Originally posted by terwilligerjones
OK...I'm a certifiable idiot, and an old one at that, but this has nagged at me for decades and perhaps someone here could explain it to me:
If I were to accelerate towards the speed of light, I would perceive other, non-accelerating, frames of reference as foreshortened in the direction of my travel, right?
Let us suppose that F1 is an inertial reference frame. In fact, let us define a rectangular coordinate system in F1, whose origin is the center of mass of the universe (CMU). Now, let the axes of this coordinate system not be spinning with respect to the stars. Suppose that at the beginning of your journey, you are at rest in F1. Let |v| denote your speed in F1. Now by Newton's first law an object at rest will remain at rest, unless acted upon by an outside force. Now you turn on your engines, and there is a force on your ship, and your ship begins to accelerate in F1, according to Newton's second law.
Eventually you turn off your rocket engines, and you now are just coasting at a constant speed |v| in F1. Now, let L0 denote the length of your ship in F1 at the beginning of your journey (when you were at rest in F1). According to the theory of relativity, your length in F1 is now less than L0. The formula for your current length L in reference frame F1 is:
L = L_0 \sqrt{1-v^2/c^2}
Now, let us suppose that there is a clock located at the CMU, which ticks at regular intervals. Suppose that an identical clock was on your ship at the beginning of your journey (when you were at rest), thus at the beginning of your journey, your clock ticked at the same rate as the CMU clock.
Now, according to the theory of relativity, your clock no longer ticks at the same rate as the CMU clock, it ticks slower. The formula which relates an amount of time of an event in the CMU frame, to the amount of time for the same event according to the ships clock is:
\Delta t = \frac{\Delta t'}{\sqrt{1-v^2/c^2}}
So suppose that the units of the CMU clock are seconds, and that the time of an event in the CMU frame is exactly one second. The time of the same event in the ships frame can be found from the following equation:
1 = \frac{\Delta t'}{\sqrt{1-v^2/c^2}}
Thus, an event which takes one second in the CMU frame now takes:
\Delta t' = \sqrt{1-v^2/c^2}
So for example, suppose that delta t' is 1/4 of a second. It follows that your ship accelerated to a final speed |v| given by:
1/16 = 1-v^2/c^2 ---> v^2/c^2 = 1 -1/16 = 15/16
---> v^2 = 15c^2/16
which implies that
v = \sqrt{15c^2/16}
Which as you can see is less than the speed of light c=299792458 meters per second.
So the point is, that the rate at which your ships clock ticks, is no longer the same as the rate at which the CMU clock ticks, and in fact your clock is ticking slower, the CMU clock is ticking faster. If you were to reach the speed of light, your clock would stop ticking, and so according to the theory of relativity, no ship can be accelerated from rest to the speed of light.
Ok so we begin with all of that. Now your question is kind of vague. You are asking about other reference frames being shortened. Here is how I understand you (correct me if I am wrong later).
Suppose that you started out at rest, and then accelerated towards our sun. Once you turned off your engines, you were at rest in an inertial frame F2. Now in F2, the sun is coming at you with speed |v|. And so I think your question is about the distance which the sun is away from you. You are asserting that it has length contracted in frame F2. So that if the distance from the sun to the origin of F2 is 1000*299792458 meters in a "sun at rest" system, that the distance the sun is away from you in frame F2 is less than 1000*299792458 meters, because of distance contraction. (you are asking yes or no)
Well, suppose that the distance you are away from the sun the moment you turn off your engines in your system is also 1000*299792458 meters. Now consider a photon fired from the sun at you. In the suns frame, a clock there will tick out 1000 seconds as the time it takes the photon to reach you. But, your clock is supposed to tick slower than a clock at rest with respect to the sun. Thus, the time it takes the photon to reach you should end up being less, and so you won't come up with the same speed of light, it will travel the same distance it traveled in the suns frame, in less time, and so you will think the speed of light is greater than c in your frame. According to relativity, this is not possible, and so the distance from you to the sun in your frame must be contracted, so that in your frame that distance is less than 1000*299792458 meters. So the answer to your question is yes, if the theory of relativity is correct.
If I were traveling with a beam of light, then, would I not perceive the universe of my travel as a plane perpendicular to my "direction" of travel? And if I did, could I have any notion of motion, reflection, refraction, direction or any of the myriad other qualities that are observable about a beam of light? Stated another way, would not I perceive the beginning and end of my journey to be simultaneous with every point through which I passed, none of which I could perceive?
That just doesn't sound right...there shouldn't be one frame of reference (mine) where no motion is detectable, and others where it is.
Sorry if that's a dumb one, but...
terwilliger
It isn't a dumb question, but it sure is a confusing one. Certainly if clocks in your frame don't tick, then there is no passage of time in that frame, and so hence no motion is detectable, and yes that is very strange.
I think your first question is more interesting though, about the length contraction.
