The size of this circle is c*t where c is the speed of light and t is the time since the big bang. Now as an organism living on the paper, we would see things in a circle around us, and the size of the circle would be determined by the maximum distance from which light will have reached us, as discussed in terms of ants above. So from this small point like clump of paper, we get a full sheet of paper almost immediately. In a matter of a fraction of a second the Universe expands to a size comparable to its size today. Very shortly after the big bang the sheet grows incredibly fast (this rapid is expansion is what we call inflation). The answer to this lies in understanding the first "trick" of the universe. Thus, we can see things at various ages simply due to the fact that it takes time for the ants (light) to reach us.īut then if this is the case, and the universe is small early on, that is nearly singular, how can we see light coming from billions of light years away? Thus, all parts of the Universe are the same age, however we see things farther out as younger since the ants are more and more delayed in reaching us to tell us about them. Note that in spite of the ant coming at t=3 to tell us about the galaxy at t=0, the galaxy it started from is actually also sitting at t=3 since all the clocks are ticking. The ant that arrives at this time tells us about how the galaxy it started from looked when it set out (i.e. Thus, only after three seconds do we know about a galaxy 30 cm away, as the first light (the ant) from that galaxy takes that long to reach us. If the sheet doesn't expand an ant 10 cm away reaches us in 1 second, an ant 20 cm away arrives after two seconds etc. Suppose further that each ant can move 10 cm per second. Suppose that we choose a drawing on the paper to represent us, and suppose then that each other drawing has an ant that starts walking towards us at time =0. Run time forward a bit so that the sheet can expand to some non-singular state, and then let's pause expansion to look at some simple things.Ĭonsider an ant walking on the page to be a photon (a piece of light). Let us assume we can place a clock at each point on the grid, and of course set them all going from time=0 at the instant of the big bang. So now that we have the big bang setup, we can run time forward to address the question. We believe the universe to be infinite today, which means it was infinite in its formation so this infinite singularity is the picture to have in mind for the Big Bang. Note also we could have done this for an infinite sheet of paper, that is started with an infinite one with a nice well defined grid and then let it contract until all points on the grid were once again overlapping and thus we had an infinite singularity. Notice that each point on the grid has the big bang occurring there so to speak as all points are singular at this time. Thus, our space-time has become singular. Even though we know there is a grid on it that we can use to identify each point on the paper, it looks like the grid has become a point too (from our vantage point outside the paper) and each grid point is overlapping with the others. If we do this for a while the paper shrinks to a point. Each cell shrinks and things appear to move closer and closer. So lets us now instead run time backwards and compress the paper. This is to say comoving coordinates are fixed but physical separations change. We would see that while galaxies remain at their locations on the grid, they do seem to get further apart since the grid itself is becoming larger (each cell of our stretchable graph paper would expand). When we speak of an expanding universe this is exactly what we mean, the very space-time on which our universe is "drawn" is stretching, dragging each "drawing" with it. Now since the universe is expanding, we should be stretching the sheet of paper as time moves forward. (The coordinates on the grid will be the so called comoving coordinates, since the grid will expand with the paper, while the ruler will give physical distances since the ruler won't expand). Their coordinates can be given by their location on the grid, or we could use a ruler to specify distances to them. Now let's imagine this sheet represents the Universe today, and so let's draw some galaxies on the sheet. Let us also draw a grid on the paper (so that it is a sheet of stretchable graph paper). To begin, consider a sheet of paper, one that we can stretch and compress (i.e. To answer this let us construct an analogy to the universe. How is it possible that the light emitted billions of years ago is still reaching us today? How can objects be 13 billion years away by some few hundred thousand years after the big bang?
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