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View Full Version : What does Lorentz contraction do @ 14 Bly?

Peter Wilson
2006-Mar-22, 10:20 PM
Hopefully there are a few musicians in the house that will understand this analogy. If you look down the neck of a guitar from the "nut" (where the tuning pegs are) to the "bridge," the frets get shorter and shorter. The frets on most guitar necks peter-out at about the 23rd, but in theory, there are an infinite number in the finite length of the guitar, they just get closer and closer together.

In an analogous way, if we look out in the distant universe, "measuring sticks" get short and shorter, due to the Lorentz contraction of the receding galaxies. A galaxy as large as ours could appear paper-thin if it is receding close to the speed of light.

My question is this: Is infinity contained within the 14 Bly radius of the universe? In other words, do the measuring sticks just get shorter and shorter as you approach the 14 Bly limit, in such a way that there is an infinite number of them between here and there, like the infinite number of "virtual frets" on a guitar neck?

Ken G
2006-Mar-22, 11:42 PM
The answer, perhaps surprisingly, is no it won't be infinite. The question you are really asking is, if you have the time dependence of the scale factor, call it a(t), then is the integral over time of c/a(t). That will tell you the distance away, measured in today's rulers, of the singularity. It will be effectively infinite for any a(t) rule that grows faster, in the early times, than linearly with t. But in early times the universe is radiation dominated, and this means that a(t) scales with the square root of t (if mass dominated, it would scale like t to the 2/3, not much different in effect). That means the integral of dt/a(t) scales like the sqare root of t also, and that doesn't diverge if you let the starting point go to t=0. In fact, most of the distance traversed comes later in the life of the universe, not earlier. So we wouldn't see to an infinite distance away, even if the universe had always been transparent to light. Inflation in very early times increases this distance way beyond what this simple radiation-dominated calculation gives, but it still isn't infinite. There would have to be an epoch prior to inflation where the energy density remained constant or increased as the universe expanded. That can't happen, it would effectively require an infinite cosmological constant. So no, you'll never fit in an infinite number of frets, that would be tantamount to being able to see an infinite distance (despite a finite age of the universe) if the universe had always been transparent and there were bright enough sources to see. It's a cute question though, and note that this answer also implies that time dilation will not make the universe infinitely old even if you apply our current concept of a second rather than the proper-time concept that is normally used.