What, you think they wear it just because it's hot out? Ha!
Oh, Muffy. You've got quite a brain in that little blonde head of yours! That's like asking why the sky above us is blue, or why the water in the club pool is wet. Or why it tastes like gin.
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To answer briefly, and obviously: White reflects the entire spectrum of light, which means it absorbs little heat. It therefore keeps you cooler than colors that absorb most of the spectrum. But there's more to it than that.
You see, while the uniform developed around the turn of the century, tennis is an ancient game. We can trace it to a group of 12th-century French monks who, while at play, hit a ball to one another with the flattened palms of their hands.
By the 16th century, the French and English aristocracy played a version of this jeu de paume with a rudimentary racket. The cousin of modern tennis sealed its place in the hearts of the gentry when Henry VIII built a court for the game at one of his palaces.
Our sporty little shorts and polos did not exist then. While the king did own several outfits designed specifically for tennis, they didn't vary too much from his usual daywear. In general, he and his (male) courtiers would have played in tights and breeches.
While such restrictive clothing no doubt taxed the Nadals and Federers of old, still the game spread across Europe. When tennis-like games came to the United States in the 18th and 19th centuries, rich Americans too played in their day clothes, and were, you can imagine, a little uncomfortable doing so.
It's fair to say that, around the turn of the 20th century, a game very much like contemporary tennis had developed and become a sport. But even then the dictates of modesty prevailed over those of performance—especially for women. Through the end of the 19th century, young ladies wore full bustles, dress hats, and gloves on the court.
Nonetheless, this era saw the first tennis tournament. And, eventually, practicality won out. Competitive tennis called for new outfits. It called for tennis whites!
Library of Congress
By about this time, the rich in America (and England) had adopted summer white as a symbol of their leisure. Since white clothing dirties easily, it didn't recommend itself to factory workers and domestic servants in a dry-cleaning-less era of weekly baths. In fact, really the only people suited to wear white were people who didn't work at all—or who could at least afford to look like they didn't.
Library of Congress
Wearing white cost a lot. If you wore it, you signaled that you, too, cost a lot. Since tennis had long been a summer game for the rich, the rich wore white to play tennis. And—since the middle class likes to imitate the rich—as tennis democratized over the course of the 20th century, the middle class wore white to play, too.
Gradually, white for tennis became a rule, so that in 1890 Wimbledon mandated all-white outfits for players. Elsewhere, the all-white tennis outfit became a decorous standard by the early 20th century—just in time, by the way, for the Frenchwoman Suzanne Lenglen to aggravate rich people everywhere by playing tennis in a (white) calf-length skirt and short sleeves.
American flappers, however, saw no problem with this. As general hemlines rose, so too—and faster still—did the hemline of the tennis skirt. American Gussie Moran wore an above-the-knee skirt to compete at Wimbledon in 1949. Her ruffled white underwear showed with every movement, and photographers lay on the ground in order to best capture it. (As usual, men faced easier fashion choices. They began to wear white around the same time as women, but their move from pants to shorts over the first half of the 20th century made little stir.)
Unlucky Gussie. She was embarrassed, but a decade or so later, her short skirt was old news. We had reached the age of the white and the short, and most female athletes thought nothing of bearing their underwear while they played—or of wearing shorts instead. (Helen Jacobs was very first to do so at a major match, in 1933.) Over time, today's tennis whites became the standard outfit. Tight and colorful aberrances in the 1980s and early '90s met appropriate outrage.
Except when they're not aberrances. See, for instance, players at the London Olympics. Over the past 20 or so years, color re-entered the mainstream of the sport, by way of sponsorship deals and branded high-performance athletic gear. Pros now view hue as a way to stand out on the court and "express themselves."
Who—you ask—allowed these color-clad clowns on Wimbledon grass?
Do I look like I'm made of answers, Muffy? You know tennis is not my game. Now, if you'll excuse me, I'm going to take a dip in the pool.