http://web.cortland.edu/moataz.emam/

This page is the official companion website for the book

COVARIANT PHYSICS From Classical Mechanics To General Relativity & Beyond

Published March 2^nd 2021 by Oxford University Press^©, this textbook is currently in its first edition. The target readership includes third or fourth year undergraduate physics majors, first year graduate students, self-learners, as well as other interested parties such as engineers. The book discusses the modern methods of theoretical physics involving mainly the general theory of relativity, but also goes beyond that into topics such as the theory of classical fields, the use of differential forms in theoretical physics, and modified theories of gravity. The basis of the approach is the principle of covariance, rarely used or discussed at such a level. The book requires, intentionally, a minimum of mathematics and physics preparation. Any student with a background of multivariable calculus, a bit of linear algebra (mainly just matrices, determinants, and their properties) and some introductory physics should be able to navigate most of the book (Junior/senior level classical mechanics and electrodynamics would help in some parts of the book but are not necessary, as these parts may be skipped at first reading without affecting the general flow). It is designed to get the student to be able to read and understand most modern research papers in the field of spacetime physics, which may include Supergravity (minus the supersymmetric part), as well as prepare them for doing research under supervision.

· The site is a continuous work in progress. Listed below is the updated errata for the first edition. The author would like to extend his gratitude to the readers who discovered some of these errors post publication (Gh. Shahali, Sunjiv Varsani, Ferreño Blanco, Toby Baldwin, Kevin Coffey, Miles Angelo Sodejana, and Omar Mahmoud).

· As the reader progresses into the book, s/he will find that the exercises become increasingly longer, not more difficult, just longer. For example, once the reader calculates a few Christoffel symbols (or, even worse, components of Riemann’s curvature tensor) it becomes less instructive and more tedious to calculate more, especially as the metrics become larger and larger. At this point the reader should, and as a matter of fact is encouraged to, use the computer to work out these things. There are many ways one may do so, depending on what you are comfortable with. For example, you may write codes in Python or C++ if you are proficient in those. Or you can use preprepared symbolic manipulation packages in Mathematica, Maple, or MatLab. There are many such packages available online for free. For example, many useful such packages may be found in the Mathematica Repository. Just search for what you need.

Errata for the first edition:

Click here for a printable version

1) Page 13: In equation (1.25) the third line up from the bottom the first two terms should be: dρ²sin²φ+ ρ²dφ²cos²φ.

2) Page 27: Part 1 of Exercise 1.13 should read “where the coefficients a_ij are constants satisfying a_ij =a_ji.”

3) Page 46: The argument around equations (2.15) and (2.16) is wrong. What it should say is that the components of the vector V do not change under the transformation (xⁱ à xⁱ + aⁱ) and then elaborate further. Since this is not important to our discussion, I have decided just to recommend removing between “Although we will mostly focus …” and equation (2.16) inclusive. Similarly and for the same reason, on the bottom of page 52 remove between “As pointed out earlier …” and equation (2.41) inclusive.

4) Page 45: Equation 2.11 has the minus sign in the wrong place it should be:

5) Page 52: Footnote 8, 2^nd line: "Since each index counts from 1 to 3, each of the nine equations contains nine terms, giving a total of 27 terms altogether." the total number of terms should be 81, not 27.

6) Page 58: Footnote 11, the phrases “physical basis” and “natural basis” should be switched.

7) Page 80: Equation (3.6) the last term dx<sup>i`</sup> should be replaced with dx <sup>j</sup> (i.e. i` → j).

8) Page 81: From the top of the page (right before equation 3.7) to right before equation (3.10) should be replaced with:

9) Page 111: Exercise 3.26 should be two dimensional. The velocity v should be replaced with components (5, 2, 0). The words “except in three dimensions” to be removed. Similarly, problem 3.27 on page 113 the same words should be removed. Also change exercise 4.7-3 to make it two dimensional by dropping the z component of the velocity.

10) Page 123: Equation (4.26) the left hand side should have the same indices on U and V.

11) Page 130: Exercise 4.7-2, let v=(0.7c, 0.5c, 0.2c) instead of the given numbers.

12) Page 136: Exercise 4.13 should be entirely replaced by: An observer at rest in some orthogonal frame of reference O sees the location of an event happening at x = 500m and ct = 600m. Another observer O' on a frame of reference traveling at a speed v = 0.3639c with respect to O sees the same event at different spacetime coordinates. Draw an accurate to-scale spacetime diagram to describe the situation and find x' and ct' from it; i.e. the location of the event as measured by O'. Verify your result analytically using the Lorentz transformations. Your graphical results should be within about 20% of your analytical ones.

13) Page 170: 3^rd line: “… Galileo, who proved it experimentally over 500 years ago.” should be 400 years.

14) Page 187: The metric (5.61) is missing factors of r in each term. It should be as follows. The form (5.67) is correct.

15) Page 206: The second line, dΩ should be squared.

16) Page 214: in equation (6.18) the last term should be 2/c² instead of 2/c.

17) Page 217: Equation (6.25): the last component of the 4-velocity should be just ω, NOT Rω since we are using natural basis vectors.

18) Page 224: In the line before equation (6.52) and again the line after equation (6.54) the definition of E should be E=cP⁰.

19) Page 225: in equation (6.56) the last term has a missing factor of c, i.e. it should be -cg_i0Pⁱ.

20) Page 234: Exercise 6.25. It is enough to do the first part of this problem. The second part needs revision.

21) Page 237: Exercise 6.28, although probably obvious, let me just clarify that the second calculation should be for the case of constant angle θ, i.e. the time derivative of θ is zero.

22) Page 243: 2^nd paragraph, 3^rd line, the reference FIG [5.11] should be FIG [6.9].

23) Page 245: The first paragraph of section 6.2.7, the reference “Starting with (6.110)” should be (6.109). Same on page 246 in the line before equation (6.125).

24) Page 248: 2^nd line after Fig. [6.10], the place of aphelion and perihelion should be exchanged. Same for Fig. [6.10] on the same page and exercise 6.44.

25) Page 263: in equation (7.13) dy^κ should be dx^κ.

26) Page 265: The discussion towards the bottom of the page is incorrect, the components of the Riemann curvature tensor do diverge at the Schwarzschild radius if one uses Schwarzschild coordinates. The correct argument is that only scalar invariants, like Kretschmann’s scalar and the Ricci scalar are well behaved at r = r_s in any system of coordinates describing a Schwarzschild manifold.

27) Page 291: Second line after equation (8.10): "it evaluates the variation of x at the fixed initial and starting points." Should be initial and final points.

28) Page 317: Equation 8.142. The M subscript in the left hand side should be removed.

29) Page 318: Equation 8.146. The second term in the parenthesis should be multiplied by ¼ g_μν.

30) Page 353: The sentence “The bein language, then, is a formulation of geometry to the coordinate/metric-based formulation we have been working with, and as such the entire structure of general covariance can be rewritten in terms of this.” should be rewritten to “The bein language, then, is a formulation of geometry equivalent to the coordinate/metric-based formulation we have been working with, and as such the entire structure of general covariance can be rewritten in terms of it.”.

The author may be reached via his online presence:

Linkedin, Facebook, X (formerly Twitter), Academia.edu, Amazon, YouTube, or his Blog.