New source of answers from University of Utah

After finishing Exercise 2.07 I searched for other solutions and found on at the University of Utah for a course by Pearl Sandick in Spring 2018. They were using the book as their textbook. The home page for the course is here: http://www.physics.utah.edu/~sandick/7720spring2018/index.html. As ever I have saved the solutions in case they get deleted. I have catalogued the answers on the Answers to Exercises page.

I am glad to say that my answers to the exercise were correct.

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

Latex, Physics Forums, Desmos, Symbolab, MathJax, GrindEQ

This is somewhat out of date for me. I have now purchased GrindEQ Word->Latex, which is highly idiosyncratic and have written Word macros to greatly improve it. See XXXXX.

Latex is a script used to display equations on websites by Physics Forums, Desmos the grapher,  Symbolab the equation solver and this Blogger blog with MathJax. Latex is described on Physics Forums here and more completely by Mark Gates here. I also have some Latex tips here (MS-Word). I started by using something from Codecogs but it was not good at displaying inline equations (##\mu=0##) or adjusting the font size of indexed variables ( ##{\partial }_{\mu }y^{\alpha }## ) .

GrindEQ can save a word document containing equations as a .tex plain text file. This contains some non-Latex (for the MS-Word text) and bits of delimited Latex. So it it fairly easy to copy and paste the Latex to other websites. The only difference is in the characters that delimit the Latex. These are summarise below.

Format	Delimiter for	Delimiter
PF (website)	New line	$$
	In line	##
.tex  (file from GrindEQ)	New line	\[
	In line	$
MathJax	New line	$$
	In line	$
Symbolab		none
Desmos		none

Here's something I wrote and pasted the RHS into Desmos and Symbolab.

$$ x = \frac{t\left(\sin \left(l\right)-1\right)}{\cos \left(l\right)} $$

You can copy and paste the Latex of the RHS directly into Desmos and Symbolab equations. It is

\frac{t\left(\sin \left(l\right)-1\right)}{\cos \left(l\right)}

Then I created a test Word document pictured below

Image from MS-word

And here is the same thing produced by GrindEQ and then pasted out of the .tex file into this blog.

So the pullback operator is (##\mu## is row index, ##\alpha## is column)

$$  {\partial }_{\mu }y^{\alpha }=\left( \begin{array}{cc} {\mathrm{cos} \theta \ } & {\mathrm {sin} \theta \ } \\ -r{\mathrm{sin} \theta \ } & r{\mathrm{cos} \theta \ } \end{array} \right) $$

$${\partial }_{\mu }y^{\alpha }\equiv \frac{\partial y^{\alpha }}{\partial x^{\mu }} =\left( \begin{array}{cc} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} \\  \  & \  \\  \frac{\partial x}{\partial \theta } & \frac{\partial y}{\partial \theta } \end{array} \right)=\left( \begin{array}{cc} {\mathrm{cos} \theta \ } & {\mathrm{sin} \theta \ } \\  -r{\mathrm{sin} \theta \ } & r{\mathrm{cos} \theta \ } \end{array} \right)$$

or on the final equation using \Large , \large and \small (\normal does not work)

##{\partial }_{\mu }y^{\alpha }  \equiv  \Large  \frac{\partial y^{\alpha }}{\partial x^{\mu }} \Large =   \left( \begin{array}{cc} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} \\  \  & \  \\  \frac{\partial x}{\partial \theta } & \frac{\partial y}{\partial \theta } \end{array} \right) \small= \left( \begin{array}{cc} {\mathrm{cos} \theta \ } & {\mathrm{sin} \theta \ } \\  -r{\mathrm{sin} \theta \ } & r{\mathrm{cos} \theta \ } \end{array} \right)##

Sadly I only get ten free uses of GrindEQ nd then I have to pay 50€.

Installing the mathjax code to show equations in Blogger is simple and instructions are contained here. However the page freezes shortly after opening and the Blogger instructions are out of date, so the vital parts are worth repeating:

To enable MathJax, just drop in the following code snippet after the header (<head>) in the Blogger template (Theme →Edit HTML→Edit Template). The the £ sign should be replaced with the dollar currency sign. It is hard to get that in the text here! The unadulterated code snippet is also in this text file.

<!--Script to enable Latex from http://web.archive.org/web/20110412103745/http://mnnttl.blogspot.com/2011/02/latex-on-blogger.html -->
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js">
MathJax.Hub.Config({
 extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js"],
 jax: ["input/TeX", "output/HTML-CSS"],
 tex2jax: {
     inlineMath: [ ['£','£'], ["\\(","\\)"] ],
     displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
 },
 "HTML-CSS": { availableFonts: ["TeX"] }
});
</script>

No doubt one could use different delimiters by changing the inlineMath and displayMath lines if one wanted.

Question on canonical for of the metric

I posted a question on the canonical for of the metric to physics forums here. It is about equation (2.46) in the book.  he writes:
Quote: A useful characterisation of the metric is obtained by putting g_μν into its canonical form. In this form the metric components become

$$ g_{\mu\nu} = \rm{diag} (-1, -1,...-1,+1,+1, ... +1,0,0, ... ,0) $$

where "diag" means a diagonal matrix with the given elements. So Carroll's expression seems to imply a diagonal matrix with with a minimum size 9x9 and one extra row and column wherever one of his .'s takes a value. That's too vague!

There were two good answers and I would rewrite (2.46) as

$$ g_{\mu\nu} = \rm{diag} (-1, 0,1...-1, 0,1 ... -1, 0,1 ... -1, 0,1) $$

i.e. each diagonal element can be -1,0 or 1. 

Sadly Orodruin wrote "That would have been wrong and very unclear." He did not elucidate.
I then wrote:

Does this work?As a matrix, a non-degenerate metric in canonical form is diagonal with ±1 in each component.

Orodruin:
Technically, that would be a pseudo-metric unless you have +1 in all diagonals, but in physics we just call it metric anyway.
martinbn:
This

$$ g_{\mu\nu} =  \begin{pmatrix} -1 & 0 & 0 & 0 \\   0 & 1 & 0 & 0 \\   0 & 0 & 1 & 0 \\    0 & 0 & 0 & 1 \end{pmatrix} $$

and this

$$ g_{\mu\nu} = \rm{diag} (-1, 1, 1,1) $$ 

are the same.

I was tempted to reply Are we discussing how many angels can fit on a pinhead? But religion is not allowed.

So the canonical form of the R² metric is

$$ \begin{pmatrix} 1 & 0 \\   0 & 1  \\    \end{pmatrix} $$

and the canonical form of the Minkowski metric (which is Lorentzian) is

$$ \begin{pmatrix} -1 & 0 & 0 & 0 \\   0 & 1 & 0 & 0 \\   0 & 0 & 1 & 0 \\    0 & 0 & 0 & 1 \end{pmatrix} $$

Symbolab differential equation solver

Click picture to enlarge

Symbolab is a godsend. 

In Exercise 2.06 I had a lot of differential equations to solve and inevitably made a few mistakes. 

Here's a sample. 

Click on it to make it bigger...

Or see the full text in Ex 2.06 Helix.pdf. 

Praise be to Symbolab!

Find them at https://www.symbolab.com/solver/ordinary-differential-equation-calculator

3-D Graph plotter

The Excel has been up dated for this. It now has animation macros. It is at 3-D graph plotter.xlsm. The new animations are in 3-D Graph plotter Version 2.... Read on.

I was doing exercise 2.06 in the book which was about a helix. I plotted the helix using Excel in a rather ad-hoc way. I wanted something better and more general so that I could plot any 3-D line on a view plane from the point of some viewer outside the view plane. I did it. Here is an animation of orbiting a big cube (100x100x100) and a small cube (10x10x10). The big cube is centred on the origin and so its bottom corner is at (-50.-50,-50). The small cube's bottom corner is at the same point, so we can easily locate the bottom corner of the big cube.

Orbiting a cube

The cubes are being viewed from a distance out of 230. The view moves in three phases:

1) The viewer moves from latitude 30°, longitude from 25° to 195° when after a short pause ...

2) Her longitude is held at 195° and her latitude increases over the North pole. There is a pause at 89.9° and 90.1°. (90° cannot be shown), At this stage the Z-axis has disappeared and the X- and Y-axes flip. Latitude 150°, longitude 195° is the same as latitude 30°, longitude from 15°. The viewer is almost back where she started.

3) Next there is a jolt as the direction of view changes from the origin to the centre of the small cube. Our intrepid viewer zooms in to distance of 100 and then back out to 230. The cycle repeats.

To get a more controlled journey round the sphere, you click here. You should see a list of the little gifs that make up the one shown here. Double click one and you can go through them like a slide show at your own speed.

The spreadsheet produces one image at a time, the animation was made with MS-Paint and GIF Animator. The Excel spreadsheet is at 3-D graph plotter.xlsx. It requires some expertise in Excel and scatter charts to use. It might be best to read the first two pages of 3-D graph plotter.pdf which give some instructions. The other 18 are devoted to developing and testing the Excel formulas needed.

A helix. The bottom quarter strand is red.

And here is the helix. It has radius 70 and goes up 2π every revolution. It appears to bend slightly around the Y-axis, but this is just a perspective effect.

This was my first excursion into 3-D Cartesian geometry and I was helped by the equations for lines and planes at https://brilliant.org

Exercise 2.06 Helix and tangent vector in Cartesian and spherical polar

Question

Consider R³ as a manifold with the flat Euclidean metric, and coordinates {x, y, z}. Introduce spherical polar coordinates {r, θ, ϕ} related to {x, y, z} by

x = rsin⁡θcosϕ⁡ (1)

y = rsin⁡θsin⁡ϕ (2)

z = rcos⁡θ (3)

so that the metric takes the form

ds² = dr² + r²dθ² + r²sin²θdϕ² (4)

(a) A particle moves along a parameterised curve given by

x(λ) = cos⁡λ  ,  y(λ) = sin⁡λ  ,   z(λ) = λ (5)

Express the path of the curve in the {r, θ, ϕ} system.

(b) Calculate the components of the tangent vector to the curve in both the Cartesian and spherical polar coordinates.

Answers

It is fairly obvious that the curve is a helix. It has unit radius, the distance between each rung is 2π and so it goes up at an angle of 45°. It is shown below, compressed in the Z direction.

The helix, with some vector components

Calculating the components of the tangent vector d/dλ in polar coordinates was non-trivial for me. It involved the chain, the quotient and the cos^-1 rules of differentiation. I checked them by using the tensor transformation law from Cartesian to spherical polar components. The law is

In this case b…z and α...ω indices disappear, so it became a bit simpler but there were still almost 50 equations to step through and I had to add the tan^-1 rule of differentiation to my armoury. I discovered that this method gave a different result for dθ/dλ component of my tangent vector. The possibility of errors had became enormous. Symbolab, a wonderful differential equation calculator, came to my rescue and I used it to check everything. It discovered a couple of minor errors in my 50 equation epic, but eventually pinned down the error to the first calculation of dθ/dλ from the equation of the curve in the {r, θ, ϕ} system. It just goes to show how important it is to have an 'independent' check.

More Gains

1) I have also now realised why the metric is sometimes written in the form like

ds² = dr² + r²dθ² + r²sin²θdϕ²

and sometimes as a matrix. And how to get from one to the other.

2) In oder to draw the helix, I developed a spreadsheet to draw the 3-D curve. I am inspired to do something more general to draw any 3-D curves from a specified perspective.

The full 11 page answer is at Ex 2.06 Helix.pdf. It includes a reference to the spreadsheet and from page 6 is mostly Differentiation in baby steps and other detailed calculations.