Learning

TUESDAY, 17TH NOVEMBER 2009 - Michael Halls-Moore - 60 Comments

As I type this I have just finished working through the problems in Chapter 1 of Schaum's Outlines of Tensor Calculus. I have always had a healthy fear of Tensor Analysis ever since I was introduced to it via a course on Continuum Mechanics at Warwick. It is however a prerequesite for understanding General Relativity, itself a precursor to gaining insight into String Theory, the current theory-du-jour of how the Universe works at the most fundamental level. One of my life goals is to be able to grasp the concepts in String Theory and it is certainly an ambitious one.

String Theory is a complicated beast. From what I understand at this stage it is an entirely theoretical framework. The predictions it currently makes are few and those which it does admit are even harder to test. It requires a substantial grounding in multiple prerequesites including, but not limited to, Relativity (Special and General), Elementary Particle Physics, Quantum Mechanics, Quantum Electrodynamics, Lie Groups (in particular Matrix Groups) and Quantum Field Theory. Personally, I am not even sure if this hierarchy of subjects is correct or remotely exhaustive.


A Calabi-Yau Manifold

A relatively accessible (and I use that word in the loosest sense) book on the subject is by Zweibach. I will consider my life goal complete when I can understand and solve the problems in that text. However, I do not possess that level of mathematical or physical sophistication as of yet. Hence my journey must begin at General Relativity, as I do possess a reasonable understanding of Quantum Mechanics. I managed to battle through a good introductory text on Special Relativity at Imperial College and decided as a PhD submission "present" I would attempt to begin the sequel book on General Relativity. I rapidly became stuck - primarily based on my limited understanding of tensors.

General Relativity is almost entirely built around the mathematical theory of tensor analysis. It describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. Einstein sums it up very well with his famous field equations. In order to describe this bending and warping of space and time, it is necessary to use more sophisticated geometric tools than most are acquainted with. This is where tensors come in.

As previously stated, in the past I found tensor analysis extremely difficult. The first obstacle is that the equations use Einstein Summation to avoid the extensive use of sigma summation signs in order to make the equations logistically manageable. However, one very quickly loses geometric insight unless one is well versed in tensor algebraic manipulation. The second issue is that once arbitrary tensors (of higher orders) are brought in, it is challenging to relate these rules to any visual representation that might have helped out in the lower orders.

I am extremely motivated with this goal, however. In my current circumstances I have a reasonable amount of free time. I have decided to spend some of it re-attempting my goal to grasp String Theory. There is no fear of external consequences or deadlines, nobody to answer to except my own perceived limitations and nobody to compete with except my own discipline. Working through the problems today was an exceedingly enjoyable experience. I managed to grasp basic Einstein Summation and all of the problems I attempted were correct. Mathematics is a set of rules built on other rules. If they are followed consistently, concepts become straightforward. I had forgotten how fascinating it is to learn and fully grasp a concept which one previously considered too complicated to understand. I can think of very little else that can compare with that feeling.

I will never tire or become bored of learning new material. I feel like I am exploring the Universe in my own private sanctuary and on my own terms. There is almost nothing that compares to the clarity and purity of acquiring new knowledge. I do hope that certain elements of society do not relegate continual learning to a minor subset of the population. Humanity should not be an amalgam of reality TV, uninspiring careers and package holidays. Rather, it should be about unbounded curiosity, exploration and overcoming the hardest challenges.

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  • Quantum Game Theory -> Adding superposition and entanglement to the Prisoners' Dilemna: http://bit.ly/ahLkoD #fb

    3 hours, 24 minutes ago

  • @chrislarvin Painful! Are you planning on riding it when you're over there? ;-)

    4 hours, 53 minutes ago

  • Ever wondered what it's like inside a black hole? http://www.youtube.com/watch?v=GYKyt3C0oT4 #fb

    23 hours, 20 minutes ago