- Snapshots of modern mathematics
- Diderot Mathematical Forum 2013: “Mathematics of Planet Earth”
- Pierre de Fermat and Andrew Wiles in Czech Republic stamps
- Stefan Banach (March 30, 1892 – August 8, 1945)
- Diderot Mathematical Forums
- Guessing the Numbers
- What is mathematics for Ehrhard Behrends
- What is mathematics for Krzysztof Ciesielski
- The Three Ducks Trick
- What is mathematics for Franka Brueckler

# Plus Magazine Monthly Column – October 2017 – Black holes and ordering

*Plus http://plus.maths.org is a free online magazine about mathematics aimed at a general audience. It is part of the Millennium Mathematics Project, based at the University of Cambridge and our aim is to open a door onto the world of maths for everyone. We run articles, videos and podcasts on all aspect of mathematics, from pure maths and theoretical physics to mathematical aspects of art, medicine, cosmology, sport and more. Plus has a news section, covering news from the world of maths as well as the maths behind the mainstream news, reviews of books, plays and films, as well as puzzles for you to sharpen your wits.*

**When Black Holes meet**

1.3 billion years ago, in a galaxy far, far away, two black holes circled each other. Each black hole was around 30 times the mass of the Sun, but less than 300 kilometres wide. They spiralled around each other at about half the speed of light, until finally they merged in a collision that involved more power than 50 times the total power output of all the stars in the Universe. The catastrophic event caused gravitational waves – ripples in the fabric of spacetime – that rang out at the speed of light across the Universe.

50,000 years ago, when the Neanderthals were walking the Earth, these gravitational waves reached the edge of our galaxy. Two years ago, on 14 September 2015, they emerged at the two LIGO detectors in Louisiana and Washington, USA, and provided our first observation of gravitational waves, our first direct evidence for black holes, and the beginning of a whole new era of astronomy. And this month the detection earned Rainer Weiss, Barry C. Barish and Kip S. Thorne the 2017 Nobel Prize in Physics.

A Nobel Prize in physics is always also a Nobel Prize for mathematics, at least in part, and this year’s prize is no exception. Mathematical considerations led Albert Einstein to predict, much to his horror, the existence of black holes, and also the existence of gravitational waves, both theoretical consequences of his general theory of relativity. This was over 100 years ago. Efforts to observe gravitational waves (which Einstein thought would be too weak to ever be seen by humans) began in 1985, under the direction of Thorne, Weiss, and Ron Drever. Detecting those faint signals from outer space not only required a major technological triumph, but also put Einstein’s equations to serious use in high-powered computer simulations which told physicists what they could expect to see. Once signals come in, sophisticated statistical analyses are essential in telling the real thing from all the noise.

When the discovery came on 14 September 2015 it took everybody by surprise. The two LIGO sites at Washington and Louisiana were still preparing for their first search for gravitational waves with the new generation Advanced LIGO detectors, scheduled to begin just three days later. As the detectors were being tuned up, a gravitational wave signal hit, first at Louisiana and, seven milliseconds later, at Washington. The two waveforms exactly matched. The signal was so strong it could even be seen by eye.

“We were so surprised; we were not expecting it at all,” says Gabriela González, Professor of Physics and Astronomy at Louisiana State University and former Spokesperson of the LIGO Scientific Collaboration. Although the theory predicted that pairs of black holes could produce strong gravitational waves in their collisions, it wasn’t known if such pairs really existed. And if they did exist, how many of them there were. So the LIGO detectors had intended to instead listen for collisions of neutron stars that produce weaker signals. “We were not expecting to see any signal, and just before we started taking data 24 hours a day this huge signal came to us — huge to us, of course it was tiny, but compared to our noise it was huge — and we just didn’t believe it. We thought it was a dream, we thought it was a test. It took us at least a day to be convinced!”

Since the award of the Nobel Prize this month, LIGO physicists have been blessed with another success. They have detected gravitational waves emitted by the collision of two neutron stars, which they had been very keen to observe. “A merger of neutron stars will also produce electromagnetic waves; X-rays and gamma rays that can be seen by telescopes” says Gonzales. “The different signals are produced by different features of the system: the gravitational waves by the motion of the masses, the electromagnetic waves by electrons. [By observing them both] you can tell what everything [about the neutron stars] is doing.”

Physicists have high hopes of what gravitational waves may tell us in the future. They should be able to confirm or refute mathematical conjectures about the nature of black holes, such as Stephen Hawking’s famous no hair theorem. They may even provide glimpses of the birth of the Universe and help us along with what’s perhaps the most mathematical goal of theoretical physics: to formulate the theory describing the fundamental forces of nature in a single mathematical framework.

But what would excite physicists most would be the discovery of something completely unexpected — whether it involves black holes, the Big Bang, or something entirely new. Thorne believes that gravitational waves will deliver. “At some point there will be some giant surprises,” he says. “But I can’t tell you when and I can’t tell you what.”

To find out more about gravitational waves and their detection, see our articles and videos based on a lecture by Kip Thorne and an interview with Gabriela González (here in the cover image).

**Ordering history in computer science**

Albert Einstein brings us to another topic we recently explored in Plus: how to order events in computer science.

I turn on my laptop. I open the file for this article. I start typing: the letters appearing with each keystroke, building to words, sentences and paragraphs. I save the file. I scroll back to reread. I move the cursor to edit the text: deleting some, moving some, typing more. I save the file, I scroll back to reread, I edit some more, until I feel this article is done.

That’s my view of the events taking place on my computer as I write this article. I’m aware that behind each of these events complex operations are occurring: in the text-editing software, the computer’s operating system, the machine code that translates between software and hardware. And underlying it all, this text ultimately exists as strings of 0s and 1s, manipulated and stored as strings of binary digits, physically encoded into the computer’s circuitry.

Computers are an interface between mathematical theory and physical reality; they operate both on a theoretical level of programming languages and data, and on the nuts-and-bolts level of the computer’s hardware. They are a pinnacle of discoveries in physics and engineering, but also in the mathematics and logic of computer science. Where does the realm of one end and the other begin?

“What distinguishes computer science from physics? It’s the notion of an event.” explains Leslie Lamport, principal researcher at Microsoft Research and winner of the 2013 Turing Award. My view of the events taking place on my computer is very different to how a computer scientist, an engineer or a physicist would view what is happening inside the box.

And this isn’t just a theoretical concern. Our digital lives rely on distributed computer systems, such as the internet, or the network of banks that allow us to deposit cash in one place and withdraw it in another. A distributed system is a bunch of distinct computer processes occurring in different places, that communicate with one another by exchanging messages. But the complication of a distributed system, made up of events happening in different processes, says Lamport, is that “it is sometimes impossible to say that one of two events occurred first. The relation ‘happened before’ is therefore only a partial ordering of the events in the system.”

This ambiguity of the history of the order of events inside a distributed system could have serious consequences. Imagine if the banking network couldn’t agree on the order of events occurring at different locations: for example your pay being deposited through an online transfer, and you wanting to withdraw money from an ATM at an overseas airport. No money in your account would be a catastrophe! And one that hinges on the definition of events in computer science, and how to order them.

Surprisingly, part of the Lamport’s answer to this problem stemmed from his visceral understanding of Einstein’s special theory of relativity, in which problems of ordering also occur. Lamport’s 1978 paper Time, clocks, and the ordering of events in a distributed system introduced a new rigorous way of approaching distributed computation and is one of the most cited papers in computer science. Although Lamport recalls two contrasting reactions to the paper when it was first published: “Some people thought that it was brilliant, and some people thought that it was trivial. I think they’re both right.”

“I realised that being able to totally order the events gave you the power to implement anything you wanted in a distributed system,” Lamport says. The impact of Lamport’s work was recognised in 2013 with the Turing Award, considered the Nobel Prize of Computer Science, and we all benefit from his work every day. “The Internet is based on distributed-systems technology, which is, in turn, based on a theoretical foundation invented by Leslie [Lamport],” says computer scientist Bob Taylor, one of the pioneers of the Internet. “So if you enjoy using the Internet, then you owe Leslie.”

You can read more in a series of articles, based on an interview with Leslie Lamport at the 2016 Heidelberg Laureate Forum.

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