Simple English - Revision history

Igor at 20:42, 7 October 2013

2013-10-07T20:42:59Z

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Cosmo All at 08:58, 5 October 2013

2013-10-05T08:58:14Z

Cosmo All: All problems and pictures added. TODO: left align bullet lists inside solutions.

2013-10-05T08:49:13Z

All problems and pictures added. TODO: left align bullet lists inside solutions.

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Cosmo All at 08:35, 5 October 2013

2013-10-05T08:35:10Z

Cosmo All: Problems added.

2013-10-05T08:20:19Z

Problems added.

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Cosmo All: Figured out, how to accurately layout the page and figures.

2013-09-25T22:15:38Z

Figured out, how to accurately layout the page and figures.

Cosmo All: Intro Text added.

2013-09-25T12:32:56Z

Intro Text added.

New page

[[Category:Horizons|1]]

A vague definition of a horizon can be accepted as the following: it is a frontier between things observable and things unobservable.


'''''Particle horizon.'''''. If the Universe has a finite age, then light travels only a finite distance in that time and the volume of space from which we can receive information at a given moment of time is limited. The boundary of this volume is called the particle horizon.

'''''Event horizon.''''' The event horizon is the complement of the particle horizon. The event horizon encloses the set of points from which signals sent at a given moment of time will never be received by an observer in the future.

Space-time diagram is a representation of space-time on a two-dimensional plane, with one timelike and one spacelike coordinate. It is typically used for spherically symmetric spacetimes (such as all homogeneous cosmological models), in which angular coordinates are suppressed.

@@ Line 77: / Line 77: @@
      <p style="text-align: left;">
       Suppose, for example,  that luminous galaxies originated 10 billion years ago and the particle horizon is therefore at distance 10 billion light years. Observers A and B see each other and have overlapping horizons. Suppose A and B are separated by a distance of 6 billion light years (see Fig. \ref{Conf-Harr6}).
 {|class="wikitable" style="margin: 1em auto 1em auto;"
 |[[File:Conf-Harr6.png|center|thumb|365px| The horizon riddle.]]
 |}
-B sends out information that travels at the speed of light and takes 6 billion years to reach A. Hence A receives from B information that was sent 6 billion years ago, when the Universe was 4 billion years old. But B's particle horizon in the past at the time when the information was sent was only 4 billion light years distant. Thus B's horizon at that time did not extend beyond A's present horizon. In other words, B communicates information to A by sending it at the speed of light on A's backward lightcone. But when B sends the information, her horizon extends no farther than A's horizon, and he cannot see farther than A.
      </p>
    </div>
@@ Line 94: / Line 97: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
-      Consider two visible bodies at equal distances in opposite directions from us, as shown by world lines A and B on Figure \ref{Conf-Harr8}.
+      Consider two visible bodies at equal distances in opposite directions from us, as shown by world lines A and B on Figure \ref{Conf-Harr8}. We see these bodies, but can they see each other? Let $t$ be the time that it takes for light to travel to us from A and B. The time it takes the light to travel from A to B, or from B to A, is obviously $2t$. Hence when the Universe is older than $3t$, we not only see A and B, but they also see each other. If the Universe is younger than $3t$, and older than $t$, we see A and B, but they cannot yet see each other. There is thus a maximum distance beyond which the observed bodies A and B do not know of each other's existence. By examining Figure \ref{Conf-Harr8}, we see that this maximum distance is one third of the distance to the particle horizon. The answer to our question is that bodies at opposite directions and equal distances from us, which are larger than one third of the distance to the particle horizon, cannot at present see each other.
 {|class="wikitable" style="margin: 1em auto 1em auto;"
@@ Line 100: / Line 103: @@
 |}
-We see these bodies, but can they see each other? Let $t$ be the time that it takes for light to travel to us from A and B. The time it takes the light to travel from A to B, or from B to A, is obviously $2t$. Hence when the Universe is older than $3t$, we not only see A and B, but they also see each other. If the Universe is younger than $3t$, and older than $t$, we see A and B, but they cannot yet see each other. There is thus a maximum distance beyond which the observed bodies A and B do not know of each other's existence. By examining Figure \ref{Conf-Harr8}, we see that this maximum distance is one third of the distance to the particle horizon. The answer to our question is that bodies at opposite directions and equal distances from us, which are larger than one third of the distance to the particle horizon, cannot at present see each other.
      </p>
@@ Line 133: / Line 136: @@
 |[[File:Conf-Harr18.png|center|thumb|365px]]
 |}
+<div style="text-align:left;">
 * The world line X is at the particle horizon. Notice the existence of an event horizon.
 * The particle horizon is at world line X, and no event horizon exists.
 * Only the event horizon exists in this case.
 * There are no particle or event horizons.
+</div>
      </p>
    </div>
@@ Line 151: / Line 154: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
-      There are universes that expand forever and yet have endings in conformal time. The de Sitter is of this kind and therefore has event horizons. The figure (\ref{Conf-Harr22}) shows such a universe in a spacetime diagram of comoving space and cosmic time
+      There are universes that expand forever and yet have endings in conformal time. The de Sitter is of this kind and therefore has event horizons. The figure (\ref{Conf-Harr22}) shows such a universe in a spacetime diagram of comoving space and cosmic time.
 {|class="wikitable" style="margin: 1em auto 1em auto;"
 |[[File:Conf-Harr22.png|center|thumb|365px|A universe with an end but no beginning in conformal time, drawn in terms of cosmic time.]]
 |}
      </p>
    </div>
@@ Line 172: / Line 177: @@
      <p style="text-align: left;">
+<div style="text-align:left;">
 * The necessary condition for the existence of a particle horizon is that conformal time has a beginning (see items a) and b) of problem 7). The observer's lightcone stretches back and terminates at the lower boundary where the universe begins. When conformal time has no beginning, there is no lower boundary (see items c) and e) of problem 7). In this case the lightcone stretches back without limit and intersects all world lines in the universe. In these universes there are no particle horizons. Note that beginning in conformal time does not necessarily mean beginning in ordinary time.
 * The necessary condition for the existence of an event horizon is that conformal time has an ending. The event horizon in cosmology is thus nothing more than the observer's ultimate lightcone at the end of conformal time. All the events inside the event horizon (the ultimate lightcone) are at some time observed, and all events outside are never observed. Note that an end in conformal time does not necessarily mean an end in ordinary time.
      </p>
    </div>

@@ Line 55: / Line 55: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
-      If the Universe is eternal and galaxies shine forever, no event horizons exist. However, it does exist in a Universe that lives for some finite time (has an ``end''). For an observer in such a Universe the event horizon is the lightcone built on its worldline at the last possible moment.
+      If the Universe is eternal and galaxies shine forever, no event horizons exist. However, it does exist in a Universe that lives for some finite time (has an "end"). For an observer in such a Universe the event horizon is the lightcone built on its worldline at the last possible moment.
-\begin{figure}[hb]
+{|class="wikitable" style="margin: 1em auto 1em auto;"
-\center
+|[[File:Conf-Harr4.png|center|thumb|365px| Evolution of horizon with time.]]
-\includegraphics*[width=0.6\textwidth]{Conf-Harr4.pdf}
+|}
-\caption{Stationary Universe with an ending}
-\end{figure}
 Inside this ultimate lightcone are the events that have been seen by the end of the Universe, and outside are the events that can never been seen. The reason is that the lightcone cannot advance farther into time and all events outside of it remain unseen.

← Older revision		Revision as of 22:15, 25 September 2013
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	Space-time diagram is a representation of space-time on a two-dimensional plane, with one timelike and one spacelike coordinate. It is typically used for spherically symmetric spacetimes (such as all homogeneous cosmological models), in which angular coordinates are suppressed.		Space-time diagram is a representation of space-time on a two-dimensional plane, with one timelike and one spacelike coordinate. It is typically used for spherically symmetric spacetimes (such as all homogeneous cosmological models), in which angular coordinates are suppressed.
		+
		+	__TOC__
		+
		+	<div id="Horizon1"></div>
		+	=== Problem 1 ===
		+	Draw a space-time diagram that shows behaviour of worldlines of comoving observers in a
		+	# stationary universe with beginning
		+	# expanding universe in comoving coordinates
		+	# expanding universe in proper coordinates
		+
		+	<div id="Horizon2"></div>
		+	=== Problem 2 ===
		+	Suppose there is a static universe with homogeneously distributed galaxies, which came into being at some finite moment of time. Draw graphically the particle horizon for some static observer.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">
		+	The worldline $O$ represents our Galaxy from which we observe the universe. At the present moment we look out in the space and back in time and see other galaxies on our backward lightcone. Worldline $X$ determines the particle horizon. Objects (galaxies) beyond $X$ have not yet been observed by the present moment.
		+	{\|class="wikitable" style="margin: 1em auto 1em auto;"
		+	\|[[File:Conf-Harr1.png\|center\|thumb\|400px\| Stationary Universe with a beginning.]]
		+	\|[[File:Conf-Harr2.png\|center\|thumb\|365px\| Evolution of horizon with time.]]
		+	\|}
		+	</div>
		+	</div>