on a park bench who later became his wife. She is Laura Pettibone, a fine-arts photographer. ''She works with masks and mannequins, costumes them, disguises them, sets them in scenes. She writes, directs, produces her own little theater but instead of using live people, she makes her own.''

John Shea brings the same intensity to a conversation that he does to a role, and when he talks about acting he gives off sparks: ''I'll tell you something,'' about acting, he says with a rapt look. ''You're transported into another world. . . . I don't come back down to earth till after the curtain comes down. For two hours I am literally transported out of myself into this other character, this other world, where I have no time to even think about anything else. The audience comes because they are similarly catapulted out of themselves, and everybody gets lost in this make believe world. For two hours we live in a state of suspended animation.''