Luminosity distance

In section 8.5 we are looking at redshifts and distances. We started in an FLRW universe with metric

\begin{align}{ds}^2=-{dt}^2+a^2\left(t\right)\left[\frac{{\rm dr}^2}{1-\kappa r^2}+r^2{d\theta}^2+r^2\sin^2{\theta}{d\phi}^2\right]&\phantom {10000}(1)\nonumber\end{align}or

\begin{align}{ds}^2=-{dt}^2+a^2\left(t\right){R_0}^2\left[{d\chi}^2+{S_k}^2\left(\chi\right){d\Omega}^2\right]&\phantom {10000}(2)\nonumber\end{align}where ##k\in\left\{-1,0,1\right\}## and

\begin{align}S_k\left(\chi\right)\ \equiv\left\{\begin{matrix}\sin{\chi},&k=+1&\rm{closed}\\\chi,&k=0&\rm{flat}\\\sinh{\chi}&k=-1&\rm{open}\\\end{matrix}\right.&\phantom {10000}(3)\nonumber\end{align}(2) is Carroll's 8.106, a sort of hybrid FLRW metric.

This starts at Carroll's 8.110 where he defines distance luminosity ##d_L## which you can get if you know the absolute luminosity of a star (or galaxy) and can measure the amount of light that reaches you. We then correct that for the expansion of the universe and further correct it because the universe deviates from a perfect sphere according to ##S_k\left(\chi\right)##. We arrive at the celebrated and complicated formula at (4) which uses (5) which uses (6) .

\begin{align}d_L=\left(1+z\right)\frac{{H_0}^{-1}}{\sqrt{\left|\Omega_{c0}\right|}}S_k\left[\sqrt{\left|\Omega_{c0}\right|}\int_{0}^{z}\frac{dz^\prime}{E\left(z^\prime\right)}\right]&\phantom {10000}(4)\nonumber\end{align}where

\begin{align}E\left(z\right)=\left[\sum_{i(c)}{\Omega_{i0}\left(1+z\right)^{n_i}}\right]^\frac{1}{2}&\phantom {10000}(5)\nonumber\end{align}and there are four elements in the summation, one each for matter, radiation, curvature and vacuum energy density. ##\Omega_i## is the density parameter which we met before and was defined as 

\begin{align}\Omega_i=\frac{8\pi G}{3H^2}\rho_i&\phantom {10000}(6)\nonumber\end{align}where the ##\rho_i## are the energy densities for each of those four things. ##\Omega_i## and ##H##, the Hubble factor acquire a 0 subscript for their values in the present epoch. 

I then try to calculate (4) from known values. ##\Omega_{c0}## the pseudo-density parameter for curvature comes from Friedmann's equation and the other three density parameters. That radiation energy is negligible, matter energy density corresponds to what is usually called ordinary (baryonic) plus dark matter and vacuum energy density is dark energy. The integral in (4) must be calculated numerically and I plotted below (page 10 in the pdf). After that we have plots of luminosity distance vs redshift for the three kinds of universe determined by ##k## in (3). Luminosity distances greater 100 billion light years are quite easy to get. I thought everything was a failure until I found some distance calculators which gave more or less the same! According to Carroll, these distances are compared with those measured from absolute and apparent luminosity. Luminosity distance is mighty peculiar. Coming soon: Proper motion distance and Angular diameter distance.

I also found that there is a missing divide by sign in Carroll's equation 8.122.

Read all about it at Commentary 8.5.2 Redshifts and Distances.pdf (12 pages)

Cosmological vs Doppler redshift

Milne Universe. Flat and expanding.

I want to compare Carroll's section 3.5 where he "demonstrates the conceptual distinction between  the cosmological redshift and the conventional Doppler effect" and Orodruin's Physics Forums Insight were he concludes "I have seen many instances where people in popular texts make a very strong claim that cosmological redshift is fundamentally different from Doppler shift. The computations above clearly show that this is not the case, instead cosmological redshift and Doppler shift are two sides of the same coin, just viewed in different coordinates."

Let's see whether Carroll is one of the guilty ones writing popular texts or if his "conceptual distinction" is just a matter of reference frames.

My first attempt at this was in September 2019 and I got told by Orodruin to read his Insight on Physics Forums. I was (correctly) daunted and postponed the reading. My next attempt was in November 2019, while travelling, but time and place defeated me. At last after nearly 1½ years I have won.

I followed most of Orodruin's insight. His approximation for the mapping from Minkowski to FLRW coordinates was the most difficult part for me. I make some effort to check its validity. With that under his belt he briskly derives Hubble's law in Minkowski coordinates showing that there are (at least) two ways of looking at the expansion of the universe. The punch line comes at the end with his pathological example (a Milne universe, pictured) where he shows, without approximations, that the only distinction between cosmological and Doppler redshift is the frame of reference used. 

Carroll's thought experiment showing a "conceptual distinction" between them involved some very unreal events: Galaxies are started and stopped in a Minkowski frame and FLRW expansion is turned on and off. I have not pinned down exactly what is wrong with that but Orodruin's method is much more real. I am convinced.

On the way I learnt 

• more about Taylor series which I find peculiar.
• that coordinates are orthogonal if the metric is diagonal and I now almost understand the notation ##e_\tau=\partial_t##.
• about proper distance and simultaneity conventions, which I had never heard of before.
• I learnt about the varying speed of light!

I made notes about all these. It was an enlightening three weeks.

Read it all at Commentary 3.5 Expanding Universe Revisited #3.pdf.

Tensor Tricks

Cat gets Carroll

The file Commentary Tensor Tricks.pdf contains some useful equations for tensor manipulation which I have collected. 

The very first on tensor rank comes from early in the book (page 21) where we are told that: a tensor ##T## of type (or rank) ##\left(k,l\right)## is a multilinear map from a collection of dual vectors and vectors to ##\mathbf{R}##: 

$$T:\left(T_p^\ast\times\cdots\times T_p^\ast\right)_{k\ times}\times\left(T_p\times\cdots\times T_p\right)_{l\ times}\rightarrow\mathbf{R}$$

That is Carroll's 1.56 and I am pretty sure he has that the wrong way round. It should say: a tensor ##T## of type (or rank) ##\left(k,l\right)## is a multilinear map from a collection of ##k## vectors and ##l## dual vectors to  ##\mathbf{R}##:$$T:\left(T_p\times\cdots\times T_p\right)_{k\ times}\times\left(T_p^\ast\times\cdots\times T_p^\ast\right)_{l\ times}\rightarrow\mathbf{R}$$But it turns out that you don't really need to know what ##k## and ##l## are separately in General Relativity (because we always have a metric). You only need to know the total rank ##k+l##!

Contents

• What tensor rank?
• Multi-dimensional Chain Rule
• Partial derivative gives Kronecker delta: Coordinates, Vectors, Tensors
• Partial derivatives commute
• Metric is always symmetric 
• Contracting with metric lowers / raises index
• You can lower or raise indices on a tensor equation
• Swap indices with metric or any similar tensor
• Inverse of a matrix
• The determinant of the inverse is reciprocal of the determinant
• Determinant of a tensor in terms of Levi-Civita symbol 
• Inverse tensor
• A relationship for the derivative of the determinant
• Fully contracted symmetric × antisymmetric tensor vanishes
• Symmetrising a tensor equation
• Two formulas involving four-velocity
• Second formula
• The projection tensor on four-velocity 
• Contra / co-variant tensor transformation matrices
• Tensor contractions using matrices

Details in Commentary Tensor Tricks.pdf 

The rest of the fab four are

Commentary Important Equations.pdf 

Commentary 2.5 The Metric.pdf 

Commentary Constants and conversion factors.pdf 

Maxwell's equations have something missing

James C. Maxwell is peeved

I return to chapter one and Maxwell's equations.  Carroll's version of Maxwell's equations "in 19th century notation" at his equations 1.92 are$$
\nabla\times\mathbf{B}-\partial_t\mathbf{E}=\mathbf{J}$$

$$\nabla\bullet\mathbf{E}=\rho$$

$$\nabla\times\mathbf{E}+\partial_t\mathbf{B}=0 $$

$$\nabla\bullet\mathbf{B}=0$$

But that's not Maxwell's equations! In SI units they are

$$\nabla\times\mathbf{B}-\frac{1}{c^2}\partial_t\mathbf{E}=\mu_0\mathbf{J}$$

$$\nabla\bullet\mathbf{E}=\frac{\rho}{\epsilon_0}$$

$$\nabla\times\mathbf{E}+\partial_t\mathbf{B}=0$$

$$\nabla\bullet\mathbf{B}=0$$

Where ##\epsilon_0,\mu_0## are the electric and magnetic constants. Carroll has already announced that we were setting the speed of light ##c=1## but he does not say anything about ##\epsilon_0,\mu_0## here or later. They are just left out. I first noticed this when calculating the Energy-Momentum tensor for electro-magnetic radiation. I think that in Carrol's "natural units" (which might be a bit unorthodox) we also have ##\epsilon_0=1## and that means that, as ##c=1##, ##\mu_0=1## too.

See my reasoning in Commentary 1.8 Maxwells equations and units.pdf (two pages). 

And I amended Commentary Constants and conversion factors.pdf.

Robertson-Walker metrics

In section 8.2 we meet what Carroll calls the Robertson-Walker metrics:$$
{ds}^2=-{dt}^2+R^2\left(t\right)\left[\frac{{d\bar{r}}^2}{1-k{\bar{r}}^2}+{\bar{r}}^2{d\Omega}^2\right]=-{dt}^2+a^2\left(t\right)\left[\frac{{\rm dr}^2}{1-\kappa r^2}+r^2{d\Omega}^2\right]
$$The second version is Carroll's preferred form - 'flouting' conventional wisdom.

Four people found the equations in various teams. They are Alexander Friedmann (Russian), Georges Lemaître (Belgian), Howard Robertson (USian) and Arthur Walker (British) and the metrics are often named after one or some or all. Carroll favours the English speakers.

As usual I check Carroll's equations and (not so usual) have three minor complaints.

1) When he gives the Christoffel symbols at his equation 8.44 the third line is$$
\Gamma_{01}^1=\Gamma_{02}^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Gamma_{03}^3=\frac{\dot{a}}{a}
$$there should be an ##=## sign between ##\Gamma_{02}^2## and ##\Gamma_{03}^3##!

2) Just after equation 8.35 he says that a flat 3-manifold could be described by 'a more complicated manifold such as the three-torus ##S^1\times S^1\times S^1##'. How can a torus, something generated by circles, be flat? I think he should have said a flat thee-torus. It can be generate from three circles but not as simply as the usual doughnut which is not flat.

3) Before that, just after equation 8.29, he says ##k## sets the curvature and therefore the size of the spatial surfaces. One might think that ##k=0## sets the size as infinite (flat curvature) but the aforementioned flat three-torus is a counter example with ##k=0## and finite size. So ##k=0## does not set the size as you might have expected. Similar arguments apply when ##k<0##.

I learned a few other things too: I suspected that if the metric is diagonal then the Ricci tensor must be diagonal. Not true! But on the plus side I found Win's supercharged formula for calculating Ricci tensor components in that case. It avoids the need to calculate Christoffel and Riemann components. I also learned a bit more about smooth isometric embedding.

I also noticed something slightly mysterious that Carroll does not draw to our attention: In the closed universe case (finite size?) the radial coordinate ##\bar{r}## is constrained by ##{\bar{r}}^2<1##. So the universe might be even more closed than we thought.

Read it all at Commentary 8.2 Robertson-Walker Metrics.pdf (7 pages + 5 of calculations)

 

Einstein-Rosen bridges: Wormholes in Schwarzschild spacetime

 

Nearing the end of section 5.7 Carroll discusses wormholes connecting regions IV and I of the Kruskal diagram. These wormholes are also called  Einstein-Rosen bridges.

It was supposed to be impossible to travel between regions I and IV of the Kruskal diagram and here Carroll shows us how it can almost be done. He is very brief, his diagram is wrong, but luckily I found a paper by Peter Collas and David Klein which goes into much more detail and helped me understand. I was even able to plot diagrams of the wormhole which takes you from the depths of region IV to the depths of region I. Sadly there is never enough time and my plot is, admittedly, a bodge. The real calculations would be too complicated.

There is another way for an intrepid explorer from region I (where we live) to get a glimpse of region IV. After they cross the event horizon (the dashed line ##r=R_s##) they could look 'down and to the left' and they could see light coming in from region IV. They could even meet another explorer from region IV. However they could never tell us back in region I what they learnt and would eventually perish in the singularity.

Read it here Commentary 5.7a Wormholes.pdf (3 pages).

Big Bang!

 Now we want to do a conformal diagram for an expanding universe. The metric equation is$$
{ds}^2=-{dt}^2+t^{2q}\left({dr}^2+r^2{d\Omega}^2\right)
$$and ##0<q<1\ ,0<t<\infty\ ,\ 0\le r<\infty##. It should be pretty easy because we did most of the heavy lifting when we did the conformal diagram for flat spacetime. However I think Carroll made another mistake!

We introduce the coordinate ##\eta## with ##{dt}^2=t^{2q}{d\eta}^2## and we get a metric$$
{ds}^2=\left[\left(1-q\right)\eta\right]^{2q/\left(1-q\right)}\left(-{d\eta}^2+{dr}^2+r^2{d\Omega}^2\right)
$$The part on the right is the same as the flat metric with ##t\rightarrow\eta## so we can use all the work we did before to transform that into$$
{ds}^2=\omega^{-2}\left[-{dT}^2+{dR}^2+\sin^2{R}{d\Omega}^2\right]
$$with$$
\omega^{-2}=\left(\frac{\left[\left(1-q\right)\eta\right]^{q/\left(1-q\right)}}{\left(\cos{T}+\cos{R}\right)}\right)^2
$$and a bit of work on that gives $$
\omega=\left[\left(1-q\right)\sin{T}\right]^{q/\left(q-1\right)}\left(\cos{T}+\cos{R}\right)^{1/\left(1-q\right)}
$$But Carroll says that$$
\omega=\left(\frac{\cos{T}+\cos{R}}{2\sin{T}}\right)^{2q}\left(\cos{T}+\cos{R}\right)
$$I'm pretty sure that Carroll is wrong, even though his formula is more attractive. I also worked out how he went wrong. Carroll writes "The precise form of the conformal factor is actually not of primary importance" (because you throw it away for the diagram). Perhaps that's why he did not check it very carefully.

And here's the diagram

At the singularity very near ##t=0## space can apparently be as big as you like. Never fear: ##r## might be big but ##t^{2q}## will be very small, so distances are very small too.

Read all the details at
Commentary App H Conformal Diagram Expanding Universe.pdf (6 pages including a diversion on values of ##q##)

Conformal Diagrams

Continuing my studies of conformal transformations and diagrams I move on to appendix H, follow Carroll's logic carefully and attempt to plot his conformal diagram of Minkowski space which he shows in Fig H.4 and I have copied above in the centre. My effort is on the right. The diagrams are similar except that the curves of constant ##t##, the Minkowski coordinate, have gradient 0 nowhere on his diagram and twice on mine. And for lines of constant ##r## the score is 1,3 (gradient ##\infty##). I was distressed. Carroll does not give explicit equations for the curves so there is quite a long chain of calculation to get them plotted. I triple checked it and could find no error so I ransacked the internet and found the short paper from from 2008 by Claude Semay, title "Penrose-Carter diagram for an uniformly accelerated observer". The first part is only about an inertial observer and Semay draws a conformal diagram for her with lines of constant ##t,r## just like mine. I have reproduced half his diagram on the left. So I think Carroll has made a mistake in his Figure H.4 - perhaps he just guessed at the curves!

Carroll lists the important parts of the diagram

##i^+=## future timelike infinity (##T=\pi,R=0##)

##i^0=## spatial infinity (##T=0,R=\pi##)

##i^-=## past timelike infinity (##T=-\pi,R=0##)

##J^+=## future null infinity (##T=\pi-R,0<R<\pi##)

##J^-=## past null infinity (##T=-\pi+R,0<R<\pi##)

Carroll use a symbol like ##\mathcal {J}## not ##J## which he calls "scri". It is hard to reproduce.

Conformal diagrams are spacetime diagrams with coordinates such that the whole of spacetime fits on a piece of paper and moreover light cones are at 45° everywhere. The latter makes it easy to visualize causality. Since Minkowski spacetime has 45° light cones, if coordinates can be found which have a metric which is a conformal transformation of the Minkowski metric, the job is done. The first part of appendix H is devoted to finding conformal coordinates for flat Minkowski spacetime, expressed in polar coordinates - presumably to ease our work later in spherically symmetrical manifolds such as Schwarzschild. So we start from that metric:$$
{ds}^2=-{dt}^2+{dr}^2+r^2\left({d\theta}^2+\sin^2{\theta}{d\phi}^2\right)
$$

On the way to finding conformal coordinates we tried coordinates$$
\bar{t}=\arctan{t}\ \ ,\ \ \bar{r}=\arctan{r}
$$which certainly pack spacetime into the range$$
-\frac{\pi}{2}<\bar{t}<\frac{\pi}{2}\ ,\ 0\le\bar{r}<\frac{\pi}{2}
$$as you will see below if you press the button. Carroll says it might be fun to draw the light cones on that, so I made a movie:

Light cone at various ##\bar{r}##

Carroll's Figure H.2 is also quite confusing. It does not show the ##u,v## axes and I naturally assumed that the ##u## axis pointed down and to the right. It does not. It does the opposite.

Read all the details at Commentary App H Conformal Diagrams.pdf (12 pages) 

Einstein's equation

Einstein age 18. Credit.

We are now on section 4.2 which has some very shady approximations. However we do get to Einstein's equation for general relativity if we tolerate that. The equation is$$
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu}
$$where ##R_{\mu\nu},R## are the Ricci tensor and scalar which tell us about the curvature of spacetime, ##g_{\mu\nu}## is the metric, ##G## is Newton's constant and ##T_{\mu\nu}## is the energy-momentum tensor. So the equation tells us how the curvature of spacetime reacts to the presence of energy-momentum (which includes mass). Newton is not forgotten altogether😊.

The equation can also be written as $$
R_{\mu\nu}=8\pi G\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)
$$where ##T=g_{\mu\nu}T_{\mu\nu}## and in empty space where ##T_{\mu\nu}=0## that gives us$$
R_{\mu\nu}=0
$$The equation is a field equation for the metric and the Newtonian gravity field equation is Poisson's equation$$
\nabla^2\Phi=4\pi G\rho
$$where ##\Phi## is the gravitational potential and ##\rho## the mass density.

The section starts by plausibly guessing that GR field equation must be of the form $$
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu}
$$where ##\kappa## is a constant we must find. The GR field equation must be the same as Poisson's equation in almost-flat spacetime. So we use a small perturbation ##h_{\mu\nu}## on the flat metric: $$
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}
$$and discarding second order terms in ##h_{\mu\nu}## eventually work out that ##\kappa=8\pi G## to bring the two equations into line.

However! Carroll's 4.38 is wrong. It says ##T_{00}=\rho## and in fact ##T_{00}=\rho\left(1-h_{00}\right)## and  if 4.38 were right then 4.39 would be wrong, but in fact it is right. Carroll is sort of having it both ways and we only get ##\ \kappa\approx8\pi G## at best. It contains first order terms in ##h_00##. Hopefully the next section using the Lagrangian formulation will do better!

See Commentary 4.2 Einsteins equation.pdf (5 pages) for the details.

Physics in curved spacetime

I've now started chapter 4 on Gravitation just in time for 2020. Very exciting!

Fools straight line

Carroll first states two formulas of Newtonian Gravity his 4.1 and 4.2$$
\mathbf{a}=-\nabla\Phi
$$where ##\mathbf{a}## is the acceleration of a body in a gravitational potential ##\Phi##. And Poisson's differential equation for the potential in terms of the matter density ##\rho## and Newton's gravitational constant ##G##:$$
\nabla^2\Phi=4\pi G\rho
$$I had a long pause thinking about the various formulas for the Laplacian ##\nabla^2## here.
How to these tie up with the old-fashioned laws? Newton's law of gravity is normally stated as$$
F=G\frac{m_1m_2}{r^2}
$$which combined with Newton's second law ##F=m\mathbf{a}## gives us the acceleration of a mass in the presence of another as$$
\mathbf{a}=G\frac{M}{r^2}
$$In exercise 3.6 we were given 'the familiar Newtonian gravitational potential'$$
\Phi=-\frac{GM}{r}
$$A bit of rough reasoning shows these are equivalent.

At his 4.4 Carroll states that the next equation gives the path of a particle subject to no forces$$
\frac{d^2x^i}{d\lambda^2}=0
$$If we solve it in polar coordinates for ##r,\theta## instead of ##x,y## Carroll says we get a circle and he cheekily suggests that we might think free moving particles follow that path. But the solution is $$
r=m\theta+k
$$where ##m,k## are constants. We can plot that and, obviously if ##m=0## we get a circle of radius ##k## but if ##m\neq0## we get other more interesting lines which are equally wrong. See above. Another error by Carroll, but only minor 😏. The next one is a corker.

Then we examine the equations in a near Newtonian environment and equation 4.13 ##g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}## is wrong. The actual equation is obviously$$
g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}
$$Properly 4.13 might be
$$
g^{\mu\nu}=\eta^{\mu\nu}-h_{\rho\sigma}\eta^{\mu\sigma}\eta^{\nu\rho}
$$which is true to first order and gives ##h^{00}=-h_{00}## which is used in the next section. If  one accepts the approximation that ##\eta## can be used to raise and lower indices on any object of order ##h## then that also gives us$$
g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}
$$which says ##h^{\mu\nu}=0##. Oops! But it turns out it turns out that the sign on ##h^{\mu\nu}## is immaterial in this section. There is a full analysis in the pdf.

We soon arrive at the conclusion that in the near Newtonian environment we have the time, time component of the metric is $$
g_{00}=-1-2\Phi
$$which is also what we were given in Exercise 3.6.

More at Commentary 4.1 Physics in curved spacetime.pdf (5 pages)

Symmetries and Killing vectors

Sean Carroll, my guide and nemesis

I'm now reading section 3.8 on symmetries and Killing vectors. It's not too hard to follow but there are a few stumbling blocks.

After equation 3.161 for the geodesic in terms of 4 momentum ##p^\lambda\nabla_\lambda p^\mu=0##  Carroll says that by metric compatibility we are free to lower the index ## \mu##. Metric compatibility means that ##\nabla_\rho g_{\mu\nu}=\nabla_\rho g^{\mu\nu}=0## so I tried to show that, given that, ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu##. Here was my first attempt:
Lower the index with the metric, use the Leibnitz rule, use metric compatibility$$
\nabla_\lambda p^\mu=\nabla_\lambda g^{\mu\nu}p_\nu=p_\nu\nabla_\lambda g^{\mu\nu}+g^{\mu\nu}\nabla_\lambda p_\nu=0+\nabla_\lambda p^\mu
$$Then I tried painfully expanding ##\nabla_\lambda g^{\mu\nu}p_\nu## and got the same result. So then I asked on Physics Forums: Why does metric compatibility imply  ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu##? I got my wrist slapped by martinbn who pointed out that  ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu## made no sense because there are different types of tensors on each side of the equation. (The ## \mu## is up on one side and down on the other). I was embarrassed😡. 

What Carroll is really saying is that metric compatibility means that$$
\nabla_\lambda p^\mu=0\Rightarrow\nabla_\lambda p_\mu=0
$$which is quite different and easy to show:$$
\nabla_\lambda p^\mu=0
$$$$
\Rightarrow g^{\mu\nu}\nabla_\lambda p_\nu=0
$$$$
\Rightarrow g_{\rho\mu}g^{\mu\nu}\nabla_\lambda p_\nu=0
$$$$
\Rightarrow\delta_\rho^\nu\nabla_\lambda p_\nu=0
$$$$
\Rightarrow\nabla_\lambda p_\rho=0
$$I posted something very like those steps and there was silence which usually means they are correct. The first step uses, ##\nabla_\lambda p^\mu=g^{\mu\nu}\nabla_\lambda p_\nu##, which can be done in several ways
1) ##\nabla_\lambda p^\mu## is a tensor so you can lower (or raise) an index with the metric as usual.
2) ##\nabla_\lambda p^\mu=\nabla_\lambda\left(g^{\mu\nu}p_\nu\right)=p_\nu\nabla_\lambda g^{\mu\nu}+g^{\mu\nu}\nabla_\lambda p_\nu=g^{\mu\nu}\nabla_\lambda p_\nu## as in (1) use the Leibnitz rue and metric compatibility
3) ##\nabla_\lambda p^\mu=\nabla_\lambda\left(g^{\mu\nu}p_\nu\right)=g^{\mu\nu}\nabla_\lambda p_\nu## using Carroll's third rule for covariant derivatives: That they commutes with contractions.

The Leibnitz rule was the second rule of covariant derivatives and I discussed all four in Commentary 3.2 Christoffel Symbol. The third caused angst and another question on PF. I now think that the third rule is just saying that because the covariant derivative is a tensor you can raise and lower indexes on it. 3 and 1 above are really the same. I have suitably amended Commentary 3.2 Christoffel Symbol.

Sometimes I hate Carroll! 

There was also another post on the thread ahead of the first two which referred to a similar question on Stack Exchange. MathematicalPhysicist was asked to show that $$
U^\alpha\nabla_\alpha V^\beta=W^\beta\Rightarrow U^\alpha\nabla_\alpha W_\beta=W_\beta
$$The proof for this is very similar to the above:$$
U^\alpha\nabla_\alpha V^\beta=W^\beta
$$$$
\Rightarrow U^\alpha g^{\beta\gamma}\nabla_\alpha V_\gamma=g^{\beta\gamma}W_\gamma
$$$$
\Rightarrow U^\alpha g_{\mu\beta}g^{\beta\gamma}\nabla_\alpha V_\gamma=g_{\mu\beta}g^{\beta\gamma}W_\gamma
$$$$
\Rightarrow U^\alpha\delta_\mu^\gamma\nabla_\alpha V_\gamma=\delta_\mu^\gamma W_\gamma
$$$$
\Rightarrow U^\alpha\nabla_\alpha V_\mu=W_\mu
$$Once again there are three ways to do the first step. Metric compatibility is not essential.

See Commentary 3.8 Symmetries and Killing vectors.pdf first two pages. Then I run into another problem with Killing.

Wolfram Mathworld great circle equation error

On rereading the Wolfram article, their equation (9) is ##\phi=\delta=\pi / 2 - v##, so ##v## is the colatidude. I can only forgive myself because they earlier say ##\delta=90°-\phi##. It would be helpful if they wrote "where ##a## is the radius of the sphere, ##c_1,c_2## are constants of integration, ##u,v## are respectively longitude and colatitude" under their equation (19).

According to www.mathworld.wolfram.com/GreatCircle.html (19) the geodesic equation on a sphere (great circle) is given below. It is derived from a somewhat specialised equation for a geodesic on a surface (http://mathworld.wolfram.com/Geodesic.html (30)), which itself is derived by considering a minimised line integral. Wolfram's (19) is given as
\begin{align}
a{\mathrm{cos} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }{\mathrm{sin} c_2\ }+a{\mathrm{sin} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }{\mathrm{cos} c_2\ }-\frac{a{\mathrm{c}\mathrm{o}\mathrm{s} v\ }}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(1) \\
\end{align}where ##a## is the radius of the sphere, ##c_1,c_2## are constants of integration, ##u,v## are respectively longitude and latitude. In the next equation it recasts that in Cartesian coordinates as\begin{align}
x{\mathrm{sin} c_2\ }+y{\mathrm{cos} c_2\ }-\frac{z}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(2) \\
\end{align}"which shows that the geodesic giving the shortest path between two points on the surface of the equation lies on a plane that passes through the two points in question and also through center of the sphere." (2) is indeed the equation of a plane which contains the origin, but it also implies that\begin{align}
x=a{\mathrm{cos} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }\ \ ,\ y=a{\mathrm{sin} u\ }{\mathrm{s}\mathrm{i}\mathrm{n} v\ }\ \ ,\ z=\ a{\mathrm{c}\mathrm{o}\mathrm{s} v\ } & \phantom {10000}(3) \\
\end{align}This is very wrong. It would be correct if ##v## was the colatitude (angle measured from the pole). The colatitude is normally called  ##\phi ## and  ##\phi ={\pi }/{2}-v##, as they say in their #7. Alternatively one can swap all ##{\mathrm{sin} v\ },{\mathrm{cos} v\ }##. I guessed that the equation is therefore\begin{align}
a{\mathrm{cos} u\ }{\mathrm{cos} v\ }{\mathrm{sin} c_2\ }+a{\mathrm{sin} u\ }{\mathrm{cos} v\ }{\mathrm{cos} c_2\ }-\frac{a{\mathrm{sin} v\ }}{\sqrt{{\left(\frac{a}{c_1}\right)}^2-1}}=0 & \phantom {10000}(4) \\
\end{align}This is correct as I have proved by other means. (Here). Numerically it can be shown with my great 3-D graph plotter (here and here). The incorrect equation is obviously not a great circle, whereas the correct one looks plausible;

The schematic shows great circles between cities. The right hand one shows the London-Peking great circle according Wolfram Mathworld. Perhaps this is what happened to the British Airways pilot who flew from London to Edinburgh instead of Düsseldorf in a month ago. (On the BBC here)

I have not been able to trace the source of the error in the Wolfram Mathworld proof. It may go as far back as their (7) where they might have intended to introduce ##\phi ##.

This error on Wolfram Mathworld caused me a lot of grief!

And another small error

There is also a typo on http://mathworld.wolfram.com/Geodesic.html between equations (11) and (12). It reads "Starting with equation (##\mathrm{\Diamond }##)" which should be "Starting with equation (5)"

Here we meet the covariant divergence and prove a thing or two about it. (Carroll said it was 'easy'.) We also have the curved-space version of Stokes's theorem using the covariant derivative and finally the exterior derivative and commutator, where Carroll seems to have made a very peculiar typo. Sir George Gabriel Stokes, 1st Baronet was a very clever but he did not discover the theorem. He popularised it.



Here's the very impressive Stokes's theorem, which applies to the diagram
$$\int^{\ }_{\mathrm{\Sigma }}{{\mathrm{\nabla }}_{\mu }V^{\mu }\sqrt{\left|g\right|}}d^nx=\int^{\ }_{\mathrm{\partial }\mathrm{\Sigma }}{n_{\mu }V^{\mu }\sqrt{\left|\gamma \right|}}d^{n-1}x
$$
At Carroll's (3.36) he says "if ##\mathrm{\nabla }## is the Christoffel symbol, ##{\omega }_{\mu }## is a one-form, and ##X^{\mu }## and ##Y^{\mu }## are vector fields, we can write
$${\left(\mathrm{d}\omega \right)}_{\mu \nu }=2{\partial }_{[\mu }{\omega }_{\nu ]}=2{\mathrm{\nabla }}_{[\mu }{\omega }_{\nu ]}
$$The phrase "if ##\mathrm{\nabla }## is the Christoffel symbol" is bizarre and it is easy to prove the equation without it, assuming the Christoffel connection is torsion-free (##{\mathrm{\Gamma }}^{\lambda }_{\mu \nu }={\mathrm{\Gamma }}^{\lambda }_{\nu \mu }##). I think our author meant "if the connection is torsion-free".

Read more at Commentary 3.2 Properties of covariant derivative.pdf (7 pages)

Corrections

I have sent this list of eight corrections I have found to date to the author, Sean Carroll. There is one extra (p.148) which I only found by peeking at a reprint of the question.

Chapter 1

p.24 just before equation (1.68) it reads "There is also the Levi-Civita symbol a (0,4) tensor." Surely it is not a tensor.

Chapter 2

p. 80 Equation (2.64) should be λ = -cot^-1 t (https://www.general-relativity.net/2018/12/commentary-27-causality.html part 2)

p.81 Figure 2.26 fig 2.26 shows a singularity at a point p and the text discusses a point p that is in the future of the singularity. (https://www.general-relativity.net/2018/12/commentary-27-causality.html part 2)

Possible error

p. 82 Equation (2.66) could be much simpler if μ'₁ μ'₂ ...μ'_n = 01...(n-1). (It's more streamlined.)
This is trued but if the simplified for was given, the next equation would not work! (GK)

Chapter 3

p.96 equation (3.10) for the connection transformation law. The + sign should be -. (https://www.general-relativity.net/2019/03/the-christoffel-symbol.html)

Possible error

p.99 definition of torsion-free (connection symmetric in lower indices) given as Γ^λ_μν = Γ^λ_(μν). Surely Γ^λ_μν = Γ^λ_νμ. would be clearer? (https://www.general-relativity.net/2019/03/the-christoffel-symbol.html)

p.148

Exercise 6(a) should end "Which clock ticks faster?" I picked this up from http://web.physics.ucsb.edu/~phys231A/231A/Homework_files/hw4_solutions.pdf.

Appendices

p. 427 equation (A.11) ϕ is used to mean two different things: a map and a polar coordinate. (https://www.general-relativity.net/2018/09/commentary-on-appendix-mapping-s2-and-r3.html)

The Christoffel Symbol

Christoffel

At last we move on to chapter three on curvature and immediately we find the Christoffel symbol Γ which is upper case γ. Elwin Bruno Christoffel was a mathematician, born in Prussia in 1829, and he studied at the University of Berlin. "He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provide the mathematical basis for general relativity."

I followed equations (3.5)-(3.10) carefully because I fell into the same tramp as I had before on one step and found an error of a sign in Carroll's  (3.10) which is the important equation for the transformation of the connection. This is fairly obvious because it comes from (3.9) and a + term has gone to the other side of the equation without changing sign. It was also confirmed by notes I found at  Physics 171 and the proof about Carroll's (3.26), see  below. I struggled with that proof (and part 2 of exercise 1) for too long. Eventually I found it after I found another erroneous proof on another website which nevertheless gave me a great new indexing trick (note d). I have corrected the error here.

I still do not understand Carroll's third rule for covariant derivatives that they commute with contractions  but he never seems to use it. Its meaning  provoked a discussion on physics forums which did not help me. In another discussion my false assumption about commuting partial derivatives was exposed.

The three most important equations here are$$
{\mathrm{\Gamma }}^{\nu '}_{\mu '\lambda '}=\frac{\partial x^{\mu }}{\partial x^{\mu '}}\frac{\partial x^{\lambda }}{\partial x^{\lambda '}}\frac{\partial x^{\nu '}}{\partial x^{\nu }}{\mathrm{\Gamma }}^{\nu }_{\mu \lambda }-\frac{\partial x^{\mu }}{\partial x^{\mu '}}\frac{\partial x^{\lambda }}{\partial x^{\lambda '}}\frac{{\partial }^2x^{\nu '}}{\partial x^{\mu }\partial x^{\lambda }}
$$ and $$
{\mathrm{\Gamma }}^{\nu '}_{\mu '\lambda '}=\frac{\partial x^{\mu }}{\partial x^{\mu '}}\frac{\partial x^{\lambda }}{\partial x^{\lambda '}}\frac{\partial x^{\nu '}}{\partial x^{\nu }}{\mathrm{\Gamma }}^{\nu }_{\mu \lambda }+\frac{\partial x^{\nu '}}{\partial x^{\lambda }}\frac{{\partial }^2x^{\lambda }}{\partial x^{\mu '}\partial x^{\lambda '}}
$$ which are alternatives for the transformation of a connection. The first one was Carroll's (3.10) (corrected.)

The third is Carroll's (3.27). He writes it is "one of the most important equations in this subject; commit it to memory." It is for a torsion-free (##{\mathrm{\Gamma }}^{\lambda }_{\mu \nu }={\mathrm{\Gamma }}^{\lambda }_{\nu \mu }##) metric-compatible (##{\mathrm{\nabla }}_{\rho }g_{\mu \nu }=0##) connection and is$$
{\mathrm{\Gamma }}^{\sigma }_{\mu \nu }=\frac{1}{2}g^{\sigma \rho }\left({\partial }_{\mu }g_{\nu \rho }+{\partial }_{\nu }g_{\rho \mu }-{\partial }_{\rho }g_{\mu \nu }\right)
$$For some reason Carroll writes ##{\mathrm{\Gamma }}^{\lambda }_{\mu \nu }={\mathrm{\Gamma }}^{\lambda }_{\nu \mu }## as ##{\mathrm{\Gamma }}^{\lambda }_{\mu \nu }={\mathrm{\Gamma }}^{\lambda }_{(\mu \nu )}## which is the same but more complicated. The brackets are the symmetrisation operator.

Read all 12 pages at Commentary 3.2 Christoffel Symbol.pdf.

Important Equations for General Relativity

Here are some important equations for General Relativity.

Babylonian equations

They are in Commentary Important Equations.pdf along with references and some notes.
Mathematics

• Definition of \partial\mu (Carroll 1.54)
• (Anti)symmetrisation operator (Carroll 1.79)
• Vector as derivative (Carroll 2.16)
• Commutator (Carroll 2.20/2.23)
• Tensor transformation equation (Carroll 2.30)
• Basis vectors (Physics Forums)
• Covariant derivative / Christoffel symbol (Carroll section 3.2)
• Torsion Tensor (C Eq 3.22)
• The Christoffel connection Γ  (C Eq 3.27)
• The geodesic equation (C Eq 3.44)
• Directional covariant derivative (C Eq 3.38)
• The parallel transport equation (C Eq 3.39, 3.40)
• Riemann tensor (c Eq 3.112/3.113)
• Bianchi identity (c Eq 3.140)
• Ricci tensor and scalar, Weyl tensor (c Eq 3.144-3.147)
• Einstein tensor and 'contravariant' derivative (c Eq 3.1452,2)
• Killing's equation, Killing vectors (c Eq3.174)
• Geodesic deviation equation (c Eq3.208)

Tensor tricks

• What tensor rank?
• Multi-dimensional Chain Rule
• Partial derivative of components gives Kronecker delta
• Coordinates
• Vectors (C1.152)
• Tensors
• Partial derivatives commute
• Metric is always symmetric (C section 2.5)
• Contracting with metric lowers / raises index
• You can lower or raise indices on a tensor equation
• Swap indices with metric or any similar tensor
• Inverses and determinants
• Inverse of a matrix
• The determinant of the inverse is reciprocal of the determinant
• Determinant of a tensor in terms of Levi-Civita symbol (C Eq 2.66)
• Inverse tensor
• A relationship for the derivative of the determinant
• Fully contracted symmetric × antisymmetric tensor vanishes
• Symmetrising a tensor equation
• Two formulas involving four-velocity
• Second formula
• The projection tensor on four-velocity (C Eq 1.21)

Physics

• Electro Magnetic Field Tensor (C Eq 1.69)
• Maxwell's equations (C Eq 1.96-1.98)
• Energy Momentum tensor for a perfect fluid (C Eq 3.93 and 1.114)
• Energy Momentum tensor for dust in SR (C Eq 1.110)
• Energy-momentum tensor from action for matter (C Eq 4.75)
• Energy-momentum conservation equation (C Eq 3.92 & 4.8)
• Einstein's equation x 3 for general relativity (C Eq2.44-4.46)
• Friedmann equations (C Eq 8.67)

More Maths

• Differential and Integration on Web
• Pullback / Pushforward operators (Carroll A.9, A.10)
• Levi-Civita symbol and tensor (Carroll section 2.8)
• p-forms (Carroll section 2.9)
• Exterior derivative (Carroll 2.76)
• Wedge product (Carroll 2.73)
• Hodge star operator (Carroll 2.82)
• Stokes's theorem (C Eq 3.35)
• Euler-Lagrange Equation

All in Commentary Important Equations.pdf along with references and some notes.

Image from Wikipedia Babylonian equations.

The rest of the fab four are

Commentary Tensor Tricks.pdf 

Commentary 2.5 The Metric.pdf 

Commentary Constants and conversion factors.pdf 

Carroll's (1.68) where the Levi-Civita symbol is defined says "the Levi-Civita symbol is a ##(0,4)## tensor. It is NOT a tensor as he reminds us elsewhere.

Commentary 2.8 Tensor Densities

The opening sentence of section 2.8 starts "Tensors possess a compelling beauty and simplicity". That fills me with fear.

We are told that the Levi-Civita symbol, which is not a tensor, is defined as$${\widetilde{\epsilon }}_{{\mu }_1{\mu }_2\dots {\mu }_n}=\left\{ \begin{array}{ll}
+1 & \mathrm{if}\mathrm{\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ even\ permitation\ of}\ 01..(n-1)\  \\
-1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ odd\ permitation\ of}\ 01..\left(n-1\right) \\
0 & \mathrm{otherwise} \end{array}
\right.$$and (Carroll's (2.66)) that given any ##n\times n## matrix ##M^{\mu }_{\ \ \ \mu '\ }##, the determinant ##\left|M\right|## obeys $${\widetilde{\epsilon }}_{{\mu '}_1{\mu '}_2\dots {\mu '}_n}\left|M\right|={\widetilde{\epsilon }}_{{\mu }_1{\mu }_2\dots {\mu }_n}M^{{\mu }_1}_{\ \ \ \ \ {\mu '}_1}M^{{\mu }_2}_{\ \ \ \ \ {\mu '}_2}\dots M^{{\mu }_n}_{\ \ \ \ \ {\mu '}_n}$$We are invited to check this for 2×2 and 3×3 matrices which we do and do discover some beauty comparing traditional methods for calculating the determinant of a matrix using cofactors or using the Levi-Civita symbol.

Setting ##{\mu '}_1{\mu '}_2\dots {\mu '}_n=01\dots (n-1)## we get the even simpler$$\left|M\right|={\widetilde{\epsilon }}_{{\mu }_1{\mu }_2\dots {\mu }_n}M^{{\mu }_1}_{\ \ \ \ \ 0}M^{{\mu }_2}_1\dots M^{{\mu }_n}_{\ \ \ \ \ (n-1)}$$Other combinations of   ##{\mu '}_1{\mu '}_2\dots {\mu '}_n## either give ##0=0## or the same as ##01\dots (n-1)## or other cofactor expansions of the determinant.

However if the equation had been simplified the next equation (2.67) would not work and (2.67) is the punch line.

We also prove that the determinant of the metric under a coordinate transformation is given by
$$g\left(x^{\mu '}\right)={\left|\frac{\partial x^{\mu '}}{\partial x^{\mu }}\right|}^{-2}g\left(x^{\mu }\right)$$ See the details  at

Commentary 2.8 Tensor Densities.pdf (6 pages)

Commentary 2.7 Causality

I think I found two errors in the book here, They are highlighted in red below.

Part 1 Causality jargon

This section was introducing a ton of jargon and, as ever, Carroll confused me with his brevity and power sentences. Early on we had an achronal hypersurface which is one where no two points are connected by a timelike curve. Carroll gives any edgeless spacelike hypersurface in Minkowski space as an example. I was having a bit of trouble imagining an 'edgeless spacelike hypersurface in Minkowski space' when I found Fig 1. That made it obvious. The HYPERSURFACE OF THE PRESENT therein is achronal.

Fig 1

I was confused by the long noun phrase 'edgeless spacelike hypersurface in Minkowski space'. Thank heavens it was not in German, it might have been Minkowskistumpfraumsechthyperfläche. As a reminder we classify the following straight lines through the observer:

timelike	inside the light cone - massive particle
null (aka lightlike)	on the light cone - photon
spacelike	outside the light cone

On a flat (x,t) spacetime diagram any line is a hypersuface, if it's edgeless it continues forever and it is spacelike if its gradient m is always limited: -1 < m < 1 - it is more parallel to the space axis than the time axis.

Fig 2 Various types of hypersurface. Both spacelike surfaces are achronal.

The upper right 'hypersurface' is timelike, null and spacelike because it is steep at the left hand end and then shallow. The spacelike hypersurfaces are also achronal, they do not contain any points that are connected by a timelike curve.

Thinking about an achronal hypersurface S, Carroll defines one + four (or eight) new terms.

Causal curve		One which is timelike or null everywhere. Two are shown.
Causal future of S	J+(S)	Set of points that can be reached from S by following a future directed causal curve
Chronological future of S	I+(S)	Set of points that can be reached from S by following a future directed timelike curve
Future domain of dependence of S	D+(S)	Set of all points that p such that every past moving inextendible* causal curve through p intersects S. Points predictable from S (see below).
Future Cauchy horizon of S	H+(S)	Boundary of D+(S). Limit of predictable points (see below).

The other four definitions follow by exchanging past and future and + and -.

* inextendible means the curve goes on forever.

We'll now concentrate on the spacelike hypersurfaces S and T, which are achronal, in fig 3.

Fig 3

The futures of T are easy to visualise. It is also easy to see that p is in D+(S) and that q is not, so D+(S) is the interior of the 'triangle' with the wavy bottom bounded by two null surfaces.

Before all the definitions, Carroll had mysteriously said that "We look at the problem of evolving matter fields …".

Light dawned: The evolution from events (the initial conditions) in S can only completely specify future events in D+(S) its future domain of dependence. Events beyond its future Cauchy horizon cannot be predicted from the initial conditions.

There were some more terms

Cauchy surface	Closed achronal surface Σ whose domain of dependence D+(Σ) is the entire manifold
Globally hyperbolic	A space time that has a Cauchy surface
Partial Cauchy surface	? Cauchy surface whose domain of dependence D+(Σ) is not the entire manifold
Closed time like curve	See below

From information on a Cauchy surface on we can predict what happens throughout the entire manifold / entire universe / all spacetime.

Part 2. Cylindrical spacetime 

We then have a simple example: Consider a two-dimensional geometry with coordinates ##\{t , x\}##, such that points with coordinates  ##(t , x)## and ##(t , x+1)## are identified. The topology is thus ##\boldsymbol{\mathrm{R}}\times S^1##. We take the metric to be
$${ds}^2=-{\mathrm{cos} \left(\lambda \right)\ }{\mathrm{d}t}^2-{\mathrm{sin} \left(\lambda \right)\ }\left[\mathrm{d}t\mathrm{d}x+\mathrm{d}x\mathrm{d}t\right]+{\mathrm{cos} \left(\lambda \right)\ }{\mathrm{d}x}^2$$where$$\lambda ={{\mathrm{cot}}^{-1} t\ }$$which goes from ##\lambda  =0## (##t = - \infty ##) to ##\lambda  = \pi## (##t =  \infty ##).

##\lambda ={{\mathrm{cot}}^{-1} t\ }## is the same as ##\lambda ={\mathrm{tan}}^{-1} ( 1 / t )## and so
$$t=\ 1 /{\mathrm{tan} \lambda \ }$$That's a problem. As ##\lambda \to 0, t \to \infty ## not ##-\infty ##. So we really want
$$\lambda =-{{\mathrm{cot}}^{-1} t\ }$$To find the light cone we want a null vector ##V^{\mu }##, at various times. We can get this from the metric and Desmos plotted various light cones from 0 to ##\pi## as shown below. Details are in the pdf. I had arrived at the same diagram as Carroll!

Fig. 4. Shows the light cones in red  in the distant past (λ = 0) to distant future (λ = π). In our diagram we are identifying points with coordinates (t,x) and (t, x+~100), so that we can better see the strange light cones for large t or λ≈π .

Our light cones rotate the same way as Carroll's  Fig 2.25.

Carroll says "When t > 0, x becomes the timelike coordinate." (Because x, not t, is in the light cone. Moreover light and particles can only move in the positive x direction and they can move in + and - t directions.) We can now draw two causal curves from a point p as shown. One reaches the surface S, the other does not. Therefore p is outside the future Cauchy horizon of S. This applies to any point p with t > 0. As he says "There is thus necessarily a Cauchy horizon at t=0." Surely it's worse than that. There is a 'global' Cauchy horizon at t = 0. Perhaps that is what he meant.

In plainer language: "Nothing at t > 0 is predictable by things at t < 0".

I don't see why we had to have the cylindrical coordinate system. The closed causal curve guarantees that the causal curve from p is inextendible, but we could have had a curve that waved around forever keeping its t coordinate always >0.

Part 3 A singularity

There seems to be another error in the book here. His fig 2.26 shows a singularity at a point p and the text discusses a point p that is in the future of the singularity. So I will repeat the paragraph and the diagram using separate p's.

Fig 5

It starts clearly enough, "Singularities are points that are not in the manifold even though they can be reached by travelling along a geodesic for a finite distance. Typically they occur when the curvature becomes infinite at some point; if this happens, the point can no longer said to be part of space time." Now I take over.

Fig 5 shows a singularity at s and an achronal surface Σ that extends indefinitely in the plus and minus x directions. The point p cannot be in D+(Σ), future domain of dependence of Σ, because there are causal curves from p that end at s. Therefore there is a future Cauchy horizon at H+(Σ) as shown. H+(Σ) also extends indefinitely in the plus and minus x directions.

I am not sure if the right branch of the past light cone from p should escape the influence of the singularity, but it does not matter for this argument.

Part 4. A Diversion

From fig 4 it is clear that photons and particles from t > 0 can travel backward in time to t = 0 or nearby. Sadly I was not able to find the equations of motion. I need to know more about geodesics perhaps.

There is more detail about that and the equations in part 2 here: Commentary 2.7 Causality.pdf

Commentary on Appendix A: Mapping S2 and R3

Sometimes I almost despair reading this book. I cannot understand part of it or find a solution. When I do, the feeling of joy is superb. There is much air punching. This is a case in point.

Why am I here?

I'm still thinking about commutators and wanted to understand Appendix B on Diffeomorphisms and Lie Derivatives. Lie (pronounced lee) brackets are the same as commutators. Appendix B starts by saying "we continue the explorations of the previous Appendix", so I was thrown back. Appendix A is fairly comprehensible and there is a good example at the end. I'll restate the first part here:

The problem part 1

Consider the two-sphere embedded in ℝ³, thought of as the locus of points a unit distance from the origin. If we put coordinates θ, ф on S² (the sphere) and x,y,z on ℝ³, the map  is given by

ψ (θ, ф) = (sinθ cosф, sinθ sinф, cosθ)          (1)

In the book this equation is written as

    ф(θ, ф) = (sinθ cosф, sinθ sinф, cosθ)            (A.11)

ф is used in the equation to mean a map and a coordinate. I will avoid this.

Sticking the sphere into  in this way induces a metric on , which is just the pull-back of the flat-space metric. The simple-minded way to find this is to start with the metric

             ds² = dx² + dy² + dz²                               (1a)

and substitute (1) into this expression yielding a metric dθ² + sin²θ dф² on S².

This was not so simple and I struggled with the substitution for days. (1) is easy to show algebraically or geometrically. Getting to the metric in polar coordinates involves various differentiation rules: (trig functions and the product rule), but then they need to be applied to infinitesimals (dx, dθ etc) rather than proper derivatives (dx/dy etc). That aspect was very novel to me.

Along the way I made an interesting observation about writing the matrix for ∂_xy_α. It is the transpose of the matrix for ∂_xy_α. I will add this to my up coming opus on Tensors, Matrices and Indexes.

Five pages. You can read it here: Commentary App A Mapping S2 and R3.pdf

It contains a very useful corollary

It contains a very useful corollary on how to get the metric of a non-Cartesian system.

If we can transform non-Cartesian components p ^j into Cartesian x ⁱ with equations

x ⁱ = f ⁱ (p ^j)                                                (20)

i = 1..m,   j = 1..n 

Then we can construct the (n x m) pullback operator

Multiplying this by its transpose gives the metric of the non-Cartesian system. 

Of necessity this is a (n x m) x (m x n) = (n x n) matrix, as it must be.