Paul P. Mealing

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Sunday, 8 November 2009

Einstein’s Code and Kerr’s Solution

When I started this blog, over 2 years ago now, I never anticipated picking up ‘followers’ and now I feel the need to maintain some sort of standard. For those who do follow this blog, it is obvious that I don’t comment on a regular basis (although I do on other people’s blogs) but that I only write when something especially attracts my attention. It’s becoming increasingly a blog where I want to share rare intellectual discoveries rather than express my opinions, though I do that as well.

Two recent such discoveries, are Cracking the Einstein Code by Fulvio Melia and What is Life by Erwin Schrodinger. The second book is a classic that I’ve wanted to read for a long time, while the first was an unexpected discovery. This post will focus on Melia’s book, subtitled, Relativity and the Birth of Black Hole Physics, and Schrodinger’s tome will probably be a subject for a future post.

Melia’s book is largely concerned with a little-known aspect of Einstein’s General Theory of Relativity (yes, it deserves capitals): a Kiwi called Roy Kerr, in 1963, unlocked the code inherent in Einstein’s 6 field equations that gave a description of space-time for a rotating body, which is the normal reality for massive bodies in the universe, from planets like ours to entire galaxies.

Now I need to say at the start, that whilst I write on esoteric topics, my knowledge is limited in the extreme, unlike the authors whom I read. Anyone following recent comments on this blog will notice that a generous intellect, called Timmo, has made critical comments on 2 of my former posts (Quantum Tunneling, Oct. 09 and Nature's Layers of Reality, May 09). I wish to acknowledge Timmo’s contribution and I welcome someone who really does know what they’re talking about when it comes to physics.

What I especially like about Melia’s account is that he acknowledges all the other people who contributed to the success of relativity theory (specifically, the General Theory), most of whom I’d never heard of. Like many people, I thought that Einstein’s theory had come effectively fully-fledged from his own mind. I wasn’t aware that there was a history of significant contributions from its conception right up to 1963, almost a decade after Einstein’s death.

Firstly, there is David Hilbert (who is extraordinarily famous in mathematics) and who had a correspondence with Einstein and helped him to develop his field equations. In fact, according to Melia, Hilbert actually published the equations on 20 November 1915, 5 days before Einstein, which led to an argument over priority. However, Einstein wrote a letter of reconciliation on 20 December in the same year, which Melia quotes from.

But even Hilbert “could not overcome a serious problem – how to demonstrate that energy is conserved in Einstein’s theory.” From a conceptual point of view, this had always troubled me about relativity, and it wasn’t until I read Feynman’s account in Six Not-So-Easy Pieces, that I believe I understood it. What I didn’t know, before reading Melia’s account, is that a woman, Emmy Noether, who worked with Hilbert at Gottingen University, was the one who resolved this issue by introducing symmetries in connection with conservation laws, specifically conservation of momentum and energy. To paraphrase Melia, Newton’s second law relates changes in momentum to a force; Noether’s Theorem shows how a change in our frame of reference and Newton’s second law are effectively the same thing. (Different frames of reference refer to different observers moving about at different velocities – with no absolute frame of reference, conservation of energy and momentum becomes an issue.)

Einstein’s General Theory of Relativity is premised on the ‘Principle of Equivalence’. Standing in a stationary elevator car in earth’s gravity is equivalent to being accelerated in an elevator car ‘vertically’ in gravity-free space (vertical, in this context, means being pulled from above our heads so our feet are pressed against the floor of the car). Gravity is felt as a force, by us on earth, only because we are stopped from falling. In free fall, no one feels a force being exerted on them, whether they are in a space ship orbiting the earth or jumping off a cliff. This is the key conceptual point to grasp about Einstein’s theory of gravity (which is the General Theory of Relativity). In free fall there is no force, even though this is counter-intuitive when you are earth-bound, because we rarely experience free-fall for any meaningful period of time without dying.

I don’t claim to understand Noether’s Theorem, but I understand its significance. Noether died relatively young in America at age 53, 2 years after escaping Nazi Germany, and Einstein wrote a moving tribute to her in the New York Times (1935). Melia quotes physicists, Leon M. Lederman and Christopher T. Hill, from their book, Symmetry and the Beautiful Universe: “..certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics…” And I had never even heard of her.

Likewise, I’d never heard of Roy Kerr before reading Melia’s book, yet his contribution to relativistic physics is arguably no less significant. According to Melia, Kerr’s Theorem is the fundamental methodology used to investigate black holes (theoretically) to this day.

Kerr completed his undergraduate course at Canterbury University in Christchurch NZ (enrolling at the age of 16 and going straight into 3rd year mathematics). Canterbury is also where Ernest Rutherford started his academic career (Rutherford uncovered the secrets of the atom: that it was mostly empty space, amongst other things). Kerr then went on to Cambridge to study pure mathematics. His doctoral thesis supervisor was Professor Philip Hall, “one of the century’s greatest mathematicians”, according to Melia, and “Britain’s greatest algebraist”. Hall realised that Kerr’s abilities were singularly impressive but his knowledge incomplete. He set him 3 problems in ‘group theory’, including the ‘Axiom of Choice’, which is a fundamental component of ‘set theory’. Kerr dealt with this and the second problem with relative ease, but the third problem, called the ‘Burnside Conjecture’ was beyond him. Following his admission of defeat, Hall apparently lectured him on the subject for an hour but didn’t tell him that the problem had never been solved. In fact, a decade later, someone managed to prove that the conjecture was false by counterexample.

Unaware of this (at the time), Kerr decided that pure mathematics wasn’t his forte and so went into applied mathematics instead, specifically relativistic physics. It is well known (amongst people who take an interest in physics) that Karl Schwarzschild was the first to provide a solution to Einstein’s field equations for the simplest, idealised scenario of a completely symmetrical sphere in a vacuum. He was a Professor of Potsdam University but formulated his solution whilst serving on the Russian front in WW1. He became ill soon after and died after returning home, but his name remains forever associated with black holes, which are a natural theoretical consequence of his solution.

Kerr’s solution (known as Kerr’s Theorem) was not realised until 1963 when he was at the University of Texas, Austin, which had major ramifications for relativistic physics, in particular black hole physics, that are still with us today. Kerr’s monumental breakthrough was overshadowed by the discovery of quasars, a source of radio waves of unprecedented energy. In 1963, the Parkes radio telescope (in Australia) was used to employ a method postulated by British astronomer, Cyril Hazard. His method was simple but ingenious: to use the moon eclipsing the radio signal to exactly pinpoint the source in the night sky. This allowed astronomers to locate the ‘light’ source of the radio waves and thus use spectroscopy to determine its distance from us.

Spectroscopy analyses the exact wavelengths of light emitted by a distant star, and from the Doppler shift we are able to determine how fast they retreat from us and thus how far away they are. There is a direct proportional relationship between how fast stars retreat and how far away they are using Hubble’s constant, named after Edwin Hubble who first discovered this phenomenon.

The first quasar, 3C273, was discovered by Maarten Schmidt at the Palomar Observatory in California, but because they were only seen as radio sources, spectroscopic analysis was not possible until a light source could be found to be directly associated with the radio source. Hence Hazard’s brilliant idea, subsequently employed at Parkes, to pinpoint quasar 3C273. And it was Schmidt who did the spectroscopic analysis, revealing that the light was red-shifted by an enormous 16% making it much further away then anyone had imagined.

During the 1960s, Australia was at the forefront of radio astronomy, and I remember in 1966, when satellites first linked up to produce the world’s first global televised transmission, the Beatles sang All You Need is Love to a worldwide audience simultaneously. Australia’s major contribution was to show the furthest known object in the universe being tracked by the Parkes’ radio telescope. This telescope featured in the movie The Dish, an Australian-made comedy starring another Kiwi, Sam Neill, which was a comedic rendition of Parkes’ role in broadcasting the first pictures from the moon landing in 1969. It also played a role in the Apollo 13 rescue mission, being the means to communicate the time to fire its retro rockets which allowed the astronauts to return to earth rather than bouncing off the atmosphere or diving too deep and burning up. The angle of descent was critical to their survival, and the ‘dish’ played a small, and little-known, but crucial role. (Unlike the rest of this exposition, nothing in this paragraph comes from Melia's book)

The curious aspect of Schmidt’s discovery is that, at the very first Texas Symposium on Relativistic Astrophysics in 1963, Kerr gave a 10minute lecture on his Theorem, which was virtually ignored because all the participants were far more interested in the recently discovered quasar. Yet Kerr’s Theorem gave the only relativistic solution to spinning black holes, which is exactly what quasars are.

Melia is meticulous in his coverage of all the people who contributed to our understanding of relativistic physics, both theoretically and experimentally. Not only the well known ones like Karl Schwarzschild, John Wheeler, Roger Penrose and Stephen Hawking, but unknown heroes like Noether and Kerr. He also mentions an Australian, Brandon Carter, who used Kerr’s Theorem to show that a ‘time loop’ (or 'time machine') could theoretically be generated beyond the event horizon of a rotating black hole. (But it only works in a vacuum, which makes it a catch-22 time machine.)

The significance of Kerr’s solution is that every significant physical body we know of in the universe is rotating, so Schwarzschild’s solution would almost certainly never be applicable to reality. Kerr’s solution reveals that there are, in fact, 2 event horizons for a rotating black hole. The event horizon is where the escape velocity from a black hole becomes the speed of light so nothing can escape from it. But a spinning black hole literally drags space-time around with it, which creates an inner and outer event horizon – don’t worry, I don’t understand it either. When a body crosses the first event horizon, the parameters of space and time are reversed: space becomes time-like and time becomes space-like. This is because time freezes at the event horizon for an outside observer and the external time becomes infinite from the inside. Time becomes space-like in that it becomes static and infinite, if I interpret it correctly. When an object crosses the second event horizon they reverse again so that ‘inside’ a spinning black hole, space and time become theoretically normal again. Of course, no one knows how true this really is. The other problem is that these theoretical considerations all assume a vacuum which is not the case if something is actually ‘crossing over’. To this day, there are no solutions to Einstein’s field equations for a non-vacuum – that is, for example, inside an object like the earth or the sun – only for outside in space.

That effectively is the limit of my understanding of this subject, even after reading Melia’s book. The story of Kerr himself is no less interesting. One gets the impression that, despite his obvious talents, he was not cut out for high level academia. He did not publish everything he uncovered, and he was not competitive in the sense that he sought to outdo his rivals at every opportunity, nor was he one for self-promotion. He left America in 1971 to take up a position of Head of Mathematics in the University of Canterbury in Christchurch, New Zealand. After his close friend and associate, Alfred Schild, died in 1977, Kerr virtually stopped visiting the US. He received the Hughes Medal from the Royal Society in 1984, the highest accolade he has received.

Roy Kerr writes his own afterward in Fulvio Melia’s book (they are good friends), in which he talks about the difficulties in attempting to get the advanced education he badly needed in 1950s New Zealand. But after going to Cambridge in the UK and the University of Texas in Austin, he believes he was fortunate in that he never had any mentors, as they would have undoubtedly led him away from the path that led to his groundbreaking discovery.

8 comments:

Timmo said...

Hi Paul,

Noether’s Theorem shows how a change in our frame of reference and Newton’s second law are effectively the same thing. (Different frames of reference refer to different observers moving about at different velocities – with no absolute frame of reference, conservation of energy and momentum becomes an issue.)

Not really. You remember from Feynman's book his discussion about symmetries. If there is something we can do to an object such that, after we have done it, leaves the object looking as it did before, then we say it is symmetrical. A vase is symmetrical because one can turn or rotate it and not change its appearance. Symmetries in the physical laws are the following: if there is something we can do to a physical situation or phenomena that leaves it unchanged, that physical situation is symmetrical.

For example, the laws of physics are the same everywhere in space. If I build an apparatus here, and then build an identical apparatus there, then both will function correctly and in the same way. We say that the laws of physics are symmetrical under a translation in space: if I shift all of my co-ordinates by a constant amount, then the mathematical form of the laws is unchanged.

Translation in space is an example of a continuous symmetry. I can build the new apparatus at point x, or a little bit further along x + y, or a little bit further along x + 2y, or a little bit further along x + 3y… I can shift the position of the experiment through a continuum of possible positions. Reflection in space is a discrete symmetry. I can make a mirror image of the situation.

Noether's theorem states that for every continuous symmetry, there is a conserved quantity. Some examples:

Translation in Space --> Momentum
Translation in Time --> Energy
Rotation through a fixed angle --> Angular Momentum.

So, there is a profound and beautiful connection between the symmetries of the physical laws and the conservation principles. The mathematical formulation of Noether's theorem allows one to go further than to say that there is some conserved quantity or other, but allows you to explicit construct what that quantity happens to be. The connection between symmetry and conservation is not obvious, and I don't know any intuitively compelling way to explain or motivate it, I'm afraid.

Timmo said...

Kerr dealt with this and the second problem with relative ease, but the third problem, called the ‘Burnside Conjecture’ was beyond him. Following his admission of defeat, Hall apparently lectured him on the subject for an hour but didn’t tell him that the problem had never been solved. In fact, a decade later, someone managed to prove that the conjecture was false by counterexample.

Are you sure about this? I think the Burnside conjecture about simple groups has been proved.

The significance of Kerr’s solution is that every significant physical body we know of in the universe is rotating, so Schwarzschild’s solution would almost certainly never be applicable to reality.

One has to be careful with remarks like this. The Schwarzschild results can be used as one model; Kerr developed a more sophisticated model. Some very apt statements from a text, Experimentation: An Introduction to Measurement Theory and Experiment Design:

It is pointless to agonize over whether a theory is "true," correct," "wrong," or whatever. As was
we should avoid using such terms... It is far better to categorize a theory or model as "satisfactory" or "good enough," or some similar phrase, because all such decisions are relative to the purposes we have in mind…  if we are sure that the properties of the model and system are in disagreement by an amount clearly in excess of the measurement uncertainty; we can say definitely that the model is no in correspondence with the system.  We can say that we have "proved" the theory to be "wrong" -- although, even in this case, it would be better to call it "unsuitable" or "inadequate."


Whether a Schwarzschild or Kerr or some other model is adequate depends, like all models, upon the system you are trying to study and what features of that system you are interested in trying to understand.

Timmo said...

I should add one more thing. Noether's theorem applies in the classical, relativistic, and quantum regimes.

Paul P. Mealing said...

Hi Timmo,

Thanks once again for all your comments.

I revisited Feynman's Six Not-So-Easy Pieces; he gives the same rules of symmetry that you do, but in relation to quantum mechanics: symmetry for space, momentum is conserved; symmetry for time, energy is conserved; and symmetry for rotation, angular momentum is conserved.

He also starts his discussion on the Special Theory of Relativity by taking Newton's second law and just adding the relativistic equation for mass, to differentiate it from rest mass.

Then he says, in one of his typical cavalier throwaway lines: "For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity - it just changes Newton's laws by introducing a correction factor to the mass."

I won't quote Melia's passage that made me think that Noether's Theorem was related to Newton's second law and different frames of reference, but I will quote this:

"Noether's profound insight into the connection between symmetries and conservation laws permitted her to define a new representation of the total energy of a system containing a gravitational field, demonstrating that its value is constant in time when one includes the contributions from all space."

In regard to the 'general Burnside Conjecture' Melia said: "Years later Kerr would discover that Hall had assigned him one of the 'great unsolved problems' of algebra... A decade later it was actually shown by way of counterexample that the conjecture was false."

Penrose, in The Emperor's New Mind, spends an entire chapter discussing his own categorisation of theories: TENTATIVE, USEFUL and SUPERB (his capitalisation) with SUPERB not necessarily being the last word (e.g. he includes Newton's equations amongst the SUPERB).

I didn't go into detail in my post, but Melia gives the impression that the great breakthrough that Kerr's solution gave was that it was 'shear-free', meaning that light travelling along the geodesics didn't distort. That may be simplistic, but that's the way Melia presented it for laypeople's consumption.

Regards, Paul.

Timmo said...

Hi Paul,

Good to hear from you.

You are right that Feynman mentions them only in the context of quantum mechanics. However, Noether's theorem that every continuous symmetry implies a conserved quantity applies more broadly to Newtonian, relativistic, and quantum mechanics. However, there some symmetries in, say, quantum mechanics that don't exist in the Newtonian or relativistic mechanics. Feynman gives the example of a shift in overall quantum mechanical phase: if we take a problem that has a certain number of arrows and rotate all of those arrows by the same fixed angle, then the probabilities we calculate are unchanged.

I have always found that line about relativistic mass by Feynman rather puzzling. It is true that if you are just working the with dynamics, trying to find out what the result of a collision is or what is the trajectory of some body under the influence of a force, then you can probably get away with just knowing how the relativistic mass varies with velocity. At the same time, it is important to know how the theory of relativity highlights the importance of symmetry in physics. If you are trying to guess a new law, then symmetry principles, especially Lorentz symmetry, impose very tight constraints on the possible forms that law might take. This isn't as important for me in my field -- the theory of relativity altogether is besides the point for me! -- but, it is important for my colleagues down the hall working on nuclear and high-energy physics!

Timmo said...

I stand corrected on the Burnside problem: http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Burnside_problem.html

The Atheist Missionary said...

I have to go untie my brain.

larryniven said...

"When a body crosses the first event horizon, the parameters of space and time are reversed: space becomes time-like and time becomes space-like. This is because time freezes at the event horizon for an outside observer and the external time becomes infinite from the inside. Time becomes space-like in that it becomes static and infinite, if I interpret it correctly."

Ah - how interesting. This, if correct, would come as a great surprise to many philosophers. I dunno if you've encountered this particular line, but there's this school of thought that says that "actual infinities" are impossible. Assuming that your interpretation is both correct and truthful, this would pretty much mark the end of that school of thought (or should, anyway).

And, for what it's worth, I intuitively drifted towards a similar interpretation myself - it was actually a slight relief to read that last sentence. So, if it's a mistake, you're not the only one making it.