Can you accelerate to almost the speed of light on a smooth, curved path? So, many of the time dilation thought experiments have periods of accelerating away from the origin, stopping, turning around, and returning, which precipitates an observable time differential between the traveler and those remaining behind.


*

*Is there a requirement that the path be straight line?

*Is is possible to accelerate to almost the speed of light (or a large fraction thereof) along a curved path (say a circle or ellipse) rather than "out and back"?


I'm happy to put aside the realities of the movement of the universe (the planet won't be in the same place to return to later on) for this thought experiment insofar as it simplifies the question.
My intuition says it must be straight, but I cannot articulate why, my physics is rusty.
 A: An element of acceleration takes a body from one inertial frame to another.  With some assumptions, widely accepted, all the relativistic effects are simply the differences in Lorentz contraction, time dilation and relativity of simultaneity between the two frames.
Conventionally, the starting frame is the rest frame.
An observer in a frame moving relative to that 'rest' frame, where that relative motion is not in line with the acceleration, will see the accelerating object follow a curved path. 
Things change again if the accelerating observer's acceleration changes direction.
A: The path does not have to be a straight line. Any smooth curve in 4-dimensional spacetime can be decomposed into a number of arbitrarily small segments, each of which is a straight line in the limit when the number of segments approaches infinity. So the full effects of such a motion can be calculated by integrating over these segments.
A: If one wishes to nitpick (and I do), any acceleration is a curved path in four-dimensional space-time. If there is curvature in an object's path in three-dimensional space, that means there is a component of acceleration perpendicular to the object's path.
It certainly is possible to accelerate perpendicular to one's path, but the curvature that results is proportional the acceleration divided by the square of velocity. This means that for an object traveling close to light speed, the acceleration need for even a modest amount of curvature would be massive. 
How massive? Well, the event horizon of a black hole consists of the surface such that if a light ray were traveling along it, it would be kept on that surface by the force of gravity. So, basically, if you have an object that's going close to light speed, then to keep it in a circle, you need a force comparable to a that of a black hole whose event horizon is the same size as that circle.
A: To create the effect illustrated by the twin paradox, there is no requirement for the relative motion to be linear, or, for that matter, for it to be limited to smooth acceleration. However, the more complicated the motion, the more complicated the calculations, so linear motion is is often considered as it is the simplest case to model and may be the simplest to envisage.
