[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
[gnuastro-commits] master 00cbca4 3/5: Tidy up 2D curvature gnuastro.tex
From: |
Mohammad Akhlaghi |
Subject: |
[gnuastro-commits] master 00cbca4 3/5: Tidy up 2D curvature gnuastro.texi fixes - s/sphericalplane/sphereandplane/ |
Date: |
Wed, 18 Oct 2017 07:52:03 -0400 (EDT) |
branch: master
commit 00cbca4167e1a772777dc39f1d2cb5fadead02aa
Author: boud <address@hidden>
Commit: boud <address@hidden>
Tidy up 2D curvature gnuastro.texi fixes - s/sphericalplane/sphereandplane/
Updating from `sphericalplane.XXX' to `sphereandplane.XXX' required changes
in plotsrc/Makefile and plotsrc/all.tex . I'm not going to try the web page
generation because (i) it sounds heavy and (ii) I don't want to try
uploading
to the GNU web server, and I'm probably not authenticated for that anyway.
---
doc/gnuastro.texi | 55 ++++++++++++++++++++++++++++------------------------
doc/plotsrc/Makefile | 2 +-
doc/plotsrc/all.tex | 2 +-
3 files changed, 32 insertions(+), 27 deletions(-)
diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index c9d7c91..fe5365c 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -16150,11 +16150,14 @@ 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
@mymath{\sin^2\theta+\cos^2\theta=1}, 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 @mymath{z}:
+solving for @mymath{z}, we find:
+
address@hidden(1\pm\sqrt{1-{r^2\over R^2}}\right).}
+
+The @mymath{\pm} can be understood from @ref{sphereandplane}: For each
address@hidden, there are two points on the sphere, one in the upper hemisphere
+and one in the lower hemisphere. An infinitesimal change in @mymath{r},
+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
@@ -16165,18 +16168,20 @@ instead of @mymath{dz} in the @mymath{ds_s^2}
equation above, we get:
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
@mymath{K=1/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}.
+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{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}.
@cindex Proper distance
A very important issue that can be discussed now (while we are still
@@ -16198,13 +16203,13 @@ radians). using the @mymath{\sin\theta} 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)address@hidden is just an arbitrary
-constant and can be directly found from @mymath{K}, so for cleaner
-equations, it is common practice to set @mymath{R=1}, which gives:
address@hidden(r)=\sin^{-1}r}. Also note that if @mymath{R=1}, then
address@hidden Generally, depending on the the curvature, in a
address@hidden universe the proper distance can be written as a
-function of the coordinate @mymath{r} as (from now on we are assuming
+l(r)=R\sin^{-1}\left({r\over R}\right)}
address@hidden is just an arbitrary constant and can be directly found from
address@hidden, 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, 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
@mymath{R=1}):
@dispmath{l(r)=\sin^{-1}(r)\quad(K>0),\quad\quad
@@ -16254,7 +16259,7 @@ position can make different choices of splitting up a
map into
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
+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
diff --git a/doc/plotsrc/Makefile b/doc/plotsrc/Makefile
index 3759b9c..f377300 100644
--- a/doc/plotsrc/Makefile
+++ b/doc/plotsrc/Makefile
@@ -42,7 +42,7 @@ all.pdf: all.tex ./tex/*.tex ./conversions.sh
cp tikz/all-figure0.eps ../gnuastro-figures/iandtime.eps
cp tikz/all-figure1.eps ../gnuastro-figures/samplingfreq.eps
cp tikz/all-figure2.eps ../gnuastro-figures/flatplane.eps
- cp tikz/all-figure3.eps ../gnuastro-figures/sphericalplane.eps
+ cp tikz/all-figure3.eps ../gnuastro-figures/sphereandplane.eps
# Make all the conversions:
./conversions.sh ../gnuastro-figures/
diff --git a/doc/plotsrc/all.tex b/doc/plotsrc/all.tex
index 8d6351d..1700910 100644
--- a/doc/plotsrc/all.tex
+++ b/doc/plotsrc/all.tex
@@ -148,6 +148,6 @@ appropriate directory.
\input{tex/flatplane.tex}
-\input{tex/sphericalplane.tex}
+\input{tex/sphereandplane.tex}
\end{document}
- [gnuastro-commits] master updated (aea662d -> e965a70), Mohammad Akhlaghi, 2017/10/18
- [gnuastro-commits] master 00cbca4 3/5: Tidy up 2D curvature gnuastro.texi fixes - s/sphericalplane/sphereandplane/,
Mohammad Akhlaghi <=
- [gnuastro-commits] master 22df182 1/5: gnuastro.texi: 2D curved space: main change in Minkowski spacetime ending, Mohammad Akhlaghi, 2017/10/18
- [gnuastro-commits] master 56cc992 2/5: 2D curvature doc: deanthropomorphisation, s/sphericalplane/sphereandplane/, Mohammad Akhlaghi, 2017/10/18
- [gnuastro-commits] master e965a70 5/5: Merged updates to Distance on a 2D curved space section, Mohammad Akhlaghi, 2017/10/18
- [gnuastro-commits] master 7500ac6 4/5: Edits for recent updates to the 2D curved space section, Mohammad Akhlaghi, 2017/10/18