While I’m working out some other things to ramble about, thought I’d finally do a run-through of a few books I’ve read lately.
First off, partly out of my ongoing fascination and uncertainty regarding the recent LIGO announcements, I began a number of months ago to feel very insecure regarding my knowledge of general relativity (GR). Who doesn’t, right?
In truth, though I’d learned about special relativity in high school and had a book of Einstein’s papers sitting perpetually on my desk, I have generally been terrified of the concept of tensors–the formidable coordinate-independent mathematical creatures with which GR is written–and so my understanding of GR has been limited to the usual hand-wavy, rubber-sheet-Brian-Greene-texture-of-reality variety.
I’ve tried to remedy this over the years, but constantly found myself frustrated by a kind of stair-step division in the books and publications on the subject. On the one hand, you have the popular “rubber sheet” stuff, which tells you almost nothing about the mathematical justification or structure, just vague conceptual gosh-wow like “matter warps space and time!” and “Einstein was amazing because he saw acceleration and gravity are alike!”.
I don’t want to underplay the value of the conceptual. John Archibald Wheeler’s “A Journey Into Gravity and Spacetime” is among the finest examples of this type of GR book, and reading it did I think help put the overarching ideas in the right general places.
But to me, most physical theories are ultimately like machines built with math, and so telling just the conceptual side of GR versus knowing at least the skeleton of the math behind it is kind of like the difference between telling a story over the campfire about a wondrous hovering creature with a spinning head, and showing me the technical drawings and specs of a helicopter.
These deficiencies naturally leads one to search out the other class of GR books, which unfortunately seem mostly to be written in the manner of a congratulatory capstone, a sort of welcome-to-the-club that assumes your successful surmounting of several years worth of arduous academic arcana. Opening one of these to any page, you will likely be greeted by the smell of decades-old book glue; then, as you gaze down, you will have the strange vertiginous sensation of being about to crack your head on a huge, neverending, treadmilling cliff-face of equations.
Stuck between these extremes, I’ve looked for a book that would lay out the mathematical lineaments of GR in a non-trivial way, while still allowing someone with a bit of background in calculus and linear algebra to (mostly) see what is going on. For years, I didn’t have much luck.
But my post-LIGO perplexities led me to restart the search, leading me to “A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity”. Here, Peter Collier has put together exactly what I was hoping for: the missing link of GR pedagogy, a volume intended for those of us badly in need of, as he puts it, “something in-between, a Goldilocks volume […] neither too easy nor too difficult”, that outlines the geometric theory of gravity.
Much of the book is “backstory”, for in the interest of completeness Collier dedicates the first 100 pages or so to bringing us up to speed on the foundational mathematics, forms of notation, and also good old-fashioned Newtonian gravitation. But I found these very clearly presented and it’s a good refresher course–I was surprised how effectively years of calculus can be shoehorned into a few dozen pages of core principles and operations, without losing very much (though practice surely makes perfect).
Perhaps inevitably given the interest of tractable length and scope, the book leaves out a great many derivations. Mostly these are not missed, as making constant detours away from the main points of GR would be far more confusing than simply assuming certain needed results and letting the reader research the proofs to their taste; still, in a few cases these elisions left me frustrated. The most notable example is that Collier gives no rationale for how Einstein arrived at and justified his final field equations. (Why is that factor of 1/2 in there in front of the Ricci scalar and the metric tensor?) Collier notes that Einstein was on completely the wrong track for years before hitting on the much-loved final form, and he gives some sample problems that suggest that equation makes sense in some toy models, but stops there. Maybe this is the best one can do at this level of explanation–it’s formidable enough to be able to say what the metric tensor etc. even is–but I wish we could get just a little more of a peek into how the correct field equation comes about.
Unfortunately for the LIGO addict/skeptic, “A Most Incomprehensible Thing” also stops short of even outlining the theoretical case for gravitational radiation, except for a passing reference to them on the way to explaining the stress-energy tensor. But it lays a foundation–or at least the foundation of a foundation–for the dedicated autodidact to move on to such an understanding, and makes the LIGO papers and simulations considerably more intelligible.
There are other trends in physics that come into a new light upon reading this volume, too. For example, many of us have likely heard the accelerating expansion of the universe reported as one of the most shocking revelations in cosmology since the cosmic background radiation itself. But on page 317 of “A Most Incomprehensible Thing”, we see that acceleration is really far from an unforeseen possibility or a fundamental shock to cosmology, for there are well-established models approximating such an acceleration that date from decades before the discovery of said acceleration in the late 1990s. In particular, it seems we’re just in the phase of the universe’s life where it is best modeled by a dark-matter dominated de Sitter universe (discovered c. 1917).
Though Collier notes that the book is something of a work in progress and welcomes readers to submit corrections, the writing is crisp, providing in most cases a fine interplay with the equations and derivations. There is a punctuation error or misspelling here or there, but nothing glaring as far as I could tell. Altogether, for anyone seeking a worthy jumping-off point into a more detailed and mechanistic understanding of cosmology or fundamental physics, I’d recommend grabbing this book and clearing your reading list for a couple months: a fun but demanding ride awaits.
More reviews to come…