The Predictive Power of Maths

This week, That’s Maths in The Irish Times ( TM042  ) is about the remarkable capacity of mathematics to anticipate physical phenomena that have not yet been observed.

The mathematical equations that express the laws of physics describe phenomena seen in the real world. But they also allow us to anticipate completely new phenomena.

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Conical Refraction

Early in his career the Irish mathematician William Rowan Hamilton used the equations of optics to predict an effect called conical refraction, where light rays emerging from a biaxial crystal form a cone. This had never been seen before but, within a year, it was observed by his colleague Humphrey Lloyd, thrusting Hamilton into scientific prominence.

The essence of a good scientific theory is its predictive power. We develop the theory using observations of the world around us. Then we use it to account for new observations and also to predict entirely new phenomena. Prediction is the acid test of theory.

Discovery of Neptune

The planet Neptune was mathematically predicted before it was directly observed. Newton’s law of gravitation describes the motions of the planets around the Sun. But, about two centuries ago, the orbit of Uranus was found to deviate from its predicted course.

What was wrong? Were the observations inaccurate, or were Newton’s equations faulty? Neither of these; mathematical analysis showed that the orbital perturbations could be explained by another planet orbiting outside Uranus.

Controversy over who first predicted Neptune: a French cartoon showing Adams spying on Leverrier.

Controversy over who first predicted Neptune: French cartoon shows Adams spying on Le Verrier.

In 1845 astronomers Urbain Le Verrier and John Couch Adams independently calculated the position of such a planet. Within a week, Neptune was found less than one degree from Le Verrier’s computed location by an astronomer at the Berlin Observatory. This was was a dramatic confirmation of Newton’s theory of gravitation.

General Relativity

Le Verrier did not stop there. In 1859, he report that the slow precession of Mercury’s orbit was not in agreement with Newtonian mechanics. But this time no new planet was found. The solution was more sensational: Newton’s law of gravitation was imprecise. A new theory, Einstein’s general relativity, was needed. This completely changed our view of space and time.

LEFT: Urbain Le Verrier. French physicist François Arago described him as “the man who discovered a planet with the point of his pen”. RIGHT:  Albert Einstein (1879-1955) photographed in 1921 [Image Wikipedia].

LEFT: Urbain Le Verrier, “the man who discovered a planet with the point of his pen”.                     RIGHT: Albert Einstein (1879-1955) photographed in 1921 [Image Wikipedia].

General relativity implied an orbit for Mercury in precise agreement with astronomical measurements. The new theory also predicted completely novel effects. One was the deflection from a straight path of star-light grazing the Sun’s disk. The observation of this effect during a solar eclipse in 1919 was a triumph for relativity theory.

Gravitational Waves

Another prediction made by Einstein in 1916 was that celestial bodies orbiting one another emit gravitational waves, ripples in the metric of space-time. Indirect evidence of these waves was found in binary star systems that are spiralling inwards. Further evidence emerged on St. Patrick’s Day this year, when astronomers at the Harvard-Smithsonian Center for Astrophysics released the “first direct image of gravitational waves” in the infant universe.

This evidence supports the theory of the Big Bang and cosmic inflation. However, the word “direct” is something of a misdirection: we still have not detected the energy of a gravitational wave passing the Earth. The search for this is ongoing.

When galaxies merge, their central black holes will combine, generating gravitational waves. (NGC 2207 and IC 21634) [NASA, ESA and Hubble Heritage Team].

When galaxies merge, their central black holes combine, generating gravitational waves [Image NGC 2207 and IC 21634. NASA, ESA and Hubble Heritage Team].

Another prediction is that radiating energy will eventually cause the Earth to drop into the Sun. Relax: this should not happen for many trillions of years. Long before that – within just a few billion years – the Sun will become a red giant and swallow us up.

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