General Relativity: The Theoretical Minimum

The Theoretical Minimum

by Leonard Susskind, André Cabannes

The latest volume in the New York Times bestselling physics series explains Einstein's the general theory of relativity

He taught us classical mechanics, quantum mechanics, and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein's general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations, and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler).

An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe's real structure.

Genres: Physics, Science, Nonfiction, Mathematics, Textbooks, Stem, Education

400 pages, Paperback

Published May 20, 2025

About the author

Leonard Susskind is the Felix Bloch Professor of Theoretical Physics at Stanford University. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology. He is a member of the National Academy of Sciences, and the American Academy of Arts and Sciences, an associate member of the faculty of Canada's Perimeter Institute for Theoretical Physics, and a distinguished professor of the Korea Institute for Advanced Study.

read more: http://en.wikipedia.org/wiki/Leonard_...

Reviews

[Chris · Rating 5 out of 5]

Totally excellent, highest possible rating. This is the fourth in Susskind's "The Theoretical Minimum" series of shorter, friendlier textbooks intended for enthusiasts of science - fans, amateurs, and professionals - who want to actually understand physics. This can only be achieved by using mathematics, as Susskind has done here. There is no other way, despite the pile of popular science books on the subject which are full of misleading words and graphics ... but no equations.

The targeted readers may fall into the following categories:

1) Those curious fans of science who are tired of the frustrating hand-waving and verbose textual explanations of physics that, without using math, can only give an illusion of understanding - often false;

2) Those who learned the material in college years ago and want a quick refresher; and

3) Current physics students who want an alternative "easy" summary to clarify more intensive and lengthy course materials.

Each of the proceeding three books in the series have been equally good, and I'm happy to see that the author has announced within that there will be two more volumes covering Cosmology and Statistical Mechanics.

[Colm · Rating 5 out of 5]

Excellent book, even better if treated as a companion to the Lectures, and a genuinely fun read.

It does have a weird structure, though, it goes through the required Riemannian Geometry, Special relativity, and Classical Mechanics, then takes a tangent - going off (very interestingly) on black holes for a few lectures before returning to derive and formalise the Einstein Field Equations in the last 2 lectures.

It potentially could have been better structured if the field equations were done right after the ground work, followed by Black Holes (all of which, historically, came after the Field Equations were derived)

[Daniel Duque · Rating 4 out of 5]

Magnificent material, as usual.

Then, why the 4 stars? (I have given 5 to the previous three volumes).

The editing. I don't know it André Cabannes is the one to blame, but this reads as almost a verbatim transcript of the lectures. These are also great, but a book should be different. Concepts are repeated so much that you find yourself saying "all right, I got it already" -- attracting glances from fellow public transit commuters.

I am also not too happy about the ordering, which again reflects the lectures. The Schwartzschild solution is discussed extensively, but never deduced. Indeed, its deduction could only have happened at the very last section, devoted to the field equations. The equations of general relativity are known to be very hard to solve, but the other important solution, gravity waves, is really not deduced. I recall this was actually not so very hard, and that one of its important features (the waves traveling at the speed of light, with no dispersion) came as a pleasant surprise.

To summarize, lots of space is devoted to repeating the same concepts over and over, while some important developments are swept under the rug. It feels that most of the heavy mathematical machinery (which is very well explained) is therefore wasted.

[Saranya ⋆☕︎ ˖]

Yes...

[Nattapon Chotsisuparat · Rating 5 out of 5]

This book is a (friendly) textbook, not a popular science book. The fourth book in the The Theoretical Minimum series is this one. The concepts of equivalence principle, tensor, curvature, black hole, gravity, and general relativity (not special reletivity) are explained throughout the book. I adore physics textbooks, and this series has captured my heart.

[Luca Signorelli · Rating 5 out of 5]

After a hiatus of few years, Lenny Susskind comes back to his successful "Theoretical Minimum" series of user-friendly physics textbooks, aimed at a public of science enthusiasts and or college/undergrad students.

As noted by another reviewer, this is a textbook, not a "science popularization" book. And rightly so. While it is possible to explain Special Relativity with a minimum of high school math, it's absolutely impossible to give a proper idea of Einstein's GR without proper mathematical treatment, and a prerequisite grasp of linear algebra and calculus. Beware of anyone telling you GR can be explained without math: he's lying (or he doesn't really know the subject).

Thus said Susskind's (and co-author Andrè Cabannes) treatment of the subject is really gentle and crystal clear, beginning with the basic concepts (ie the Equivalence Principle) and then moving to an explanation of tensors, the basic GR mathematical "language", before going into the Einstein Field Equation and its application. The book is a companion of Susskind's own (free) series of Stanford University lectures, available on YT and should be read while watching his lectures on GR, in particular the 2008 11 lessons series.

I strongly believe Susskind's lessons are the easiest "real" introduction to physics, at least if you want to know the real deal and not been told a bunch of confusing metaphors. This is the fourth book on the series. Reading book 1 (on Classical Mechanics) and 3) (on Special Relativiy and classical field theory) is strongly recommended, while, unless you're interested in the topic, you can skip book 2, on Quantum Mechanics.

[Paul B · Rating 5 out of 5]

This book cuts the clutter and complexity and succeeds in explaining what General Relativity is all about. There's plenty of maths in there, but you're led through it and it makes sense. It is a brilliant summary, brilliant because other authors get lost - or get me lost - in the long and highly detailed calculations. There's no dumbing down, just a focus on understanding what's new versus calculating stuff.

[Matthew · Rating 5 out of 5]

This covers the same material as the lectures Leonard Susskind gave at Stanford that are are available online https://theoreticalminimum.com/course.... The book is accessible to anyone familiar with calculus. It is particularly good at explaining the physical significance of the mathematics. The book contains simple examples such as an accelerating elevator or polar coordinates to help understand curvature. The explanation of black holes is particularly good.

[Harry Channing · Rating 4 out of 5]

Hyperbolic coordinates are hard

[Barry Shitpeas · Rating 5 out of 5]

lenny susskind proper geezer

[Richard Wagner · Rating 4 out of 5]

good stuff! could have used some editing on the way. looking forward to volume 5

[Thomas Ray · Rating 3 out of 5]

General Relativity: The Theoretical Minimum, Leonard Susskind (1940-) & André Cabannes, 2023, 373 pages, Dewey 530.11, ISBN 9781541601772

Susskind sheds light on a weighty subject. Clear and concise. The book is largely about coordinate transformations. Requires first reading books 1 and 3 in the series, or equivalent on matrix algebra, vector calculus, Lagrangian mechanics, special relativity and fields.

10 lectures (Susskind gave in a continuing-studies class for adults):

1 Acceleration is equivalent to uniform gravity; tensor analysis.

2 Tensor math, to change frames of reference among various states of acceleration or gravity.

GRAVITY is Riemannian geometry in Minkowskian spacetime:

3 Flatness & curvature

4 Geodesics & gravity

5 Metric for a gravitational field.

BLACK HOLES are point masses in Minkowskian spacetime:

6 Black holes

7 Falling into a black hole

8 Formation of a black hole

9 Einstein field equations

10 Gravitational waves


Special relativity: Moving clocks run slow. Particles that decay in less than a second may make it here from the sun, over 8 minutes in our frame of reference. p. xi.

General relativity: Acceleration and uniform gravity are equivalent. Masses warp space and time. p. xi.

FICTITIOUS FORCES

In accelerated frames of reference, there are fictitious forces that have real effects. Consider a man of mass m in an elevator undergoing upward acceleration g: In the stationary frame of reference of the building, the forces on him are 2mg upward from the floor, and 1mg downward from Earth's gravitational pull, for a net acceleration of g upward. In the accelerated frame of the elevator, there's also a fictitious downward force of mg, yielding 0 net acceleration in the elevator frame. Then if the man in the elevator drops an apple, it accelerates downward at 2g, from the perspective of the elevator frame, while as seen from the inertial, building, frame, it's seen to accelerate downward at 1g. p. 10.

EQUIVALENCE OF ACCELERATION AND GRAVITATION

A uniform gravitational force field can't be distinguished from the effects of an accelerated reference frame. To the man in the elevator, accelerating upward at 1g on Earth, or not accelerating at all on a planet with twice Earth's gravity, or accelerating upward at 2g far from any planet, all feel the same. p. 11.

EFFECT OF GRAVITY ON LIGHT pp. 15-17

Emit a pulse of light at speed c horizontally from one side of the elevator to the other, a distance x away. Seen from the inertial frame, it moves horizontally, arriving at time t = x/c. Seen from the frame of the elevator accelerating upward at g, what was a horizontal line in the inertial frame is a parabola: height = -.5gt^2.
Substituting for t, height = -.5gx^2/c^2.
We have assumed the light pulse is emitted when the elevator's speed is zero, at time t=0.

Gravitation has the same effect as acceleration. Light bends in a gravitational field.

RIEMANN GEOMETRY

If coordinates are curved, it's not easy to say whether vectors at separate points are equal. p. 96.

If coordinates are curved or not perpendicular, the distance dS between neighboring points is given by
(dS)^2 = the sum over all pairs of coordinates, of (a factor depending on the coordinates)*(the small change in the one coordinate)*(the small change in the other coordinate)
Written
(dS)^2 = g_mn(X) dX^m dX^n
where the summation over all coordinates m and n is implied. Superscripts m and n are merely indices. As are subscripts. Superscript 2 does mean squared. For example, a small distance dS along the locally horizontal surface of the Earth is given by
dS^2 = (R dtheta)^2 + (R cos theta dphi)^2
Here there are no dtheta*dphi terms because lines of latitude theta and longitude phi are everywhere perpendicular.
R is Earth's radius.
Latitude theta and longitude phi are measured in radians, to make the above distance formula true.

In a flat geometry there is a set of coordinates (X1, X2) such that dS^2 = dX1^2 + dX2^2. p. 88.

The g_mn is called the metric tensor. The above one is just a 2-dimensional matrix. The metric tensor is symmetric: g_mn = g_nm. p. 80. Its eigenvalues are always greater than zero. p. 82.

Where coordinates are perpendicular, as above, the elements of tensor g are nonzero only on its diagonal, where m=n.

CONTRAVARIANT vs. COVARIANT VECTORS
A displacement (of some small distance in some direction) is a /contravariant/ vector: if its coordinates were in centimeters, and we /divide/ the unit by 10 to millimeters, the coordinates /multiply/ by 10. By contrast, a gradient, such as degrees of temperature change per unit distance in some direction, is a /covariant/ vector: if we /divide/ the unit by 10, the components of the vector along the axes each also /divide/ by 10. p. 45.

[James F · Rating 3 out of 5]

General Relativity is the fourth book of the Theoretical Minimum series, and is based on a course of ten lectures of the same name. The previous books included brief explanations of derivatives, integrals, partial derivatives, vectors and so forth, and were apparently aimed at the layman interested in physics and wanting to go beyond the usual popularizations and understand the physics with the relevant mathematics, which would describe my own situation. In the preface to this book, however, it is stated that the target audience of the series is people who studied physics "at the undergraduate or graduate level" but "went on to do other things" and want essentially an up-to-date refresher. I think this represents a certain change, and the book is a bit more difficult than the earlier ones. It is not a "stand-alone" book; to be comprehensible at all, it requires that the reader have read books one and three (classical physics and special relativity; the second book is on quantum theory and uses an entirely different set of mathematical tools, so it is more of a "stand-alone") or have learned that material in other courses or from textbooks. I can't claim to have "studied physics at the undergraduate level" although I did have two semesters of introductory physics, which the course description claimed were "calculus-based" because they used simple calculus, but were not in the way these books are (starting from Lagrangians and least action.) In fact, since this book just came out, about six and seven years after I read the first and third books, I was quickly lost until I found my notes on those. (And yes, this is the kind of book you need to take notes on.)

The first thing to be clear about is that these are physics books, not math books. To make an analogy, my high school ATA course (basically today's Precalculus with a short introduction to calculus) taught vectors in some depth; my high school physics class, which did not require Precalculus, also taught how to use vectors, but without as much explanation of the "why's". These books similarly teach the necessary math from the standpoint of how to manipulate the symbols and do the calculations in the physics rather than from the standpoint of explaining it as math. I was actually surprised how much they did explain, and in the previous books and about the first two lectures of this one I was generally able to fill in the "why's" myself; but from the end of lecture two here, that is from the point at which Susskind moves from tensor algebra into tensor calculus with the covariant derivative, I essentially understood the "what's" and most of the "how's" but not the "why's" (of the math). Which is what the book is about, doing physics, not mathematics. It is however frustrating that the text is mostly presented abstractly and the actual calculations are left to "exercises" with no examples and no answers given.

The book begins, like most popularizations, with Einstein's thought experiment of the elevator and the equivalency principle. It then goes on to explain coordinate transformations and Riemannian geometry. The second lecture is devoted to tensors (without ever really explaining clearly what a tensor actually is, although after a while I figured it out) and the algebra of vectors and tensors. The third lecture continues on into tensor calculus and finishes the math "toolbox" with the curvature tensor. Then, after a bit more math (Minkowski geometry, hyperbolic coordinates), the book returns at the end of lecture four and in lecture five to the physics with gravity as curvature (the Schwarzchild metric), and proceeds through an explanation of black holes in lectures six through eight, a general explanation of Einstein's field equations in lecture nine, and ends up with gravity waves in lecture ten. Contrary to the claim in the subtitle, this is not all you need "to do physics" but it will give an idea of what it is really about if you spend the effort to understand it. At the end, Susskind announces the next two volumes, book five on cosmology and book six on statistical mechanics.

When I read the first two volumes of the series (and Roger Penrose's The Road to Reality) some seven years ago, I was motivated to begin working my way through math starting with what I already knew and going on a bit further. I got off to a good start with a book on logic and set theory, a couple histories of math, and rereading my high school ATA text; then I got diverted to other subjects and haven't read a lot of anything mathematical for another four or five years. Perhaps this book will motivate me to get started again, now that I am retired and have more time to read and study.

[Nadav · Rating 4 out of 5]

This is the fourth book in Susskind's excellent "The Theoretical Minimum" series. This installment is about Einstein's General Theory of Relativity - after the Special Theory of Relativity was already explained in book 3. Like the previous books, this book too is aimed amateurs who want to learn physics - it is not a regular textbook. The book motivates and describes the general theory of relativity, it describes in detail the mathematical machinery needed to formulate it - such as Riemann and Minkowski spaces, tensors, curvature, metrics, Christoffel symbols, etc., and derives its equations (Einstein's Field Equations). It also goes on a long tangent in the middle of the book describing the Schwarzchild metric and black holes, which Susskind describes not as an esoteric corner case of general relativity, but rather as its simplest solution, analogous to the case of a point mass in Newton's gravity.

I rated this book with "only" four star, one star fewer than the previous book, because I felt it was somewhat less perfect than the previous ones. As usual, this book had excellent parts. I was impressed by its introduction of tensors - which I felt was better than what the previous book (which also relied heavily on tensors) did. I was also impressed by how it not only explained, but also derived, Einstein's field equations - something that I have not seen done in any other book (if you don't read textbooks). I was slightly less impressed by the introduction and motivation of general relativity (I feel that others have done a better job at this), and was quite thrown off by the long tangent on black holes in the middle of the book, which felt repetitive and unmotivated - after all, Einstein and his contemporaries understood general relativity before they understood black holes (I guess I'll need to read Susskind's "The Black Hole Wars" next to understand why he's so fascinated with them).

I was also disappointed that a few times in this book, Susskind didn't actually complete a calculation, and wrote something like "after two more pages of calculations, you end up with ...". This sort of excuse makes a lot of sense for a lecture (which is the source material for this book), but in a book? What prevents you from actually adding two pages to the book, with the complete calculation? You can tell the reader they can skip this calculation, but it will be there - in case a curious reader wants to see "how a real physicist thinks" (which was always the goal of this series, in my mind). In one place they even say that something can be easily calculated in Mathematica (the software) - so why not include an example instead of saying "it can be done"? This book is smaller (in height and thickness) than the 3rd book, so had place to grow.

Two other places where I felt that the book should have given significantly more details - that perhaps a lecture couldn't but a book can. One was the definition of the stress-energy tensor, which I think was better motivated and explained in the previous book (where it was needed to derive Maxwell's equations). The other was deriving Einstein's field equations from the action principle at the very end of the book - I felt it was done too quickly, with hardly any details, as if the publisher ran out of paper to print the book :-) Given the lengths that the previous book went into deriving Maxwell's equations from the action principle, it would have natural to see here a complete derivation of Einstein's field equations from the action principle.

[William Schram · Rating 5 out of 5]

Leonard Susskind and André Cabannes are here for the fourth installment of The Theoretical Minimum. Special Relativity resulted from several minds toiling away and Einstein coming along and finishing it. General Relativity was Einstein’s baby. Sure, he had assistance with the differential geometries and tensors, but no one else worked on it as he did.

The Theoretical Minimum introduces advanced physics concepts to the layperson. It came from a set of lectures with a question-and-answer segment at the end. So we find many misconceptions that Susskind has to tackle.

Susskind and Cabannes spend several chapters developing a toolkit to foster our understanding. We go through Christoffel symbols, tensors, manifolds, and more. Susskind and Cabannes write in the conversational style typical of the series. Susskind and Cabannes explain all the notation and standards used, including the Einstein Summation Convention. Einstein introduced it to physics for brevity. He was sick of writing out all the terms every time.

Finally, Susskind and Cabannes devote a considerable portion to black holes. They discuss their formation, what would happen if you got stuck in a black hole from two frames of reference, and more. I learned some of this before, but Susskind and Cabannes take a different tack and cover all the equations.

Thanks for reading my review, and see you next time.

[Andrea Piazza · Rating 4 out of 5]

Finally an introduction to general relativity which is not too divulgative or too technical. The vast majority of topics is accessible if you have a solid mathematical/physical background.

The first part, explaining the main motives behind Einstein's research, is outstanding: in particular, the explanation of the equivalence principle and the parallelism between "real" gravitational fields and curved geometries. The mathematical tools are also explained in an (almost) palatable way: contravariant and covariant tensors, parallel transport, metric, curvature. Finally, I have a good grasp of these concepts, and why they were introduced in the theory.

Some improvements could however be made: first, the book needs a bit of editing; right now, it resembles too much an oral lecture, and sometimes you are left wondering "What is the goal here? Where are we getting to?". Secondly, I feel the discussion about black holes is too long and leaves little space for Einstein's equations (and their solutions); it got me confused in the end, it is sometimes too advanced and doesn't focus enough on the gravitional field produced by a spherical object (not a black hole). Nonetheless, it is still very interesting and original: I never thought black holes were so central in the theory!

[John Robin · Rating 3 out of 5]

Having read the previous three books, and finding them excellent ways to understand physics at the ground level, I was excited to fill in the gaps with general relativity.

However, this book was flawed due to the nature of which previous books were made: relying on a student’s notes, this book, by a different student, was not well written, with too many leaps and very sparing in mathematical rigour.

General relativity does require more intensive calculations, but I would have gladly worked through the pages if calculations with annotation and descriptions, if the benefit was clearly seeing how things like Christophel symbols work.

The topics covered are fascinating and left me with much insight, but it feels skeletal. I am hoping that Susskind’s course on Stanford YouTube will fill in some of these gaps.

[Des Bladet · Rating 5 out of 5]

Actually I've barely opened the book; I watched the YouTube lectures of which it is mostly a transcript instead. Susskind says many times in the lectures that general relativity is fine, unless you have to calculate things and he mostly *doesn't* calculate things, and that has strengths and weaknesses. The obvious weakness is I still don't really know how to calculate anything, but there are other books for that I guess.

[Simone Scardapane · Rating 5 out of 5]

The fourth book in the series is still very good, despite feeling a bit more "draft-y" than the previous ones (lots of very short paragraphs, not sure if it's an editorial choice or a lack of iteration over the manuscript). Also in some parts it's unclear why a series of computations are made until the very end. Nonetheless, highly accessible overview of general relativity which skips over a lot of passages compared to previous volumes due to the density of the main equations.

[Daniel · Rating 5 out of 5]

Another fantastic volume in TTM. Similar to the layout of the quantum volume, the authors build up GR in an uncommon yet superior way, the Einstein field equations introduced at the end as a natural result of everything done before. It's as smooth and painless a treatment, while still tackling the mathematics, that you're likely to find. Leaves the reader in a great place to start a more advanced treatment.

It's also enjoyable to read The Black Hole War at the same time as this one.

[Martijn Huynen · Rating 4 out of 5]

Personally, I found this the weakest book in The Theoretical Minimum series, but this could very well be because I understand it the least. I find that it suffers from overly repeating certain concepts and formulas while skipping over other crucial parts. Especially towards the end, difficult calculations are hand-waved away, which I do get but it still leaves me slightly frustrated.

[Yassine Alouini · Rating 5 out of 5]

As usual in this series, a great book.
I have finally filled, to some extent, a hole in my understanding of physics which was RG.

The last page sums up RG (even though the concepts aren't that hard):

"After a few hours of calculations, you will end up with Einstein field equations".

[Menglong Youk · Rating 5 out of 5]

As fascinating as the third book in this series. Can't wait for the next books on cosmology and statistical mechanics

[Hassaan Naeem · Rating 3 out of 5]

Awesome book on GR but a bit confusingly put forward. I am a huge fan of Prof. Susskind and his Theoretical minimum lecture series, but for some reason did not quite resonate with this book.

[ayds · Rating 5 out of 5]

I have become master.

[Nar · Rating 4 out of 5]

very interesting but hard to follow towards the end especially if this is your first foray into general relativity

[Pieter van der Zwet · Rating 4 out of 5]

I HAVE FINISHED THIS BOOK!!!!

Het is te begrijpen voor mensen met mijn bèta universitaire achtergrond. Allerlei regels voor differentiëren, coördinaat transformaties en variatierekening is ooit , 40 jaar terug, langs gekomen en het blijkt toch dat je veel opslaat wat je tijdens je studie opslaat. Het enige wat ik nooit gehad heb is tensor analyse en daar raak ik nog wel eens de weg kwijt. Contractie en het omhoog/omlaag brengen van indices is onbekend terrein en Süskind is daar ook wat makkelijk in.
Om te voorkomen dat het een boek van 1000 blz wordt snijdt de auteur wel vaak wat bochtjes af. De Schwarzschild metriek in H5, formule 32 komt een beetje uit de lucht vallen, en ook in de afleiding van de Einstein Gravity Field theorie wordt er wel wat magie gebruikt. Ook hier geldt waarschijnlijk dat een echt formele afleiding vele blz bloedsaaie formules eist.

Want: General Relativity is een formule moloch. In de basis heb je 16 parameters, die ieder weer zijn opgebouwd uit symbolen, die weer opgebouwd zijn uit scheiden van de metriek waarmee de ruimte beschreven wordt.

Begrijp ik alles? Absoluut niet. Het grote verband tussen alle stappen, en de aannames die bij iedere stap horen heb ik nog niet scherp. Maar dit boek geeft je wel alle elementen die je daarvoor nodig hebt en dat is al heel wat