Why Has More Than One Person Solved Einstein's Equations?

Keystone/Getty Images
Keystone/Getty Images

Viktor T. Toth:

For the same reason more than one person solves most other types of equations.

Unlike simple equations such as, say, the quadratic equation you learn about in high school, most equations do not have nice, simple, general solutions. Rather, specific solutions exist for specific values, or specific sets of values, of the equations' parameters.

Einstein’s field equations are like this. Spelled out in full, they represent a set of 10 coupled second-order differential equations in 10 unknown functions. That is not something that you just whip up a solution for.

The solutions that do exist are solutions representing special cases. The most famous among these is perhaps the Schwarzschild solution. This is a solution that represents a highly symmetric scenario: a vacuum solution (with no matter present), which does not depend on time, and which is spherically symmetric, so it depends only on the radial coordinate. In the end, this turns out to be a solution of only two unknown functions, in the form of two very simple differential equations that can be readily solved.

Other solutions are not this simple. In most cases, nice, elegant, closed form solutions do not exist, so the equations need to be solved numerically. And even that is a challenge, as it is difficult to specify initial values for the unknown functions that correspond to physically meaningful, stable configurations of matter. There is an entire discipline, numerical relativity, devoted to this topic alone.

Bottom line: most equations do not have nice, simple, generic solutions, and Einstein’s field equations are no exception.

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What's That Thing That Hangs Off a Turkey’s Face?

NCHANT/iStock via Getty Images
NCHANT/iStock via Getty Images

That thing is called a snood. And it's there to let the other turkeys know that its owner is kind of a big deal.

When a male turkey—known as a tom—wants to mate, he faces two hurdles. One is his potential mates, the female turkeys (a.k.a. hens). In the realm of turkey mating, the hens wield the power of choice and the toms have to get a hen's attention and win the opportunity to reproduce. Come mating season, a tom will strut around, gobble, puff out his chest, fan his tail, and drag his wings to attract the hens, who then pick which of the toms they’ll mate with.

The second problem for a tom looking for love is the other toms in the area. They’re all competing for the same limited number of hens. Sometimes a good mating display isn’t enough to win a mate, and toms will attack and fight each other to secure a hen.

This is where the snood comes in. That goofy-looking piece of dangling flesh helps a tom both with choosy hens and with competition from rival males. Having a long snood almost always means that a hen will want to mate with him and that another tom will back down from a fight.

Dudes and their snoods

When two toms are trying to establish dominance, they’ll size each other up. Then they'll either fight, or one will flee.

In the late 1990s, Dr. Richard Buchholz, an animal behaviorist who focuses on turkeys, wanted to figure out which, if any, characteristics of a tom turkey could predict how they fare in dominance fights. That is, did bigger turkeys tend to win more scuffles? Did older ones? He also wanted to see if the turkeys used any of these predictive cues when sizing each other up. He looked at various characteristics of dominant toms that fight and win, and compared them to those of subordinate toms that lose fights or run from them. Of all the characteristics he looked at, only “relaxed snood length” seemed to be a reliable predictor of how a tom would do in bird-vs-bird combat. The dominant males, the ones who won fights and got a choice mate, had longer snoods.

With that in mind, Buchholz looked at how toms reacted to other toms with snoods of varying sizes. The birds tended to avoid confrontation with other males with longer snoods, and wouldn’t even feed near them. A big snood, this suggests, says to the other turkeys that this is a tom you don’t want to tangle with. Buchholz noted that snood length correlates with age, body mass, and testosterone, so, to competitors, the snood could be a good indicator of a tom’s aggressiveness, age/experience, size, and overall condition and fighting ability.

In the snood for love

Once the males have established who’s going to have a chance to mate, the final choice goes to the hen. While the mating display is the main draw for getting a hen to check him out, a tom’s snood helps him out again here.

Like it did for the other males, a tom’s snood signals a lot of information to a female assessing potential mates—it indicates how old and how big he is, and even says something about his health. In another study, Buchholz found that longer-snooded toms carried fewer parasites. If a hen wanted to choose a mate with good genes that might help her offspring grow large, live long, and avoid parasites, a tom’s snood is a good advertisement for his genes. In that study, hens showed a clear preference for toms with longer snoods. In another experiment years later, Buchholz found that healthy hens again showed a strong preference for long snoods and that hens with their own parasite problems were less picky about snood length and checked out more potential mates—perhaps, Buchholz thinks, because the hens recognized their own susceptibility to infection and were willing to invest more time searching for a tom with genes for parasite resistance that would complement their own—but still showed some preference for longer ones.

While a snood might look goofy to us, for a turkey, it’s integral to the mating game, signaling to other toms that they should get out of his way and letting hens know that he’s got what they’re looking for.

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What's the Difference Between Yams and Sweet Potatoes?

Julia_Sudnitskaya/iStock via Getty Images
Julia_Sudnitskaya/iStock via Getty Images

This Thanksgiving, families across the country will enjoy a traditional meal of turkey, stuffing, and sweet potatoes ... or are they yams? Discussions on the proper name for the orange starchy stuff on your table can get more heated than arguments about topping them with marshmallows. But there's an easy way to tell the difference between sweet potatoes and yams: If you picked up the tuber from a typical American grocery store, it's probably a sweet potato.

So what's a sweet potato?

Sweet potato and yam aren't just different names for the same thing: The two produce items belong to their own separate botanical categories. Sweet potatoes are members of the morning glory family. Regular potatoes like russets, meanwhile, are considered part of the nightshade family, which means that sweet potatoes aren't actually potatoes at all.

Almost all of the foods most Americans think of as yams are really sweet potatoes. The root vegetable typically has brown or reddish skin with a starchy inside that's orange (though it can also be white or purple). It's sold in most supermarkets in the country and used to make sweet potato fries, sweet potato pie, and the sweet potato casserole you have at Thanksgiving.

Then what's a yam?

Yams.
Yams.
bonchan/iStock via Getty Images

Yams are a different beast altogether. They are more closely related to lilies and grasses and mostly grow in tropical environments. The skin is more rough and bark-like than what you'd see on a sweet potato, and the inside is usually white or yellowish—not orange.

They're a common ingredient in parts of Asia, Africa, and the Caribbean. Because the inside of a yam is less moist than the inside of a sweet potato, they require more fat to make them soft and creamy. They're also less sweet than their orange-hued counterparts. In many regions in the U.S., yams aren't sold outside of international grocery stores.

Where did the mix-up come from?

Also sweet potatoes.
Also sweet potatoes.
Kateryna Bibro/iStock via Getty Images

So if yams and sweet potatoes are two totally different vegetables that don't look or taste that similar, why are their names used interchangeably in the U.S.? You can blame the food industry. For years, "firm" sweet potatoes, which have brown skin and whitish flesh, were the only sweet potatoes grown in the U.S. In the early 20th century, "soft" sweet potatoes, which have reddish skin and deep-orange flesh, entered the scene. Farmers needed a way to distinguish the two varieties, so soft sweet potatoes became yams.

Nearly a century later, the misnomer shows no signs of disappearing. Many American supermarkets still call their orange-fleshed sweet potatoes yams and their white-fleshed ones sweet potatoes, even though both items are sweet potatoes. But this isn't a strict rule, and stores often swap the names and make things even more confusing for shoppers. So the next time you're shopping for a recipe that calls for sweet potatoes, learn to identify them by sight rather than the name on the label.

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