Paul P. Mealing

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Tuesday, 24 November 2015

The Centenary of Einstein’s General Theory of Relativity

This month (November 2015) marks 100 years since Albert Einstein published his milestone paper on the General Theory of Relativity, which not only eclipsed Newton’s equally revolutionary Theory of Universal Gravitation, but is still the cornerstone of every cosmological theory that has been developed and disseminated since.

It needs to be pointed out that Einstein’s ‘annus mirabilis’ (miraculous year), as it’s been called, occurred 10 years earlier in 1905, when he published 3 groundbreaking papers that elevated him from a patent clerk in Bern to a candidate for the Nobel Prize (eventually realised of course). The 3 papers were his Special Theory of Relativity, his explanation of the photo-electric effect using the newly coined concept, photon of light, and a statistical analysis of Brownian motion, which effectively proved that molecules made of atoms really exist and were not just a convenient theoretical concept.

Given the anniversary, it seemed appropriate that I should write something on the topic, despite my limited knowledge and despite the plethora of books that have been published to recognise the feat. The best I’ve read is The Road to Relativity; The History and Meaning of Einstein’s “The Foundation of General Relativity” (the original title of his paper) by Hanoch Gutfreund and Jurgen Renn. They have managed to include an annotated copy of Einstein’s original handwritten manuscript with a page by page exposition. But more than that, they take us on Einstein’s mental journey and, in particular, how he found the mathematical language to portray the intuitive ideas in his head and yet work within the constraints he believed were necessary for it to work.

The constraints were not inconsiderable and include: the equivalence of inertial and gravitational mass; the conservation of energy and momentum under transformation between frames of reference both in rotational and linear motion; and the ability to reduce his theory mathematically to Newton’s theory when relativistic effects were negligible.

Einstein’s epiphany, that led him down the particular path he took, was the realisation that one experienced no force when one was in free fall, contrary to Newton’s theory and contrary to our belief that gravity is a force. Free fall subjectively feels no different to being in orbit around a planet. The aptly named ‘vomit comet’ is an aeroplane that goes into free fall in order to create the momentary sense of weightlessness that one would experience in space.

Einstein learnt from his study of Maxwell’s equations for electromagnetic radiation, that mathematics could sometimes provide a counter-intuitive insight, like the constant speed of light.

In fact, Einstein had to learn new mathematics (for him) and engaged the help of his close friend, Marcel Grossman, who led him through the technical travails of tensor calculus using Riemann geometry. It would seem, from what I can understand of his mental journey, that it was the mathematics, as much as any other insight, that led Einstein to realise that space-time is curved and not Euclidean as we all generally believe. To quote Gutfreund and Renn:

[Einstein] realised that the four-dimensional spacetime of general relativity no longer fitted the framework of Euclidean geometry… The geometrization of general relativity and the understanding of gravity as being due to the curvature of spacetime is a result of the further development and not a presupposition of Einstein’s formulation of the theory.

By Euclidean, one means space is flat and light travels in perfectly straight lines. One of the confirmations of Einstein’s theory was that he predicted that light passing close to the Sun would be literally bent and so a star in the background would appear to shift as the Sun approached the same line of sight for an observer on Earth as for the star. This could only be seen during an eclipse and was duly observed by Arthur Eddington in 1919 on the island of Principe near Africa.

Einstein’s formulations led him to postulate that it’s the geometry of space that gives us gravity and the geometry, which is curved, is caused by massive objects. In other words, it’s mass that curves space and it’s the curvature of space that causes mass to move, as John Wheeler famously and succinctly expounded.

It may sound back-to-front, but, for me, Einstein’s Special Theory of Relativity only makes sense in the context of his General Theory, even though they were formulated in the reverse order. To understand what I’m talking about, I need to explain geodesics.

When you fly long distance on a plane, the path projected onto a flat map looks curved. You may have noticed this when they show the path on a screen in the cabin while you’re in flight. The point is that when you fly long distance you are travelling over a curved surface, because, obviously, the Earth is a sphere, and the shortest distance between 2 points (cities) lies on what’s called a great circle. A great circle is the one circle that goes through both points that is the largest circle possible. Now, I know that sounds paradoxical, but the largest circle provides the shortest distance over the surface (we are not talking about tunnels) that one can travel and there is only one, therefore there is one shortest path. This shortest path is called the geodesic that connects those 2 points.

A geodesic in gravitation is the shortest distance in spacetime between 2 points and that is what one follows when one is in free fall. At the risk of information overload, I’m going to introduce another concept which is essential for understanding the physics of a geodesic in gravity.

One of the most fundamental principles discovered in physics is the principle of least action (formulated mathematically as a Lagrangian which is the difference between kinetic and potential  energy). The most commonly experienced example would be refraction of light through glass or water, because light travels at different velocities in air, water and glass (slower through glass or water than air). The extremely gifted 17th Century amateur mathematician, Pierre de Fermat (actually a lawyer) conjectured that the light travels the shortest path, meaning it takes the least time, and the refractive index (Snell’s law) can be deduced mathematically from this principle. In the 20th Century, Richard Feynman developed his path integral method of quantum mechanics from the least action principle, and, in effect, confirmed Fermat’s principle.

Now, when one applies the principle of least action to a projectile in a gravitational field (like a thrown ball) one finds that it too takes the shortest path, but paradoxically this is the path of longest relativistic time (not unlike the paradox of the largest circle described earlier).

Richard Feynman gives a worked example in his excellent book, Six Not-So-Easy Pieces. In relativity, time can be subjective, so that a moving clock always appears to be travelling slow compared to a stationary clock, but, because motion is relative, the perception is reversed for the other clock. However, as Feynman points out:

The time measured by a moving clock is called its “proper time”. In free fall, the trajectory makes the proper time of an object a maximum.

In other words, the geodesic is the trajectory or path of longest relativistic time. Any variant from the geodesic will result in the clock’s proper time being shorter, which means time literally slows down. So special relativity is not symmetrical in a gravitational field and there is a gravitational field everywhere in space. As Gutfreund and Renn point out, Einstein himself acknowledged that he had effectively replaced the fictional aether with gravity.

This is most apparent when one considers a black hole. Every massive body has an escape velocity which is the velocity a projectile must achieve to become free of a body’s gravitational field. Obviously, the escape velocity for Earth is larger than the escape velocity for the moon and considerably less than the escape velocity of the Sun. Not so obvious, although logical from what we know, the escape velocity is independent of the projectile’s mass and therefore also applies to light (photons). We know that all body’s fall at exactly the same rate in a gravitational field. In other words, a geodesic applies equally to all bodies irrespective of their mass. In the case of a black hole, the escape velocity exceeds the speed of light, and, in fact, becomes the speed of light at its event horizon. At the event horizon time stops for an external observer because the light is red-shifted to infinity. One of the consequences of Einstein’s theory is that clocks travel slower in a stronger gravitational field, and, at the event horizon, gravity is so strong the clock stops.

To appreciate why clocks slow down and rods become shorter (in the direction of motion), with respect to an observer, one must understand the consequences of the speed of light being constant. If light is a wave then the equation for a wave is very fundamental:

v = f λ , where v is velocity, f is the frequency and λ is the wavelength.

In the case of light the equation becomes c = f λ , where c is the speed of light.

One can see that if c stays constant then f and λ can change to accommodate it. Frequency measures time and wavelength measures distance. One can see how frequency can become stretched or compressed by motion if c remains constant, depending whether an observer is travelling away from a source of radiation or towards it. This is called the Doppler effect, and on a cosmic scale it tells us that the Universe is expanding, because virtually all galaxies in all directions are travelling away from us. If a geodesic is the path of maximum proper time, we have a reference for determining relativistic effects, and we can use the Doppler effect to determine if a light source is moving relative to an observer, even though the speed of light is always c.

I won’t go into it here, but the famous twin paradox can be explained by taking into account both relativistic and Doppler effects for both parties – the one travelling and the one left at home.

Saturday, 14 November 2015

Two miracles that are fundamental to the Universe and our place in it

The 2 miracles I’m referring to will not be found in any scripture and God is not a necessary participant, with the emphasis on necessary. I am one of those rare dabblers in philosophy who argues that science is neutral on the subject of God. A definition of miracle is required, so for the purpose of this discussion, I call a miracle something that can’t be explained, yet has profound and far-reaching consequences. ‘Something’, in this context, could be described as a concordance of unexpected relationships in completely different realms.

This is one of those posts that will upset people on both sides of the religious divide, I’m sure, but it’s been rattling around in my head ever since I re-read Eugene P. Wigner’s seminal essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. I came across it (again) in a collection of essays under the collective title, Math Angst, contained in a volume called The World Treasury of Physics, Astronomy and Mathematics edited by Timothy Ferris (1991). This is a collection of essays and excerpts by some of the greatest minds in physics, mathematics and cosmology in the 20th Century.

Back to Wigner, in discussing the significance of complex numbers in quantum mechanics, specifically Hilbert’s space, he remarks:

‘…complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers in this case is not a calculated trick of applied mathematics but comes close to being a necessity in the formulation of the laws of quantum mechanics.’

It is well known, among physicists, that in the language of mathematics, quantum mechanics not only makes perfect sense but is one of the most successful physical theories ever. But in ordinary language it is hard to make sense of it in any way that ordinary people would comprehend it.

It is in this context that Wigner makes the following statement in the next paragraph following the quote above:

‘It is difficult to avoid the impression that a miracle confronts us here… or the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.’

Hence the title of my post. The key that links the 2 miracles is mathematics. A number of physicists: Paul Davies, Roger Penrose, John Barrow (they’re just the ones I’ve read); have commented on the inordinate correspondence we find between mathematics and regularities found in natural phenomena that have been dubbed ‘laws of nature’.

The first miracle is that mathematics seems to underpin everything we know and learn about the Universe, including ourselves. As Barrow has pointed out, mathematics allows us to predict the makeup of fundamental elements in the first 3 minutes of the Universe. It provides us with the field equations of Einstein’s general theory of relativity, Maxwell’s equations for electromagnetic radiation, Schrodinger’s wave function in quantum mechanics and the four digit software code for all biological life we call DNA.

The second miracle is that the human mind is uniquely evolved to access mathematics to an extraordinarily deep and meaningful degree that has nothing to do with our everyday prosaic survival but everything to do with our ability to comprehend the Universe in all the facets I listed above.

The 2 miracles combined give us the greatest mystery of the Universe, which I’ve stated many times on this blog: It created the means to understand itself, through us.

So where does God fit into this? Interestingly, I would argue that when it comes to mathematics, God has no choice. Einstein once asked the rhetorical question, in correspondence with his friend, Paul Ehrenfest (if I recall it correctly): did God have any choice in determining the laws of the Universe? This question is probably unanswerable, but when it comes to mathematics, I would answer in the negative. If one looks at prime numbers (there are other examples, but primes are fundamental) it’s self-evident that they are self-selected by their very definition – God didn’t choose them.

The interesting thing about primes is that they are the ‘atoms’ of mathematics because all the other ‘natural’ numbers can be determined from all the primes, all the way to infinity. The other interesting thing is that Riemann’s hypothesis indicates that primes have a deep and unexpected relationship with some of the most esoteric areas of mathematics. So, if one was a religious person, one might suggest that this is surely the handiwork of God, yet God can’t even affect the fundamentals upon which all this rests.