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[gnuastrocommits] master 7500ac6 4/5: Edits for recent updates to the 2
From: 
Mohammad Akhlaghi 
Subject: 
[gnuastrocommits] master 7500ac6 4/5: Edits for recent updates to the 2D curved space section 
Date: 
Wed, 18 Oct 2017 07:52:03 0400 (EDT) 
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
overall 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
nonzero uniform curvature. For a relativistic alternative to dark energy
(and maybe also some part of dark matter), nonuniform 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 noncurved (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), nonuniform 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 nonzero 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={Rz\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^22Rz+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 1Kr^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 finitearea (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 finitearea (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 doesnâ€™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 timedependent (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
`leftright' and `updown'. This model, well supported by twentieth and
twentyfirst 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^2a^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 timedependent (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 `leftright' and `updown'. This model is well supported by
+twentieth and twentyfirst 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^2a^2(t)(d\chi^2+r^2d\phi^2).}
@node Extending distance concepts to 3D, Invoking astcosmiccal, Distance on a
2D curved space, CosmicCalculator
 [gnuastrocommits] master updated (aea662d > e965a70), Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master 00cbca4 3/5: Tidy up 2D curvature gnuastro.texi fixes  s/sphericalplane/sphereandplane/, Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master 22df182 1/5: gnuastro.texi: 2D curved space: main change in Minkowski spacetime ending, Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master 56cc992 2/5: 2D curvature doc: deanthropomorphisation, s/sphericalplane/sphereandplane/, Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master e965a70 5/5: Merged updates to Distance on a 2D curved space section, Mohammad Akhlaghi, 2017/10/18
 [gnuastrocommits] master 7500ac6 4/5: Edits for recent updates to the 2D curved space section,
Mohammad Akhlaghi <=