Spacetime and Geometry:
Errata (by date)

If you find a mistake that isn't listed here, please email me: seancarrollatgmaildotcom, appropriately translated.

October 2003

• pp. 86-87: Maxwell's equations are referenced as (1.88) and (1.89), while they are actually (1.96) and (1.97). [Thanks to D. Hartmann and J. Olson, 10/14/03.]
• p. 145: At the end of the paragraph containing (3.204), the last sentence should include the condition "(when t is the proper time)". [10/14/03.]
• pp. 437, 469: The Exercises in the Appendices are numbered incorrectly; the correct version should be obvious. [10/14/03.]
• p. 444: In the line between (D.10) and (D.11), the equation for zeta should be an equation for xi. [10/14/03.]
• p. 190: In the very first line, "will suppressed" should be "will be suppressed". [Thanks to T. Keech, 10/20/03.]
• p. 2: In the last full paragraph, "the one sitting a table" should be "the one sitting on a table." [Thanks to D. Starr, 10/26/03.]
• p. 451: Each appearance of "Codacci" should be "Codazzi." [Thanks to M. Godina, 10/29/03.]
• p. 87: A phrase was mistakenly dropped from the sentence containing (2.87), which should read "The other one of Maxwell's equations, (1.96), can be expressed as an equation between three-forms: ..." [10/31/03.]
• p. 379: In (9.9), the exponent should be "i(omega t + alpha_0)", with parentheses, rather than simply "i omega t + alpha_0". [Thanks to T. Keech, 10/31/03.]

• p. 22: In the first line of the last full paragraph, "tangent vectors" should be "dual vectors." [Thanks to D. Taylor, 11/10/03.]
• p. 42: In the sentence containing (1.160), "stregth" should be "strength." [Thanks to D. Hartmann, D. Taylor, and J. Collins, 11/10/03.]
• p. 45: In Exercise 1, "mirror" should be "wall." [Thanks to D. Taylor, 11/10/03.]
• p. 1: Most embarassing mistake yet. Equations (1.1) and (1.4) should both have minus signs; apparently my grasp of Newtonian gravity is a little shaky. Note that (4.1) is okay. [Thanks to T. Damour, 11/12/03.]
• p. 43: In the sentence after (1.163), "tricker" should be "trickier." [Thanks to J. Collins, 11/15/03.]
• p. 74: Near the beginning of the first full paragraph, "to see a sketch a proof" should just be "to sketch a proof." [Thanks to J. Collins, 11/21/03.]
• p. 403: The last term in (9.130) should be 1/alpha^2, rather than alpha^2. [Thanks to J. Fric, 11/26/03.]
• p. 32: (1.106) should have a minus sign. [11/28/03.]
• p. 49: In the middle of the page, "accelerating at a constant velocity" should be "accelerating at a constant rate," of course. [Thanks to D. Hartmann, 11/28/03.]
• p. 108: The sentence containing (3.56) was mangled. After the semicolon should read "in fact, looking back to the expression (1.106) for the Lorentz force in special relativity, it is natural to guess that", and then the equation. [Thanks to D. Hartmann, 11/28/03.]
• p. 111: In the first sentence of the first full paragraph, "convenient a way" should just be "convenient way". [Thanks to D. Hartmann, 11/28/03.]

• p. 28: 7th line from the bottom, "give us tensor" should be "give us tensors." [Thanks to D. Taylor, 12/6/03.]
• p. 79: Near the end of the first paragraph, "a curve of zero length is chronal but not causal" is exactly backwards, and should be "a curve of zero length is causal but not chronal." In the same sentence, the word "in" should be inserted between "not necessarily" and "its own." [Thanks to D. Taylor, 12/6/03.]
• p. 80: In figure 2.24, the line from D^+(Sigma) should extend into the interior of the past light cone. [Thanks to D. Taylor, 12/11/03.]
• p. 84: In line 5 of the first paragraph, "the the" should just be "the". [Thanks to D. Taylor, 12/11/03.]
• p. 443: In (D.2), the index on the derivative should be "nu". [Thanks to D. Snead and I. Sahabandu, 12/21/03.]
• p. 439: In third line of second paragraph, "the the" should be "the". [Thanks to D. Snead, 12/23/03.]
• p. 439: Each appearance of "phi[M]" in the second paragraph should be "phi[S]". [Thanks to D. Snead and G. Chinnici, 12/23/03.]

• p. 126: To be perfectly clear, at the end of the first paragraph of section 3.7 we should say "reduce the number of independent components" rather than just "reduce the independent components". [Thanks to R. Riklund, 1/7/04.]
• p. 225: "Kruskal-Szekres" should be "Kruskal-Szekeres". [Thanks to R. Riklund, 1/7/04, and apologies to Prof. George Szekeres.]
• p. 87: After (2.86), "phi" should be capitalized for consistency with Chapter One. [Thanks to D. Taylor, 1/10/04.]
• p. 102: In (3.36), the second and third terms should be multiplied by a factor of 2, given the definition of the exterior derivative. [Thanks to D. Taylor, 1/10/04.]
• p. 112: In the fifth-to-last line of the (only) full paragraph, the "mu" in "the sought-after coordinates x^mu" should have a hat. [Thanks to D. Taylor, 1/10/04.]
• p. 57: I goofed on the cone, when I said it was a non-smooth manifold. The metric on the cone, inherited from its embedding in Euclidean space, is not smooth at the vertex, but it is perfectly possible to give the cone a smooth atlas (just project it down to the plane, and use the conventional atlas there). The irony is that I knew all of this perfectly well, but had somehow convinced myself that there was a more natural atlas for the cone, one that was not smooth. Upon reflection, I'm not sure where that conviction came from. [Thanks to P. Shepard and N. Jeevanje, communicated by R. Bousso, 1/29/04.]

• p. 375: Problem 2 was completely fumbled. First off, it should be "e^{2Ht}", not "e^{Ht}" (if we want H to really be the Hubble parameter). Second and worse, comoving observers are precisely the wrong ones to ask this question about; for them, t actually is an affine parameter. The question should have asked about non-comoving observers. [Thanks to D. Garfinkle, 2/2/04.]
• p. 92: Problem 9 doesn't make sense in "Euclidean three-space", it should be in ordinary four-dimensional Minkowski space. Only the ball in part (d) is in Euclidean three-space. [Thanks to D. Starr, 2/10/04.]
• p. 122: In the second line after (3.111), "the last term is simply the torsion tensor" would be better rendered as "the antisymmetrized connection coefficients in the last term are simply one-half times the torsion tensor". [Thanks to D. Taylor, 2/11/04.]
• p. 130: After (3.148), instead of "the Appendices" we should have "Appendix G". [Thanks to D. Taylor, 2/11/04.]
• p. 136: The long sentence at the bottom of the page was apparently mangled in the typesetting process; delete the word "using" at the beginning of the sentence, and change the comma after "generalizes to additional indices" to a colon. Not my most elegant sentence in any event. [Thanks to D. Taylor, 2/11/04.]

• p. 52: In the paragraph beginning "Consider", each appearance of "box" should be "rocket", to agree with Figure 2.2. [Thanks to D. Taylor, 4/3/04.]
• p. 96: After (3.11), it's not precisely correct to say that the transformation properties of the tilded connection coefficients are the same as those of the original Gammas; in the analogue of (3.10), there would be a plus sign instead of a minus sign. [Thanks to D. Starr, 4/3/04.]
• p. 99: In (3.20), the index on V in the second and third terms should be a nu, not a lambda. [Thanks to G. Chinnici, 4/3/04.]
• p. 141: In (3.191), the nu in the last term should be an ordinary subscript, not a sub-subscript. [Thanks to D. Starr, 4/3/04.]
• p. 179: In the sixth line, "out" should be "our". [Thanks to D. Taylor, 4/3/04.]
• p. 187: After (4.144), "minus the square root of" should be "the square root of minus".
• p. 191: In Exercise 5, "the Appendices" could be more usefully rendered as "Appendices D and F". [Thanks to D. Taylor, 4/3/04.]
• p. 202: After (5.39), delete "following". [Thanks to D. Taylor, 4/3/04.]
• pp. 221, 223: In Figures 5.10 and 5.11, one side of the middle light cones should be perfectly vertical, to line up with the event horizon at r=2GM. [Thanks to D. Taylor, 4/3/04.]
• pp. 255-273: Late in the game I decided to use rationalized (Heaviside-Lorentz) units for electromagnetism. In these units there are no 4pi's in Maxwell's equations (see p. 29) but there are in the expression for the electric field of a point charge (p. 457). The one place the change didn't happen is in the discussion of charged (RN) black holes in Chapter Six. I believe that every appearance of the charges Q or P in that chapter should be divided by 4pi to make it okay. [Thanks to E. Abers, 4/3/04.]
• p. 304: There should not be a minus sign in (7.134). [Thanks to E. Abers, 4/3/04.]
• p. 343: Three lines below (8.94), "the the" should just be "the". [Thanks to R. Riklund, 4/3/04.]
• p. 384: In (9.34), that should be a time-ordered exponential, similar in spirit to the path-ordering discussed in Appendix I. [Thanks to R. Riklund, 4/3/04.]
• p. 388: The left-hand-side of (9.57) really shouldn't use mu as a dummy index twice. [Thanks to R. Riklund, 4/3/04.]
• p. 390: There should not be a comma in the square root in (9.70). [Thanks to R. Riklund, 4/3/04.]
• p. 425: The left arrow in Figure A.2 should be labeled "V(p)". [Thanks to D. Starr, 4/3/04.]
• p. 426: There should be parentheses around the last p in the caption to Figure A.3. [Thanks to D. Starr, 4/3/04.]
• p. 432: In the caption of Figure B.3, at the end of the second line "T a point" should be "T at a point" . [Thanks to D. Starr, 4/3/04.]
• p. 432: The pullbacks in (B.9) and (B.10) should have upper asterisks, not lower ones. [Thanks to G. Chinnici, 4/3/04.]
• p. 436: In (B.24) and (B.25), the vector V should be thought of as infinitesimal; alternatively, think of V as finite and multiply each appearance by an infinitesimal parameter epsilon. [Thanks to D. Starr, 4/3/04.]
• p. 436: Below (B.25), the definition of the energy-momentum tensor is actually (4.75), not (4.73). [Thanks to D. Starr, 4/3/04.]
• p. 439: In the definition of a submanifold, the condition on the inverse map should be that it is C^infinity, not that it is one-to-one (which it always will be). Also, it should have been made clear that this definition, following Wald's GR book, differs from one common in the math literature; Wald was careful enough to point this out, while I was not. In much of the math literature, what I called an "immersed submanifold" would simply be an "immersion"; this makes sense, since if the image is self-intersecting it's not really a manifold. What I called an "embedded submanifold" would only be an "immersed submanifold", and the definition of "embedded" would include the additional condition that the image not come arbitrarily close to intersecting itself (technically, that the map phi defines a homeomorphism between S and its image). [Thanks to D. Starr, 4/3/04.]
• p. 440: In the third line of the first full paragraph, "m function" should be "m functions". [Thanks to D. Starr, 4/3/04.]
• p. 441: In the first complete sentence after (C.4), "fit together to define a submanifold" should be "fit together to define a set of submanifolds". [Thanks to D. Starr, 4/3/04.]
• p. 190: In Exercise 1, just to be clear, let's assume that the A_mu J^mu term does not contribute to the energy-momentum tensor. It won't, for example, for charged point particles, since J^mu contains an implicit factor of 1/sqrt{-g}, as in the delta-function defined in (4.157). [Thanks to H. Haber, 4/5/04.]