Difference between revisions of "Simple English"

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.

Stationary Universe with a beginning.

Evolution of horizon with time.

At present the observer $O$ sees no farther than $X$ (see Fig. \ref{Conf-Harr12}). At some later moment he sees beyond $X$ up to some worldline $Y$. The particle horizon thus recedes in the static universe and as time passes the part of the Universe we observe grows ever larger.

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.

Evolution of horizon with time.

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.

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}). \begin{figure}[!htb] \center \includegraphics*[width=0.6\textwidth]{Conf-Harr6.pdf} \caption{\label{Conf-Harr6} The horizon riddle.} \end{figure} 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.

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}. \begin{figure}[!hb] \center \includegraphics*[width=0.6\textwidth]{Conf-Harr8.pdf} \caption{\label{Conf-Harr8} Two observers unaware of each other's existence by the time seen by some third observer.} \end{figure} 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.

See Figure \ref{Conf-Harr15-18} \begin{figure}[htb] \center \includegraphics*[width=0.48\textwidth]{Conf-Harr15}\hfill \includegraphics*[width=0.48\textwidth]{Conf-Harr16}\hfill \includegraphics*[width=0.48\textwidth]{Conf-Harr17}\hfill \includegraphics*[width=0.48\textwidth]{Conf-Harr18} \caption{\label{Conf-Harr15-18} Spacetime diagrams for universes with and without beginnings and endings.} \end{figure}

• 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.

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 \begin{figure}[hb] \center \includegraphics*[width=0.5\textwidth]{Conf-Harr22} \caption{\label{Conf-Harr22} A universe with an end but no beginning in conformal time, drawn in terms of cosmic time.} \end{figure} The event horizon is as shown. As the moment now advances into the infinite future, the observer’s backward lightcone approaches the event horizon more and more closely. For example, the observer O at moment $O$ sees event $a$. As the moment of observation advances into the unlimited future, the lightcone moves upward and approaches more and more slowly but never reaches the event horizon. The event horizon is the observer’s lightcone in the infinite future. Events outside this horizon can never be observed.

• 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.

Yes, it is possible. The corresponding spacetime diagram is shown on \ref{Conf-Harr10}. Worldlines branch out radially in all directions from the ``big bang. Spatial slices of constant cosmic time are represented by spherical surfaces perpendicular to the world lines, while time is measured along the radial worldlines. An arbitrary worldline is chosen as the observer and labeled $O$. At some instant in time -- let it be ``now -- the observer's lightcone stretches out and back and intersects other worldlines such as $X$ and $Y$. Because of the expansion of space, the lightcone does not stretch out straight as in a static universe, but contracts back into the big bang. All worldlines and all backward lightcones converge into the big bang. \begin{figure}[hb] \center \includegraphics*[width=0.48\textwidth]{Conf-Harr10} \caption{\label{Conf-Harr10} The reception and emission distances of galaxies UNIQ-MathJax30-QINU and UNIQ-MathJax31-QINU. Although galaxy UNIQ-MathJax32-QINU has a greater reception distance, its emission distance is smaller than that of UNIQ-MathJax33-QINU. Thus UNIQ-MathJax34-QINU, which is now farther away than UNIQ-MathJax35-QINU, was closer to us than UNIQ-MathJax36-QINU at the time of emission (which is different for UNIQ-MathJax37-QINU and UNIQ-MathJax38-QINU) of the light we now see.} \end{figure}

A spacetime diagram of comoving space (in which all worldlines are parallel) and cosmic time is shown on Fig. \ref{Conf-Harr13}. Some universes have particle horizons and in their case the lightcone stretches out and back to the beginning at a finite comoving distance indicated by the world line $X$ . In universes without particle horizons, the lightcone stretches out to an unlimited distance and intersects all world lines. \begin{figure}[hb] \center \includegraphics*[width=0.48\textwidth]{Conf-Harr13} \caption{\label{Conf-Harr13} Lightcones in universes with and without particle horizons.} \end{figure}