[gnuastro-commits] master 6ada9d2 1/2: curvature is intrinsic; GWave speed now known

[Mohammad Akhlaghi]

branch: master
commit 6ada9d28a1c6e82aae9adb6421e6be715d370148
Author: Boud Roukema <address@hidden>
Commit: Boud Roukema <address@hidden>

    curvature is intrinsic; GWave speed now known
    
    2D Curvature section of gnuastro.texi: (i) bug correction - the curvature
    of a pseudo-Riemannian manifold is intrinsic - embedding it in a higher
    dimensional flat space can help model it or think about it, but it is not
    required for curvature to make sense; (ii) update - since a few days ago we
    have the spectacular result confirming standard GR, giving
    |(c_{grav waves} - c_{light})/c_{light} | < 10^{-14}
    https://arxiv.org/abs/1710.05834.
---
 doc/gnuastro.texi | 54 +++++++++++++++++++++++++++++++-----------------------
 1 file changed, 31 insertions(+), 23 deletions(-)

diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index 36633bc..0e4ad51 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -16095,11 +16095,18 @@ 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.
+curvature of a 3D volume. We might try to do that by embedding the 3D space
+in a address@hidden' means in this context that the Pythagorean
+theorem is true for every triangle in a space.} 4D space, but that would
+require good intuition of the 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. Curvature is something to think of mathematically as
+intrinsic to a space itself, but to make things easier, we will do what
+mathematicians often do, and think of our 2D surface (we can also call it a
+space) embedded in a flat 3D space. Once the concepts have been
+created/visualized here, we will extend them, in @ref{Extending distance
+concepts to 3D}, to a real 3D spatial `slice' of the 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
@@ -16299,9 +16306,10 @@ 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
+that this conversion constant is the speed of light or gravitational waves
+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
 
 @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).}
 
@@ -16309,22 +16317,22 @@ can thus parametrize the change in distance on an 
expanding 2D surface as
 @node Extending distance concepts to 3D, Invoking astcosmiccal, Distance on a 
2D curved space, CosmicCalculator
 @subsection Extending distance concepts to 3D
 
-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{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:
+The concepts of @ref{Distance on a 2D curved space} are here extended to a
+3D space that @emph{might} be curved. We can start with the generic
+infinitesimal distance in a static 3D universe, but this time not in
+spherical coordinates instead of polar coordinates.  @mymath{\theta} is
+shown in @ref{sphereandplane}, but here we are 3D beings, positioned on
address@hidden (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:
 @dispmath{ds_s^2=dx^2+dy^2+dz^2=dr^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2).}Like
-our 2D friend before, we now have to assume an abstract dimension
-which we cannot visualize. Let's call the fourth dimension @mymath{w},
-then the general change in coordinates in the @emph{full} four
-dimensional space will be:
address@hidden(d\theta^2+\sin^2{\theta}d\phi^2)+dw^2.}But we
-can only work on a 3D curved space, so following exactly the same
-steps and conventions as our 2D friend, we arrive at:
+our 2D friend before, we now have to assume an abstract dimension which we
+cannot visualize. Let's call the fourth dimension @mymath{w}, then the
+general change in coordinates in the @emph{full} four dimensional space
+will be:
address@hidden(d\theta^2+\sin^2{\theta}d\phi^2)+dw^2.}But we can
+only work on a 3D curved space, so following exactly the same steps and
+conventions as our 2D friend, we arrive at:
 @dispmath{ds_s^2={dr^2\over
 1-Kr^2}+r^2(d\theta^2+\sin^2{\theta}d\phi^2).}In a non-static universe
 (with a scale factor a(t), the distance can be written as: