First a warning. This is a long post.

So I realized that additional mathematical manipulation by me is just for my own edification and since the math will not play a prominent role in this post, I will finally answer.

First, let me say that the conundrum posed by Mac was a rather good one, the solution of which is non-trivial. While he was using equations inappropriately, the concept of accelerating reference frames in special relativity is quite complex. In the end, I was best able to see what was going on by using 4D (well actually 2D in this case….radius and time) spacetime diagrams. Figuring things out took a fair bit of head scratching.

Special relativity does not allow for paradoxes. It does not allow for things to happen in one frame, but not in another. That is the most fundamental tenet of relativity. Any conundrum that appears to violate this tenet stems from a misinterpretation of the problem….most notably because our intuitions treat space and time as independent things, rather than the contiguous spacetime that we really have. Relativity tells us that different observers will perceive objects and occurrences at different places in spacetime, but it does not move the objects in spacetime.

The relativity transform equations are linear in space and time, which is to say that in a new reference frame you can convert via simple equations of the form

xnew = a * x + b * t

tnew = c * x + d * t

where a,b,c,d are numbers that reflect the velocity between the two. Further, all objects in inertial frames move in straight lines in any reference frame (x,y) or (xnew,ynew). It doesn’t matter.

Personally, I rather like Brian Greene’s more recent book “The Fabric of the Cosmos” which does a pretty good job with this and explains things in a rather clear and cartoon-driven manner. There is a particular passage in there in which Chewbaca walks at low speeds but due to the velocity makes it impossible for him to see something happen at a different position. (I cleverly forgot my copy at home, so I cannot post the page number right now, but it’s in the first chapter on relativity.) (Minor editorial comment…I find the appearance of Scully, Mulder, StarWars characters and the entire Simpson clan a bit distracting, but if you can get past that, the explanations of spacetime are really quite lucid.)

Now for the acceleration. Everything I said above is true, except that accelerating objects follow curved paths in space time. Note that spacetime is still flat and Euclidean. It is only the paths that are curved.

So in the example I posed, which I repeat for clarity:

“A particle accelerator is placed a 100 lyr from Earth. Said accelerator has no motion in the radial direction. An electron, initially at rest in the accelerator, is then accelerated with a constant force until it is moving at 0.8 c, moving radially outwards. The acceleration distance is (say) on the order of meters, in the accelerator's rest frame. Discuss the velocity of the Earth, as viewed by the electron. Since the electron sees the universe contracted, you are concerned that the Earth will appear to leap towards the electron at superluminal speeds.”

There are a few things that we can figure here. First is an unappreciated fact. In order for a point in a particular accelerating reference frame to maintain a fixed distance from another point, the other point must accelerate at a different rate. This means if you have a string and try to accelerate one end, the other end of the string needs to be accelerated at a different rate in order to maintain the length of the string to be a fixed distance. Otherwise, the string will stretch and eventually break. This is in essence the tidal forces surrounding a black hole. Essentially, there is an event horizon behind the accelerating electron. The distance between the accelerating electron and the event horizon is dependent on the actual acceleration, but in the case mentioned here, the event horizon appears about an accelerator-length behind the electron. For lower accelerations, the event horizon forms further back.

Now, while the Lorentz-Fitzgerald contraction does hold, the super-luminal velocities that bother you do not occur inside the event horizon. And after the acceleration period, the event horizon recedes at the speed of light, but when finally see the Earth again, you see it is moving away at a speed of 0.8 c. Thus you never see anything moving at superluminal speeds.

Now while it is true that you do not see the Earth, because it is far away and outside the event horizon, there is a train of light in transit from the Earth. The light that was inside the event horizon continues to travel through space and arrives at the electron, with a Doppler shift that is appropriate for the velocity. There is nothing bizarre going on there.

So now we jump a bit over into philosophy. It is true that space is contracting quickly, as perceived by the accelerating electron. (Remember that in spacetime there is no problem. The proper distance (rest time times the speed of light) is the same in all reference frames.) But you don’t see the distant object. And whenever you actually can see it, you find that it is well-behaved. This is all consistent with relativity. Relativity simply tells you how different observers will perceive a situation. (And perceive in this context isn’t some wishy-washy thing. It means time and position.)

So the bottom line is the electron will never see the Earth approach at superluminal speeds.

Now the math to justify this is substantial. Rather than typing it in, I did some web snooping to find out if somebody else has typed it in. I ended up finding some sites of interest. I append the URLs below:

This is a constant acceleration simulator in special relativity that illustrates the event horizon.

http://www.astrophysicsspectator.com/to ... onSim.html

Discussion of some of the physics listed above:

http://www.astrophysicsspectator.com/to ... rated.html

Some useful mathematics for the expert or tenacious amateur

http://www.ph.utexas.edu/~gleeson/NotesChapter13.pdf
I hope this answers the question. It also should refute the comment tendered that “I actually feel you are over complicating the issue by attempting to produce a specific mathematical result.” In fact, with relativity, it is only through diligent use of mathematics that we can overcome our underdeveloped 4D intuitions.

On the other hand Mac, the query you posed was good enough to stump a number of my colleagues. The amount of time needed to figure this out was substantial….more than I had originally anticipated. Most physicists don’t have the time to put into answering these conundrums…I certainly didn’t….but since I committed, I needed to follow it through. Others won’t.

But I’m finally happy that relativity does not predict (in this case and probably ever) anything that contradicts theoretical self-consistency and perhaps even experiment, although we remain open-minded about that.

You know Mac…as much as you have thought about this, you would be well-advised to see if you could take a relativity class at a local university, preferably one of a respectable caliber. A thorough understanding of relativity would make you all the more intellectually dangerous....or maybe you’d just come over to the dark side…