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update space and time
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17 changes: 17 additions & 0 deletions old_snippets.tex
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The more an observer's speed is close to the speed of light in vacuum $\yc$, a universal physical constant:
%
\marginpar{\vspace{-\baselineskip}%
\footnotesize\color{mpcolor}The SI symbol for the speed of light in vacuum is \enquote{$c_{0}$} \parencites[item~6-35.2]{iso2008}. For simplicity we shall omit the subscript \enquote{${}_{0}$} in these notes.%
}%
\begin{equation}
\label{eq:c}
\yc = \qty{299792458}{m/s}\quad\text{(exactly).}
\end{equation}







Let's say that we want to measure the \enquote*{distance} of a moving object; a couple of problems appear. One problem is that when we speak of the distance of an object, traditionally we mean the distance between us and the object \enquote{at the same instant of time}. But as we have seen, it does not make sense for us to ask \enquote{what is the time for the object, right now?}. We could bypass this problem by specifying \enquote{when our clock shows time $a$ and the object's clock shows time $b$}, instead of saying \enquote{now}. Another problem appears, though. Distance is traditionally measured along a \emph{straight line} between us and the object. But spacetime is curved. The closest notion to a \enquote{straight line} is that of a \furl{https://mathworld.wolfram.com/Geodesic.html}{\emph{geodesic}}. It turns out, however, that there may be several geodesics connecting us at our time $a$ and the object at its time $b$, and the distances measured along these geodesics will generally be different. The very notion of \enquote*{distance} therefore becomes tricky and has different and non-equivalent definitions.


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63 changes: 34 additions & 29 deletions seven-wonders.tex
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\pdfinclusioncopyfonts=1
%% Author: PGL Porta Mana
%% Created: 2015-05-01T20:53:34+0200
%% Last-Updated: 2025-01-09T11:23:35+0100
%% Last-Updated: 2025-01-09T12:51:07+0100
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\newif\ifanon
\anonfalse
Expand Down Expand Up @@ -1976,20 +1976,13 @@ \section{Time}
% }
% %

But that's an imaginary world. In our world, what occurs is the more complicated situation with time discrepancies described initially. Only one conclusion can be drawn from these experimental results: \textbf{Time is not some sort of universal quantity. It is, so to speak, \enquote{local} to a person or clock, or to a group of persons or clocks that stick together.} This also means that \emph{it does not make sense} to ask questions like \enquote{what is the time for far-away Charlie, \emph{right now}?}, or \enquote{what is happening at some other place \emph{right now}?}. The notion of \emph{now} is also local.
But that's an imaginary world. In our world what occurs is the more complicated situation with time discrepancies described initially. Only one conclusion can be drawn from these experimental results: \textbf{Time is not some sort of universal quantity. It is, so to speak, \enquote{local} to a person or clock, or to a group of persons or clocks that stick together.} This also means that it does not make sense to ask questions like \enquote{what is the time for far-away Charlie, \emph{right now}?}, or \enquote{what is happening at some other place \emph{right now}?}. There is no universal \enquote*{now}; the notion of \emph{now} is local.


The time measured by a specific observer is called the \textbf{proper time} of that observer. Luckily we know more about how much the proper times of separated observers can differ when the observers meet again. According to our current understanding, it turns out that the time differences depend, roughly speaking, on how fast the observers are moving with respect to one another and with respect to the distribution of energy in the universe, and on how much energy is contained in the regions they travel across. The general theory of relativity gives us the equations determining any such proper-time differences. The more an observer's speed is close to the speed of light in vacuum $\yc$, a universal physical constant:
%
\marginpar{\vspace{-\baselineskip}%
\footnotesize\color{mpcolor}The SI symbol for the speed of light in vacuum is \enquote{$c_{0}$} \parencites[item~6-35.2]{iso2008}. For simplicity we shall omit the subscript \enquote{${}_{0}$} in these notes.%
}%
\begin{equation}
\label{eq:c}
\yc = \qty{299792458}{m/s}\quad\text{(exactly).}
\end{equation}
The time measured by a specific observer is called the \textbf{proper time} of that observer. Luckily we know how to calculate how much the proper times of separated observers will differ when the observers meet again. According to our current understanding, it turns out that the time differences depend, roughly speaking, on how fast the observers are moving with respect to one another and with respect to the distribution of energy in the universe, and on how much energy is contained in the regions they travel across. The general theory of relativity gives us the equations that determine the proper-time differences.

The situation depicted in our initial thought-experiment is real. Time discrepancies can be measured, for example, comparing initially synchronized clocks that have been put in aeroplanes flying in different directions. The first measurement of this kind was made by Hafele and Keating in 1971. They synchronized four caesium atomic clocks with a reference clock, and then flew the four atomic clocks around the world on commercial jet flights, first eastward, then westward. At the end of the eastward trip, the clocks showed a time \emph{behind} the reference one by around \qty{6e-8}{s}. At the end of the westward trip, their time was \emph{ahead} the reference one by around \qty{3e-7}{s}. These measurements were in agreement with General Relativity's prediction, within experimental error.
% The situation depicted in our initial thought-experiment is real.
Time discrepancies can be measured, for example, comparing initially synchronized clocks that have been put in aeroplanes flying in different directions. The first measurement of this kind was made by Hafele and Keating in 1971. They synchronized four caesium atomic clocks with a reference clock, and then flew the four atomic clocks around the world on commercial jet flights, first eastward, then westward. At the end of the eastward trip, the clocks showed a time \emph{behind} the reference one by around \qty{6e-8}{s}. At the end of the westward trip, their time was \emph{ahead} the reference one by around \qty{3e-7}{s}. These measurements were in agreement with General Relativity's prediction, within experimental error.
%
\marginpar{\vspace{-4\baselineskip}\centering\includegraphics[width=\linewidth]{images/hafele_keating_time.jpg}\\\footnotesize\flushleftright\color{mpcolor}Hafele, Keating, and their clocks aboard aeroplane (from \furl{https://time.com/vault/issue/1971-10-18/page/93}{\emph{Time}, October 1971})%
}%
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\section{Space and distance}
\label{sec:space}

Together with the notion of time, also the notions of space and distance lose some of their traditional intuition. Traditionally, when we speak of the distance of a moving object at a given time, we mean the distance at the same instant of time, measured on a straight line. But we have seen that it does not make sense for us to ask \enquote{what is the time for the object, right now?}; and moreover, spacetime is curved. Owing to the path-dependence of time and to curvature, we can define and measure several \enquote*{distances}, which are not equivalent to one another.
Together with the notion of time, also the notions of space and distance lose some of their traditional intuition. Traditionally, when we speak of the distance of a moving object at a given time, we mean the distance at the same instant of time, measured on a straight line. But we have seen that it does not make sense for us to ask \enquote{what is the time for the object, right now?}; and moreover, spacetime is curved. Owing to the path-dependence of time and to curvature, we can define and measure several \enquote*{distances}, which are \emph{not} equivalent to one another.

\medskip

Expand All @@ -2037,18 +2030,34 @@ \section{Space and distance}
\label{eq:physical_distance}
d \defd \tfrac12 c \Dt
\end{equation}
where $\yc = \qty{299792458}{m/s}$ is the constant speed of light in vacuum, which we encountered before.
where $\yc$ is the speed of light in vacuum, a universal physical constant:
% %
% \marginpar{\vspace{-\baselineskip}%
% \footnotesize\color{mpcolor}The SI symbol for the speed of light in vacuum is \enquote{$c_{0}$} \parencites[item~6-35.2]{iso2008}. For simplicity we shall omit the subscript \enquote{${}_{0}$} in these notes.%
% }%
\begin{equation}
\label{eq:c}
\yc = \qty{299792458}{m/s}\quad\text{(exactly).}
\end{equation}

%
\marginpar{\vspace{-\baselineskip}\centering\includegraphics[width=0.75\linewidth]{images/laser_meter.jpg}%
\marginpar{\vspace{-2\baselineskip}\centering\includegraphics[width=0.75\linewidth]{images/laser_meter.jpg}%
\\[\jot]\footnotesize\flushleftright\color{mpcolor}A laser distance meter (the light beam is not visible in reality).%
}%
The \furl{https://www.nist.gov/si-redefinition/meter}{metre}, SI unit of length, is based on the measuring procedure above. Common laser distance meters also work by the same procedure, and therefore yield radar distance. Radar distance is considered to be the \enquote{physical} distance.
The \furl{https://www.nist.gov/si-redefinition/meter}{metre}, SI unit of length, is based on the measuring procedure above. Common laser distance meters also work by the same procedure, and therefore yield radar distance. When we speak of \enquote*{physical} distance, we typically mean radar distance.

Radar distance, however, makes sense only if the time lapse $\Dt$ is small enough, so that the relative motion between you and the object is approximately uniform. For this reason this distance cannot be used if the object is too far away: the farther away it is, the longer it takes for a light beam to travel to and fro. Radar distances can be used between the Earth and other Solar System planets; but they cannot be used for galaxies or other distant cosmological objects.

The value of the radar distance \emph{depends on the relative motion} between you and the object. Imagine that a friend of yours is located very close to you at time $t$, but is moving with respect to you. Upon measuring object~B's radar distance, your friend will generally find a value different from yours. The discrepancy between you and your friend's measured values will be the larger, the higher is the relative velocity between you two. Several observers in motion with respect to one another will generally disagree on the dimensions of an approximately rigid objects in their vicinity.


%
\marginpar{\vspace{-3\baselineskip}\centering\includegraphics[width=\linewidth]{images/tuebingen.jpg}%
\\[\jot]\footnotesize\flushleftright\color{mpcolor}How a street in T{\"u}bingen would look like (except for colour and some other features) if we travelled through it at around \qty{240000000}{m/s} (from \textit{\furl{https://www.spacetimetravel.org/}{Relativity visualized}})%
}%
The dependence on relative motion also affects, at high speeds, how we \emph{see} objects, which appears more and more deformed. You can find beautiful visualizations, both static and animated, at \textit{\furl{https://www.spacetimetravel.org/}{Relativity visualized}}.


\medskip

Given a coordinate time it is possible to define an alternative distance, which we can call \textbf{spatial coordinate distance} or just \enquote*{coordinate distance} for short. It is the length of the path joining you and the object at coordinate time $t$ for both; all points of the path are also understood to be at coordinate time $t$ as well. The path should be a \emph{straight line} between you and the object; but we must remember that spacetime is curved. The notion closest to a \enquote*{straight line} in a curved space is that of a \furl{https://mathworld.wolfram.com/Geodesic.html}{\emph{geodesic}}. It turns out that there may be several geodesics connecting you and the object at coordinate time $t$. This fact may lead to some complications. For instance there may be different paths having shortest distance; therefore saying \enquote{the path of shortest distance} may be ambiguous.
Expand All @@ -2065,16 +2074,7 @@ \section{Space and distance}

The distances we'll use in these notes can be interpreted either as coordinate distances or as radar distances.

\bigskip

%
\marginpar{\vspace{-7\baselineskip}\centering\includegraphics[width=\linewidth]{images/tuebingen.jpg}%
\\[\jot]\footnotesize\flushleftright\color{mpcolor}How a street in T{\"u}bingen would look like (except for colour and some other features) if we travelled through it at around \qty{240000000}{m/s} (from \textit{\furl{https://www.spacetimetravel.org/}{Relativity visualized}})%
}%
The peculiarities of space and time also affect, at high speeds, how we \emph{see} objects. You can find beautiful visualizations, both static and animated, at \textit{\furl{https://www.spacetimetravel.org/}{Relativity visualized}}.



\medskip

% %
% \marginpar{\vspace{0\baselineskip}\centering%
Expand Down Expand Up @@ -2112,15 +2112,20 @@ \section{Space and distance}
\section{Coordinate systems}
\label{sec:coords}

\marginpar{\vspace{0\baselineskip}\footnotesize\color{mpcolor}\enquote{\emph{%
\subsection{Spacetime}
\label{sec:spacetime}

\marginpar{\vspace{\baselineskip}\footnotesize\color{mpcolor}\enquote{\emph{%
% The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical.
Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.}}\sourceatright{\cites{minkowski1908b}}
}
%
From our discussion about time and space, we conclude that physical events happen in spacetime, and there is no unique way to attribute a universal time and a universal position in space to a physical event.
From our discussion about time and space we conclude that physical events happen in spacetime, and there is no unique way to attribute a universal time and a universal position in space to a physical event.
% There is one absolute: whoever locally measures the physical speed of light in empty space, will find the value $\yc$. This value, exact by definition, serves as a way to define a local notion of space and distance.

In the previous sections we used the word \enquote*{event}, informally taking its meaning for granted. Let's be more precise now: we call \textbf{event} or \textbf{spacetime point} a very small region of space that lasts for a very short lapse of time -- so small and short that it can be considered as a point in a four-dimensional space. The word \enquote*{event} is used because typically we identify such a spacetime point by means of a physical phenomenon of limited spatial extension and duration. How \enquote{limited} should these extension and duration be? It depends on the kinds of physical phenomena we're interested in. The sudden burst of a soap bubble can be considered as an event in comparison to geological distances and times; but it cannot be considered as an event if we're studying subatomic particles.
In the previous sections we used the word \enquote*{event}, informally taking its meaning for granted. Let's be more precise now. We call \textbf{event} or \textbf{spacetime point} a very small region of space that lasts for a very short lapse of time, so that it can be considered as a point in a four-dimensional space. Note that it doesn't matter which definitions of time or distance we're using: a short lapse of time coordinate means also a short lapse of proper time, and similarly for short radar distance or coordinate distance.

The word \enquote*{event} is used because typically we identify such a spacetime point by means of a physical phenomenon of limited spatial extension and duration. How \enquote{limited} should these extension and duration be? It depends on the kinds of physical phenomena we're interested in. The sudden burst of a soap bubble can be considered as an event in comparison to geological distances and times; but it cannot be considered as an event if we're studying subatomic particles.

%
\marginpar{\vspace{-2\baselineskip}\centering\includegraphics[width=\linewidth]{images/net_rotations.png}\\[-\jot]\footnotesize\color{mpcolor}\url{https://xkcd.com/2882}%
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