My Book Review for "A Most Incomprehensible Thing: Notes Toward A Very Gentle Introduction To The Mathematics of Relativity" ~ by Peter Collier.
This is the Must-Have Book on Both Special & General Relativity. I would give it 10 Stars if I could. Read this when the first time you studied Special Relavity (SR) was a haze. Review SR the second time with this book it becomes crystal clear. This is my first time and book on General Relativity (GR). General Relativity (GR) was to me the Most Incomprehensible Mountain! After reading through this book, GR is now A Comprehensible Mountain for me (and you too). The author takes you by the necessary/essential equations (i.e. your core guide line/rail) gently climbing the equations up from the bottom to the top summit without skipping an equation (i.e. step). Don't worry! He has strung and weaved together elegantly all the needed equations so that if you get the former equation/step you will understand the next equation/step and so on up to GR and a bit beyond. I vouch for it.
The core guide line/rail snakes from SR all the way high up to GR and also extends beyond to your final sweet treat - Applying GR. What a better way to grasp GR that actually applying it to understanding the common Schwarzschild Black Holes and Relativistic Cosmology. The First Equations for the Cosmos/Universe. The Big Bang, Big Chill, Big Crunch, or Big Bounce. Yes, I did finished this book down to the last page and last equation. Whew! I must admit I snail'ed slowly through the latter half of the book. I am the type who likes to ant scrutinize closely every equation in a book. The concept clarity and math rigor is satisfyingly enough. It took me a whole summer 3 months as a side read when I can. This is your foundational GR book to branch off and then pursue advanced GR topics. I will see what I like to learn next.
The author introduces the foundation mathematics in Chapter 1. You can skim through it if you have taken 1 yr Calculus and 1/2 yr Differential Equations. I would skim through Chapter 2 though for a quick review on Newtonian Mechanics even if you have taken 1 yr Physics. Do start reading SR in Chapter 3. The author does present 4-dimensional curvature math in spacetime really well as a prelude/interlude to GR. He presents all the required advanced math as he goes and as you will need but no more than necessary, such as Tensors, to keep the book in good length before easing you comfortably into the GR equations. No need for another side advanced math book, but this book does provide the advanced math topics for you to delve deeper in more pure subject-matter math textbooks if you want.
Have you ever wonder how Action-at-Astronomical Distances can be Instantaneous?! This 'appears' to be happening. How can the attraction between the planets and the Sun make the planets orbit the Sun at such vast distances? This Action-at-a-Distance attraction seems instantaneous, but light has finite speed! Nothing can connect the planet to the Sun in zero no time, not even light itself! Yet, something is connecting the planet to the Sun instantaneously for sure. Newton's Law of Universal Gravitation just describes and quantifies the attraction that's all, but doesn't give the underlying why or reason that gives rise to this attraction. General Relativity gives the thesis for this Action-at-Astronomical Distance attraction. To understand the why/reason literally requires us to go up into another higher dimension. 4-dimensional spacetime. That's right. 4-dimensions for us 3-dim creatures.
General Relativity provides the why/reason that gives rise to this attraction as the Sun's heavy mass causes curvature in the surrounding 4-dimensional spacetime. The 4-dim spacetime curvature extends all the way out to the planets and beyond. Think of the sun as the heavy bowling ball resting on at the center of a taut vast sheet fabric. That vast sheet fabric would represent our hypersurface called 4-dim spacetime. A marble like the earth would continually fall in its orbit around the heavy bowling ball our Sun. To understand Action-at-Astronomical Distance that seems Instantaneous requires us to think beyond 3-dimensions and into 4-dimensions. But, we are only 3-dimensional creatures and can't see beyond that! Fortunately, we have developed 4-dimensional math as our main eyes to think in 4-dim and see in 4-dim with the help of 3-dim analogies. Ref. the Riemann Curvature Tensor.
Throughout the book, the author not only provides the concepts and equations, but the historical contexts and characters that contributed to our current understanding. Galileo, Newton, Minkowski, Riemann, Einstein, Schwarzschild, Kerr, Robertson/Walker, Friedmann, Hubble, and others I missed mentioning from the book.
Space and Time are not separate physical entities, but fused together as one single 4-dimensional Spacetime Fabric. 4-dim spacetime is not a human fabrication, but an experimental physical reality we should start to think in terms as one. "Matter tells Spacetime how to curve. Spacetime tells Matter how to move". This is one written 'Art Piece' of Knowledge!
By K. K. L. on September 3, 2018
NASA's Timeline of the Universe:
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