Does length contraction "break the speed limit"?

Question

I've been trying to learn some special relativity, and while trying out a thought experiment I hit a paradox that I don't understand how to resolve. Can someone help me understand where I'm going wrong?

I'll start with the part I'm pretty sure I understand. I'm on Earth and want to visit Zorbulax, a planet 100 light-years away which happens to be perfectly at rest compared to Earth. I take off, my powerful spaceship's engine applying enough force to accelerate me to, say, .1% the speed of light in just one second, and continuing to apply that force steadily.

From the perspective of Earth and Zorbulax, time slows down for me as I accelerate, which means the energy I'm outputting accelerates me less and less, and I approach, but never pass, the speed of light. Let's say I continue accelerating until Earth sees my time dilation to be 10x, which I think should take an hour or so.

Now the part I'm less sure about: What about my perspective? If time is supposed to be much slower for me, and Zorbulax and Earth are moving at near-light speed relative to me, should it seem to be moving faster than light from my reference frame? My understanding is that this is resolved by length contraction. Since, from my perspective, Earth and Zorbulax are moving at near-light speed, their lengths are contracted - Not only the lengths of the planets, but the distance from Earth to Zorbulax as well (since I'm moving in that direction), so I don't actually see them moving faster than light - if I'm experiencing time at ~1/10th speed from the perspective of the planets, then from my perspective I see their distance to be ~1/10th as far, so they are still moving a little under light speed.

Now this is where I get really confused: If the distance from Earth to Zorbulax has been contracted to a small fraction of what it was when I shared their reference frame, and the Earth is behind me, doesn't that mean that Zorbulax is now only 1/10th of its original distance from me? But doesn't that mean I've changed from having Zorbulax 100 light years away to less than 10 light years away (in my reference frame) in only an hour or so? I know these are only snapshots of location and time, but surely some version of the mean value theorem must then imply that, from my perspective, it had a velocity greater than the speed of light?

I know I've introduced acceleration into my reference frame here - so is there something going on here with general relativity (and if so, what)? Or am I misunderstanding something more fundamental? Or is my handwavy math incorrect and leading me astray? Any help that can lead me into some insight would be greatly appreciated!

Answer 1

Stand on the earth, facing in a direction so that Zorbulax is 100 light years behind you.

Now, over a period of say, a tenth of a second, spin 180 degrees in place, so that Zorbulax is 100 light years ahead of you.

Did Zorbulax just move 200 light years in a tenth of a second? Did it violate the cosmic speed limit?

Answer: Of course not. Zorbulax did nothing unusual at all. All that happened was that you changed frames so that your description of Zorbulax's location changed dramatically.

When you spin in place, you change frames. When you change from one velocity to another, you change frames. In either case, the new distance from you to Zorbulax can be dramatically different in Frame B than it is in Frame A. If relativity denied that possibility, it would have been obviously wrong from the beginning, because for centuries before Einstein, people had clearly understood that frame changes (e.g. the consequences of spinning in place) can happen arbitrarily fast.

Answer 2

As WillO's answer says, all that happened is that your frame of reference changed, just like what would happen if you turned around. But in the linear acceleration case, the rotation is happening in the $xt$-plane rather than the $xy$-plane.

There are two events to consider: $A$ is the event where you leave Earth and $B$ is the event where you land on Zorbulax. From the stationary observer's frame, these events are separated by a vector with components $B - A = (\Delta x, \Delta y, \Delta z, \Delta t)$. From the traveller's frame, they are separated by $B' - A' = (\Delta x', \Delta y', \Delta z', \Delta t')$.

In this case spatial distance $\Delta x^2 + \Delta y^2 + \Delta z^2$ is not preserved, instead what is preserved is the whole spacetime interval $\Delta s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$.

Due to Lorentz contraction, the moving observer sees $\Delta x'^2 + \Delta y'^2 + \Delta z'^2 \lt \Delta x^2 + \Delta y^2 + \Delta z^2$, so preservation of $\Delta s^2$ implies that the observer also sees a corresponding decrease in $\Delta t'^2$ (which is why the trip is shorter from the moving observer's point of view).

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"But isn't the traveller seeing Zorbulax moving faster than light during the acceleration phase, in order to achieve Lorentz contraction?"

Even though Zorbulax seems to visually get closer to the Earth faster than light, it is not the "same" Zorbulax, as it is also moving through time (the Zorbulax that is seen by the accelerating observer is aging much faster than the Zorbulax seen from the Earth). If you were looking at an alien living on Zorbulax at the start of your trip, you'd see him age faster while you accelerate and eventually you'd see his entire life go by, and by the time you reached Zorbulax, at least 100 years will have passed so that alien might already be dead. That alien living on Zorbulax never met you, he's separated from you by time, so you cannot apply the Mean Value Theorem to conclude that the alien moved faster than light at any point.

Answer 3

The claim that nothing can travel faster than light means that, for each inertial reference frame, nothing travels faster than light in that reference frame. Each reference frame has a coordinate system, and the change of the $x$ coordinate in that coordinate system divided by the change of $t$ in that coordinate system is less than $c$. That is, $\frac{x_f-x_i}{t_f-t_i}<c$.

If we try applying inertial reference frames to your situation, then we aren't comparing $x_f$ to $x_i$. We're comparing $x'_f$ to $x_i$. That is, we're comparing what we end up recording your $x'$-coordinate being in one coordinate system to what your $x$-coordinate started out being in another coordinate system.

Imagine at time $t=0$ sec you start measuring the temperature in Celsius at you get 20, and then at time $t=10$ sec you measure the temperature in Fahrenheit and you get 70. Was the temperature rising at 7 degrees a second? Of course not. It's the same thing with relativity. Just because you measured a distance of $100$ ly in one coordinate system at the start and then $10$ ly later on, that doesn't mean you traveled $90$ ly.

Another analogy: suppose you're in a boat going across a river. You look straight ahead, and look at how far the other side of the river is, and measure the distance as being $1000$ m. You then turn your boat, and look straight ahead again. But now "straight ahead" is a different direction, so rather than measure the distance perpendicular to the shore, you measure a diagonal distance, and get $1500$ m. Did you travel $500$ m? Of course not.

When calculating velocity, you need to use the distance traveled in one reference frame, divided by time in that same reference frame. You can't use time from a different reference frame, and you can't use subjective time experienced (which for an accelerating observer doesn't correspond to time in any inertial reference frame).

If you were to analyze this situation in a co-moving reference frame, then that reference frame wouldn't be inertial, so the $c$ speed limit wouldn't apply. You be using Rindler coordinates, and you can get velocities greater than $c$ in Rindler coordinates.