The curve she shows is not a closed timelike curve. There are no CTCs in special relativity. A particle going faster than light moves on a spacelike curve. I'm surprised she'd get that wrong.
FWIW I'm not buying her argument that the comoving frame represents a preferred frame in SR either. She glosses over quite some fundamental questions there. I'd love to see this written up as a proper paper.
No OP is correct, her use of the term CTC is not what is usually meant. The curve she draws is not timelike along the faster than light segments in the sense of having tangent 4-vector with timelike norm with respect to the underlying metric. She would not disagree I think.
You are incorrect on this point. A spacetime interval is defined as:
ds^2 = c * dt^2 - dx^2 - dy^2 - dz^2
If ds^2 > 0 then the interval is timelike.
If ds^2 = 0 then the interval is lightlike.
If ds^2 < 0 then the interval is spacelike.
Let's just focus on an object moving strictly along the x-axis faster than the speed of light, this results in a spacetime interval as follows:
ds^2 = c * dt^2 - dx^2
Since the object is moving faster than light, the dx^2 term would exceed the c * dt^2 term, resulting in a ds^2 < 0 and hence spacelike rather than timelike curve.
I disagree. A curve can be closed that is either timelike or spacelike. The interest in timelike curves is that it hypothetically can result in time travel whereas spacelike curves do not describe the trajectory of physical objects. Closed spacelike curves can be used to describe the boundary between causally connected events.
Within the context of special relativity, an object travelling faster than light would be travelling along a spacelike curve, not a timelike curve. The significance of general relativity is that there are solutions that allow for the trajectory of an object to move along a closed timelike curve without exceeding the speed of light.
Uhh, no, they are closed and timelike. It's in the name.
The whole point of CTCs is that in exotic spacetime geometries, in particular wormhole-like structures, there can be curves that are entirely timelike, meaning that the object never moves faster than light locally, and are closed at the same time, meaning the object ends up at the same point in time from where it started.
In SR, a closed curve requires at least some spacelike sections of the curve on which the object moves faster than light.
FWIW I'm not buying her argument that the comoving frame represents a preferred frame in SR either. She glosses over quite some fundamental questions there. I'd love to see this written up as a proper paper.