…
.
NEW CORRECTIONS (incorporated into new paperback editions from 2007)
Note: line counts include figure captions and footnotes (exercises)
[corrections to captions and footnotes are meant to be in smaller type]
viii+9 …Clifford–Hopf bundle
xxvi–7 …the corresponding
xxviii–11 …refers to time-reversal. Sans serif
4+1 …the Daemon might…
16+3 …Note 4.10).
32–13,12,11 Proof that the angles of a triangle ABC sum to p…
49–16 …referred to in Note 1.2,…
50–7 et al. (2002) (concerning BOOMERanG). See also…
56+8 …Joseph L. Lagrange…
69–20 …Russell (1927),…
85+21 …‘shorthand’ to…
89+4 …insist -p<q£p,…
92–7 …to logbw. We…
[i.e. reduce the space between “b” and “w”.]
108, Fig.6.4 [In the top part of the figure (i.e. in Fig.6.4a) the black marked point should be moved back (to the left) along the sloping straight line by about a millimeter, so that it more clearly marks the actual contact point between line and curve.]
109+15 …whenever f¢¢(x) vanishes.)
109, Fig.6.5f There is a minus sign missing just before the “2” in the equation depicted in the figure, which should read: y=-2+4q(x)
111–5 …undergraduate myself.[6.2]) Despite its…
113–17 …(see Exercise [6.2]) and therefore do
121-12 Priestley (2003).
124–14 …such cancellation is not permitted…
129–8 …by integrating z-1eiz around…
[The minus sign in the first exponent is essential here.]
129–5 [The first “p” appearing on this line should be “p2” (so the sum of the series comes out as “pi squared over six”).]
135–7 Bernhard Riemann, who introduced…
[i.e. remove the “t” in Riemann’s first name.]
139+19 …matter, involving what…
140+9 …figure that is being exam-
146-9 …or by 1/p.) [8.8] It can…
149-8 …paragraph.5
[i.e. delete “)”]
157+1 …b2, a3, b3, a4, … in the decomposition
[i.e. we need a triple of dots at the end of the sequence of letters.]
160–4 …when f(x) is not
164, Fig.9.9 [In the figure, there is a letter “t” and a numeral “1”. These are poorly placed. The “t” should be moved downwards to about the middle of the thick vertical line (just to the left of the centre of this line), while the “1” should be moved to the right through about the same distance, so as to be placed just below the centre of the horizontal radial line segment from the centre of the circle to its circumference.]
166–17 …with p<0. Euler might
168-16 …the point C (= –il/2p)…
175+12 morphic functions on R -`g, reduced…
175+13 on R. It is quite…
[i.e. we need to transpose “R” and “R -`g” between these two lines of text.]
177+5 …of a distribution15 is as an…
182+3 …(see Fig.6.2a)
183+24 …for which it would be quite…
185–11 …the coordinate change from
196–14 …form of a hyperfunction), so that f+ig…
206+16 rotations through p instead of…
(220–1 [12.17]. [This is corrected in the second set of printings, and is only relevant for those who require the earlier set of corrections, where the line on the page is incorrectly given as “220–15”.] )
229–1 …=aÙ…ÙgÙlÙ…Ùn where j=aÙ…Ùg, c=lÙ…Ùn.
[We need a wedge “Ù” in two corresponding places, just before “n”.]
230+8 …integrals over spaces of different
236+3 ‘non-pathological’, then…
[Quotes and spelling!]
241–5 …lower half of the figure, antisymmetrizers…
249+3 operations being performed…
251-4 …for all g in G.
251–1 …in mind Note 13.2).
253+4 of G; but when S is normal…
257-13 …vector space V, the latter
[This should be bold capital sans-serif “V”.]
261+6 normalized according to
262–8 …= etrace A,
[The comma should not be at the superior level.]
263+13 …i.e. Tabvb…
[This (second) “T” should not be bold.]
265+7 …components of ej (the
265+8 …en themselves) are,…
267+19 …where e is taken as ‘small’…
[This “e” should be italic.]
271+9 …§13.3). If A
[The “A” should be upright (and bold).]
271–1 …(see Fig.6.2a)
[The icon at the beginning of the line is too small, and should be slightly larger.]
274+14 …constants’ gabc of
[The indices on “g” must be staggered, as shown.]
281+4 sets of inequalities…
283+8 `vb =`va¢…
[The first lower index should be “b”, not “a”.]
283–10 …use this here. The asterisk…
[Remove “—in bold type”.]
284-2 …= v*•w is given…
300–13 more so in the diagrammatic notation…
310+17 …Q measures how…
[The “Q” is placed at an inferior position on the line, which it should not be; it should be at the normal height (just like the second “Q” on the line).]
312+3 …the sphere S2 in the case …
312+4 …Fig.13.2). But, for…
[Full stop.]
317–1 Bernhard Riemann in…
[Remove “t” in Riemann’s first name.]
319+19 …Bernhard Riemann (1826–66),
[Remove “t” in Riemann’s first name.]
321+3 =Rcdab; ka is a Killing…
[The “k” should be lightface (italic). Note that, with regard to 323–4, we need two additional items in the Bibliography: Simmonds & Mann, and Nayfeth; see below]
323–18 subsequent equations …
[Remove one “the”.]
325+4 …introduced in Chapters 12 and 14 is sufficient…
329–3 the same region…
334+1 15.4 The Clifford–Hopf bundle
334+8,9 …Chapter 11) William Clifford. I call S3 geometrically fibred in this way a Clifford (or Clifford–Hopf) bundle.
[I am trying not to spill over into a third line. This should be OK.]
334+10 …to obtain the Clifford–Hopf bundle is…
335+1 Fig. 15.8 The Clifford–Hopf bundle. Take…
335+5 …the base space M.
336+11 …this particular (Clifford–Hopf) bundle…
337–3 …S3 as the (Clifford–Hopf) bundle…
353+10 …involved in the Clifford–Hopf
[I have not inserted “–Hopf” in every relevant location. This is deliberate!]
380–14 16.1 See Howie (1989)…
[Delete Stephenson reference.]
380–10,9 ever fails—or whether n-point projective planes (necessarily non-Desargueian and non-Pappian) exist for n¹q2+q+1 with q a prime power.
381–6 16.15 See Penrose (1989, 1994, 1997a,b).
383-1 …with that of §14.8.
412+11(counting headings, etc.) …in E4. Of course we…
414–5 or sections…
[This should fit on the line, with a small reduction in the space sizes.]
416+14 …see Fig.18.3.The…
425–6 …conformal (‘Poincaré’) model…
430+9 angles (§22.11, Fig…
438–5 Hartle (2003). For an…
469+14 19.4. See Weyl (1928)…pp. 100-1); also Weyl (1929). This observation…
511+10 …of arrival at…
533–17 …R part, dertermining, in…
535+13 scalar product of the bra áf| with…
576–18 22.31 Penrose (1994, 200b); Zimba…
582+15 …H30. For…
[This sanserif “H” should be bold.]
582–1 …(1935), are…
583+1 …as Einstein–Podolsky–Rosen (EPR)
592+18 …colleague on Titan. If…
602–15 …(on Titan, in this
610–8 …antiparticles lead us away…
619+12 The repeated “x”s, at the bottom, should be “x”, “y”, “z” (´2):
…
.
653+16 …Huang (1952) for an alternative picture; or, for an interesting
655-1 we have encountered in Chapters 24 and 25.
659, Fig.26.1 In very first line of the figure, the alignment is poor. In the left-hand portion of the line of type, the “a” should be superior, whereas the entire right-hand portion of the line of type has been set a bit too low. The two portions of lines of type should be at the same height on the page. (Also, close up the space between “á0|” and “y*” somewhat.) Thus:
ya = |yñ = y|0ñ …… ya = áy| = á0|y*
679–21 freedom of the strong force (the 2004 Nobel-Prize work of Gross, Polizer and Wilczek). For very…
689–17 …with the Second Law. For this…
689–8 …that the Second Law is an…
690–19 …the First Law is an equality, the Second Law is…
692-6 We shall have S = k log VR , by Boltzmann’s formula, where VR is the
694-2 …Stirling’s formula n!»(n/e)n(2pn)1/2…
[No minus sign here.]
699–17 the Second Law, which…
702–10 …of the Second Law!…
702–9 …familiar Second Law,
702–2 …the Second Law asserts,…
703–10,9 …with the Second Law and with…
704+6 …the Second Law. The
[All these capitalizations are for consistency with elsewhere.]
709+18 …not more than 1.6M⊙).
[italic “M”]
709–1 Schwarzschild…
712+12 …3 ´ 106 M⊙, black…
[italic “M”]
715+9 …the Second Law. These ten…
722+2 (b) K=0, (c) K<0, and then…
733+1,2 …(1917). Though he is rarely given credit for these observations, he also headed the team in which Clyde Tombaugh finally discovered Pluto!
733–3 27.25…Carter (1971); Hawking…
734–15 …introduced by Carter (1966a,b; see also Gibbons and Hawking (1977)).
734–14 …Carter (1966b); Penrose and…
744+14 …past light cones of q and r will not…
744 -11 …3-surface S at time ~10-12s, along…
744-1 …the ‘freezing out’ at ~10-12s.
[The above two minor alterations are for consistency with elsewhere; there seems to be some flexibility in the literature as to the exact value of this number.]
748–3 …Figs.28.7 and 28.8. Find
749–2,1 …owing to the cut (dotted line in Fig.28.9a), this incompleteness…
752-13,12 …one second (horizontal) or one centimeter (vertical). The ‘radius’…
769+12 nothing that gets close to a consensus as…
769-3 Minkowskian version. The other advantage
773–3 …this ‘vacuum energy' ought to be…
[Close quotes.]
775–13,12 …the time of decoupling, and it is…
[delete “recombination”]
777+7,8 …WL and Wd are, just at the present epoch, of the same general order of size presents a somewhat puzzling conundrum.
780+1 …which are argued to
780+23 …electric to gravitational attraction
802+3 …‘here’ (Earth) and ‘r’ refers to ‘there’…
802+5 …=`yarybr, while that…
[missing bar over the first “psi”]
807+5 The entire 5th line in the caption of Fig.29.8, should be in a smaller font (to match the line above it):
+ z ´ |dead catñ |dead cat’s environmentñ |perceiving dead catñ.
811+8 conflict between…
817+18 20th century gave us two fundamental…
[italic ‘two’.]
817+21 great schemes for…
819–8 …in the penultimate para-
821+20 …fact that the Second Law…
824+10 …= 1) for the entropy…
824–14 …black holes of a plausible size. For…
824–6 …that expression, nor had…
829, Fig.30.5 [In each of Figs. 30.5a and 30.5b, there ought to be an italic Capital “O” just to the left of the central point. Please insert this O, but as small and as close to the central point (just to the left of this point) as can be reasonably achieved.]
830-6 … which when increased by 2p…
[Only one “increased by”.]
834+3,4 …regions B and B¢. Comparing…
[Prime on the second “B”.]
834–18 …is past-pointing in…
837–18 …(as measured from…
839–2 …view that in the final…
841+12 …actually have been crossed…
841+13 …find it inconceivable that …
841–15 …destroyed in the singularity itself.
846–17 …state triggers a device…
850+19 …certainly be committing a violation…
857+4 second (or even one second, as suggested in Fig. 30.24). In the meantime…
864–2 …with a mountingly…
872–10 …been around for over half a century, now, the
881+8 known German mathematician Theodor Kaluza…
890–18 …might then be needed…
906–7 …the singularities in…
909–8,7 …powers of z-1; see §7.4—although…
919–23 …specific technical difficulty…
923–5 …timelike structure of 1+q
947+14 these requirements was to…
953–8 …fundamental difficulty that the
982–5 …I had better see to this!
985+1,2 …would be the sum of two…
996+2,3 …vector field is (¶f/¶w0)¶/¶w1-(¶f/¶w1)¶/¶w0, with f homogeneous…
1003+3 statutory warning…
1004–20 …strength of its aesthetic
1006+15 be able to accommodate…
1009+13 …Hodges (1982, 1985, 1990a,b, 1998).
1009–7 …some reassessment is needed…
1018+3 …have been discussed in…
1024–16 directions of theoretical scientific…
1027–19 …theory’s insistence on higher…
1028–13 cause more embarrassment…
1028–9 …invariably describe things…
1029–8,7 …Truth are intertwined, the…
1033–7 …road to understanding the nature…
1034–10 largely unaddressed…
1038+12 many places elsewhere in…
1043–9 …deep underlying physics…
1044+18 …1628 BC. It devastated the civilized…
1044–16 …an asteroid collision
1046+4 34.16. Despite recent claims…
1046–15 …As far as I am aware…
1053-9,7 Bern, Z.…[http://relativity.livingreviews.org/Articles/lrr-2002-5/].
[The above link has apparently changed]
1054+7 Birrell, N. D.,…Quantum Fields in Curved Space.
1056+6 We need Carter’s Cambridge PhD thesis:
Carter, B. (1966b) Cambridge Ph.D. thesis; the relevant pages are accessible on http://luth2.obspm.fr/~carter/Thesis/4p25A-25B.html
Also we need to amend the previous reference in the list to:
1056+4 Carter, B. (1966a) The Complete Analytic Extension of the Reissner-Nordstrom Metric in the Special Case e2=m2, Phys. Letters 21, 423-4; Complete Analytic Extension of the Symmetry Axis of Kerr's Solution of Einstein's Equations, Phys. Rev. 141, 1242-7.
[So, we run these two together into the one reference: Carter (1966a), [the thesis being Carter (1966b)].
1057–11 Crumeyrolle, A. (1990) Orthogonal and Symplectic Clifford Alebras Abstract: On Spinor Strutures. Kluwer [ISBN 0-7923-0541-8].
1059–10 We also need to add the following two new references:
Dyson, F.J. (1979) Time Without End: Physics and Biology in an Open Universe, Rev. Mod. Phys, 51, 447-460
1062+26 Gibbons, G.W. and Hawking, S.W. (1977) Phys. Rev. D15, 2738-51.
1066+13 …Mass eigenstates in twistor theory,
1066+27 Hollands…Local Wick Polynomials and Time Ordered
1066–1 Huang, K. (1952) On the Zitterbewegung of the Dirac Electron. Am. J. Phys. 20(8), 479-484.
1068+14,15,16 Delete this reference to “Kamberov, G. et al. (2002)…”, since it appears already (correctly) above in 1068+7,8,9.
1070 This page needs some sorting out. The Lasenby, Lasenby and Doran paper appears twice, in different places on this page. The second one should be deleted. Also Linda should be Linde (same author as the one below it). Please make sure the alphabetical ordering is sorted out.
Moreover, on this page, we need:
1070+1,2,3 Lange, A. E. et al. (2001). First Estimations of Cosmological Parameters from BOOMERanG, Phys. Rev. D63, 042001. [astro-ph/0005004]
1070–23 Lemaître, G. (1933) L’univers en expansion Ann. Soc. Sci. Bruxelles I
A53, 51-85 (see p. 82).
1071+20,21 and 1071+5,6 The reference to “McLennan, J. A., Jr. (1956)” appears twice! Please remove one of these, according to editorial conventions with regard to “Mc”.
1071+19 Majorana… Teoria asimmetrica dell’ elettrone e del positrone. …
1073+1 Nayfeh, Ali H. (1993). Methods of Normal Forms, John Wiley & Sons, New York.
1073+8 Netterfield, C.B. et al. (2002) A measurement by BOOMERANG of multiple peaks in the angular power spectrum of the cosmic microwave background Astrophys.J. 571, 604-614 [astro-ph/0104460].
1073+10 Newman, R. P. A. C. (1993) Please put this reference correctly in ALPHABETICAL ORDER; also, immediately preceding this reference (and after “Newman, E. T.”, we need:
1073+? Newman, M.H.A. (1942) On a string problem of Dirac, J. Lond. Math. Soc. 17, 173-7.
1073-24…October 2003. http://alphaserv3.aei.mpg.de/events/stringloop.html.
[The link has apparently changed]
1073-8 Page, D. N. (1995) Sensible…
1073-6 Page, D. N. (1987) Geometrical…
1075-16 Penrose, R. (1988)… Delete this reference. It is the same as the one just below it, namely “Penrose, R. (1988a) Topological…”, and only the “(1988a) Topological…” version should be retained.
1077+9 Petiau, G. (1936) Contribution à la théorie des équations d’onde corpusculaires.
1078–13 Sachs, R. (1962) Asymptotic… Delete this reference. It is the same as the one a few lines below it, namely “1078–5 Sachs, R. K. (1962b) Asymptotic……”, and only the “Sachs, R. K. (1962b)” version should be retained.
1079–26 Schrödinger, E. (1952) Science and Humanism: Physics in Our Time. (Cambridge Univ. Press, Cambridge).
1080+24 Simmonds, J.G. and Mann, J.E. (1998). A First Look at Perturbation Theory, Dover Publications, New York.
1081+13 Steenrod, N. E. (1951) The Topology of …
1083+11 Trèves F. (1967) Topological Vector Spaces, Distributions and Kernels, (Academic Press, New York).
1084+17 Weyl, H. (1929a) Z.Phys. … Delete this reference. It is the same as the one immediately below it, namely (with the removal of “b”)
1084+18 Weyl, H. (1929). Elektron…[so retain this one, without the “b”]
1085–22,21,20 Woodin, W. H. (2001) The Continuum Hypothesis Part I Notices Amer. Math. Soc. 48, 567-76, Part II, ibid. 681-90; also (Part I) online at http://www.ams.org/notices/200106/fea-woodin.pdf and (Part II) ditto, but with /200107/.
1086 asymptotic freedom 679
1088 confinement 679
1089–21 … FAPP (‘for all practical purposes’) 803
1090+7 … Gross, David 679
1092–19 Kirschner, Robert 774
1094+8 … Perlmutter, Saul 774
1094+27 O(e) order symbol 307, 317
1099–4 … Politzer, David 679
1096–16 … Schmidt, Brian 774
1099–23 … Wilczek, Frank 679
Note concerning index page-numbering: Pages in the index entries referring to Chapter 29 need to be incremented by 1
Note: The following fonts have been used in the above
Times New Roman
Symbol
Nimbus Script
Lucida Sans Unicode
Arial
Corrections/criticisms by reviewers (minor amendment)
With regard to the K0 and`K0 mixing (p.650), he raises a point of significance, and although what I say in the book is technically correct (as I have been assured by Bruce Winstein, an acknowledged expert in weak-interaction physics) it is open to some confusion of interpretation, owing to the brevity of my description. Where I said that the “‘rotation’ between the strong-… and weak-interaction basis states takes place through an (abstract) angle referred to as the Cabibbo angle…”, I should perhaps have said “‘rotation’ between the strong-… and weak-interaction basis states is governed, generally, by a number of ‘rotation’ parameters. The most significant of these (not the important one for K0 and`K0 mixing, but the one most relevant to the decay) is an (abstract) angle referred to as the Cabibbo angle…”.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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============================================================================================
Corrections/criticisms by reviewers
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Some reviewers have pointed out errors (or apparent errors) in early printings of The Road to Reality. As I believe that it is important that the book be free of serious error, and also to be seen to be free of such error (as far as this is possible), I propose to use this location to point out where corrections to these printings are required (and where they are not required).
I shall also endeavour to make sure that the actual errors are indeed corrected in later printings.
Towards the very end of his review, entitled “Treks of Imagination”, in the February 2005 issue of Science, Frank Wilczek refers to what he calls “several serious blunders” in my descriptions of particle physics and quantum field theory, saying that:
“The Cabibbo angle does not govern the mixing of K0 and`K0 to make the long- and short-lived Ks. There are not alternative directions of electroweak symmetry breaking. And no associated disorder arises at that symmetry-breaking transition, any more than at the analogous transition in ordinary superconductors.”
With regard to the K0 and`K0 mixing, he is certainly correct, and this resulted from a carelessness in my description. Where I said that the “‘rotation’ between the strong-… and weak-interaction basis states takes place through an (abstract) angle referred to as the Cabibbo angle…”, I should have said “‘rotation’ between the strong-… and weak-interaction basis states is governed, generally, by a number of ‘rotation’ parameters. The most significant of these (but not the important one for K0 and`K0 mixing) is an (abstract) angle referred to as the Cabibbo angle…”.
As for his comments concerning “alternative directions of electroweak symmetry breaking” and “no associated disorder arises…any more than at the analogous transition in ordinary superconductors.” I believe that Wilczek has misunderstood my point (which has important relevance to §28.3), and that his comments are not correct. There are indeed “alternative directions” in electroweak symmetry breaking, since the photon (g) can be rotated into various possible combinations of g, W+, W-, and Z0 via the U(2) (or SU(2)´U(1)) symmetry. The chosen “direction” of g is not, however, a direction in ordinary space, but in a fibre lying above each spacetime point—as described in §§15.7,8 and in §25.5. (No doubt my descriptions should have been more explicit here.) This kind of “directional freedom” is essential, for example, for the conventional discussion of cosmic topological defects (such as cosmic strings and monopoles, as described in §28.2). The point that I raise about “domains” arising is a somewhat different one from that usually discussed for topological defects, though equally valid, and it seems not to have been much discussed in the literature. And his comments on “associated disorder…in ordinary superconductors” are not appropriate here for at least one (and perhaps two) reasons. The U(1) group of electromagnetism, which is of importance for superconductors, does not rotate an abstract “direction” in this sense, but only a phase, owing to the 1-dimensionality and, in any case, when a superconductor is constructed, there would be ample time for any “domains” to get ironed out within the constraints provided by the speed of light. Neither of these points applies to the electroweak symmetry breaking that I am concerned with here.
Finally, I should briefly comment on a review by Jeff Forshaw, who again worried about my descriptions in relation to K0 and`K0 mixing, but this time by pointing out that this is not the only known example of CP-violation. He is correct here, as I was not aware of the latest experimental information. This error is now rectified in later printings. (See the corrections to and in the accompanying CORRECTIONS document). However, his assertion that I am mistaken in asserting that weak interactions only affect the left-handed part of particles, seems to be a misunderstanding on his part, and what I say in the text (in §§25.2,3) is in accordance with the conventional understanding (though more usually expressed as “V-A coupling”).
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Modified by Nick Iles
Corrections to 1st printing of The Road to Reality (with missing and incorrect fonts inserted)
INSTRUCTIONS
Correction changes are in RED; text to remain unchanged is in BLACK; descriptions of text changes in BLUE.
There are a few changes indicated by PINK letters; these are to indicate the involvement of special fonts,
such as “v” for the particular “rounded” italic l.c. “vee”. (For a correct example of this rounded “vee” font, see the
displayed equation at the bottom of p.193.) Where further explanation of a font is needed, this will be explained in
blue text.
Page and line numbering in GREEN where
428+16 …you are only perceiving about…
means that on page 428, on line 16 (where the counting starts just below the bold line across the top,
and includes the section heading “18.5 The celestial…”) the mis-spelled “percieving” should be corrected to “perceiving”.
121–13 …convergence; see §4.4. See
tells us that the section indicated as “§2.5” in end-note 6.10 should actually be “§4.4”. The absence of dots (…) at the end indicates that we have reached the end of the line. Note that page numbers with a minus sign refer to counting up from the bottom of the page, such as in the following, where the red passage is to be inserted:
120–17 been infinity there and it is zero elsewhere. Yet the delta…
The absence of dots (…) at the beginning indicates that we start at the beginning of the line. The counting from the bottom of the page is to include any exercise footnote (but not the continuous separating line), and also any figure caption. In particular, this applies to the following, where there is a correction to the connecting dots and a suffix:
802–3 D = p1|1ñá1| + p2|2ñá2| + … + pn|nñán|,
so the suffix “3” ought to be an italic “n” and the three dots should be at the level of ordinary full stops (periods). Note that a single line of displayed equation counts as one line, whether or not there is more than one line in the equation as displayed in a fraction, whereas the blank lines below or above do not count. For an example:
113+20 …(see Exercise [6.1] and therefore…
As another example:
720–21 …seem to be a fair description of the history…
Here, the RED in “…description of the…” refers to the fact that the plural “s” should be removed in “…descriptions of the…”
As mentioned above, the lines of text in a figure caption are also counted, either when counting from the bottom, as applied within the caption to Fig. 2.20
44–8 …of a spherical triangle, on a sphere of radius R, with angles
or from the top, such as:
90+9,10 …equals opposite over adjacent’. One should also note that tan-1…
________________________________________________________________
____________________________________________________________
_____________________________________________________________
xxiii–13 Gerard ’t Hooft, Paul …
(Note that the apostrophe appears before the “t”, and that it is the awkward way around for many programs. In other words it must be
’
so we have
’t Hooft
Moreover, there is a space between “t” and “H”.
xxvi+8 cosq, eiq, or…
(the “i” in the exponent should not be italic)
xxvi+15 …written with lightface italic
xxvi–15 …should be clear from the text.
xxviii–14 …(see Chapters 25, 30). This
3–16 …terrible Daemon
(for consistency with lower down the page)
4+11 father and his father’s father before him and his father’s father’s father before that. He
4–1 …why the Gods had…
(for consistency with earlier pages)
_________________________________________________________
CHAPTER 1
11–20 …provide geometries…
11–12 …a century and a half after Pythagoras. Plato
15+3 …and 1042 (and caps 300, 300, 200, 200; see Note 4.1).
21–18 …many readers that,…
22+12 …§§34.2,5,9 par-
23–25 …ratios 24 : 27 : 30 : 32 : 36 : 40 : 45 : 48;
23–7 integer solutions to the…
(where “positive” is to be deleted)
24–1 Penrose (1989, 1994, 1997a, 1997b).
(so, “1996” is to be deleted)
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CHAPTER 2
35+3 bolic triangle illustrated. …
37–18 …they provide ways of visualizing hyperbolic …
38–11 The entire line “Though less appealing……viewpoint!” should have been deleted, so the text should read:
…artistry is still evident.)
There is a more directly…
43–16 …hyperbolic geometry’s physical
(i.e. italics here)
44+1 …known one due, in 1603, to Thomas…
44+12 …Chapter 4 and §18.4, Fig. 18.9), and it indicates…
44–8 …of a spherical triangle, on a sphere of radius R, with angles
45+3 geometer Nicolai Ivanovich…
(so, delete “artillery man”)
47–1 …compared with Fig. 2.11).
49–21 …Thomas (1939). Compare also Schutz (1997), who gives a nice axiomatic account of Minkowski’s 4-dimensional spacetime geometry (§17.8, §18.1).
49–11 …= a (see Note 1.2). Moreover…
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CHAPTER 3
50+11 Fauvel and Gray (1987), Gray (1979).
50+12 …half-plane’ representation (with metric form (dx2 + dy2)/y2; see §14.7). Its constant negative curvature was actually noted by Riemann (1854)!
50+17 …should be clear from the text.
50–5 …MAXIMA), Halverson……DASI), and Bennett et al. (2003) (concerning WMAP).
56–17 …natural number, or rational greater than 1, e.g.
57+19 …numbers that do turn out…
57–14 …Greek ‘dactylos’) was
60–7 …later discussions; e.g. §§31.1,6…
68–6 …discussed in §§2.4,5. On the…
69+15 …Notes 1.2 and 2.4. In this…
69–20 …Russell (1903), Chap. 4.
69–18 …Stachel (1995).
69–10 collection of all finite ordinals.
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CHAPTER 4
72–3 …this. (Alternatively, can you check it by multiplying both sides by (c - id)?)
73+18 …in the next, I shall endeavour…
73–7 …Chapters 21-23, 26, and 31-33). For…
73–6 some of the mathematical magic…
77–20 the ‘answer’ to the sum of…
77–2 …successively up to 4 terms…
77–1 …this expression, in a formal algebraic way?
78–1 …, 1 + x2 + x4, 1 + x2 + x4 + x6 (dashed lines) of
79+17 …§§26.7,9 and §31.13). It is a very…
79–3 …five terms, similarly to before, together…
84–14 the entire black region.10
85–6 …for example, Hardy (1949).
85–4 …Priestley (2003)…
85–3 ham (1997), pp. 67, 264. (It’s just the added space. Then we need an additional Note, as follows.)
Section 4.5
4.10 In computer-produced pictures of the Mandelbrot set (such as Fig. 1.2), one cannot, of course, compute indefinitely to ensure that an apparently bounded sequence is actually bounded. It is usual to ‘cap’ the iteration at some suitable number of steps. Simply adopting a larger cap does not, however, necessarily improve the accurate appearance of a picture, because the filaments tend to get lost.
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CHAPTER 5
89+1 …to polar [r, q],
(i.e. delete “we have z = x + iy = reiq,”)
90+9,10 …equals opposite over adjacent’. One should also note that tan-1…
91–8 …as was indicated in §3.4, §4.1. First, the…
97–11 …be added to log w, which
102–8 5.5 Roger Cotes (1714) had the equivalent…
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CHAPTER 6
113+20 …(see Exercise [6.1] and therefore…
113–1 …given towards the end of §6.5.
115+10 …given immediately above this one is
117+3 (b), where the area bounded…
117+4 bottom curve is the difference,
120–17 been infinity there and it is zero elsewhere. Yet the delta…
121–13 …convergence; see §4.4. See
_________________________________________________________
CHAPTER 7
123–11 …theory; see §§31.5,13,14).
128–1 justification. Hint: Look at the origin-shifted Cauchy formula.
129–3 …Use Exercise [7.4], finding the poles…
131–13 will fix the continuation provided that the interiors…
131–11 out consistently, the resulting function is uniquely…
132–5 of §6.3, Fig. 6.7, which suddenly…
134+6 The upper index v on the first integral sign (“∫”) should be a rounded italic lower-case v (“vee”). For a correct example of this font, see the displayed equation at the bottom of p.193. (The lower index on the integral sign (“∫”), here, is correctly an ordinary italic lower case u.)
Then: …f(z)dz, where p(u) is…
(so the “d” is upright, the “z” is italic, and the parenthesis should be removed
134+7 path and p(v) is its
(so this is a rounded italic lower-case “vee”).
134–12 …not intended here is simply any ‘connected open
134–10 …f is defined, which in general need not be necessarily…
134–9 connected (though here it is taken to be open).
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CHAPTER 8
144 Fig. 8.7 There are several font errors in Fig. 8.7, including in the top line, where what looks like an italic “ell” or “one” should be an ordinary upright numeral “one”.
147–2 hyperbolic plane, in the conformal representation of Fig. 2.12, with…
148–5 …(see Note 8.2), bounded…
150+6 w = ½ (z +…
In this displayed formula, the “one-half” should not be written with a solidus (and normal-sized numerals?), as presently printed, but should be like the “standard-fraction” “one-half” that is presently correctly printed in the first line of the caption to Fig. 8.15 near the top of the page. The remainder of the displayed formula is correctly set.
152+2 hypersurface; see Note 27.36.
_________________________________________________________
CHAPTER 9
156–2,1 There is a bad break at the end of the second line up. The “one-third” symbol should appear on the next line, attached to the “cos” that follows it.
159–13 gence for F- is A and……for F+ is B-1 . (b) The…
163+6 The “one-half” in the displayed formula (though correctly displayed with a normal fraction bar rather than a solidus) should have small (not normal-sized) numerals.
163–2 As with the previous correction, this formula should be written as
t = tan ½c.
but where the “½” should be an ordinary-looking (small) fraction, with an ordinary fraction bar, not a solidus.
166+8 …Thus zr = eirc/N = eicp; so…
(Thus, there is an italic l.c. letter “r” missing in the exponent, where indicated.)
166–8 8,7c). This…
171+11 Fig. 8.8). In terms…
173–2 …actual hyperfunction, on g is (f, g) modulo…
178–6 …the ‘product non-existence problem’ to a non-uniqueness…
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CHAPTER 10
191+4 dF of a scalar F… (The “d” should be upright.)
197+9 …See Woodhouse (1991).
_________________________________________________________
CHAPTER 11
211+14 …(see §§22.8–11, §§23.4,5, §§24.6,7,…
215+7 …+QÙP if p, or q, or both,…
215–1 …that PÙP =0, if p is odd…
(Bold italic (´2).)
216+4 …not ‘spynor’. See Note 24.12.
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CHAPTER 12
220–15 [12.17].
223+21 …d(fg) = fdg + gdf, da =…
(This just means “reduce space” in these two places.)
223–7 …scalar product of a with x; see §10.4), where,
223–4 a · (Fx) = F (a · x).
(All this just means that there should be a (just significant) space between the “F” and the “(a · x)”, so that it doesn’t look as though F is acting as a function on
(a · x), rather than just multiplying it, the latter being what is intended.)
226+15 …direction of the rth (local) coordinate axis’.
226+18 …axes (see Fig. 12.9a).
226–4 along the rth (local) coordinate axis’…
226–1 the (local) coordinate axes except for the (local) xr-axis.
227+7 …Fig. 12.9b).[12.5]
231–1 …coordinates (r,q). (See §5.1.) Hence prove G…
(Italic “r”. Full stop inside second bracket.)
232–1 …result of Exercise [12.11], prove…
234+17 …infinite sequence (so that points of the sequence get
237–17 see §5.3. However, I shall not…
237–5 …is the scalar product (§12.3), which combines
240–11 which can have q lower and p upper indices…
240–7 …multilinear19) of q vectors A,…, C and p covectors…
242 Fig. 12.18 There should be a full stop just before the first occurrence of the word “Antisymmetric” (in the third line of diagrammatic symbols). A bit more serious is to correct two errors in line 5 of the caption, as follows.
242+5 …vectors (here p = 3 and q = 2). Symmetric and
246+4 …(1963). (In that book, what…
246–19 quantum gravity (see §§31.5,14, §32.2, §§33.11,12) and…
246–14 12.16 See Penrose (1968a), pp…
246–12 12.17 See Penrose (1968a); Penrose and Rindler…
246–6 …Appendix; Penrose (1971b); Cvitan…
__________________________________________________________
CHAPTER 13
248–6,5 …reflective symmetries are given, in C, by
C (complex conjugation), Ci, – C, and – Ci.
These four “capital Cs” should be sans-serif (lightface), as on p.249 (and in the actual Fig. 13.1).
249–9 total number of distinct elements…
253–2 …finite subgroup S of a finite group G, is the order
254+7 (§31.14), while various…
254–9 …vector space V, we have, defining
The letter V is (correctly) sans-serif; but it should be bold sans-serif.
255 Fig. 13.4 There is a l.c. italic letter v in the diagram, but this should be the rounded font l.c. italic “vee” (as in the corrections to 134+6 and 134+7 above; see the displayed equation at the bottom of p.193.).
254+12 …transformation of V is a transformation that takes V to
Just as in 254–9, the letter V should be bold sans-serif (´2).
255–5 §13.8 (see also §13.1).
259 Fig. 13.6 The third line in the actual figure, one finds the very strange
i.e. x ↦ Tx
which is in quite a wrong collection of fonts (& sizes etc.). This should be:
i.e. x ↦ Tx
which is the same as the formula correctly printed at the beginning of the second line of the figure caption of Fig. 13.6.
259+4 of a non-zero vector v such that [13.7]
This v, again, is the rounded l.c. italic “vee” used in 134+6 and 134+7 above; here it is a bold rounded “vee”; see the displayed equation at the bottom of p.193.
259–1 …= T-1T. Hint: see Fig. 13.8.
262–4 …Establish this expression. Hint: Use…
263+3 The displayed equation is very confusing as written because of the use of normal-type-sized fractions with a solidus (which would seem to imply that the A2, A3, etc. are in the denominators (which they are not)! This is easily remedied by putting all the fractions (presently written “1/2”, “1/6”, “1/24”) in the standard fraction notation with numerators and denominators in small type and with a standard horizontal fraction bar between them (as in line 8 or 14 or 27 of page xvii).
263–16 …characterizes v as an eigenvector
Again, v is the rounded italic “vee” of 134+6 and 134+7, but it’s the bold rounded “vee” as in 259+4.
263–1 them out for n = 2 and n = 3.
265+4 …unique T sending each ej to
(so the italic lightface suffix should be a “jay”)
265+9 …components of ej are
(so the bold “e” should be italic)
265+20 find a basis (e1, e1,…, en) for…
(so each bold “e” should be italic)
266–17 It turns out that quantum …
(so no comma)
272 Fig. 13.13 There are three “barred arrows” “↦” used in the actual figure, which are unclear. The font used in the first line of the figure caption is much clearer here. (See 259 Fig. 13.6 above, for this “↦” symbol.)
272–6 V (with T depicted as a white triangle)…
Just as in 254+12 and 254–9, the letter V should be bold sans-serif.
272–2 Qabcde ↦ Sa¢a Sb¢b Tcc¢ Tdd¢ Tee¢ Qa¢b¢c¢d¢e¢.
First, there is sign “µ” (proportionality sign) placed in error here; this should be the “barred arrow” “↦” of 272 Fig. 13.13 (see 259 Fig. 13.6 above), as shown. Second, there are serious spacing errors, which are corrected in the red expression above.
274+7 …(See §12.6, Fig. 12.12 for
275–2 …(i.e. gT = g) gab = gba, and gabgbc =…
(so the superior “T” is “sans-serif” and the “ab” should be superior, not inferior)
276–4 …vector space V leaves V pointwise fixed,
(so the V (´2) should be bold sans-serif)
277+17 §13.7). In index-notation…
277–16 there may be some terms…
278+5 for all x ¹ 0. In index…
(so the “x” should be upright bold)
279–3 …form ea…c, i.e. ea…c Tap…Tcr =
(the indices on “Tap…Tcr” should be staggered EXACTLY as shown).
281+12 §31.13, §32.3). But…
282–8 …separate vector space`V . This viewpoint is
This is a bold CAPITAL sans-serif “vee” with a bar over it (not to be confused with the rounded “vee-bar” occurring on the line above).
287+9 …bilinear form s), in the sense that
287+10 sab Tac Tbd = scd , i.e. T…
(staggered EXACTLY as shown)
287–5 …linear transformations Tab that preserve both the
(staggered EXACTLY as shown)
287–2 `Ta¢b¢ Tab ha¢a = hb¢b,
(staggered EXACTLY as shown)
289+18 dimensions m(m + 2), m(2m + 1), m(2m + 1), and m(2m – 1), as
291+9 Hawkins (2000),
section 6.2. However…
(I think that using the “§”
sign here confuses with the sections of the present book.)
291–12 and gab, so va = sabvb, va = sabvb (see §13.8); but,…
There are 4 instances of “v”, which should all be rounded italic (as in 134+6 and 134+7).
291–3 …Levi-Civita’s e… in terms of …
these dots on “e” should be inferior.
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CHAPTER 14
292+13 §22.3), vector…
297–11 …sections (§14.7, §§15.7,8, §19.4). We can see
300–1 …(see Fig. 12.17). The (anti)symmetry factors give the ‘12’.
301+17,18 Bad spacing. Move the word “form”, at end of the lower line, to the line above, so we have:
…noted in §10.2, in the form
¶2f/¶x¶y = ¶2f/¶y¶x) that coordinate…
304–9 …= 1, i.e. taÑau = 1
(so “Ñ” is lightface upright)
305+8 parameters l and m. Choose…
305+9 an affine distance l = e along l from p…
305+16 …M to satisfy LÑM = 0 along l). Now,…
Here I use the pink notation “LÑ” to signal the fact that “L” is to be placed immediately beneath the “Ñ”, and not as at suffix level (see the placing of “t” in three of the displayed equations on page 304). All three letters Ñ, L, and M, are in bold type, with Ñ upright and with L and M italic.
305+18 from q by an affine distance m = e along m¢. However,…
305+20 …out from r an affine distance l = e along l¢…
Note: in each of 305+8, 305+9, 305+20, we had the wrong font for l. These should all be an ordinary italic l.c. “ell”.
305–1 …according to Ñ.
This is a bold Ñ. (There are two more of these in the figure itself, tilted at an angle, which should each also be bold.)
314+13 tensor (i.e. a vector) h. We wish to…
314+14 connection Ñ. The required…
This is a bold Ñ.
315+11 (Ñ being torsion-free);…
This is a bold Ñ.
315+13 The first symbol of this displayed equation should not be “Ñ”, with a “x” beneath it, as shown, but it should be a (bold upright) “pound symbol” £, with a “x” (a bold italic “xi”) beneath it. In other words, simply replace “Ñ” by “£” in the initial symbol in the expression (but leave the rest of the expression unchanged, so the remaining instances of “Ñ” should be left as they are).
315–3 …formula of Exercise [14.9]?
323–15 Penrose 1968a; Penrose and …
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CHAPTER 15
329+15 …confusing, so to get a better…
329+18 …origin 0 marked). Such a
330–20 …S1 – p. The distinction
330–16 different points of S1; then we…
332–5 …function at each point
334+8 …William Clifford. I
(delete date)
334–1 …Riemann sphere S2. Accordingly, we can
341+3 …manifolds (see §14.8,
348+16 …circumnavigated (Fig. 15.16).
350+5 …the explicit connection Ñ = ¶/¶z – A,…
This is a bold Ñ.
350+7 if A = ik`z (with suitable k),…
Thus there should be bar over the “z” (but there should be no space between the “k” and the “`z”, which is an artefact of the word system that I am using).
355+7 1.58 See Penrose (1987a); Penrose and…
355–11 onov and Anandan 1987; Shankar (1994); also, Woodhouse 1991…
355–6 …(co)tangent vectors so that Ña can
This is a lightface (upright) Ñ.
355–4 …case of XÑ, we can use the commutator
As in 305+16, I just use the pink notation “XÑ” to signal that “X” is placed immediately beneath the “Ñ”. The change here is simply that the “X” is to be a (bold italic small type) CAPITAL “ex”).
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CHAPTER 16
360+20 every distance 1, 2, 3,…, q+1 can be…
364+10,11…the idea of a 1–1 (i.e. a ‘one-to-one’) correspondence.4 We say…
372–1 This is the wrong icon here. I should be the little character with a straight mouth, and finger in the mouth, looking up to the left. For an example, see bottom of page 374.
377+23 …derive falsehoods) and are not too limited, then we…
381+10,11…Pitkäenen (1995).
(so delete “and applications…stuff”)
381+17…than is worth while for our
(Fowler says two words here!)
381–10 …(1989), Chap. 2, for
381–6 …Penrose (1989, 1994, 1997a).
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CHAPTER 17
387+4 Attractor (§27.11)?
387–7 …spacetime G is a fibre bundle…
390–5 …of this (one part in 103).
394+12 …ground is not executing an inertial
394+19,20 …insect or astronaut are, according to Newton, not inertial.
400+9 …The method he proposed (in 1638),
400+11 …crude. But he had no…
403 Fig. 17.11 At the far left of the figure, the pair of words “Time like” should be joined into the one word “Timelike”
404–13 …a preferred time-direction at each event p…
(note italics)
404–12,11 …element), and this preferred time-direction gives us…
406 Fig. 17.15 In (a), the expression “t = òds” should be moved a little to the left (and up) so as to be closer to the point on the “curly bracket”. Likewise (and more
importantly) the expression “ds=0” should be moved much closer to the point on the “curly bracket”.
407–19 …outside the constraints imposed by the
408–16 …tic of the metric of M.
Here, the “M” stands for a shell capital “em” (of the same general style as “ℍ” or “ℕ”; see 2 lines down (408–14)).
410–23 17.4 See Trautman (1965); Arnol’d (1978); Penrose (1968a), p.126.
410–20 …See also Penrose (1987c), p.49.
410–16 …in aeroplanes in parabolic flight!
410–13 …Query 30; Penrose (1987c), p.23.
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CHAPTER 18
414–5 sections of ℂE4. We can single out any one of these…
(Note: the “E” above is meant to be in “shell” font, like the “ℂ”, and it is correctly printed in the book, but I do not have this font for “E” on my PC! So please ignore the discrepancy here.)
414–3 …chosen real section pointwise…
414–2 …Find C explicitly for……Think of how C is to act on.
Each “C”, here should be sans-serif lightface (´2), as we see 2 lines above (414–4).
414–1 w, x, h, and z. Modify standard complex conjugation with signs, in the cases…
415–6 plex-dimensional group—clearly, because…
416–7 …Lie algebra—see §13.6—and check that…
418+1 …new real section, the Lorentzian…
418 Fig. 18.4 The two instances of the superior sign “┴” should be upright, not italic (but not important).
419+5 …(see Fig. 17.13).[18.5] However…
419+11 …develop the full details of…
419+12 relativity nor to see why,…
426–20 (last line of main text) …unit pseudo-radius—§§2.4,6—
427+7 …= (er - e-r) ¸
(I think the solidus is very confusing here, and the “¸” would be much clearer.)
428+16 …you are only perceiving about…
429+15 very small (about 20 seconds of arc, in fact). Neverthe-
430+9 angles (§2.11, Fig. 22.16) by the formula z =…
432+6 …great many constituent particle
432+18 The momentum p of our particle…
(i.e. delete “mass”)
432+24 …recall from §§17.2,3 that a relativity…
434+4 E = mc2,
434+5 …system and m is its total mass,…
438–5 Hartle (2002). For an axiomatic goemetric approach, see Schutz (1997).
439+6 …of x Ù p are x[ipj] = …
(second upper index letter is italic “jay”)
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CHAPTER 19
440+12 …a crucial ingredient of
442+11 of these equations, such as…
442–8 …namely x0 = t, x1 = x, x2 = y, x3 = z
(Indices should be superior (´4).)
444 Fig. 19.1 In the second line of diagrammatic formulae, the quantity “*Fab” appears. The second of these (but not the first) should be “Fab” (so the indices should be raised). Also, the final mathematical expression in the figure “Jabc” should be “*Jabc” (so we need an asterisk attached just before the “J” symbol).
445–8 …§§33.6,8,11—see §18.3, exercise [18.5](ii)), it is…
452–13 Earth (direct route AC) and
452–2 …Figs. 15.16 and 15.19; the basic bundle
453+9 the same type (see §23.7). In particular…
453–18 …often being referred to as y—
(so the italic Greek “psi” should be lower case here)
453–16 ment y ↦ eiqy is still referred…
(again the italic Greek “psi” (´2) should be lower case)
454 Fig. 19.6b The script letter “R” should appear towards the upper right of the picture.
455+8 electromagnetism is the y ↦ eiqy symmetry)…
(lower case italic Greek “psi” (´2) again)
457+7 …= 0; see §14.7), which
457–17 Tab satisfies ÑaTab = 0.[19.14]…
458–9 time-curvature corrections…
462–9 L is constant (and Ñg =0)…
This should be a bold upright “Ñ”.
465+6 integrated energy densities, as measured…
469–10 19.8 See Stachel (1995), p. 353-64. Among the many excellent texts on general relativity are Synge (1960) Weinberg (1972); Misner, Thorne and Wheeler (1973); Wald (1984); Ludvigsen (1999); Rindler (1977, 2001); Schutz (2003); and Hartle (2003).
470+815 …density of spin (see §22.8); see Kibble (1961), Sciama (1962), and the accounts by Trautman (1972, 1973).
470–7,8 19.15 Bondi (1960), Bondi et al. (1962); Sachs (1961, 1962a). This work was partially anticipated by Trautman (1958).
(Basically this is just transposing the two sets of references.)
470–2 …Penrose and Rindler (1986) and §32.3,…
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CHAPTER 20
472–1 In the equation at the bottom, the denominators are displaced too far to the right, and the expressions would look better if the “d” at the bottom is almost directly under the “d” at the top (with perhaps the lower ones displaced only very very slightly to the right of the upper ones).
474–5 space, given in §14.7, §17.9, and…
476+7 In this displayed formula, the italic “q” at the bottom (in the denominator) should have a dot above it. (See the correctly printed formula at the bottom of the page; i.e. 476–2.)
477+10 Replace the full stop at the end of this displayed equation by a comma.
477+11 written as {H, } in §20.4. This provides a ‘flow’ on…
Note that “H” is the same script capital “aitch” as is found elsewhere on this page, such as twice in the line (477+10) just above this one. (Note the gap that correctly appears between the comma and the final curly bracket “}”.) Unfortunately, the notation in the figure itself uses the wrong kind of bracket. The incorrect “[H, ]” should be “{H, }”:
477 Fig. 20.5 The lettering on the right-hand side should be
Hamiltonian vector field {H, }
477–2 energy conservation. (See Note 27.36 for the term ‘hypersurface’.)
479+4 …mechanics (§22.13). The theory…
479+9 …glimpse of in §22.13. The Lagrangian theory
480–3 At the end of this line, there is a mathematical expression containing two instances of “q” with a “dot” directly above it. Writing this dotted “q” as “q”, we find, following the “½” symbol, “mh2q2(h2 – q2)-1 +” The second instance of the dotted “q” is incorrect, and there should be no dot. Thus, this part of the formula should read, following the (correct) “½” symbol,
mh2q2(h2 – q2)-1 +
485+4 …lies on a (2N – 1)-dimensional surface….
485+7 …of H, is (2N – 2)-dimensional, see Fig. 20.8…
485+10 …resulting (2N – 2)-manifold is again…
488+10 …Ö(-det g). The action…
so replace the present “gij” by “g”, (with no indices) which is upright bold
492+15 …such non-holonomicity does not occur…
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CHAPTER 21
493+7 general relativity. In fact, what quantum theory…
499+14 In this displayed equation, the symbols inside the bracket “H(……)”, which are expressed as fractional quantities, should be slightly re-expressed, with the “iℏ” (´2) not being placed in the numerator, but as an initial multiplying factor—as in the displayed formula (in 499–17) 5 lines below this one.
503–4 …(see §25.4, Fig. 25.10).
504+21 …Although the spots appear at the
508–4 Ae-B…… (i.e. simply remove the “=” sign, and close up)
510–9 …direction of the momentum, where…
518–10 vector space W of wavefunctions……space ℙW of idealized
Upright sans-serif “W” (´2) should be bold (the shell “ℙ” being correctly printed).
523–3 …of Exercises [21.11], [21.13], and [21.14] to…
525–4 §§29.2,9.
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CHAPTER 22
528–11 have about the physical reality…
534–14 …to the Hilbert space H,
This upright sans-serif “H” should be bold
537+11 evolution |fñ ⇝ |fTñ = UT|fñ, with…
This arrow is the same “wiggly” one that we find in the displayed equation 3 lines above this one (537+8).
538+12 …’what is your position?’, etc.
539+2 …linear operator (as with the examples of position…
539+10 … One can ensure this (assuming normalizability of eigenvectors) by 539+18 harmonic oscillator (§22.13), etc.12 An…
539+24 and any pair of (normalizable) eigenvectors of…
542–6 operators are (thoroughly) degenerate.
We say that Q is degenerate,…
(so a new paragraph should be started here)
543+16 …appropriate projector E, namely the…
This italic capital “E” should be bold.
544+11(i.e. 2nd line of main text) …, namely the factor whereby the ‘norm’…
547+9(i.e. last line of caption) …in Figs. 22.12 and 22.13.
550+7 coordinate (§§21.1,2) so also…
(just close gap)
550+11 rotations (§§13.6-8), i.e. the Lie…
(close gap and replace “,10”)
550–8 transformations of H (see §§13.6-8, §14.6).
553+6 this being the ‘squared length’ of the 3-vector…
554+3 space variables x. Then…
(just delete “of”)
554–6 …this expression from what has been said above.
555+2 Fig. 8.7). For the spin states…
556+12 …the length A¢C divided by the sphere’s…
559+8 …down to the sphere’s equa-
559–11 …matter which one), so q = 0, ¥ give |+ñ, |-ñ, respectively. Let us consider…
563+4 There is rather too much space between “sin q ” and “cos f ”, and then between “sin q ” and “sin f ”.
565–12 shall be seeing in §§27.7,10,11, §28.4, and especially §28.10.
567+19 …Fig 18.16). (In this section, c =1 is not assumed.)
571+22 …the matrix W = PQ (i.e. of…
bold upright “W”; bold upright “PQ”
574–8 …Geroch (1984), Deutch (2000).
575+15 22.12. See Dirac (1982a). Complex…
575+25 Robert Dicke (cf. 1981). It has some…
576+6 …more explicitly in §22.12.
576+12-15 …and somewhat more ‘physical’ (albeit more complicated) derivation of the angular momentum algebra is given in Shankar (1994).
22.23. See Penrose and…
So, delete the rest, namely: “; though in my opinion there is……this approach!”.
576+17 22.25 Witten (1959); Penrose (1960, 1968a); Geroch (1968, 1970); Penrose and Rindler (1984, 1986), O’Donnell (2003).
576+23 22.27 See Shankar (1994).
576+30 …see Swain (2004). See also Note 22.31.
576+31 …Penrose (1993, 1994); Zimba and…
577–9 …Dirac (1982a).
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CHAPTER 23
581+16 …the case of an actual quantum
581+19 …instead, which is stupendously larger!
581–13 This discrete model is not really such an absurdity…
591+14,15 answer, on those occasions, to a à measurement if just one or the other of us switches to this, so as not to violate (1) or (2). Yet…
591+16 …colleague might both happen to perform
594+9 Recall (§23.2) our entertaining of… fr th
594+16 …characteristically different from those of our…
594+18 two quite different such procedures!…
597+13 …opposite (see §24.8, Fig. 24.2). fr th
601–15 …(Recall Fig. 22.10) In…
603–1 …pictured in Figs 22.10, 22.13, 22.14), this…
607–5 …Schrödinger (1935); also…
607–3 …Jozsa and Linden (2002)
607–2 23.4. See Bell (1987). Perhaps the neatest and…
608+4 and B, D for the other. Highly relevant, also, are Gleason (1957) and Kochen and Specker (1967). See Redhead (1987).
608+5 23.5 See Bohm (19); Redhead (1987); Afriat (1999). For some recent extrordinary experimental confirmation of EPR effects, see Tittel et al.(1998).
608+6 For various specific examples relevant to this (Heywood and Redhead 1983; Stairs 1983), see Kochen and Specker (1967); Peres (1991, 1995); Conway and Kochen (2002); Penrose (1994), Section 5.3; Zimba and Penrose (1993).
608+7 23.7. See, for example, Hannabuss (1997). See Nielsen and Chuang (2000)…
608+15 …adopted in my (1994) book Shadows…
608+18 …Penrose (1989).
608–7 …Penrose (2002).
608–5 …Penrose (1998a).
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CHAPTER 24
611+15 …Chapters 21-23; but we shall see
618–3 …fermions (see §23.7) of Nature, namely…
620–10 = ((ð - iM)(ð + iM)
(Here I am merely using the symbol “ð” to stand in for the “¶-slash” symbol that correctly appears in the book, and there is to be no change to this symbol. The change is simply to put a “minus” sign in place of what is currently a “plus”, and a “plus” sign in place of what is currently a “minus”.)
620–8 (ð + iM)y = 0, i.e.
(Again, I use the symbol “ð” to stand in for the (correct) “¶-slash” in the book; the change is merely the removal of the “minus sign”.)
621+10 …Lorentz transformations.[24.7]) It came as
621–1 …Explain this. Hint: The equation given in Exercise [26.3] may inspire!
622–16 …is now explained, to the above explicit precision, from
623+5 …described in Fig. 22.10). Pauli…
624+13 …positron, however (and compare §25.2). Things are…
626+9,10 …appropriate; see Bailey et al. (1982).
626+17,18 …Dirac (1983).
(i.e. delete “Schrödinger’s……erotic!!”)
626+20 of the CPT theorem that…
The “CPT” should be sans-serif lightface.
626–15 Waerden (1933); Laporte and Uhlenbeck (1931); Penrose (1960); Penrose and Rindler (1984, 1986) O’Donnell (2003). In the 2-spinor notation of §22.8,…
626–6 …1984, pp.389,90).
626–1 Missing icon at the beginning should be the happy-faced “easy” one.
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CHAPTER 25
628+8 …period of about fifteen minutes. One of the
628+14 …from these beginnings of an understand-
634–19 …shown in Fig. 25.6. What is the significance…
638+16 …called C-invariant. The operation of
638–4,3,2 A very few observed physical effects are known to violate CP-invariance. The most long-standing example (a ‘non-ordinary’ weak process, first observed by Fitch and Cronin in 1964) is a particle decay seen to be non-invariant…
The first new (red) material starts the paragraph, and replaces the passage “It should be remarked……one physical process” (and note that “CP” is sans-serif lightface, upright). The second new (red) material replaces “that is known”.
639+1 …is the decay of the K0
639+14 …within the context of Feynman
646+8 of SU(3);[25.3] see also §§5.4,5).
646+12 …Gell-Mann christened ‘quarks’ (three types)…
650–20 certain ‘rotation angles’ between the basic…
652+13 …prevents the ‘colour-charged’ quarks…
652+15 …earlier work by ’t Hooft and Weinberg.23)…
(Note the way around the apostrophe should be, namely
’
See xxiii–13 above.)
653+12 Rindler (1984), pp. 221-23.
(delete reference to Zee)
653+14,15 25.4 See Infeld and van der Waerden (1933); Laporte and Uhlenbeck (1931); Penrose and Rindler (1984).
653+16 25.5 See Schrödinger (1930); Dirac (1982a); Huang (1949)…
653–9 Dirac (1982b). Pauli had…
654+10 …strong symmetry group SU(3), the full
(i.e. delete “/Z3”)
654+24 …theory; V. E. Barnes’s paper on the
654–14,13 …theory in 1954, although the basic…
(so, delete “(Phys. Rev. 96, 191-5)”)
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CHAPTER 26
661+22 …described in §§9.2,3,5, we can express…
656+4 …of that magnificent and imposing
656+15 …Veltman, and ’t Hooft, among others, has led us to a very
666–17 …recall from §21.2, §23.1, and Chapter 24,
666–12 (see §20.5). How do we construct…
666–2 earlier, and was made particularly explicit in §22.2. Here, we think of that
678+2 …coming to later—in §27.10—the electron’s measured
678–10 …Gerard ’t Hooft, who supplied…
678–7 …University of Utrecht, ’t Hooft
679+12 relevant at energies even remotely accessible…
679+13 a ‘quantized gravity’, the relevant extreme energies are…
680+15 was discussed in §4.3, where…
(“we” to be replaced by “was”)
684+15 (1987); Berry and Robbins (1997).
(Correct Berry reference and delete reference to Shankar and Berry phase)
684–10 …Schwinger’s competing approach…
684–8 …use the more intuitive picture…
685+5,6 26.20 See, for example, the classic Bjorken…
(delete: “This may be……‘old fashioned’;”)
685+15 26.23 See Note 25.6.
685+16 …‘S-matrix’ (due, basically, to Heisenberg and the highly original…
685–19 …discussed briefly in §33.1; see Connes
__________________________________________________________
CHAPTER 27
686–8 …bodies about a centre of force are
687–18 …confront later; see §§27.8,9, §28.8,
689+9 …asserts:
The end-note marker “2” does not belong here (line 9), but 5 lines down (line 14):
689+14 temperature.2 This is the expectation…
690+6 …new (§18.6, §20.4,§21.3). But…
707–5 …from ground level (§17.3, §18.6).
710–8 spherical polar coordinates (§22.11). Explain how…
712+12 to be a massive 3 ´ 106 M⊙ black hole…
713+9 detail in §28.8.
715+15 graining box is indeed stupendous, and black…
717+10 3 ´ 106 M⊙, which may well be…
718–17,16 very slight temperature deviations, at the tiny level of only a few parts in 105, isotropy is well supported.31
720+23 …would seem to be a fair description of the history of our
721–18 …points of S3 (see §2.7, §§15.4-6); it is hard to…
721–13 …in §§2.4-7 and §18.4. See Fig. 2.22a,b
724+20 described in Fig. 27.11). In Fig. 27.17, I have…
724+24 …depicted in Fig. 27.16c, the
724–7 …Minkowski 4-space of Fig. 27.16b explicitly, by taking…
728+6 …Recall (Fig. 27.10) that whereas maximum entropy is described, in the
728+9 ate, namely a black hole. With gravitation…
(primaily for spacing purposes)
728+11 …leads us to black holes. This is
728–18 in Fig. 27.13a, which is the time-reversal…
729+11 similarity to the actual universe…
730–4 that of Fig. 27.20d in the case L =0,…
(i.e. delete “comma”)
730–3 …case of 27.20c, would be…
731–11 …illustrated in Fig. 27.20c—or in the
731–4 Second Law, but it gives…
733+18 …the atmosphere. See §34.10.
733–8 27.22. See Penrose (1969a, 1998b); Belinskii…
733–7 27.23. See Penrose (1969a, 1998b).
733–3 27.25. See Israel (1967); Carter (1970); Hawking (1972); Robinson (1975).
734+6 …(as we shall see in §31.15, and see Note 27.26), there is…
734–26 …Greeks were right (Fig. 1.1) and the
734–22 n-manifold. Here, Tt is a spacelike 3-surface.
(Start new sentence with “Here,” and delete the following “a”. The “Tt” is unchanged from the existing text.)
734–18 grams’. I first made use of them in my Warsaw…
734–15,14,13 27.39 See Penrose (1964, 1965a); Carter (1966); Penrose and Rindler (1986), Chapter 9.
Note that “Hawking and Penrose (1996) make use……interpretation” is being deleted here.
734–12 …the Big Bang (or, rather, the Big Crunch) is indeed conformally…
________________________________________________________
CHAPTER 28
741–14 …outrageous nature. Indeed, one of the most
741–13 plausible models of galaxy formation—…
741–12 …support—proposes that they are largely ‘seeded’ by the supermassive…
741–11 that now reside at their centres.7…
744–10 §27.12, Fig. 27.18b. Look at the schematic…
746–1 Fig. 28.5a tells us that……at distant points u
Note that the full stop after “Fig. 28.5” is to be removed. The “q” at the end of the line should be “u”.
747+1 and v, both observed from…
To replace the l.c. italic “r”, we need the rounded l.c. italic “vee”, as in 134+7; see the displayed equation at the bottom of p.193.
747+3 because the points u and v (where we take S to be at the time of ‘decoupling’,
747+10 …of Fig. 28.5b. The spacelike 3-surface…
747+12 …pasts of u and v now do intersect…
Here the “v” is rounded l.c. italic “vee”, as in displayed equation at the bottom of p.193. (But “u” is ordinary italic l.c.)
747+15 temperatures at u and v can come about…
Again, the “v” is rounded, as displayed at the bottom of p.193.
748+17 time is given by constant…
748–6 …de Sitter space in Fig. 28.9a, and also for the portion
748–5 …steady-state model (the dotted
(delete “in Fig. 28.9b”)
749–8 space in Fig. 28.9c (where the top…
(delete semi-colon)
749–7 be identified) and in Fig. 28.9d, and…
749–3 in Fig. 28.7, retaining only…
754+9 …position; see §§30.10–14. Or must we resign…
757–5 …by Robert Dicke in 1957 and Brandon Carter in 1973,24 when
761+2 Exercise [27.19]). However, it is…
761+14 …in his remarkable book The Life…
762–15 as a result of an inflation of that…
764–12 the curve (c)(b), rather than the observed (d)(b), inflation or no inflation!
(No space at all between “(c)” and “(b)” or between “(d)” and “(b)”.)
765–15 …illustrated in Fig. 17.9a (and the fact that
765–13 …contrasted with Fig. 17.9b, in which
765–1 …Why not? Hint: Explain Note 28.34.
768+22 represented in Fig. 27.21. As a result, there would…
770–1 collapse to black holes (or Big Crunch).
771–5 …positive-frequency condition §§9.3,5), but it…
772+1 …‘wrong signature’; see §13.8, and §18.2. I do not see
(delete “§§9.3,5,”)
772–17 What is this bearing of…
775–17,16 by a ‘horizon scale’ at decoupling…and imagine points u and v moved around on the decoupling surface until…
775–15 this is the ‘horizon scale’).45 It is at this…
775–12 decoupling, and it is here…
776+11 is not small, and seems to be reasonably clear. The…
776+24 mainly to Vahe Gurzadyan et al. (1992,…
778+1 …(see Fig. 27.21) as unacceptable…
778+2 …anthropic principle (§§28.4,6), although, as we
779+19 28.7. For a promising-looking theory of this kind, see Silk and Rees (1998). See Haehnelt (2003) for…
779–4 28.16. See Barbour (1989, 2001a, 2001b); Barbour, Foster, and O’Murchadha (2002); Sciama (1959); Smolin (2002). An example…
780+1,2,3 …issues, however, which are argued to support the case that Einstein’s theory is ‘Machian’ after all. See Barbour (2004); Barbour et al. (2002); Raine (1975).
780+5 account: Livio (2000).
780+9 …the Second Law—thermal processes…
780+10 …universe out. See Misner (1969).
(delete “for the original paper”)
780+18 …of Hoyle’s Cambridge lectures
780+22 28.25. Roughly: the cube root of the age of the universe in Planck units is remarkably closely the square root of the ratio of the electric to gravitational attraction between a proton and an electron.
(i.e. replace “Dirac (1937)” by the above)
780–22 …Barrow and Tipler (1988) break this down…
780–21 28.28. For more information concerning this story, see Barrow and Tipler (1988); Smolin (1997), p.111. For the profound implications for the contents of our universe, see Hoyle et al. (1956); Burbidge et al. (1957).
780–18 …Prize essay of 1966 (see Penrose 1966, 1968), I put forward…
780–5 28.35. Penrose (1969a); see Penrose 1998b for a general overview…
780–4,3 …Newman 1998; Tod and Anguige (1999a,
1999b); Anguige (1999) A particularly…
781+5 …by Renate Loll and her collaborators suggests that…
__________________________________________________________
CHAPTER 29
786+4 completeness, independence, or…
787+11 normal, see §22.5) operator Q.…
787–15,14 …sum to the identity I on H:
…+ Er = I.
(Here the symbols “I” (´2) are to be italic bold; the “H” is upright bold sans-serif—similar to the two examples in 786–3.)
798+20 because my colleague’s decision as to…
801+4 …aimed at Earth and a spin ½(n - 1)
(Simply make the “n” italic; the “one-half” is correctly printed.)
802 Fig. 29.5 The final (fourth) mathematical quantity, on the top line of the figure, which resembles “Dsar” (but without the staggering of indices) is incorrect, and it should have just one inferior index “r” (i.e. no “a”), where the “D” should have a tilde “~” directly above it. This is correctly printed in the final line (802+5) of the caption (immediately following “used ‘there’ is”, and just before the final “=” sign). There is also a minor error in the third mathematical quantity on the top line of the figure, as the capital “D” should be in italics. This, also, is correctly printed in the final line of the caption, as the first mathematical quantity (just before the first “=” sign there) appearing in this line (802+5).
802–3 D = p1|1ñá1| + p2|2ñá2| + … + pn|nñán|,
so the suffix “3” ought to be an italic “n” and the three dots should be at the level of ordinary full stops (periods).
803+8 …Further, we cannot then deduce that the state is one of
804+9 …to these ideas later, in §§30.4,7,8,14.
805–5 ingly (if we discount the consciousness…
807+11 There is a slighly confusing error in the way that this displayed equation is written, and it should be slightly reorganized. To be fully consistent with the figures, we require it to be:
|Yñ = w|live catñ |perceiving live catñ + zïdead catñ |perceiving dead catñ
Moreover, we require a corresponding change to the final two lines (807–2,1) of the caption to Fig. 29.9:
Ö8 |Yñ = {|live catñ + |dead catñ} {ïperceiving live catñ + |perceiving dead catñ}
+ {|live catñ - |dead catñ} {ïperceiving live catñ - |perceiving dead catñ}
Thus, we must interchange the two curly-bracketed expressions in the top line, and similarly interchange the two curly-bracketed expressions in the second line. Note also that the “normalization factor” was given incorrectly as “2” on the far left here, whereas it should be “Ö8”. This also applies to the number “2” at the far left of Fig. 29.9 itself, which is also incorrect and should be “Ö8”. Accordingly, the diagrammatic equation of Fig. 29.9 should start out, on the left, as:
Ö8 |Yñ = …
809+4 …find (see Exercise [29.12]) that the…
809+14 …also why the wondrous and
809–10 now asserts that this matrix rapidly becomes extremely closely diagonal
(Note that “the conclusion” is now deleted.)
812+7 …heroic) proposals for an “objective R” lie in their unnatural…
(Note that “R” is bold.)
812–15 29.3. See Everett (1957); Wheeler (1957); DeWitt and Graham (1973); Deutsch (2000).
(No gap in “DeWitt”)
812–5 For (d), see Gell-Mann and Hartle (1995); for (e), see Bohm and Hiley (1994). The list (a), (b), (c), (d) is representative only,…
813+20 Note 29.7. The projectors in the successive pairs (E,E¢), (F,F¢), …
(Observe that “§29.3” is being replaced by “Note 29.7”)
813–3,2,1 …2N-form S of §20.4, which featured…
so that fS can legitimately…
actually òfS = 1.
(Each “S” should be bold italic.)
814+19 29.19. See Hawking (1976b); Preskill…
814+23 …point out that my own position fundamentally…
814–11 29.24. See……of such thoughts in the context of ‘consistent histories’ (d).
814–3 …is to be |Añ + |Bñ + |Cñ and suppose that…
(Vertical bar “|” missing from “|Cñ”)
814–2 measured to be |Añ + |Bñ - |Cñ. This is possible because |Añ + |Bñ + |Cñ
(Vertical bar “|” missing from “|Bñ” and another from “|Cñ”))
__________________________________________________________
CHAPTER 30
818+8 …‘C’ stand respectively for space reflection
818+14 …, as described in §27.13 (and graph-
(i.e. delete full stop)
818+15 …illustrated in Fig. 27.21). Once such ‘generic’…
818+17 …the ‘anthropic’ perspective, §28.6, un-
820–14 …application of the R procedure. There is an
bold “R”
822–3 reversed one has a ‘teleological’ aspect…
824+14 …(Kerr geometry; see §27.10), we find
825+3 27.16c (this being strict if…
827+7 …information on ℐ- gets divided…
Superior “minus”, not “plus”.
827–5 …Kerr solution—see §§27.8,10) that represents
828+21 …(see Fig. 30.5a,b; see also Fig. 18.2). It is…
(delete “b”)
829–9 quantity that is smooth at the Euclidean…
831+12 …part of Fig. 30.7 lying above and…
831+13 indicated outer boundary line of the actual…
831–1 …expect exponential time-behaviour? (Consider eigenmodes of ¶/¶t.)
833–7 Fig. 30.8 (a) The spacetime K viewed globally…
833–5 …and into the past. (b) Any space
836–20 …negative energies can occur for real
838–16 process; see Fig. 30.13. It has commonly…
838–5 are any black holes of mass less…
840 Fig. 30.14 This figure looks rather small and, simply as far as the spacing goes, it could have been larger.
841+7 …As the horizon is crossed, nothing particular
841–6 one must in some way explain how this information…
843+14 …‘symmetrical’ case Fig. 30.17 involves
843+17 black hole had. I have never seen such…
843+18 …situations like those of Fig. 30.17 are
847 Fig. 30.20 For some reason, the reproduction of this figure is of very poor quality, though its meaning is clear enough.
847–4 10.5) that the ‘invariant’ way of thinking…
853+20 …each nucleus of about 10-51 Joules
The capital “J” is used on p.524, so a capital here would be consistent with that.
859+8 …the current ‘record
(i.e. delete “of”)
860–7 …field’ (see §29.6 and §30.14).
So, delete “full stop” (period) after “see”.
861–19 …applied to field quantities (see §§26.2,3, §26.9), that, if one
861–9 Planck scale of 10-33 cm would be…
864–8 …reduction. The ‘minimalist’ proposal for
864–1 …fall. Already the BOOMERanG,
Yes, the original capitalization is “BOOMERanG”!
865–3 …arrange that the initial photon is always…
866+24 30.11. See Kay (2000); Kay and Wald (1991); Kay, Radzikowski, and Wald (1996); Hollands and Wald (2001); Haag (1992)
866+27 …Wald (1984); Synge (1950); Kruskal (1960); Szekeres (1960).
866–10 …Thorne (1995a); Davies (2003).
866–7 …Penrose (1969a); Floyd and Penrose (1971).
866–1 …(1992), or, for a different outlook Kay (1998a, 1998b, 2000).
867+6 …with supersymmetry; see §31.2) de-
867+8 …treated by the S-matrix. See Hawking (1976b). It should be remarked that, as of 2004, Hawking has (regrettably, in my view) retracted his LOSS position, in favour of RETURN!
Note that “LOSS” and “RETURN” are in SMALL CAPITALS.
867+16 30.28. See Penrose (1979).
(delete Hawking and Preskill references)
867–2 …Moroz et al. (1998). For ‘expectation value’, see Note 22.11.
868+14 …Christoph Simon. See Penrose (2000a), Marshall et al.
__________________________________________________________
CHAPTER 31
871–21 …John Wheeler (e.g. 1973) who later emphasized…
871–15 did in 1906, but published way back in 1881. However, there is…
873+9 …see §14.4 and Note 19.10) to produce
875–8 possess superpartners. Otherwise…
878+8 …such a way that if we take the ‘ordinary’…
(no comma)
878+9 …(with no es involved) then we just get…
(no comma)
878–14 ambient space—see Note 27.36), where e is to be…
883–24 …(in fact U(1), cf. §13.9) symmetry…
891+12 …the appearance of tachyonic behaviour
893–6 …—since the basic unexcited string itself
894+14 consider the string theorist’s…
(delete “instead”)
899+6 …in §12.6, Fig. 12.13.) How…
905–17 …an unacceptable flood of
906+4 …a space Z that is singular, so general…
912–20 metric) ‘bosonic strings’ of §§31.5,7 that were to…
913+17 do with physics, but with counting the number of rational…
(delete “it had”)
916–8 …is no actual event horizon
919–15 …Hawking’s TBH = 1/8pm
920+11 …states of a (quantum) string field theory defined on…
920–13 …Sitter space (see §28.4, Figs.28.8
920–12 and 28.9b—but here there are four…
922+8 theory (§31.4), the spacetime symmetry…
922+15 Fock spaces (see §26.6, Note 26.12) will be completely different…
922–5 …his field equation (§19.7)76.
(simply insert full stop at the end)
923+11 …in Fig. 15.1 and Fig. 31.3. What we…
(delete “a”)
927+7 …several ‘tour guides’ (§31.6) throughout its over-
927–10 …1990, which has had the status, among
929–20 31.4. But recall ’t Hooft’s remark,…
(see xxiii–13 [and 652+15, etc.] above, for this correction)
929–15 …(1982); Seiberg and Witten (1994), on supersymmetric…
(just insert comma)
929–14 theory, led to great…
930+7 31.15. For a rapid route to these counts, see Penrose and Rindler (1984), p.389.
930+17 …normal 4-dimensional spacetime. See §31.31,
930+18 §33.14 and Note 31.81.
930+20 31.19. Compare with Note 12.14.
(Note missing full stop after “31.19”.)
930+21 31.20. See Schwarz (2001) for a…
930–13 31.26. Authoritative works that…
(just delete “Greene (1999)” reference)
931+20 …Bryant et al. (1991); also Gibbons and Hartnoll…
931–11 …sense that somewhere along each one k[aRb]cd[ekf]kckd ¹ 0,
All that is needed is the insertion (on the right-hand end of the left-hand side of the equation) of the quantity “kckd”. where all letters are italic, the “c” and “d” being superior indices. Also, a couple of commas have been removed in the sentence.
931–5 …and §28.5). For ‘compact hypersurface’, see §12.6 and Note 27.36.
933–21 …are to be found in Penrose (1968a).
933–20 31.76. Einstein (1917).
(so, replace the entire “See Nair……et al. (2004)” by this above Einstein reference)
933–5 31.81. See Nair (1988); Witten (2003); Cachazo et al. (2004a, 2004b, 2004c); Brandhuber et al. (2004).
_________________________________________________________
CHAPTER 32
936+7 §33.7). What does this mean?…
936+9 …as we shall see in §33.8, is most manifest…
936+21 …illustrated in Figs. 17.8a, 17.9a, 28.15, and 28.16. Now if we rotate…
936+25 ellipse of distortion in Fig. 28.15, and we noted in §31.9 that
936–7 …with q2s = z/w replacing q2 = z/w). Thus…
937–6 not problematic. But it turns out that…
(no “(” parenthesis)
945+10 …spin-network theory; Penrose 1971a, 1971b.)
948 Fig. 32.10 There should be a numeral “0” labelling the downward-pointing line at the far right (which makes it consistent with the other three pairs of downward-pointing lines at other places in the figure). Also the numeral “1”, which seems to be incorrectly labelling the vertex that lies at the top end of this particular downward-pointing line, should be displaced slightly up and to the right—so that it now correctly clearly labels the sloping line just above it, instead of the vertex.
948+4 …(as in Fig. 23.2). If B and D combine to form a single
948+7 and C. But these probabilities are not…
948+11 B and D together and allow…
948+14 C together, then the possible…
952+4 …one seemed to need to take the curious value
952+6 …parameter. Although certainly a strange value,
952+7 such one choice then correctly…
952–16 …—this being dependent on a
952–15 single h choice (given above as h = log2/pÖ3) for the Barbero…
952–13 perhaps such a real value is actually…
952–8 as that displayed above, the chiral aspect…
952–5 mixture of intrinsic and extrinsic parts, whose meaning…
955+19 …sh cl ea fr th
956+4 …for example. See DeWitt (1967) for the Wheeler-DeWitt equation
(note the lack of a space in “DeWitt”, as well as the capitalization)
957+8 32.17. See Penrose (1988a). It may be noted that the new twistor-string ideas put forward by Witten (2003) do provide developments of this general kind; see Note 31.81.
957–15 32.20. See Penrose (1971a), Moussouris (1983).
957–13 …Ashtekar et al. (2000). However, it has recently been shown that this displayed value for h is in error, and h has now been assigned a more complicated-looking value. See Domagala and Lewandowski (2004); Dreyer, Markopoulou, and Smolin (2004). The remainder of my discussion in §§32.6,7 seems not to be affected.
957–9 …excellent review article by Ashtekar and…
957–4,3 32.27. See Barratt and Crane (1998); Baez (1998); Reisenberger (1997, 1999).
957–2 32.28. Barrett and Crane (2000); Baez (2000); Reisenberger and Rovelli (2001, 2002); Perez (2003).
________________________________________________________
CHAPTER 33
959–10 …for the lattices—somewhat like that of Fig. 33.1)…
959–8 …Finkelstein,5 the octonionic (§11.2, §16.2)
961–18 …= iħL1, L3L1 - L1L3 = iħL2).
963+11 …teleportation, as described in §23.9,
966–18 …operators defined at spacelike-separated
966–17 events to commute. If the very…
(delete “§26.11”)
970–5,4 …in more detail, explaining the pointwise identification of ℐ+ with ℐ- in ordinary spacetime terms? Can you see why…
The part “What is the ordinary description…of ℐ-” is to be deleted. Note that “ℐ” is a special script “I” font—the same as that used in the second line of the caption for
Fig.33.9 (i.e. 970+2).
971–2 Hint: See §18.4 and Fig. 18.9.
971–1 …[33.6] Why? You may assume the result of Exercise [33.5].
972+5 O(1,3); see E in Fig. 18.9.
(this “E” is ordinary italic lightface)
973+7 …Lorentz group (§13.8). Although this…
975–3 The icon at the far left of this line should be that of the happy face, with raised finger.
976+6 into the complex 4–spaces…
981+4 this defining a complex projective
981–16 …speed of light—is represented in E3 by a
Just insert “in”. (The “E” is “shell” capital, and is correctly printed.)
984+19
¡ = Za…
This italic letter (a “rounded” version of capital Greek “upsilon”) should be bold; otherwise it is correctly printed. (See 996–1 for a correct version of this symbol.)This is exactly what is required for the following two corrections also:
984+21 …and 9, particularly.) As Euler showed, ¡
984–10
¡f = uf,
(bold, italic ¡ as above)
989 Fig. 33.18 At the top of the page (in the “(a)” part of the figure), one finds the combination of symbols “º-”. Here the minus sign “-” is incorrect and should be deleted, so just the triple-barred “equivalence” sign “º” should remain.
991–9 being just a (topological) product, cf. §15.2. (See Fig.15.3)…
993+3 …theory? Recall, from §§9.2,3, the way…
993–14 on M#) are represented as first cohomology…
This “M” should be a “shell” capital “em”.
996–1 What’s the commutator of F with ¡ (§33.7)?
Just insert “(§33.7)”. The preceding symbol is fine.
997–17 …, that represents R (Fig.
This “R” should be a “script” capital “R”. See the line (997–18) above for an example.
997–16 33.26b,c). We can take it that the topology…
997–9 …hemispheres (Fig. 33.26d). We now ‘shunt’…
997–7 …(Fig. 33.26e,f).
997–6 …to the p space (Fig. 33.26f),
998–6 …cross-sections (Fig.33.26f,g). It turns out
998–3 …that two points P* and Q* of
998–2 …corresponding lines P* and Q* in
998–1 …intersect (Fig. 33.26g).
1003–6 …physical requirement for space-asymmetry (except
1005+2 …equations (§§33.9,11), however…
1006+9 …the twistor representations of
1006+17 …hyperfunctions, introduced at the end of
1006–19 …; and Snyder (1947).
1007+19 …1964, 1965a, 1986).
(delete “b”)
1007+21 …and Rindler (1986), particularly the Appendix.
1007+24 33.21. See Penrose and Rindler (1986); Huggett and Tod (2001).
1007–17 (1984); Huggett and Tod (2001).
1007–16 33.23. See Penrose (1967, 1975, 1987b); Penrose and Rindler (1986).
1007–11 33.26……(1938, 1940); Penrose (1965a).
1007–10 …and Rindler (1986); Penrose (1965a); Penrose
1007–8 33.28. See Penrose (1968b, 1969b, 1987b); Huggett and…
1008+5 …to this realization. See Penrose (1979b) for the original rough arguments for twistor cohomology and Eastwood et al. (1981) for a thorough treatment.
1008+8,9 of getting at the cohomology concept. See Wells (1991); Ward and Wells (1989); Griffiths and Harris (1978).
(so delete references to Penrose and Rindler, and to Huggett and Tod)
1008+10 33.33. See Gunning and Rossi (1965); Ward and Wells (1989); Wells (1991); also Penrose and Rindler (1986).
(so delete “provides a more extensive discussion”)
1008+14 33.36. See Penrose and Penrose (1958).
1008+16 33+38. See Gunning and Rossi (1965); Griffiths and
(so delete reference to Penrose and Rindler)
1008–16 a holomorphic fibration. See Penrose (1976b).
1008–15 …circumstances; see Huggett and Tod (2001), Ward and Wells (1989); Penrose and Ward (1980);
(so this involves the deletion of reference to “Penrose (1976b)”)
1008–9 …as referred to in §32.5; see Note 32.17.
1008–4 Witten (1978). See also Penrose and…
(so delete “et al.”)
1009+6…Bramson (1975); Penrose (1992).
1009+7 33.53 See Penrose (2001).
1009+10 33.55. For some of the older references, see Penrose and MacCallum (1972); Penrose (1975). For more recent work, see Hodges (1982, 1985, 1990, 1998).
(note that “Penrose and Rindler…p. 149” is deleted)
1009–19 33.59. See Note 31.81.
1009–7 …does not allow for the violation…
1009–2 …path integrals. (see §26.6). But see end of §32.5 and Note 32.17.
----------------------------------------------------------------------------------------------------
CHAPTER 34
1011+2 …no reason to believe (§§31.11,12) that, in
1012+14 …like 10120 larger (or, according to some proposals, possibly only 1060
1012–14 distances of up to 15 kilometers2 are…
1014+6 …justified) conception that a finite (as
1016–9 Chapter 6).
1019+21 But not being comfortable with the 2-spinor…
1019+23,24,25 equations, which tend now to be called such things as the ‘Duffin-Kemmer-Petiau’ equation (1936-1939, for spins 0 and 1) or the ‘Rarita–Schwinger equation (1941,……their work (using tensor/4-spinor methods), in this area,
1031–2 …scientific proposal that this region
1022+3 model of cosmology (§27.11), although not …
1022+8 …, or K = 0; see §27.11). If our
1027–7 been made over two and one half millennia…
1030+12 …such matters in detail elsewhere, and
1032–21 world—fundamentally makes use of the actual OR…
1039+2 (See also §16.3.)
1041+2 when the number 317 206 375, as obtained…
This numeral is incorrectly displayed. Elsewhere in the book (see pp. 913, 914, particularly) commas are not used (incorrectly done here—on p. 1041—in any case), but there is a small space to separate each successive pair of digit triples.
1042–18 taken to have genus 0 (see §8.4)…
1043–6 …pointed out in 1944, in his very…
We have to give the original 1944 publication date here, because of the historical nature of this comment, and the reference in the Bibliography is in accord with this.
1044–2 …most distant object visible to the
1045–24 …Sarkar (2002); or, for an alternative pespective, Magueijo
1045–16,15 34.6 See Rovelli (1998).
Delete “in Gravity and……and J.Narlikar)”; this specific information goes in the Bibliography.
1046+8 …Hoyle 1948; Bondi and Gold 1948, was emphatically Popper-
(replace full stop by comma)
1046+11 …presence of the 2.7K microwave background
1046+12,13 …of the Big Bang; see §§27.7,10,11,13 and §§27.4,7,10.
1046+17 34.21. See Lange et al. (2001).
1046–22 34.24. See Penrose (1989, 1994, 1997a, 1997b).
1046–15 …I am aware, Duggins’s investigations are not
1046–13 34.33 See Penrose (1987a, 1994); Hameroff…
1046–6 …elsewhere in this book (see Notes 13.4 and 23.4), has taken an…
1047+6 34.37. See Penrose (1988b).
1047+8 …relativity; see Sachs (1962a, 1962b); Penrose
1047+13,14,15 …and Keating (1999). Another relevance of the (Euler) z-function to physics is what is called “z-function regularization”. A quick search of the LANL arXiv showed 142 hits at last count.
1047+18 34.43. See Note 31.81.
1047–3 34.47 Reprinted in Schrödinger (1967).
1047–2 34.48. See Davies (1997), pp. 89-94 for a graphic description of this event (from which I have borrowed in the Prologue). For the legend, an old favourite is riprinted in Hamilton (1999).
Delete “(reprinted, obviously!!)”.
__________________________________________________________
Bibliography
1050+2 to 24
In addition to the great advances in physical understanding that have been achieved in the 20th century—from highly refined experiment and sophisticated mathematical theory—modern technology and innovation have vastly improved the capabilities for disseminating and retrieving information on a global scale. Specifically, there is the introduction of arXiv.org, an online repository where physicists and mathematicians, biologists and computer scientists, can publish preprints (or ‘e-prints’) of their work before (or even instead of!) submitting it to journals. Indeed, arXiv.org has made it possible for scientists to communicate new ideas at an incredibly high speed, and as a consequence the pace of research activity has been accelerated to an unprecedented (or, as some might consider, an alarming) degree.
I have tried to take advantage of this important new trend by supplying, wherever possible, the arXiv.org links for items in the Bibliography. Finding a paper on arXiv.org is very simple. First, use your favourite web-browser to go to www.arxiv.org. Then either search for the paper or enter ‘www.arxiv.org/’ followed by the identification code provided in brackets in the Bibliography. For example, to call up Lee Smolin’s 2003 paper ‘How far are we from the quantum theory of gravity?’, one would enter as the web address: www.arxiv.org/hep-th/0303185.
This feature will be especially helpful for those readers who have access to the World Wide Web but are far away from a university library that might stock the specialized journals where scientific papers are often published. I hope that this new feature in the Bibliography encourages readers to study the many fine papers on arXiv, whether referenced here or otherwise.
Bibliography Additions:
ADD THE FOLLOWING (appropriately in alphabetical order):
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Bateman, H. (1944). Partial Differential Equations of Mathematical Physics. Dover, New York.
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Cartan, É. (1923). Sur les variétés à connexion affine et la théorie de la relativité generalisée I. Ann. École Norn. Sup. 40, 325-412.
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Bibliography Corrections:
CORRECT THE FOLLOWING, WHICH ARE ALREADY IN THE BIBLIOGRAPHY , BUT WHICH NEED ATTENTION (Bear in mind that both lists must be amalgameted together, all appropriately in alphabetical order):
Adams, J. F. and Atiyah, M. F. A. (1966). On K-theory and…
Barrow, J. D. and Tipler, F. J. (1986). The Anthropic Cosmological…
Barrow, J. D. and Tipler, F. J. (1988). The Anthropic Cosmological…
Somehow this reference has appeared twice, with two separate printings. PLEASE DELETE the first “(1986)” version and RETAIN only the second “(1988)” version.
Cachazo, F., Svrcek, P., and Witten, E. (2004a). MHV vertices and…
Now we have two other references by the same three authors (see above) so we need the “a”.
Green, M. B., Schwarz, J. H., and Witten, E. (1977). Superstring Theory…
Green, M. B., Schwarz, J. H., and Witten, E. (1978). Superstring Theory…
Again, somehow this reference has appeared twice, with two separate printings. PLEASE USE THE FIRST VERSION, but WITH THE SECOND “(1978)” DATE so it reads:
Green, M. B., Schwarz, J. H., and Witten, E. (1978). Superstring Theory, Vol. I & II. Cambridge University Press, Cambridge.
Hartle, J. B. (2003). Gravity: An introduction to Einstein’s…
There are three references starting with the name “Károlyházy”, whose alphabetical listing is such that it appears before all the other “Ka”s. Is it normal practice to consider “á”to occur before “a”, alphabetically? Have we been consistent with this elsewhere? My own expectation would have been to consider “á” to be on a par with “a” (i.e. to ignore the acute accent) for the purposes of alphabetical listing, in which case the three “Károlyházy” references would come in following “Kapusta”. But I do not have strong feelings about this and would be happy with whatever is done, provided that it is consistent.
Lange, A.E., et al. (2001). A measurement by BOOMERanG of multiple peaks in the angular power spectrum of the cosmic microwave background. Astrophys. J. 571, 604-614. [astro-ph/0104460]
So: replace “Langs”, in the present Bibliography, by “Lange”, and change the reference, as above (noting the correct(!) capitalization “BOOMERanG”).
Lasenby, J., Lasenby, A. N., and Doran, C. J. L. (2000). A unified…
Mott, N. F. (1929) The wave mechanics of a-ray tracks. Proc.Roy. Soc. Lond. A126, 79-84. Reprinted in: Quantum Theory and Measurement (ed. J.A. Wheeler and W.H. Zurek) Princeton Univ. Press, Princeton, New Jersey, 1983.
This is the better way around, as in Bohm (1951), above.
Needham, T. (1997). Visual Complex…
Peitgen, H.-O. and Richter, P. H. (1986)…
Penrose, R. (1966). An analysis of the structure of space-time. Adams Prize Essay, Cambridge University, Cambridge (unpublished; but much of it is in Penrose 1968a).
Penrose, R. (1969a) Gravitational collapse: the role of general relativity, Rivista del Nuovo Cimento Serie I, Vol. 1; Numero speciale, 252-276.
Note spelling: “RIVISTA”, which is wrong in the existing version.
Penrose, R. (1979a). Singularities and Time-Asymmetry…
Penrose, R. (1987c). Newton, quantum theory, and reality…
Penrose, R.(1992). H-space and Twistors, in Recent Advances…
This “H” is a script capital “aitch”, just like the first symbol in the displayed equation in 476–2.
Penrose, R.(2000a). Wavefunction collapse as a real…
Penrose, R.(2003). On the instability…
Penrose, R.and MacCallum, M. A. H. (1972). Twistor…
This would surely be the correct alphabetical ordering; these two are reversed in the present printing.
Smolin, L. (1991). Space and time in the quantum…
Smolin, L. (1997). The Life of the Cosmos…
Smolin, L. (1998). The physics of spin networks…
Smolin, L. (2001). The exceptional Jordan…
Smolin, L. (2002). Three Roads to Quantum Gravity…
Smolin, L. (2003). How far are we from the quantum…
This would surely be the correct alphabetical ordering; the first two of these are wrongly placed, in the present printing.
’t Hooft, G. (1978a). On the phase transition towards…
And I am a little unclear about the rationale for the correct alphabetical ordering here. Does the presence of the apostrophe entail this reference coming before all the other “T”s? Or is it the space after the “t” of ’t Hooft that has this effect? The only thing that concerns me here is that the ordering should be correct. I am happy with any consistent decision
Thirring, W. E. (1983). A Course in Mathematical…
Thomas, I. (1939). Selections Illustrating the History…
This would be the correct alphabetical ordering.
Wells, R. O. (1991). Differential analysis…
Werbos, P. (1989). Bell’s theorem: the forgotten loophole…
This would be the correct alphabetical ordering.
Wheeler, J.A. (1957). Assessment of Everett’s…
With this date correction—previously “(1975)”—this reference must now appear just before “Wheeler, J. A. (1960). Neutrinos, Gravitation and Geometry…”, in order to preserve alphabetical ordering.
Wigner, E. P. (1960). The Unreasonable Effectiveness of Mathematics in the Physical Sciences. Commun. Pure…
Index Corrections:
Corrections in RED (or PINK, for special fonts), surrrounding correct text in BLACK, comments in BLUE, page numbers in GREEN, as before.
1082 Column 2 DELETE the following two items:
Chen Ning Yang 211
…
Chien-Shiung Wu 211
These two have the wrong word order here, and they are already correctly given on p.1094.
1083 column 1
CPT theorem 638–9, 818–19
Here, the letters “CPT” are in sans-serif lightface type (as in p.638)
1083 column 2
dark energy 436, 777
…
differentiation 103–4, 106–7
C¥-smooth functions 110–12, 113–14
…
Dowker–Kent critique 809–10
1086 column 1
Hariot, Thomas 44
(just one “r”)
1087 column 2
Mach –Zehnder interferometer 514–15, 515
1089 column 1
normal modes of vibration 479–80, 482
normal operator 539
normal subgroups 251–2
1093 column 2
’t Hooft, Gerard 678–9
…
Delete: “Tsung Dao Lee 211” (wrong location due to wrong word order; correct is already on p.1087; similar to corrections for p.1082)