[addition after timeout:]
Above, in the first question, should read “… multiple shorter back and forth trips.”
Since reading Asymptotically’s post partially cited above, I am operating under the image of the rocket simply killing time at ~0.999 c as physics goes about its mysterious business. I’m not comfortable with that, though.
It may well be that the author of your textbook simply doesn’t understand special relativity. This is unsurprising, since a great many people, even physicists, don’t understand it, but at the same time a shame, since it really isn’t fundamentally difficult to understand SR (unlike some other modern physics theories I won’t name). Overall, though, relativity forms a small enough part of most introductory physics courses (it’s usually the last topic covered, and the first dropped when other topics inevitably run over schedule) that this is perhaps forgivable, if the author is good enough at explaining the other topics. Though on the other other other hand, this is probably a large part of why relativity is so often misunderstood in the first place.
Not just calculus – differential equations. {shudder}
There’s no inherent limit to the minimum radius of a circular path from special relativity or the minimum distance of any path. Mass doesn’t directly affect reference frames as we are still in the realms of special rather than general relativity.
Perhaps it’s better if I explain why spatially-extended run into trouble by examining the problems encountered.
Firstly above I explained how you can approximate the frame of an accelerated observer by breaking it down into a series of inertial frames and summing their contributions and you can describe the experience of an accelerated observer exactly by taking the limit of increasingly better approximations and further that the tools of calculus means we don’t even have to construct these approximations to derive the limit (see the definition of an integral to see why this is). We do this by taking the inertial frames that are co-moving at any given moment (MCRFs for short- momentarily co-moving rerference frames- the frames that at a given instant our accelerated observer is at rest in) and integrating over their contributions.
Now consider what exactly it is an accelerated observer sees when observing, there is no problem working this out as we can work out the amount of time elapsed from the p.o.v. of an accelerated observer between receiving light from distant events. So actually constructing the experience of an accelerated observer is ‘no problemo’, but a ‘frame of reference’ is distinct from the actual observations of an observer as to construct a reference frame we factor in the time delay and the distance between the emission and receiving of light. To take an example : Terrell rotation is an observed effect in inertial frames of reference , though the object that is Terrell-rotated is not actually rotated in the inertial frame of reference. In a spatially extended reference frame we take into account the delay between the emission and receiving of light from distance events and use this to assign those events temporal and spatial coordinates (e.g. "I wasn’t at the start of the party, but it started at 8pm [temporal coordinate] at 53 West Street [spatial coordinate(s)]). In inertial frames of reference as light axiomatically has a constant velocity there is no problem assigning distant events temporal and spatial coordinates, however as the constancy of the speed of light only applies to inertial frames of reference we can’t make the same assumption in accelerated frames of reference.
One way we might think of constructing a spatially extended frame of reference is to use the spatial and temporal coordinates from the MCRFs and indeed this is a good way of assigning spatial and temporal coordinates to events that are local or ‘nearby’ (how nearby exactly depends on the details of the motion of the accelerated observer), however for more distant events we run into a problem with using MCRFs due to the relativity of simultaneity as there may be a significant change in velocity between receiving and emitting the light which means that multiple spacetime coordinates can be assigned to the same event which can be drastically different and the different coordinates may not even agree on the ordering of distant events. So this is why we say that accelerated reference frames are local: i.e. because an accelerated observer can always be approximated well by an inertial observer when the time and distance intervals are appropriately small (the definition of ‘small’ depending on exactly how the observer is accelerating).
“Okay then,” we may say “perhaps there is a better way of constructing a frame of reference for an accelerated observer?” and there are other ways, however they all suffer from the problems of either unwanted coordinate artifacts and/or non-uniqueness/arbitrariness. For example, a spatially-extended inertial frame of reference can be seen as being constructed out of a set of inertial observers who are all at rest wrt to each other, similarly we may wish to eliminate the bad behavior in the time coordinate by constructing a coordinate system out of accelerated observers who are accelerating in the same way so that they are always at rest in the MCRFs of the accelerated observer whose reference frame we are constructing. However straight away we run into a problems 1) if the acceleration is not linear then the direction of the accelerated observers relative to each other will change 2)even if the acceleration is linear, even though from the point of view of any inertial observer these accelerated observers are always travelling with the same velocity relative to each other and maintaining a constant distance, due to length contraction and the fact that the observers are constantly changing MCRFs the distance between them in the MCRFs is changing (the effects of this can be seen in Bell’s spaceship paradox), so we still have the problem of which distance coordinate to use.
Wolfgang Rindler came up with a very important method of constructing a spatially-extended spacetime coordinate system which can be used to define the frame of reference of a linearly accelerated observer called Rindler coordinates. The is constructed out of observers who are linearly accelerating such that their distance doesn’t change at any given time from the p.o.v. of the MCRF of the accelerated observer (whose reference frame we are constructing) at that time. From the p.o.v. of an inertial observer the Rindler observers, apart from when they are all at rest are travelling with different velocities (as they have different accelerations) and therefore they each have different experinces of the passage of time, however the time coordinate is chosen to be defined by the Rindler observer whose frame we are constructing. However Rindler coordinates have ‘problems’: the Rindler observers that trailing (from the pov of an inertial observer) the Rindler observer whose frame we are constructing have to accelerate increasingly more the further behind they are and at a finite distance behind this observer the required acceleration diverges (i.e. goes to infinity). At this point a singularity occurs in Rindler cooridnates called the Rindler horizon (which actually is very similar to the event horizon of a black hole) at this point the coordinate system ‘fails’ (technically Rindler coordinates are a coordinate chart rather than a coordinate system as they fail to cover the whole of spacetime).
Diffy-Q was the first time I failed to get an A in a math course. It was devastating. Fortunately, I went into electrical engineering, where we cheat whenever possible.
Oh, and before I forget:
If your space ship can accelerate from a standing stop to 0.99c “more or less immediately”, you’re going to turn everybody on board into pancakes.
After nearly 4 months, I don’t think you’re going to forget that thought.
My brain uses supercooled silicon crystals that take thousands of years to complete a single thought. I figure I’ll have this sentence finished some time before the heat death of the universe.