[gnuastro-commits] master 7500ac6 4/5: Edits for recent updates to the 2D curved space section

[Mohammad Akhlaghi]

branch: master
commit 7500ac6cd8073b4c55579f4a32b04a3634078382
Author: Mohammad Akhlaghi <address@hidden>
Commit: Mohammad Akhlaghi <address@hidden>

    Edits for recent updates to the 2D curved space section
    
    In this branch, Boud had made some good corrections to the "Distance on a
    2D curved space" section of the book. To make it fit better into the
    over-all style of the book, some minor corrections/edits were made with
    this commit.
---
 doc/gnuastro.texi | 214 ++++++++++++++++++++++++++----------------------------
 1 file changed, 103 insertions(+), 111 deletions(-)

diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index fe5365c..e3b1dfc 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -15920,8 +15920,8 @@ One line examples:
 ## Add noise with a standard deviation of 100 to image:
 $ astmknoise --sigma=100 image.fits
 
-## Add noise to input image assuming a background magnitude (with zeropoint
-## magnitude of 0) and a certain instrumental noise:
+## Add noise to input image assuming a background magnitude (with
+## zeropoint magnitude of 0) and a certain instrumental noise:
 $ astmknoise --background=-10 -z0 --instrumental=20 mockimage.fits
 @end example
 
@@ -16044,38 +16044,33 @@ interested readers can study those books.
 @subsection Distance on a 2D curved space
 
 The observations to date (for example the Planck 2015 results), have not
-measured the presence of significant curvature in the universe, when the
-observations are interpeted under the assumption of uniform
-curvature. However to be generic (and allow its measurement if it does in
-fact exist), it is very important to create a framework that allows
-non-zero uniform curvature. For a relativistic alternative to dark energy
-(and maybe also some part of dark matter), non-uniform curvature may be
-even be more critical, but that is beyond the scope of this brief
-explanation.
-
-As 3D beings, it is difficult for us to mentally create (visualize) a
-picture of the curvature of a 3D volume embedded in a 4D space without
-sustained training.  Hence, here we will assume a 2D surface and discuss
-distances on that 2D surface when it is flat, or when the 2D surface is
-curved (and thought of embedded in a 3D non-curved (flat) space). Once the
-concepts have been created/visualized here, in @ref{Extending distance
-concepts to 3D}, we will extend them to the real 3D universe we live in and
-hope to study.
address@hidden observations are interpeted under the assumption of
+uniform curvature. For a relativistic alternative to dark energy (and maybe
+also some part of dark matter), non-uniform curvature may be even be more
+critical, but that is beyond the scope of this brief explanation.} the
+presence of significant curvature in the universe. However to be generic
+(and allow its measurement if it does in fact exist), it is very important
+to create a framework that allows non-zero uniform curvature. As 3D beings,
+it is difficult for us to mentally create (visualize) a picture of the
+curvature of a 3D volume embedded in a 4D space. Hence, here we will assume
+a 2D surface and discuss distances on that 2D surface when it is flat and
+when it is curved (embedded in a flat 3D space). Once the concepts have
+been created/visualized here, in @ref{Extending distance concepts to 3D},
+we will extend them to the real 3D universe we live in and hope to study.
 
 To be more understandable (actively discuss from an observer's point of
 view) let's assume there's an imaginary 2D creature living on the 2D space
-(which @emph{might} be curved in 3D) who is trying to learn geometry. So
-here we will be working with this creature in its efforts to analyze
-distances in its 2D universe. The start of the analysis might seem too
-mundane, but since it is difficult to imagine a 3D curved space, it is
-important to review all the very basic concepts thoroughly for an easy
-transition to a universe that is more difficult to visualize (a curved 3D
-space embedded in 4D).
+(which @emph{might} be curved in 3D). Here, we will be working with this
+creature in its efforts to analyze distances in its 2D universe. The start
+of the analysis might seem too mundane, but since it is difficult to
+imagine a 3D curved space, it is important to review all the very basic
+concepts thoroughly for an easy transition to a universe that is more
+difficult to visualize (a curved 3D space embedded in 4D).
 
 To start, let's assume a static (not expanding or shrinking), flat 2D
-surface similar to @ref{flatplane} and that our 2D creature is observing its
-universe from point @mymath{A}. One of the most basic ways to parametrize
-this space is through the Cartesian coordinates (@mymath{x},
+surface similar to @ref{flatplane} and that the 2D creature is observing
+its universe from point @mymath{A}. One of the most basic ways to
+parametrize this space is through the Cartesian coordinates (@mymath{x},
 @mymath{y}). In @ref{flatplane}, the basic axes of these two coordinates
 are plotted. An infinitesimal change in the direction of each axis is
 written as @mymath{dx} and @mymath{dy}. For each point, the infinitesimal
@@ -16094,13 +16089,13 @@ the same radius.
 plane.}
 @end float
 
-Assuming a certain position, which can be parameterized as @mymath{(x,y)},
-or @mymath{(r,\phi)}, a general infinitesimal change change in its position
-will place it in the coordinates @mymath{(x+dx,y+dy)} and
address@hidden(r+dr,\phi+d\phi)}. The distance (on the flat 2D surface) that is
-covered by this infinitesimal change in the static universe (@mymath{ds_s},
-the subscript signifies the static nature of this universe) can be written
-as:
+Assuming an object is placed at a certain position, which can be
+parameterized as @mymath{(x,y)}, or @mymath{(r,\phi)}, a general
+infinitesimal change in its position will place it in the coordinates
address@hidden(x+dx,y+dy)} and @mymath{(r+dr,\phi+d\phi)}. The distance (on the
+flat 2D surface) that is covered by this infinitesimal change in the static
+universe (@mymath{ds_s}, the subscript signifies the static nature of this
+universe) can be written as:
 
 @dispmath{ds_s=dx^2+dy^2=dr^2+r^2d\phi^2}
 
@@ -16108,7 +16103,7 @@ The main question is this: how can the 2D creature 
incorporate the
 (possible) curvature in its universe when it's calculating distances? The
 universe that it lives in might equally be a curved surface like
 @ref{sphereandplane}. The answer to this question but for a 3D being (us)
-is the whole purpose to this discussion. So here we want to give the 2D
+is the whole purpose to this discussion. Here, we want to give the 2D
 creature (and later, ourselves) the tools to measure distances if the space
 (that hosts the objects) is curved.
 
@@ -16123,32 +16118,34 @@ cannot visualize the third dimension or a curved 2D 
surface within it, so
 the remainder of this discussion is purely abstract for it (similar to us
 having difficulty in visualizing a 3D curved space in 4D). But since we are
 3D creatures, we have the advantage of visualizing the following
-steps. Fortunately our 2D friend knows our mathematics, so it can follow
-our reasoning.
+steps. Fortunately the 2D creature is already familiar with our
+mathematical constructs, so it can follow our reasoning.
 
 With the third axis added, a generic infinitesimal change over @emph{the
 full} 3D space corresponds to the distance:
address@hidden is very important
-to recognize that this change of distance is for @emph{any} point in the 3D
-space, not just those changes that occur on the 2D spherical shell of
address@hidden Recall that our 2D friend can only do measurements in
-the 2D spherical shell, not the full 3D space. So we have to constrain this
-general change to any change on the 2D spherical shell. To do that, let's
-look at the arbitrary point @mymath{P} on the 2D spherical shell. Its image
-(@mymath{P'}) on the flat plain is also displayed. From the dark triangle,
-we see that
+
address@hidden
 
 @float Figure,sphereandplane
 @address@hidden/sphereandplane, 10cm, , }
 
address@hidden sphere (centered on @mymath{O}) and flat plane
-(light gray) tangent to it at point @mymath{A}.}
address@hidden spherical shell (centered on @mymath{O}) and flat plane (light
+gray) tangent to it at point @mymath{A}.}
 @end float
 
+It is very important to recognize that this change of distance is for
address@hidden point in the 3D space, not just those changes that occur on the
+2D spherical shell of @ref{sphereandplane}. Recall that our 2D friend can
+only do measurements on the 2D surfaces, not the full 3D space. So we have
+to constrain this general change to any change on the 2D spherical
+shell. To do that, let's look at the arbitrary point @mymath{P} on the 2D
+spherical shell. Its image (@mymath{P'}) on the flat plain is also
+displayed. From the dark gray triangle, we see that
+
 @dispmath{\sin\theta={r\over R},\quad\cos\theta={R-z\over R}.}These
-relations allow our 2D friend to find the value of @mymath{z} (an abstract
-dimension for it) as a function of r (distance on a flat 2D plane, which
-it can visualize) and thus eliminate @mymath{z}. From
+relations allow the 2D creature to find the value of @mymath{z} (an
+abstract dimension for it) as a function of r (distance on a flat 2D plane,
+which it can visualize) and thus eliminate @mymath{z}. From
 @mymath{\sin^2\theta+\cos^2\theta=1}, we get @mymath{z^2-2Rz+r^2=0} and
 solving for @mymath{z}, we find:
 
@@ -16172,41 +16169,38 @@ change in a static universe can be written as:
 
 @dispmath{ds_s^2={dr^2\over 1-Kr^2}+r^2d\phi^2.}
 
-Therefore, we see that a positive @mymath{K} represents a real @mymath{R}
-which signifies a closed 2D spherical shell like @ref{sphereandplane}. When
address@hidden, we have a flat plane (@ref{flatplane}) and a negative
address@hidden will correspond to an imaginary @mymath{R}. The latter two cases
-may be infinite in area (which is not a simple concept, but mathematically
-can be modelled with @mymath{r} extending infinitely), or finite-area (like
-a cylinder is flat everywhere with @mymath{ds_s^2={dx^2 + dy^2}}, but
-finite in one direction in size).  However, when @mymath{K>0} (and
-curvature is the same everywhere), we have a finite universe, where
address@hidden cannot become larger than @mymath{R} as in @ref{sphereandplane}.
+Therefore, when @mymath{K>0} (and curvature is the same everywhere), we
+have a finite universe, where @mymath{r} cannot become larger than
address@hidden as in @ref{sphereandplane}. When @mymath{K=0}, we have a flat
+plane (@ref{flatplane}) and a negative @mymath{K} will correspond to an
+imaginary @mymath{R}. The latter two cases may be infinite in area (which
+is not a simple concept, but mathematically can be modelled with @mymath{r}
+extending infinitely), or finite-area (like a cylinder is flat everywhere
+with @mymath{ds_s^2={dx^2 + dy^2}}, but finite in one direction in size).
 
 @cindex Proper distance
-A very important issue that can be discussed now (while we are still
-in 2D and can actually visualize things) is that
address@hidden is tangent to the curved space at the
-observer's position. In other words, it is on the gray flat surface of
address@hidden, even when the universe if curved:
address@hidden'-A}. Therefore for the point @mymath{P}
-on a curved space, the raw coordinate @mymath{r} is the distance to
address@hidden'}, not @mymath{P}. The distance to the point @mymath{P} (at
-a specific coordinate @mymath{r} on the flat plane) on the curved
-surface (thick line in @ref{sphereandplane}) is called
-(in the cosmological context that we aim at motivating)
-the
address@hidden distance} and is displayed with @mymath{l}. For the
-specific example of @ref{sphereandplane}, the proper distance can be
-calculated with: @mymath{l=R\theta} (@mymath{\theta} is in
-radians). using the @mymath{\sin\theta} relation found above, we can
-find @mymath{l} as a function of @mymath{r}:
+A very important issue that can be discussed now (while we are still in 2D
+and can actually visualize things) is that @mymath{\overrightarrow{r}} is
+tangent to the curved space at the observer's position. In other words, it
+is on the gray flat surface of @ref{sphereandplane}, even when the universe
+if curved: @mymath{\overrightarrow{r}=P'-A}. Therefore for the point
address@hidden on a curved space, the raw coordinate @mymath{r} is the distance
+to @mymath{P'}, not @mymath{P}. The distance to the point @mymath{P} (at a
+specific coordinate @mymath{r} on the flat plane) over the curved surface
+(thick line in @ref{sphereandplane}) is called the @emph{proper distance}
+and is displayed with @mymath{l}. For the specific example of
address@hidden, the proper distance can be calculated with:
address@hidden (@mymath{\theta} is in radians). using the
address@hidden relation found above, we can find @mymath{l} as a
+function of @mymath{r}:
 
 @dispmath{\theta=\sin^{-1}\left({r\over R}\right)\quad\rightarrow\quad
 l(r)=R\sin^{-1}\left({r\over R}\right)}
+
+
 @mymath{R} is just an arbitrary constant and can be directly found from
 @mymath{K}, so for cleaner equations, it is common practice to set
address@hidden, which gives: @mymath{l(r)=\sin^{-1}r}. Also note that if
address@hidden, which gives: @mymath{l(r)=\sin^{-1}r}. Also note that when
 @mymath{R=1}, then @mymath{l=\theta}. Generally, depending on the the
 curvature, in a @emph{static} universe the proper distance can be written
 as a function of the coordinate @mymath{r} as (from now on we are assuming
@@ -16230,45 +16224,43 @@ can include a multiplicative scaling factor, which is 
a function of time:
 @mymath{a(t)}. The functional form of @mymath{a(t)} comes from the
 cosmology, the physics we assume for it: general relativity, and the choice
 of whether the universe is uniform (`homogeneous') in density and curvature
-(the case under discussion here) or inhomogeneous.
+or inhomogeneous. In this section, the functional form of @mymath{a(t)} is
+irrelevant, so we can aviod these issues.
 
 With this scaling factor, the proper distance will also depend on time. As
 the universe expands, the distance between two given points will shift to
-larger values. We thus define a special set of spatial coordinates that are
-independent of time, such that from the `main' observer to a given distant
-observer, the distance, that we call the @emph{comoving distance}, is fixed
-(`comoving' with the set of fundamental observers), and represent it by
address@hidden such that: @mymath{l(r,t)=\chi(r)a(t)}. We thus shift the
address@hidden dependence of the proper distance we derived above for a static
-universe to the comoving distance:
+larger values. We thus define a distance measure, or coordinate, that is
+independent of time and thus doesn’t `move'. We call it the @emph{comoving
+distance} and display with @mymath{\chi} such that:
address@hidden(r,t)=\chi(r)a(t)}.  We have therefore, shifted the @mymath{r}
+dependence of the proper distance we derived above for a static universe to
+the comoving distance:
 
 @dispmath{\chi(r)=\sin^{-1}(r)\quad(K>0),\quad\quad
 \chi(r)=r\quad(K=0),\quad\quad \chi(r)=\sinh^{-1}(r)\quad(K<0).}
 
-So @mymath{\chi(r)} is the proper distance to an object at a specific
-reference time: @mymath{t=t_r} (the @mymath{r} subscript signifies
+Therefore, @mymath{\chi(r)} is the proper distance to an object at a
+specific reference time: @mymath{t=t_r} (the @mymath{r} subscript signifies
 ``reference'') when @mymath{a(t_r)=1}. At any arbitrary moment
 (@mymath{t\neq{t_r}}) before or after @mymath{t_r}, the proper distance to
-the object can be scaled with @mymath{a(t)}. Measuring the change of
-distance in a time-dependent (expanding) universe will require making our
-spacetime consistent with Minkowski spacetime geometry, in which different
-observers at a given point (event) in spacetime split up spacetime into
-`space' and `time' in different ways, just like people at the same spatial
-position can make different choices of splitting up a map into
-`left--right' and `up--down'.  This model, well supported by twentieth and
-twenty-first century observations, only makes sense if we can add up space
-and time. But we can only add bits of space and time together if we measure
-them in the same units, with a conversion constant, like 1000 is used to
-convert a kilometre into metres.  Experimentally, we find extremely strong
-support for the hypothesis that this conversion constant matches the speed
-of light in a vacuum, and it is almost always written either as `c', or in
-`natural units', as 1. To satisfy the linear transformations in spacetime
-required by Minkowski spacetime, the hypothesis that is extremely useful in
-modern cosmology is that we can define an infinitesimal spacetime element
-as
-
address@hidden(t)ds_s^2 =
-c^2dt^2-a^2(t)(d\chi^2+r^2d\phi^2).}
+the object can be scaled with @mymath{a(t)}.
+
+Measuring the change of distance in a time-dependent (expanding) universe
+only makes sense if we can add up space and address@hidden other words,
+making our spacetime consistent with Minkowski spacetime geometry. In this
+geometry, different observers at a given point (event) in spacetime split
+up spacetime into `space' and `time' in different ways, just like people at
+the same spatial position can make different choices of splitting up a map
+into `left--right' and `up--down'. This model is well supported by
+twentieth and twenty-first century observations.}. But we can only add bits
+of space and time together if we measure them in the same units: with a
+conversion constant (similar to how 1000 is used to convert a kilometer
+into meters).  Experimentally, we find strong support for the hypothesis
+that this conversion constant can be the speed of light in a vacuum. It is
+almost always written either as @mymath{c}, or in `natural units', as 1. We
+can thus parametrize the change in distance on an expanding 2D surface as
+
address@hidden(t)ds_s^2 = c^2dt^2-a^2(t)(d\chi^2+r^2d\phi^2).}
 
 
 @node Extending distance concepts to 3D, Invoking astcosmiccal, Distance on a 
2D curved space, CosmicCalculator