| As scientists venture to understand our world, mathematics often serves as both their language and their guide. Physicists rely on math not just to describe what they see in their laboratories but also to predict and explore phenomena that their tools cannot otherwise touch, like the insides of black holes or the moments just after the universe began. Some of the biggest breakthroughs in physics were only made possible by mathematical advances. Isaac Newton's laws of motion, for instance — which allow us to model how a planet will orbit its sun or how fast an object will fall — first required the invention of calculus.
But math isn't just useful for introducing or developing a new physical theory. Long after physicists have established a theory, mathematicians comb over it with the rigor that their field demands, intent on placing it on more solid logical footing. This cleanup work can take decades, but it's necessary to establish deeper trust in physical ideas.
There's an entire field devoted to the mathematical study of physics-inspired problems, aptly called mathematical physics. One of its core aims is to work out the precise consequences of the complicated mathematics at the heart of general relativity. In 1915, Albert Einstein showed that the shape of our four-dimensional universe — composed of three spatial dimensions plus one dimension of time — is determined by the matter that lives within it. That shape in turn gives rise to what we experience as gravity. From the mathematics of general relativity, bizarre aberrations like black holes emerge. Even a century later, many of these phenomena remain mysterious. And so mathematicians continue to pore over Einstein's equations, using them as a conceptual laboratory in which to test out new hypotheses, gain novel insights and prove ideas that physicists might take for granted.
Quanta has covered a lot of recent research on the mathematics of space-time, thanks in large part to the work of contributing writer Steve Nadis.
What's New and Noteworthy Long after physicists have accepted something as true, it's often left to mathematicians to give it a rigorous foundation and to build a complete, coherent framework around it.
Einstein's general theory of relativity predicts how the matter that fills space-time, like [MOU1] stars and galaxies, warps and curves its shape. But the equations that describe this are notoriously hard to work with. For example, physicists have long accepted that Einstein's equations imply that less matter means less warping — that is, flatter space. But it wasn't until last year that mathematicians definitively proved it. Similarly, a common assumption in physics holds that, unlike the other shapes that space-time might take, a negatively curved universe is deeply unstable: Any matter placed within it will eventually collapse into a black hole. Yet mathematicians were only recently able to verify this.
Mathematical proofs don't always go the way physicists expect. In August, for instance, Quanta reported on how a pair of mathematicians felled the great Stephen Hawking's "third law" of black hole thermodynamics. Fifty years ago, Hawking and two other physicists conjectured that "extremal" black holes, which pack so much electric charge or spin that they behave in incredibly counterintuitive ways, are mathematically impossible. But the new proof demonstrates that they can exist — at least in theory.
Of course, mathematicians do more than clean up physicists' mess. They can also provide new and important insights. By rigorously redefining long-standing models of how fundamental particles interact, mathematicians have been able to offer a better understanding of how quantum gravity might work. They've also been able to explore black holes more deeply. Even though physicists can now observe black holes in the real world, they still can't tell you whether any given patch of matter-filled space will eventually turn into one. Mathematicians can — and they're getting really good at it. It was a mathematician, Roy Kerr, who realized in 1963 that black holes could rotate, and another who, two years ago, proved that such a black hole is stable.
Mathematicians' emphasis on abstraction also lets them take black holes into weird worlds physicists might not even imagine. It turns out that in a five-dimensional universe, for instance, black holes wouldn't have to be spherical anymore. They can instead come in all sorts of exotic forms.
Whether they're providing theoretical scaffolding or exploring concepts in the nth dimension, mathematicians have been instrumental in propelling physics forward. And with gravity and quantum mechanics still at odds, space-time is one area where mathematical ideas tend to lead rather than follow.
This week, Quanta Magazine will publish "The Unraveling of Space-Time," a series that explores the fundamental nature of reality. We'll send a note to your inbox when the series goes live.
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