[gnuastro-commits] master 56cc992 2/5: 2D curvature doc: deanthropomorphisation, s/sphericalplane/sphereandplane/

[Mohammad Akhlaghi]

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 3-sphere is a flat 2-sphere, and the
    equator of the 2-sphere 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 
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.
-
-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
+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.
+
+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{gnuastro-figures/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{gnuastro-figures/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={R-z\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^2-2Rz+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^2-2Rz+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 1-r^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 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{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
+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
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 finite-area (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 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. 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 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
 
 @dispmath{ds^2=c^2dt^2-a^2(t)ds_s^2 =
 c^2dt^2-a^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 4D-sphere. 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