Rennae Stubbs is the greatest women’s doubles player Australia has ever produced—playing on the tour for nearly 20 years, reaching No. 1 in the world, winning 60 titles, 4 of them Grand Slam tournaments.

She’s a tennis analyst now for ESPN, and she was talking with me, over a late-afternoon coffee at the Australian Open, about a Grand Slam final she and her doubles partner, fellow Australian Samantha Stosur, lost to Venus and Serena Williams, at Wimbledon in 2009. Stubbs brought up, as all players (and commentators) do, Serena’s power, her depth of shot, the unreadability of her serve.

Then she paused for a moment, before adding, “And then there’s her presence. Presence. Can’t explain it exactly. But it’s unmistakable.”

Serena Williams is an international celebrity and a cultural icon to millions. Celebrities and icons have a presence brought about and sustained by the forces of modern media—the way, say, a fashion-magazine cover or Instagram post somehow both narrows and furthers the distance between a fan and a star, and how that friction between intimacy and remove sets off the sparks (glamour and attraction, hauteur and fascination) without which that sort of presence can’t be forged.

But what about Serena’s presence on the court, the presence Stubbs was trying to put her finger on, the presence of Serena across the net in a tennis match?

The subject came up often when I talked to players during the year I spent following Williams on the women’s tour for my book Seeing Serena. It surprised me. Tennis players don’t tend to think about such things—they’re trained not to think about such things.

“And then there’s her presence. Presence. Can’t explain it exactly. But it’s unmistakable.”

They think about the match, the game, the point. And they fill their heads with visualizations of their own imposing presence. They are narcissists, mostly healthy narcissists, brimming with confident belief in their greatness, and you would have to be, too, to enter an arena alone and face the prospect of gladiatorial defeat. They are imbued with what the British philosopher Colin McGinn, who’s written a fair amount about sports, has called their peculiar “muscular solipsism.”

But not, it seems, when the opponent across the net is Serena Williams. What explains it?

A number of veteran players on the women’s tour have a long and fruitless history of matches against Serena. Since first encountering Williams, in 2012, the Czech Republic’s Barbora Strýcová has never defeated her—she’s never even taken a set from her. This can be tough on those muscles working to convince you it’s all about you.

Others talk of how Williams has crowded them by positioning herself for a service return a foot or two inside the baseline, rocking and waiting to pounce. That’s a menacing presence.

For the youngest players, still in their first years on tour, it can be the challenge of looking across the net at someone you’ve looked up to since you were four or five and first picked up a racket. Dayana Yastremska, of Ukraine, was 18 when she faced Williams (aged 37) for the first time, two years ago. Serena crushed her, 6–2, 6–1, and left her in tears.

“It seemed very simple before the match,” Yastremska said afterward. “But when I entered the arena…. I felt: I was here with Serena. I could feel her energy.” Yastremska searched for the right word but couldn’t find it. The word was presence.

Gerald Marzorati’s Seeing Serena is out now from Scribner