Shadows of the Deep
Over the past week, I have been slowly learning the basics of General Relativity, a physical theory of gravitation which was “invented” by Einstein, with help from Marcel Grossman and Tullio Levi-Civita who introduced him to the necessary mathematics (Riemannian geometry and what are now called ‘tensors’). All these names and their luminous insights stand as far from my experience as the struggles of a weekend runner to run a mile stand in contrast from an ultramarathoner. This recognition of the differences, or the chasm between me and them, however doesn’t overwhelm me or turn me into a devotee of human genius and so on. Instead, I think, I am quite relieved on having no pressure to be smart, or sound smarter than I am. I plod at my own pace, perhaps for the pleasure of plodding itself. My concerns and goals are however more mundane and thus m discoveries are all too common place for those who know Relativity well.
One of the early “tricks” of general relativity that one learns is the use of indices and superscripts to subsume a whole lot of information about various dimensions that the theoretical constructs seek to manipulate. On my end, I struggle to “see” the details of the manipulations of indices and superscripts in those equations, the rapturous sweep of notational simplifications that leaves me wondering if indeed these representations explain the physical realities of our universe. It is a frequent suspicion that makes me feel small. How can I not simply believe the ineluctable beauties of inferential reasoning? Such hesitations don't last however. Notwithstanding, these slivers of doubt I press on, like a nun walking past shirtless men playing soccer. Cautiously, eager to not get distracted. There is a certain abstemiousness to all this. The real challenge however is to keep my eyes on the exercise at hand and not look for any greater meaning. This is difficult. For most non-physicists dabble in physics with the illusions of talking 'deep' stuff about epistemology and methodology. But the actual practice of physics, the laborious mathematical routines that eventually burble up as an elegant summary (whose real meaning may still elude us), has little use for grandiloquence. This recognition is, in a way, quite humbling. The deep claims of Relativity – special or general – are for those who have ascended the peaks and even they seem to struggle about what it all means. Subrahmanyam Chandrasekhar had suggested that Einstein himself did not "believe in the theory sufficiently deeply". I don't know entirely what that means, but what I do know is that believing a theory of physics - however extravagantly nature confirms it experimentally - is a counter-intuitive exercise. A leap of faith and a calculated rush of reason. The rest of us must, in the meanwhile, like all students, proceed with humility. (Admittedly, humility is easy to come by, given the material). I continue to work through the material, line by line, often not knowing where it is all adding up to. In all these exercises, there is an element of mindlessness about these discrete manipulations that a neophyte like myself must undertake to get anywhere. All the while however, amidst this forest of concatenated manipulations of symbols, there are subtle and often unexpected relationships that I have begun to take for granted. These relationships become as commonplace as sine and cos in trigonometry. They become short hands, unthinking summaries that I have stopped bothering to wonder about. No different than the invention of a symbol, like zero, that encapsulates both: notational shorthand and metaphysical emptiness . But as a beginner filled with self-doubt, often enough, I pause to wonder about things I take on faith (even if I intellectually recognize how they all tie together).
In special relativity, which is a physical theory of non-accelerating objects that began the great Eisenstein revolution, there is a factor k that is part of what is called the Lorentz Transformation. This is defined thus,
k = 1 / [(1-v/c)^(1/2)]
where v and c are velocity of an object and c is the speed of light. Notice if v = c, k = 1/0 = infinity; and when v = 0, k = 1/1 = 1. This k reappears in many modified forms and guises. It is useful shorthand. It transforms fundamental phenomena like mass or time into something more complicated under specific circumstances (when velocity v --> c) and something familiar under others (when velocity v --> 0). Niels Bohr called this ( and I simplify here) the 'correspondence principle': when a new theory is reducible to an old theory in previously understood circumstances.
Much in the manner that writers no longer focus on individual alphabets that constitute their sentences, students of relativity take this k in their stride and move on to do more complex things. Yet, to me, occasionally, the fact that this factor which is a remarkable relationship between speed of light and our daily velocities, which is derived through algebraic manipulation that even high school students and amateurs like myself can follow and yet which is somehow be fundamentally true about the universe, is kind of inexplicable. Breathtaking. I may even intellectually be able to parse through the algebra about frames of reference, time lines and so on. Yet, thinking about things like the factor k still produces a deep disquiet about the fact that I can seemingly glimpse at what lies at the edges of a black hole through algebra and an act of pure thinking that is analogous to the highs from a dose of heroin. Addictive, mysterious, briefly all powerful. Occasionally when I get at the end of an exercise or a page of counter-intuitive reasoning, it feels like watching bubbles rise up in a still pond. I can’t be sure what produces them. Perhaps, it is merely air escaping to the surface. Or perhaps, it is something more fun, more mysterious, more improbable. A stir from the shadows of the deep.