Monday, June 22, 2009

Tigers of wrath

The tygers of wrath are wiser than the horses of instruction

(William Blake)

The tigers of wrath are on the march in the streets of the world. Postelectoral uprising in Tehran is a magnificent and tragic example of the outbursts of long suppressed anger and accumulated frustration against a blind and unyielding system.

Yet what are, more generally, Blakes' tigers of wrath? And the horses of instruction? I tend to see them both as the motivations that drive us on to fight, to overcome seemingly impossible obstacles. The horses of instruction carry us along the long and arduous journey of learning, of accumulating the knowledge and strength that is needed to push forward, to keep going, to make progress. It is a hard journey, a day by day fight with hardness. It is endurance. The tigers of wrath are the sudden cry of anger and frustration that makes the burst possible, that carries the greatest momentum. They are the explosion of resentment and primitive fury, the coming of the dies irae.

But why and when is anger called for in what should after all be a peaceful progress towards knowledge? The answer is that, unfortunately, it is almost inevitable, because knowledge is more often than not hijacked by ambitions, envies, jealousies and hysterias against which the only possible response, in order not to succumb to their rising tide, is to unleash one's own tigers of wrath. Anger is what makes it possible for a wounded soul to bounce back and recover strength. Anger is what makes it possible to continue walking when one has reached beyond the limits of exhaustion. Anger is what keeps the world spinning long after its future has eroded. Anger is recovery, so let us open the gates to the tigers of wrath!

I keep fighting on too many fronts, gaining and losing ground in endless fluctuations of strength and willpower. Lately I've given it a serious try at opening up a new direction, a new front in the war that is to say, in part to escape the impossible quagmire I've inadvertedly ended up trapped into with what used to be my main line of research. So now it's a shot at the cosmos, the very early universe of dark energy and the mystery of dark matter in the universe of large structures. Funny as it may seem, I am learning quite a lot, which is the good duty of the horses of instruction to perform, but this all story has been born in the roaring of the tigers of wrath and will bring forever their indelible mark.

The trick to get a model of inflation is that you need to make gravity turn from attractive to repulsive, at some point in the early universe, and then by some regulating mechanism that acts as an internal clock of the cosmos, turn it into its normal attractive self before the universe starts to look at all like anything we know. Dark energy appears to be a related effect of that seemingly unlikely repulsive second nature that gravity can have. The cosmological constant is a likely culprit in turning the effect of gravity upside down, but besides possible models with variable cosmological constant, there are variable gravitational constant models, sometime generated by the presence of a conformally coupled scalar field, that can also provide regimes of negative gravity, and finally there are more exotic geometric models that can yield all of these things combined, like the one I am trying to grind my teeth on these days. As for dark matter, I am mostly wandering about Majorana masses and right handed neutrinos and see-saw mechanisms.

As I learn about cosmology and entertain myself with computer simulations, I also seek to entertain myself with accompanying music, so to speak, by which I really mean the reading of a number of completely unrelated books, which in some vague form resonate with my mind in this stage of early exploration. It is a way for me to concentrate, perhaps not unlike those attractors in dynamical systems that work more efficiently in the presence of a moderate amount of disorder, of background noise.

Among my side readings, near the pile of the actual cosmology and particle physics papers from the archive, there is the recent World Scientific book by James Overduin and Paul Wesson, "The light/dark universe". It is a light/not-so-light book that starts out with the entertaining story of the Olbert paradox of the darkness of the night sky and very quickly moves on to a fairly up-to-date account of the dark matter and dark energy problems. It is fairly technical, and not a generic popular book, which is good if you want to get anything out of reading it. It assumes the reader to be a scientifically trained person, with some knowledge of astrophysical matters, but it does not necessarily assume a deep and extensive background in cosmology, which is also good if this is to be your bedtime entertainment reading. The thing I liked about it is that I quickly encountered there all the things I had been playing with in the model I am exploring: primordial black holes, negative gravity due to coupling with fields, slow roll inflation. It is always nice after having spent a certain amount of time struggling with technical papers and trying a range computations, to is it all again neatly presented in an engaging narrative. It makes one want to make the effort of turning a directory full of messy computation into a readable paper.

Having acquired some familiarity with the main views of the cosmos provided to us by modern cosmology, with its fantastic supply of data from the satellite probes, it is somewhat interesting to take a very long step back in history and return to a time when none of the modern instruments, both theoretical and observational, to investigate the vastity of space were yet available, and still people tried their best, with little more than everyday common sense and a baggage of literary knowledge, to argue about the properties of the universe surrounding us. So I got back to reading Giordan Bruno's "L'infinito, universo e mondi", this time in a French translation I got hold of a few weeks ago in Paris. Seen from the eyes of today's mathematical thinking (yes, I know, one never interprets an author writing four centuries earlier in the modern light, but I am not giving an exegesis of Bruno here, I'm just having fun reading it so I am allowed the licence), Bruno's writings are extremely revealing by stark contrast between what was possible to argue about geometry before a number of major developments in mathematics had taken place, with respect to what we nare naturally inclined to think today.

In his earlier writings, when he was too busy collecting excommunications from the largest number of churches then available in Europe to have any time left for serious philosophy, Giordano Bruno mostly mimicked Ficino's neoplatonist views, and imparted vitriolic criticism of religion and society. It was only later, at the time of his major philosophical production of which "L'infinito, universo e mondi" is the centerpiece, a time when for a brief period he was only fighting against the academics of Oxford and Cambridge, he unfolded all his arguments about the infinite nature of the universe, the plurality of worlds, the fact that stars are other solar systems with other planets and other Earths, the anima mundi. On some things he happened to be right on others possibly not. As I mentioned above, the point, however, is that in reading some of the "arguments" that the characters in his dialogues present one gets a very interesting idea of how our reasonings about geometry have dramatically changed in the intervening centuries. I will give here just a very small and simple example, to illustrate my point.

Right towards the beginning of the first dialog of the "Infinite, universe and worlds" one of the characters, arguing against the finitude of the universe, remarks that

"se uno stendesse la mano oltre quel convesso, che quella
non verrebe essere in loco, e non sarebe in parte alcuna,
e per conseguenza non arebe l'essere."

In this argument there is a clear clash of two notions that became clearly distinguishable only within the context of more modern mathematics: that of a manifold with boundary and of a compact manifold. The argument of Bruno against the finitude of space is an argument about the existence of a boundary, while modern geometry allows for endeless possibilities of compact, finite volume, spaces without boundary (gluing together the opposite sides of a square to form a torus being the simplest example that comes to mind). There is no mystery today in the distinction between these two notions, but in the time of Bruno's writing it seemed that an argument against a boundary would necessarily entail an argument against compactness (or finite volume). One could argue that even today a certain amount of confusion remains, even among scientists: it is not uncommon to find statements in cosmology books on negative curvature (hyperbolic geometry) implying an infinite universe, while one knows very well that there are hyperbolic 3-manifolds that are compact. Even between two slightly different notions of finiteness for manifolds, compactness and finiteness of volume, confusions sometimes arise, while the first implies the latter, but the converse need not hold. So perhaps one should not blame Giordano Bruno for not making these fine distinctions at a time when most of the currently common tools to think of space were not available. It still remains an interesting fact that, while the era of Bruno was often portrayed (most articulately by Koyre') as the transition between the closed cosmos of antiquity to the infinite universe of modernity, in fact the debate on the finitude (today we'd better say compactness) of the universe is still raging in all the now fashionable investigations on the problem of cosmic topology, trying to statistically match circles in the WMAP sky as an indication of the gluing data of the folding up of space into a closed manifold.

Although still inconclusive, results hinting to the possible role of a space known to mathematicians as the Poincare' homology 3-sphere and to cosmologists as the dodecahedral space seem to defy precisely that argument of Bruno on the hand sticking out of the boundary of the universe and meeting...? the void? No hand sticks out anywhere in the Poincare' sphere, as there is no boundary anywhere to cross, and yet the universe folds up into a compact manifold, and not the infinite universe Bruno wished to advocate. His desire for the infinite is rescued by the accelerating expansion perhaps, whereby a spatial universe may be folded up, but its temporal unfolding picks up speed instead of recollapsing to a closed 4-dimensional manifold. It's easy to get carried away along these lines of thought when trying to read Giordano Bruno and cosmology books and papers at the same time, yet it's not for his insight into the prehistory of cosmology that one reads the Nolan philosopher, but for his uncompromising challenge to institutions, dogma, status quo, pretense and hypocrisy. Even if his cosmology does not stand the judgment of time and perhaps the anima mundi does not really exist, Bruno will remain one of the main figures in the history of thought and for that alone we read him still. For whoever wishes to get an excellent critical account of the work of Bruno in the context of the culture of the Renaissance, I highly recommend the famous monograph of Frances Yates, "Giordano Bruno and the Hermetic tradition".

As for myself, I am about to trade the lazy sunny afternoons of Southern California with the rains of Germany, an equitable trade for the sake of another conference, a student's thesis defense, some collaborations left dormant from last summer, and a lot of somber thoughts accompanying this trip.

The void has become a familiar concept to any working physicists. We have vacuum bubbles, energy of the void and quantum fluctuations: a crowd of virtual particles streaming in and out of the void. It seems almost surprising how frightening the concept had appeared to our predecessors. It is only in its psychological manifestations that emptiness retains its full capacity to stir our deepest angst, when, as in the image from the dialog of Giordano Bruno, a hand stretches beyond the barrier and instead of meeting with another human hand it only meets with a terrifying void. It is to look away from the chill of that void that I am seeking the emptiness of the early universe, in the time before nucleosynthesis and structure formation. It is to emerge unscathed from this frightening confrontation with that cold void that I now return to science, our "dangerous but irresistible pastime". It is the darkness of these thoughts about the nature of human relations that I seek to replace in my mind with the mysterious but not so scary darkness of matter and energy, more comfortable certainly than the impenetrable barriers of the void. While that icy psychological void will remain forever entrenched where it retreated, beyond the grasp of human comprehension, there is at least a chance that the physical and cosmological void of the early universe will yield answers more willingly to the inquiring mind.

Lost in thought and lost in time
While the seeds of life and the seeds of change were planted
Outside the rain fell dark and slow
While I pondered on this dangerous but irresistible pastime

(Pink Floyd, "Coming back to life" -- The Division Bell)

It's not like I expect that looking at pictures of psychedelic galaxies will suffice to put my mind at peace: it would take a whole other kind of psychedelia to come even close, but I learnt that the practice of computer generated calculations, as opposed to the more familiar habit of abstract thinking of the theorem-proving kind, can have a very soothing effect on the mind. I don't know if this means that Maple, Mathematica and the like should start advertising themselves as an alternative to aromatherapy, but indeed the kind of relaxed attentiveness that is needed to perform this type of small programming and trial and error experimental mathematics has the potential of stirring the mind away from the siege of intrusive thoughts that are too painful to entertain. Regardless of the actual scientific value, in terms of realistic models of the early universe, that I can get out of this simple kitchen table research, this hesitant venturing into the world of cosmology might have at least helped me to gain enough momentum to conjure up the tigers of wrath from inside my previously abased consciousness and unleash them towards what I hope might become a reboot of the system and thus to avoid defeat.

I took a heavenly ride through our silence
I knew the moment had arrived
For killing the past and coming back to life.
I took a heavenly ride through our silence
I knew the waiting had begun
And I headed straight...into the shining sun


Tyger, Tyger, burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?

In what distant deeps or skies
Burnt the fire of thine eyes?
On what wings dare he aspire?
What the hand dare seize the fire? [...]

What the hammer? what the chain?
In what furnace was thy brain?
What the anvil? what dread grasp
Dare its deadly terrors clasp? [...]

(William Blake -- The Tyger)

The complete works of Giordano Bruno online