General Relativity: A Geometric Approach 1st Edition by Malcolm Ludvigsen (PDF)

Description

Starting with the idea of an event and finishing with a description of the standard big-bang model of the Universe, this textbook provides a clear and concise introduction to the theory of general relativity, suitable for final-year undergraduate mathematics or physics students. Throughout, the emphasis is on the geometric structure of spacetime, rather than the traditional coordinate-dependent approach. Topics covered include flat spacetime (special relativity), Maxwell fields, the energy-momentum tensor, spacetime curvature and gravity, Schwarzschild and Kerr spacetimes, black holes and singularities, and cosmology. All physical assumptions are clearly spelled out and the necessary mathematics is developed along with the physics. Exercises are provided at the end of each chapter and key ideas are illustrated with worked examples. Solutions and hints to selected problems are provided at the end of the book. This textbook will enable the student to develop a sound understanding of the theory of general relativity.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐OK !

⭐In his (really great) book/lectures notes “Advanced General Relativity” Winitzki dedicates a few paragraphs solely commenting on this book.You can find Winitzki’s book online for free as it was released under GNU FDL, and he also keeps it on his website. Try it here […] , it should be the file “GR_course.pdf”What he wrote (page 175, you should read it from the file to understand the equations and symbols):”The book is M. Ludvigsen, General relativity: a geometric aproach (Cambridge University Press, 1999). Many explanations in that book are outstandingly clear, and I benefited greatly by reading it. Nevertheless, there are some minor gaffes:1) On p. 91, eq. 9.27 is supposedly the same as eq. 9.20 when”written in full.” However, these equations actually differ bythe choice of the permuted indices. The relation 9.27 can beobtained from 9.20 only if one assumes the identity R abcd =R cdab , which Ludvigsen actually never mentions in the book.This well-known standard identity is a consequence of 9.19and 9.20.2) On p. 103, eq. 10.14 contains ∇ a ∇ a φ = R…, whilethe preceding (unnumbered) equation on p.102 contains−∇ e ∇ a φ = R…. A minus sign has materialized fromnowhere! The answer (10.17) is correct, and the extra minussign is actually needed to compensate for an error made ear-lier. In the last paragraph on p. 101, Ludvigsen writes (in3-dimensional notation) a = ∇ φ whereas in fact a = −∇ φin Newtonian gravity. (The acceleration points down, the po-tential grows upwards.) So the correct calculation starts byintroducing a a = ∇ a φ and not −∇ a φ.3) On p. 108, top line, “Using the fact that l a n a = 1 and∇ a l b = ∇ b l a , we have…” — actually, the same result followswith merely the assumption that l a is a null geodesic. It is notnecessary to assume that l a is integrable.4) On p. 109, top line, — one cannot actually derive Eq.(11.10) as claimed. By contracting the top equation withl a m b m c and using Eq. (11.8), which already assumes that l isintegrable, one getsDσ = R(l, m, l, m) + 2la¯m b ( ∇ a mb).Now it is unclear how to show that the last term is equal to2ρσ. There is a significant freedom in ∇ m since m is chosensimply to be orthogonal to n, l and this is not sufficientto fix ∇ m. In fact, the null tetrad can be changed by thetransformation m → e iλ m. Then σ → e 2iλ σ, as shown at topof page 110. If λ is a function of position then the equationDσ = … will be changed after the transformation! Thus thisequation really depends on the choice of the tetrad, and somechoices are better than others. The equation (11.10) is perhapsobtained with a suitable choice of the tetrad, but this is notdiscussed in the book.”[I rated Ludvigsen’s 4 stars, because I haven’t read the book myself.]

⭐The book is remarkable in its approach: it uses coordinate-free geometric formulation throughout, never using non-tensors such as the Christoffel symbols, and never resorting to brute-force calculations. Beware that this is not a textbook of GR for beginners! Rather, this is a second book that explains a more advanced approach. The initial chapters (non-GR) are useful for those who already know special relativity+mechanics+EM, but not in a more abstract four-dimensional formulation (a 4-vector of *electric field*?). The beauty of the book starts to shine in the GR chapters; for example, the derivations of the Newtonian limit (although it contains a hard-to-spot sign error), the redshift, the properties of horizons, and the introduction of a null tetrad are elegantly executed. The Schwarzschild metric is derived from symmetry arguments together with the Raychaudhuri equation, rather than by a long calculation most textbooks avoid. This book sets a worthy example of lucid explanations and geometric reasoning from which especially graduate students would benefit.

⭐This is a good book, with a pronounced mathematical accent and many useful and solvable problems. It can be considered as a textbook, though there are some points which deserve corrections, for example: (a) In p. 36, the Author characterized the Lorentz transformation too shortly, and he calls it “Lorenz transformation”. This is especially strange since in the neighbouring Denmark (Ludvigsen’s address in the book is in Sweden) there was a great physicist L.V. Lorenz, 1829-1891, inventor of the Lorenz condition, creator of the electromagnetic theory of light (1867, independently of Maxwell), and co-author of the famous Lorenz-Lorentz formula — together with H.A. Lorentz of Holland to whom pertains the above transformation. Eight lines below Ludvigsen introduces “Levi-Cevita” tensor (named after T. Levi-Civita, and this is a pseudo-tensor = axial tensor). These errors are not misprints (see Index, p. 216, and the text in the pages given there). (b) The Author uses, of course, real coordinates (not the ancient imaginary time), so it is inadequate to picture the Lorentz transformation as a sheer trigonometric rotation of space-time axes (cf. figs. 4.7, 5.3). (c) In the very title of the book the subtitle (“A Geometric Approach”) seems to be artificial since general relativity practically is a synonym of geometry: the well-proportioned abundance of figures in the book is not identical to geometry. Imagine that Euclid would entitle his books as “The Elements. A Geometric Approach”… But the book by Ludvigsen is definitely a success, though it needs some editing more. I highly recommend it to students.

⭐I have taught electromagnetism at the advanced graduate level, and am reasonably familiar with classical differential geometry and general relativity (e.g. as presented by Weinberg). I am finding Ludvigsen’s book a tough read, though a worthwhile one. A considerable amount both of the physics background and the mathematics background of the subject is omitted or treated very briefly. Three paragraphs cover the Coulomb potential *and* the plane wave, for instance. Definitions tend to be ostentive – by example.Ludvigsen’s book has the virtues of brevity as well as its difficulties, however: the reader can see where he is going at all times, and the author takes the reader through the modern approach to differential geometry, and deeply into the results of relativity theory, quickly and efficiently.I hope to see a second edition of this book. A little more explanantion, here and there, would do much, if not to reduce the steepness of the learning curve, at least to provide a few handholds on the way up!

⭐good copy

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