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[gnuastrocommits] master 56cc992 2/5: 2D curvature doc: deanthropomorph
From: 
Mohammad Akhlaghi 
Subject: 
[gnuastrocommits] master 56cc992 2/5: 2D curvature doc: deanthropomorphisation, s/sphericalplane/sphereandplane/ 
Date: 
Wed, 18 Oct 2017 07:52:02 0400 (EDT) 
branch: master
commit 56cc9925cac172ed0d3dff7e61b9e45da5d8252d
Author: boud <address@hidden>
Commit: boud <address@hidden>
2D curvature doc: deanthropomorphisation, s/sphericalplane/sphereandplane/
Here are some further edits to the 2D curvature section of gnuastro.texi,
together with a name change from doc/plotsrc/tex/sphericalplane.tex to
doc/plotsrc/tex/sphereandplane.tex, because spherical plane is an oxymoron
in this context. [The equator of the 3sphere is a flat 2sphere, and the
equator of the 2sphere is a straight circle. But that's another topic...]
I reverted from friend/she/her to creature/it/its; I clarified that the
matching of the spacetime unit conversion value to the speed of light in a
vacuum is a solid experimental result, not just a coincidence; and I
reformatted paragraphs to the number of columns that my emacs editor
happened to choose automatically.

doc/gnuastro.texi  270 ++++++++++
.../tex/{sphericalplane.tex => sphereandplane.tex}  0
2 files changed, 134 insertions(+), 136 deletions()
diff git a/doc/gnuastro.texi b/doc/gnuastro.texi
index c7b901b..c9d7c91 100644
 a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ 16043,37 +16043,37 @@ interested readers can study those books.
@node Distance on a 2D curved space, Extending distance concepts to 3D,
CosmicCalculator, CosmicCalculator
@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.

To be more understandable (actively discuss from an observer's point
of view) let's assume we have an imaginary 2D friend living on the 2D
space (which @emph{might} be curved in 3D). So here we will be working
with her in her efforts to analyze distances on her 2D universe. The
start of the analysis might seem too mundane, but since it is
impossible 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 difficult to visualize more (a curved 3D space in 4D).
+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.
+
+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).
To start, let's assume a static (not expanding or shrinking), flat 2D
surface similar to @ref{flatplane} and that our 2D friend is observing its
+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},
@mymath{y}). In @ref{flatplane}, the basic axes of these two coordinates
@@ 16104,59 +16104,57 @@ as:
@dispmath{ds_s=dx^2+dy^2=dr^2+r^2d\phi^2}
The main question is this: how can our 2D friend incorporate the (possible)
curvature in her universe when she is calculating distances? The universe she
lives in might equally be a locally flat but globally curved surface like
address@hidden The answer to this question but for a 3D being (us)
is the whole purpose to this discussion. So here we want to give our 2D
friend (and later, ourselves) the tools to measure distances if the space
+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
address@hidden 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
+creature (and later, ourselves) the tools to measure distances if the space
(that hosts the objects) is curved.
address@hidden assumes a spherical shell with radius @mymath{R}
as the curved 2D plane for simplicity. The spherical shell is tangent
to the 2D plane and only touches it at @mymath{A}. The result will be
generalized afterwards. The first step in measuring the distance in a
curved space is to imagine a third dimension along the @mymath{z} axis
as shown in @ref{sphericalplane}. For simplicity, the @mymath{z} axis
is assumed to pass through the center of the spherical shell. Our
imaginary 2D friend cannot visualize the third dimension or a curved
2D surface within it, so the remainder of this discussion is purely
abstract for her (similar to us being unable to visualize 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 she can follow our reasoning.

With the third axis added, a generic infinitesimal change over
address@hidden 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 @ref{sphericalplane}. 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
address@hidden 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 Figure,sphericalplane
address@hidden@image{gnuastrofigures/sphericalplane, 10cm, , }

address@hidden spherical plane (centered on @mymath{O}) and flat plane
(gray) tangent to it at point @mymath{A}.}
address@hidden assumes a spherical shell with radius @mymath{R} as
+the curved 2D plane for simplicity. The 2D plane is tangent to the
+spherical shell and only touches it at @mymath{A}. This idea will be
+generalized later. The first step in measuring the distance in a curved
+space is to imagine a third dimension along the @mymath{z} axis as shown in
address@hidden For simplicity, the @mymath{z} axis is assumed to
+pass through the center of the spherical shell. Our imaginary 2D creature
+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.
+
+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 Figure,sphereandplane
address@hidden@image{gnuastrofigures/sphereandplane, 10cm, , }
+
address@hidden sphere (centered on @mymath{O}) and flat plane
+(light gray) tangent to it at point @mymath{A}.}
@end float
@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 her) as a function of r (distance on a flat 2D
plane, which she can visualize) and thus eliminate @mymath{z}. From
address@hidden, we get @mymath{z^22Rz+r^2=0}
and solving for @mymath{z}, we find:
address@hidden(1\pm\sqrt{1{r^2\over R^2}}\right).}The
address@hidden can be understood from @ref{sphericalplane}: For each
address@hidden, there are two points on the sphere, one in the upper
+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
address@hidden, we get @mymath{z^22Rz+r^2=0} and
+solving for @mymath{z}, we find: @dispmath{z=R\left(1\pm\sqrt{1{r^2\over
+R^2}}\right).}The @mymath{\pm} can be understood from @ref{sphereandplane}:
+For each @mymath{r}, there are two points on the sphere, one in the upper
hemisphere and one in the lower hemisphere. An infinitesimal change in
address@hidden, will create the following infinitesimal change in
address@hidden:
address@hidden, will create the following infinitesimal change in @mymath{z}:
@dispmath{dz={\mp r\over R}\left(1\over
\sqrt{1{r^2/R^2}}\right)dr.}Using the positive signed equation
@@ 16164,35 +16162,37 @@ instead of @mymath{dz} in the @mymath{ds_s^2}
equation above, we get:
@dispmath{ds_s^2={dr^2\over 1r^2/R^2}+r^2d\phi^2.}
The derivation above was done for a spherical shell of radius
address@hidden as a curved 2D surface. To generalize it to any surface, we
can define @mymath{K=1/R^2} as the curvature parameter. Then the
general infinitesimal change in a static universe can be written as:
address@hidden 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{sphericalplane}. 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 are open universes (where @mymath{r} can extend to infinity).
However, when @mymath{K>0}, we have a closed universe, where
address@hidden cannot become larger than @mymath{R} as in
address@hidden
+The derivation above was done for a spherical shell of radius @mymath{R} as
+a curved 2D surface. To generalize it to any surface, we can define
address@hidden/R^2} as the curvature parameter. Then the general infinitesimal
+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
address@hidden When @mymath{K=0}, we have a flat plane
+(@ref{flatplane}) and a negative @mymath{K} will correspond to an imaginary
address@hidden 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 @dispmath{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 @mymath{r} cannot become larger than @mymath{R} as
+in @ref{sphereandplane}.
@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
@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
address@hidden, even when the universe if curved:
address@hidden, even when the universe if curved:
@mymath{\overrightarrow{r}=P'A}. Therefore for the point @mymath{P}
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) on the curved
surface (thick line in @ref{sphericalplane}) is called
+surface (thick line in @ref{sphereandplane}) is called
(in the cosmological context that we aim at motivating)
the
@emph{proper distance} and is displayed with @mymath{l}. For the
specific example of @ref{sphericalplane}, the proper distance can be
+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}:
@@ 16215,54 +16215,52 @@ more simpler and abstract form of
@dispmath{ds_s^2=dl^2+r^2d\phi^2.}
@cindex Comoving distance
Until now, we had assumed a static universe (not changing with
time). But our observations so far appear to indicate that the
universe is expanding (it isn't static). Since there is no reason to
expect the observed expansion is unique to our particular position of
the universe, we expect the universe to be expanding at all points
with the same rate at the same time. Therefore, to add a time
dependence to our distance measurements, we can include a
multiplicative scaling factor, which is a function of time:
+Until now, we had assumed a static universe (not changing with time). But
+our observations so far appear to indicate that the universe is expanding
+(it isn't static). Since there is no reason to expect the observed
+expansion is unique to our particular position of the universe, we expect
+the universe to be expanding at all points with the same rate at the same
+time. Therefore, to add a time dependence to our distance measurements, we
+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 and the physics we assume for it: general relativity.

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
address@hidden distance},
is fixed (`comoving' with the set of fundamental observers),
and represent it by @mymath{\chi} such that:
address@hidden(r,t)=\chi(r)a(t)}. We thus shift the @mymath{r} dependence
of the proper distance we derived above for a static universe to the
comoving distance:
+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.
+
+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:
@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 ``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
address@hidden(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. This conversion constant happens to match the speed
of light, and 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
+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
+``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
@dispmath{ds^2=c^2dt^2a^2(t)ds_s^2 =
c^2dt^2a^2(t)(d\chi^2+r^2d\phi^2).}
@@ 16275,7 +16273,7 @@ The concepts of @ref{Distance on a 2D curved space} are
here extended
to a 3D space that @emph{might} be curved in a 4D space. We can start
with the generic infinitesimal distance in a static 3D universe, but
this time not in spherical coordinates instead of polar coordinates.
address@hidden is shown in @ref{sphericalplane}, but here we are 3D
address@hidden is shown in @ref{sphereandplane}, but here we are 3D
beings, positioned on @mymath{O} (the center of the sphere) and the
point @mymath{O} is tangent to a 4Dsphere. In our 3D space, a generic
infinitesimal displacement will have the distance:
diff git a/doc/plotsrc/tex/sphericalplane.tex
b/doc/plotsrc/tex/sphereandplane.tex
similarity index 100%
rename from doc/plotsrc/tex/sphericalplane.tex
rename to doc/plotsrc/tex/sphereandplane.tex
 [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 <=
 [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, 2017/10/18