This sinewy, sombre, handsomely crafted and beautifully shot western is Viggo Mortensen’s second feature as a director, an impressively authored movie in which Mortensen is also writer, composer and star. With almost anyone else that might be the recipe for narcissism, and yet the self-effacing and even reticent quality in Mortensen’s screen presence works against that danger. He is, however, certainly working within the traditional strong, silent template of the old-school western hero.

Holger Olsen (played by Mortensen) is a Danish immigrant to the United States of the 1860s, who finds himself in San Francisco – a carpenter, rough-hewn outdoorsman and military veteran. He finds himself meeting the frank, unabashed gaze of Vivienne Le Coudy (Vicky Krieps) a French-Canadian woman of modest means but independent temperament, who has just broken off an understanding with a wealthy but obnoxious man (Colin Morgan).

Olsen and Vivienne move in together in Olsen’s shack just outside a distant frontier town, and Vivienne is soon going to give birth to a son. But before their family responsibilities arise, Vivienne finds herself a job in the town’s saloon bar, where she comes into fateful contact with the town’s weaselly mayor, Rudolph Schiller (soft-spoken Danny Huston). Schiller is complicit in the unlawful affairs of the town’s crooked land baron, Alfred Jeffries (Garret Dillahunt), and always ready to turn a blind eye to the behaviour of Jeffries’s psychotically violent son, Weston (Solly McLeod). As the civil war approaches, the tensions within this frontier community come to the surface.

Laid out like this, these story ingredients may seem obvious. And yet Mortensen, working with editor Peder Pedersen, remixes them into a structure of flashback/flashforward which is intriguing, makes for a delayed revelation and certainly endows Vivienne throughout with a tragic and poignant dimension. But it takes some time to get used to, particularly at the beginning, which brings us from the end of Olsen and Vivienne’s life together, back to the circumstances of their first meeting and then even back to Vivienne’s own girlhood.

It is in these childhood sequences that we see Vivienne’s own fantasy reveries about the heroism of Joan of Arc, and the all too real brutal execution of Jeanne’s father by the British redcoats. Mortensen coolly juxtaposes this execution, in the present day, with the wrongful execution of the town’s poor innocent soul, Ed Wilkins (Alex Breaux), an injustice connived at by the mayor. It is also hints at a Freudian link in Vivienne’s mind between Olsen and Vivienne’s father.

It is a world of cynicism, brutality and bad faith, in which Olsen and Vivienne’s love blooms like a miraculous flower – and Vivienne, who was formerly employed as a florist in the big city, wants to make the rough soil around Olsen’s shack bloom with flowers. Krieps and Mortensen’s rapport is just right: romantic, besotted with each other and yet tough and without illusion.

Mortensen creates a lovely touch for his character (perhaps improvised) when they visit an art exhibition hosted by her soon to be-ex-fiance. Olsen, though no pampered art lover, reaches out to one painting that is incorrectly hung and, with a touch on the frame, puts it straight. Clearly, the carpenter in him will not permit this object to hang out of alignment: he has a craftsman’s exacting sense of what is and is not right, which the pampered city slickers and art lovers do not.

Violence and tragedy is where the story is naturally heading, and this trajectory is plain in every scene and every shot: a world where aggression must either be violently and dangerously resisted or accepted. The title comes from a plaintive question from their little boy, Vincent (Atlas Green), who wonders if the bird that Olsen has just shot is in pain. Olsen tells him no – but this film seems populated by dead people who are in quite as much pain as the living, and the film’s plangent ending does nothing to take the pain away.

• The Dead Don’t Hurt screened at the Toronto film festival, and is in UK and Irish cinemas from 7 June.