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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Celebrate the Holidays With the 2020 Harry Potter Funko Pop Advent Calendar

Funko
Funko

Though the main book series and movie franchise are long over, the Wizarding World of Harry Potter remains in the spotlight as one of the most popular properties in pop-culture. The folks at Funko definitely know this, and every year the company releases a new Advent calendar based on the popular series so fans can count down to the holidays with their favorite characters.

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Right now, you can pre-order the 2020 edition of Funko's popular Harry Potter Advent calendar, and if you do it through Amazon, you'll even get it on sale for 33 percent off, bringing the price down from $60 to just $40.

Funko Pop!/Amazon

Over the course of the holiday season, the Advent calendar allows you to count down the days until Christmas, starting on December 1, by opening one of the tiny, numbered doors on the appropriate day. Each door is filled with a surprise Pocket Pop! figurine—but outside of the trio of Harry, Hermione, and Ron, the company isn't revealing who you'll be getting just yet.

Calendars will start shipping on October 15, but if you want a head start, go to Amazon to pre-order yours at a discount.

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Are Halloween Pumpkins Edible?

Diane Helentjaris, Unsplash
Diane Helentjaris, Unsplash

When people visit their local family-owned pumpkin patch around Halloween, they aren’t usually looking for dinner. The majority of the nearly 2 billion pounds of pumpkins cultivated in the U.S. each year are carved up instead of eaten, making the squash a unique part of the agriculture industry. For people who prefer seasonal recipes to decorations, that may raise a few questions: Are the pumpkins sold for jack-o’-lanterns different from pumpkins sold as food? And are Halloween pumpkins any good to eat?

The pumpkins available at farms and outside supermarkets during October are what most people know, but that’s just one type of pumpkin. Howden pumpkins are the most common decorative pumpkin variety. They’ve been bred specifically for carving into jack-o’-lanterns, with a symmetrical round shape, deep orange color, and sturdy stem that acts as a handle. Shoppers looking for the perfect carving pumpkin have other options as well: the Racer, Magic Wand, Zeus, Hobbit, Gold Rush, and Connecticut field pumpkin varieties are all meant to be displayed on porch steps for Halloween.

Because they’re bred to be decoration first, carving pumpkins don’t taste very good. They have walls that are thin enough to poke a cheap knife through and a texture that’s unappealing compared to the squashes consumers are used to eating. “Uncut carving pumpkins are safe to eat; however, it's not the best type to use for cooking,” Daria McKelvey, a supervisor for the Kemper Center for Home Gardening at the Missouri Botanical Garden, tells Mental Floss. “Carving pumpkins are grown for their large size, not the flavor. Their flesh can be bland and the fibers are very stringy.”

To get the best-tasting pumpkins possible this autumn, you’re better off avoiding the seasonal supermarket displays. Many pumpkin varieties are bred especially for cooking and eating. These include Sugar Pie, Kabocha, Jack-Be-Little, Ghost Rider, Hubbard, Jarrahdale, Baby Pam, and Cinderella pumpkins. You can shop for these varieties by name at local farms or in the produce section of your grocery store. They should be easy to tell apart from the carving pumpkins available for Halloween: Unlike decorative pumpkins, cooking pumpkins are small and dense. This is part of the reason they taste better. McKelvey says. “[Cooking pumpkins] are smaller, sweeter, have a thicker rind (meatier), and have less fibers, making them easier to cook with—but not so good for carving.” These pumpkins can be stuffed, blended into soup, or simply roasted.

If you do want to get some culinary use out of your carving pumpkins this Halloween, set aside the seeds when scooping out the guts. Roasted with seasonings and olive oil, seeds (or pepitas) from different pumpkin varieties become a tasty and nutritious snack. Another option is to turn the flesh of your Halloween pumpkin into purée. Adding sugar and spices and baking it into a dessert can do a lot to mask the fruit’s underwhelming flavor and consistency.

Whatever you do, make sure your pumpkin isn’t carved up already when you decide to cook with it. There are many ways to recycle your jack-o’-lanterns, but turning them into pie isn’t one of them. "If one does plan on cooking with a carving pumpkin, it should be intact,” McKelvey says. “Never use one that's been carved into a jack-o'-lantern, otherwise you could be dealing with bacteria, dirt and dust, and other little critters.”