As is becoming a theatrical fashion, this production opens with its leading lady on stage as the audience piles in. In front of a saintly triptych, embellished with the words "Must a Christ perish in every age to save those that have no imagination", Gemma Arterton as Saint Joan communes with her God, raising her eyes heavenwards, mumbling prayers, her cheekbones perfectly sculpted by the spotlight that falls upon her.

She looks both rapt and entirely medieval – which makes the next scene all the more of a shock, as men stride in to a modern conference room in suits and a spluttering TV channel announces that the French egg price is falling thanks to a shortage triggered by the hens refusing to lay. It's because of Joan, of course, better known as the Maid, who is embarking on her visionary campaign to drive the English out of France and put the Dauphin on the French throne.

It is quite funny – but it doesn't make much sense. Nor does the rise in the Loire index a few scenes later, when Joan changes the direction of the wind. There's not much equivalence between the city and medieval battles, and I couldn't help feeling that director Josie Rourke could have updated George Bernard Shaw's examination of what happens when passionately held beliefs collide and a woman assumes the role of leader, without resorting to such literal minded devices. I was also profoundly irritated by the modish revolve of Robert Jones's boardroom set, constantly turning; it made me feel distracted rather than involved.

All of which is a shame because when Rourke stops worrying about applying relevance and simply lets the intensity of Shaw's arguments have full rein, this is a terrific production. Cutting the text with bold abandon allows her to bring the running time down to less than three hours (it's usually nearer four) and the increased focus makes Joan's pell mell dash from saintly heroine to death carry a heavy weight of inevitability.

Rourke underlines this by stressing the fact that Joan is always a woman alone; the way she wears men's clothes is part of the reason she is condemned by society. Her otherness is emphasised by the way that the past seeps into the room when she arrives. Howard Harrison's lighting suffuses her with light and religious paintings appear on the screens at the back of the set. Her appeal is transcendental and individual – her profound belief in her rightness is a challenge to any society.

No wonder society rejects her, slickly depicted in the towering scene where the two men who will ultimately condemn her to the stake - the English politician Warwick and the French cleric Cauchon - discuss her many failings over port. They fear that her nationalism, her radical spirit (protest-antism) and her heresy of talking directly to God will all bring down the established order of church and state.

That scene concludes the first half and from that point on, the play simply hurtles along, full of convincing and compelling performances. As Warwick, Jo Stone-Fewings is all icy calculation, dripping disdain, and Niall Buggy is a wonderfully smug Archbishop, simultaneously touched by Joan's innocence and frightened by her power. Fisayo Akinade lends the Dauphin just the right petulant cowardice and Rory Keenan is a chillingly pragmatic Inquisitor.

As Joan, Arterton finds a terrific grace, and quiet conviction. There is something withheld about her performance – she is conversational rather than rousing – but there's radiance too. I liked the way she juts her chin and pushes her shoulders back when finally facing down her challengers, like a young girl finding courage.

But it's the rich intellectual argument with which Shaw surrounds her that makes his Saint Joan such a vivid figure. It's quite a play. If only the production had trusted it a little more.

Saint Joan runs at the Donmar Warehouse until 18 February 2017.