First Course In General Relativity - What You Actually Need To Know Before Starting
You pick up the book, open to chapter one, and there is a tensor. Not just any tensor, but one that looks like it was designed specifically to make you question your life choices. This happens to almost everyone. I remember working through the geodesic equation derivation for what felt like the third time because my first attempt had a sign error that I spent four hours hunting down. The answer ended up being I dropped a negative when raising an index with the metric. Classic. First Course In General Relativity by Peter L. Szakov is honestly one of the better entry points into the subject if you have some physics background. It does not assume you know differential geometry cold. It builds that up as it goes. The problem is building it up as it goes while simultaneously introducing Ricci tensors, which means you are often encountering two new concepts at once and your brain has nowhere to park either of them.
Prerequisites - What First Course In General Relativity Actually Requires
Linear algebra. Real analysis at the level where you understand convergence without panicking. Classical mechanics with Lagrangians, preferably. Electromagnetism at the graduate level or advanced undergraduate level would help but is not strictly necessary. If you can handle a multivariable integral and understand what a coordinate transformation is without looking it up, you are in roughly the right ballpark. The book expects you to be comfortable with partial derivatives and basic matrix operations. That is about it for mathematics. The physics side is where people stall out. You need to have done some special relativity before touching this. Not just "I know E equals MC squared" level. I need you to be able to write down a Lorentz transformation in your sleep.
How The Book Is Structured And Where People Get Stuck
Chapter one covers tensors in special relativity. It sounds simple. It is not simple if you have never seen tensor notation before and you are trying to process index placement, covariant versus contravariant components, and the Minkowski metric all at the same time. Spend real time here. Do not rush. The entire rest of the book depends on you being comfortable manipulating indices. Chapter two moves into curvature and the Riemann tensor. This is where the book earns its reputation. Szakov explains the Riemann tensor through parallel transport on a sphere before going full abstract. That choice is deliberate and it works better than most people give it credit for. I have watched students who bailed on chapter two because they tried to memorize the Riemann tensor formula without understanding why it exists. That approach fails. You need the geometric intuition first. Chapter three is Einstein's field equations. The math is straightforward once you have the Ricci tensor and scalar laid out. The hard part is getting comfortable with the fact that you are essentially equating geometry to stress-energy. The proportionality constant is eighty-one point five G over c to the fourth power in SI units. Write that number down somewhere. You will need it for problem solving and you will forget how it comes together if someone asks you to derive it from dimensional analysis.
Get the Full Details
Chapter four covers the Schwarzschild solution. This is where general relativity starts producing actual predictions. Light bending, perihelion precession, gravitational redshift. The derivations are long but mechanical. The trick is keeping track of which approximation you are making and when. A student once showed me a calculation for light deflection where they used the weak field approximation on a path that went through the Sun's interior. That approximation breaks down completely inside a massive body. He got a number that was close to the right answer by accident because two mistakes cancelled each other out.
Common Mistakes When Working Through First Course In General Relativity
Index management is the biggest one. Writing an index twice on the same side of an equation when you did not mean to sum. Forgetting that repeated indices are summed over and free indices must balance on both sides. I use a system where I color-code covariant and contravariant indices when I am doing derivations by hand. Covariant in red. Contravariant in blue. It sounds silly. It caught me at least a dozen errors before I built the habit of checking it mentally. Sign conventions are another trap. The metric signature. The Riemann tensor definition. These vary between authors and even between chapters in some books. Szakov uses the mostly plus signature. If you pull a formula from another source and it does not match, check the sign convention first before you assume the formula is wrong. Half the frustration in this subject comes from incompatible conventions masquerading as mathematical errors. Another thing nobody tells you about is the computational load. Working through these derivations by hand takes serious time. A single geodesic equation derivation with nontrivial Christoffel symbols can take forty-five minutes to an hour if you are being careful. Planning your study sessions around this reality rather than hoping to power through will save you from burnout. I used to schedule two-hour blocks with nothing else on the calendar when I was working through problem sets. Anything less and I was rushing.
Problems That Are Worth Doing Versus Problems To Skip
Do every problem in chapter one. They build the index manipulation muscle you will need constantly. Do the problems on parallel transport in chapter two. Those give you direct geometric insight that later chapters rely on. The Schwarzschild problems in chapter four are non-negotiable. Deriving the orbital equation, calculating the perihelion shift, working through the light bending integral. These are the core calculations. The more abstract problems involving Killing vectors and conserved quantities along geodesics are worth doing but you can return to them after you finish the book once. The first pass is about building working knowledge. You can deepen it on the second pass. There is one specific problem type I want to flag. The problem involving the interior Schwarzschild solution for a uniform density sphere. The algebra is messy and the result is not physically realistic for most real stars. It teaches you something about boundary matching conditions which is valuable. But I would spend no more than an hour on it. If you cannot get through it in that time, look at the solution and note the method. Moving on is the better use of your time.
What The Book Does Not Cover Well
Framedragging and the Kerr metric barely get a mention. If you want rotating black holes, you will need supplementary material. Wald or Carroll for the formal treatment. Or if you want something more accessible, Thorne and Blandford. The book also skims over cosmology. You get the Friedmann equations but not much physical interpretation or observational context. Chown or Peacock would fill that gap reasonably. Numerical relativity is completely absent. The field equations in their full glory are not solvable analytically for most interesting cases. The book will not prepare you for that world. That is fine. It is an introductory text. It is not trying to be comprehensive. One more honest limitation. The treatment of gravity waves is terse. You get the linearized theory and the quadrupole formula. You will not come away with a strong grasp of gravitational wave detection physics or the astrophysical sources. That requires a dedicated course or a more advanced text.
How Long This Should Take
If you are working through it seriously with practice problems, plan for about twelve to sixteen weeks at three to five hours per week. That is a graduate semester pace. Some people compress it into six to eight weeks by pushing harder. I would not recommend that. The subject rewards slow digestion. Reading three chapters in a single weekend will leave you with the illusion of progress and the reality of confusion. If you are doing this on your own without a course structure, expect it to take longer. Six months to a year is more realistic for self-study. That is not a failure. It is just the time scale the subject actually demands.
Where To Get First Course In General Relativity
The book is widely available. Amazon carries it in paperback and Kindle. University bookstores typically stock it. Used copies circulate frequently and tend to be in decent condition since it is a standard textbook. The third edition from 2002 is the one most people use. The content is solid enough that older editions are still viable, though the problem sets differ slightly between editions. Check the ISBN before buying used to confirm you are getting the third edition. There are also solution manuals floating around online. I do not recommend using them as a first pass. Work through the problems yourself first. If you are stuck for more than twenty minutes, look at the setup in the manual, not the full solution. Then close it and finish the derivation on your own.
Final Notes On Making This Work
Keep a dedicated notebook for derivations. Not summary notes. Actual step by step work. The difference between knowing a result and being able to reproduce it under exam conditions is usually the quality of your personal derivation notebook. Mine from when I worked through this book has maybe thirty percent of the original work I am still proud of. The rest I redo from scratch when I need to remind myself how something comes together. Do not read this cover to cover before starting problems. Start problems in parallel with reading. The reading alone will not teach you general relativity. The problem solving will. The reading provides the framework. The problems provide the understanding. If you get to the chapter on advanced topics and feel like you are drowning, that is normal. General relativity is not an easy subject. It is not supposed to be easy. The people who make it look easy are the ones who have done this work multiple times. Give yourself permission to move slowly and come back to difficult sections later.
