update time space

seven-wonders.tex

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 %% Author: PGL  Porta Mana
 %% Created: 2015-05-01T20:53:34+0200
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@@ -2196,15 +2196,19 @@ \subsection{Spatial coordinates}
 
 The peculiarities of space and the curvature of spacetime can make it difficult to communicate the positions of objects and events by relying on distances. In giving indications about the location of a shop we can say \enquote{it's 200 metres down the road} without ambiguity. But in situations where much higher precision is needed and extreme motions or gravitational fields are involved, we would need to know the velocity of the person we're talking to, because the distances measured by us and by that person could be very different. Moreover, the communication of locations usually involves reference to places that are known to all parties involved in the communication.
 
-These problems are all bypassed by using a \textbf{spatial coordinate system}, or \enquote*{coordinate system} for short. A coordinate system is the assignment, by agreement, of three numerical labels to every point in space \emph{for each coordinate time}. The values of these labels are thus the same for all observers. Often these labels have physical meaning, such as the distance from some event as measured by a specific observer, but they don't need to.
+These problems are all bypassed by using a \textbf{spatial coordinate system}, or \enquote*{coordinate system} for short. A coordinate system is the assignment, by agreement, of three numerical labels to every point in space \emph{for each coordinate time}. We use symbols for these labels, such as $(x,y,z)$ or $(r, \theta,\phi)$ or others. The values of these labels are thus the same for all observers. Often these labels have physical meaning, such as the distance from some event as measured by a specific observer, but they don't need to. In situations where only two or just one dimension are relevant, we can just use two or one spatial coordinates.
 
-Whereas in the case of time we typically use a standard time coordinate~\autoref{sec:coord_time}, in the case of spatial coordinates we typically have more freedom and set up peculiar coordinate systems adapted to the physics problem. In many exercises in these notes you'll see sketches of tailor-made reference systems.
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-\includegraphics[width=\linewidth]{images/coords_curv.jpg}
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+\includegraphics[width=\linewidth]{images/coords_curv.jpg}%
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-% The assignment of coordinates on and around Earth depends on nominal values assigned to distant astronomical objects (from \cites{capitaine2010})%
+% Grid representing a two-dimensional coordinate system.%
 }%
+A coordinate system can be visualized as a grid made by a set of lines or planes, one set for each coordinate, which allow us to read the coordinates of any point. The side figure shows an example with coordinates $(x,y)$ in two dimensions. It is of course assumed that the grid can be refined as much as needed. We are all familiar with the coordinate system $(\lambda, \phi)$ of \emph{latitude} and \emph{longitude} to identify locations on Earth's surface, and used internally by mobile phones' location systems.
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+Whereas in the case of time we typically use a standard time coordinate~\autoref{sec:coord_time}, in the case of spatial coordinates we typically have more freedom and set up peculiar coordinate systems adapted to the physics problem. In many exercises in these notes you'll see sketches of tailor-made reference systems.
 
 
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