The Best of All Possible Worlds

by Nicholas Mee on March 15, 2017

Voltaire is best known today for Candide, a short novel published in 1779. The young hero Candide travels the world in a tale littered with rape, murder, pestilence, enslavement and natural catastrophe. Amidst this apocalyptic nightmare Candide’s tutor Dr Pangloss maintains a philosophical detachment, arguing against all evidence to the contrary that we live in the best of all possible worlds. The purpose of this strange tale was to demolish a theological position held by the philosopher Leibniz caricatured as Dr Pangloss, and this objective was certainly achieved. So how did Leibniz come to be identified with such a world-view?

A scene from the opera Candide by Leonard Bernstein.
(School of Music, The University of South Carolina.)

The story begins in antiquity with Hero of Alexandria (also known as Heron), a brilliant engineer and mathematician in the days of the first Roman emperor Augustus. His works survive in scattered fragments that describe ingenious devices such as vending machines, wind-powered pipe-organs and even a simple steam-powered gadget the aeolipile, named after the Greek wind god Aeolus.

A modern reconstruction of the aeolipile. Left: starting up. Right: in full motion.
(Photographs by John R. Bentley.)

Hero also wrote the Catoptrica (theory of mirrors) in which he attempted to explain why light rays follow straight lines. He proposed that a light ray travelling between two points takes the path that minimizes the time taken, and noted that if light travels at a constant speed, then the quickest route is also the shortest route – a straight line. In other words, any other path that the light ray might have followed would have taken longer than the actual path. Hero showed that this optimization principle would also explain why the angle of incidence equals the angle of reflection when an image is reflected from a mirror.

In 1662, the French mathematician Pierre de Fermat extended Hero’s idea to explain refraction. When passing between different media, such as air and water, the path of a light ray bends at the interface between the media. According to Fermat, this is because light travels slower in water than in air, so the quickest route is not perfectly straight, it has a longer straight segment in the faster medium and a shorter straight segment in the slower medium. Fermat used this argument to derive Snell’s law and explain how lenses work. In Fermat’s day, the speed of light had not been determined in any medium so his proposal was rather speculative, but it proved to be correct.

A laser beam fired from the top left into a tank of water. The beam is partially reflected from the surface of the water, with the angle of incidence equalling the angle of reflection, and the transmitted beam is refracted at an angle to the incoming beam. (Image by Towert7.)

Leibniz is credited with being the first to realise the wider significance of optimization in physics. Although this approach to mechanics was not perfected until the 19th century, Leibniz’ account survives in a letter from 1707.

The motion of a body such as a cannonball, subject to known forces, may be calculated using Newton’s laws of motion. This is often expressed in terms of changes in its kinetic energy K and potential energy V, where its total energy is E = K + V. The key to the alternative optimization approach to mechanics is the Lagrangian, L = K – V, defined as the difference between the kinetic and potential energy. Integrating the Lagrangian over a path produces a quantity known as the action. When this is calculated for the path that a body follows through space it is smaller than for any other path the body might have taken. This is the principle of least action. It means that the trajectory of a material body can be predicted by finding the path that minimizes its action.[1] Newton’s laws of motion can be derived from the principle of least action, so the optimization approach is completely equivalent to Newtonian methods. But the principle of least action is more fundamental with much wider application. For instance, Maxwell’s equations for electric and magnetic fields can be derived from an action principle, and so can the Einstein equation, which describes gravity as curved spacetime. Indeed, the action lies at the heart of theoretical physics. It is essential when formulating the incredibly successful modern theories of elementary particles, such as the Standard Model, and even the more speculative and esoteric theories of strings.

This raises the question of why such optimization principles should hold. How does a beam of light know which route is quickest? How does a cannonball know that by tracing out a parabola it is minimizing its action? Leibniz thought he knew the answer and saw it as firm evidence for God’s creative power.

Gottfried Wilhelm Leibniz
by Johann Friedrich Wentzel (c. 1700).

Leibniz agreed with Thomas Aquinas and other philosophers that even an all-powerful God must be constrained by the laws of geometry and logic. It would therefore be impossible for God to create a world containing logical contradictions. But this necessarily means the world includes some bad as well as good. For instance, although providing humans with free will is a great good, this necessarily allows humans to commit crimes, which is bad. However, because God is all-powerful and has the attribute of perfect goodness, God will optimize the goodness of the world. So, although the world cannot be flawless, weighing the good and the bad together, it must be the best of all possible worlds. Leibniz saw the optimization principles of mechanics as clear evidence of God’s role in the optimization of the universe.

However, as Bertrand Russell pointed out, it is a short step from here to an argument claiming that this must be the worst of all possible worlds. If the world was created by a malevolent demiurge, perhaps humans were given consciousness to maximize awareness of their suffering. Leibniz was lampooned mercilessly by Voltaire and since Candide, rational arguments for God’s role in the world have lost much of their force.

It is remarkable that so much physics can be described in terms of optimization principles. But this is just the first half of the story, the second half is even more surprising. We now have a far better understanding of the origin of the action principle, and it is all due to quantum mechanics.



[1] It is worth noting that in some circumstances a trajectory is determined by maximization of the action rather than minimization, and also that the optimization of the action is local and not necessarily absolute. In other words, the trajectory optimizes the action compared to neighbouring paths.

 

Further Information

The principle of least action is a theme that runs through the new book that I have written with Nick Manton – The Physical World: An Inspirational Tour of Fundamental Physics. There is more information about the book here: http://www.oup.com/localecatalogue/cls_academic/?i=9780198796114

Candide by Voltaire

A History of Western Philosophy, Chapter XI: Leibniz by Bertrand Russell

The End of Time: The Next Revolution in Our Understanding of the Universe, Chapter 7: Paths in Platonia  by Julian Barbour

The Aeolipile by John R. Bentley

 

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The Event Horizon Telescope

by Nicholas Mee on March 2, 2017

The image below shows a beautiful region of the night sky in the constellation Sagittarius. The asterism known to amateur astronomers as the ‘teapot’ forms part of the constellation. This is rather apt as the many nebulae and gas clouds located towards the centre of the galaxy appear as steam rising from the spout of the teapot. The precise centre of the galaxy is indicated by an ‘X’ in the illustration.

X marks the spot of the centre of the galaxy, a region known to astronomers as Sgr A*.

Where the Action is!

Radio astronomers have named this region is Sgr A*, an abbreviation that means the most powerful source of radio signals in the constellation of Sagittarius. The star * is added to emphasise the special nature of this object. It is where the action is in our galaxy. Our immediate cosmic neighbourhood is incredibly quiet. The Sun is surrounded by oceans of space, it is over four light years to the nearest star. By contrast, within one light year of the centre of the galaxy there are, perhaps, a million stars. These include many burnt out stellar remnants such as neutron stars and black holes, as well as many luminous blue supergiants.

The Innermost Heart of the Galaxy

German astronomer Reinhard Genzel studied the innermost heart of the galaxy in the early 1990s using the European Southern Observatory’s 3.5 metre New Technology Telescope in Chile. His observations showed that the stars at the centre of the galaxy are moving extremely fast, and the closer to the centre the faster they are travelling. This suggests that there is a very high concentration of mass right at the centre. Furthermore, the location of the point right at the centre appears to be fixed while all else whirls frantically around it.

The orbits of the stars at the centre of the galaxy as mapped out by Andrea Ghez and her team.
Credit: Keck/UCLA Galactic Center Group.

Genzel’s observations were followed up by the American astronomer Andrea Ghez and her team with the two 10 metre Keck telescopes in Hawaii. The stars right at the centre of the galaxy are moving so quickly that over the course of just a few years they were able to plot out significant segments of their orbital paths. The closest neighbours to Sgr A* are racing around at up to 5 million kilometres per hour. As well as tracking their motion across the sky, it is possible to measure their velocity towards or away from us through the Doppler shift of their light. This has enabled Ghez and her team to calculate accurate trajectories of these stars in three dimensions. One such star designated SO-2, which takes fifteen and a half years to complete its highly eccentric orbit, has been monitored over the course of an entire orbit. It will be watched eagerly as it returns for another close encounter with the central black hole next year. Ghez has also found a star known as SO-102 with an even smaller 11.5 year orbit.

A Supermassive Black Hole

The speed at which SO-2 and these other stars are moving is determined by the mass of the object that they are orbiting. This mass can be calculated using Kepler’s 3rd Law and it turns out to be around 4 million times the mass of the Sun. But the observations show that this object is smaller than the Earth’s orbit around the Sun. There is only one possible conclusion – it is a supermassive black hole. The event horizon of the black hole is thought to have a radius of around 12 million kilometres. Anything that finds itself within this radius, including light, cannot escape the clutches of the black hole. By comparison the radius of the Sun is 700,000 kilometres. So the black hole event horizon has a diameter that is around twenty times that of the Sun.

An artist’s impression of a supermassive black hole.
Credit: ESO/L. Calçada.


The Event Horizon Telescope

A computer generated image showing what the Event Horizon Telescope is expected to reveal.

The ultimate challenge is to image the event horizon of the black hole, but at a distance of around 25,000 light years this is currently beyong the resolving power of even the world’s best astronomical instruments. This could all change within the next few months as Shep Doeleman of MIT (Massachusetts Institute of Technology) is leading an incredibly ambitious international effort to assemble the Event Horizon Telescope (EHT) in order to generate the world’s first image of a black hole. Success will require at least 5,000 times the resolving power of the Hubble Space Telescope. It is comparable to imaging a cricket ball on the Moon. The galactic centre is shrouded in hot gas, which blocks the visible light emitted from the stars in this region of the galaxy. Infra-red radiation is much better at penetrating the murk, however, so the EHT will be an Earth-sized instrument operating in the far infra-red/microwave region of the spectrum. It will combine the data collected by a network of radio telescopes around the world to produce images with an unparalleled resolution. These instruments are located at sites in California, Arizona, Hawaii, Chile, Europe and even the South Pole.

Some of the sites of the telescopes that will form the Event Horizon Telescope. The images from these telescopes will be combined using Very Long Baseline Interferometry.

A second target for the Event Horizon Telescope is the supermassive black hole at the centre of the giant elliptical galaxy M87. This is a huge galaxy at the centre of the nearby Virgo cluster of galaxies. Nearby in cosmological terms, anyway. The distance to M87 is around 53 million light years, so it is about 2 thousand times as distant as the galactic centre. However, the supermassive black hole at its core is believed to be around 6 billion times the mass of the Sun, so the radius of its event horizon is about 1,500 times that of the supermassive black hole at the centre of our galaxy. This means that imaging its event horizon should be only marginally more difficult.

A jet is emanating from the centre of the giant elliptical galaxy M87 – the diffuse amber sphere in this image. The jet is thought to have been produced by the supermassive black hole at the centre of the galaxy. Credit: Hubble Space Telescope/NASA

What’s more, the M87 supermassive black hole is very active. It has spewed out an enormous jet into intergalatic space, as can be seen in the image above. (It is assumed that there is a second jet in the opposite direction, but we can only see the one pointing towards us.)

We are entering a new era for black hole physics. We might have the first direct image of a black hole some time this year.

Further Information

There is a lot more information about black holes in my book Gravity: Cracking the Cosmic Code. www.virtualimage.co.uk/html/gravity.html

The official website of the Event Horizon Telescope is at:
http://www.eventhorizontelescope.org/science/index.html

 

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