The following is from an email I sent to Sabine (which she didn't reply to):
In your video "Is faster-than-light travel possible?", you say that the causality-violation,
grandfather paradox objection to such travel is nonsense, rubbish, since it is based on a
confusion about the direction of time, that if we have a consistent direction of time, there is no such paradox. You claimed the confusion is that the paradox says that by traveling FTL you are going backward in time in some inertial coordinate frame IF, but nevertheless are getting older, so your entropy is increasing. As far as I can tell, your argument involves this Claim: By traveling backward in time in any inertial frame IF you would be going in the time direction of decreasing entropy in IF, since entropy decreases in the backwards time direction in each IF. This Claim is self-contradictory, for somewhat the same reason that if you are traveling FTL in some frame IF1, you are, according to special relativity, traveling backward in time in some other frame IF2. Specifically, if at point p1 along your FTL path you are, in frame IF1's time, earlier than you are at point p2, so the entropy (of some closed subsystem S) at p1 would be, according to Claim, less than it is at p2, in some other frame IF2's time, p1 will be later than p2, so the entropy of S at p1 would be, according to Claim, greater than it would be at p2 (contradiction).
While, as shown above, the entropy (of S), or any other (single-valued) scalar function of
all space-time points cannot be an increasing function of (frame) time everywhere in each
inertial frame, and so it wouldn't be true for all inertial frames IF that travel backward in time in IF would be travel in the direction of decreasing entropy everywhere (at the same frame time) in IF, it is true, assuming that the 2nd law of thermodynamics holds
everywhere and always along each time-like curve, and so entropy of S in a past light
cone would be less than the entropy of S in the corresponding forward light cone, that
travel into your own past light cone would involve travel to a s-t region where your own
entropy would be lower than before reaching the past light cone. However, this no more
shows that travel into the past with FTL travel would be impossible than does the fact that in such FTL travel one's own proper time is going in the opposite direction to the frame time of each frame in which one is traveling backward in time. It is assumed that along your FTL path going backward in time in some frame, if you could travel such a path, in your own proper time, as your mental time increased, i.e., as the time seemed to get later, according to a popular theory of the relation of mental time direction to the temporal direction of entropy increase, your own entropy would increase, while the entropy of most other things, those not traveling with you, would decrease. This is just one of the peculiar aspects of time travel into the past, but is not a strict contradiction of natural law even if the 2nd law applies with the time it refers to being the external frame's time, since, as Maxwell emphasized, the 2nd law is only a statistical law, holding only almost all the time, for large systems, but not always. More importantly, with FTL travel into the past, the time to use in the application of the 2nd law to the time traveler would clearly be her/his own proper time, not the time of the frame wrt which he/she was traveling backward in time, so his/her-her/his entropy could increase while the entropy of the things wrt whoever was traveling FTL was decreasing, without contradicting the 2nd law.
Your 2nd law argument that I just, I think, demonstrated to be faulty was an argument
against the causality-violation/grandfather paradox's showing that FTL travel would result
in time travel into the past in some inertial frame, so could lead to causal contradictions, and so FTL travel is impossible. The causality-violation/grandfather paradox does involve travel by something into the past light cone of something, neither necessarily a person, but the argument for it is based more-or-less rigorously on the generally agreed-upon local causal structure of space-time. It shows that FTL travel could result in travel into the past. If the argument conflicts with the 2nd law of thermodynamics, which it doesn't, so much the worse for the 2nd law.
The causality objection assumes a time-orientable (that there exists a continuous choice
of light-cones to be the future light-cone) universe, which I think you would agree this
universe is, at least locally. Also, it is stated for a region U of space-time which is causally iseomorphic (bicontinuously isomorphic) to a convex open region of Minkowski s-t. This could be time-oriented by the local temporal direction of the increase of entropy if one of the 2 Minkowski light-cone orientations is everywhere in U the same as that of the local entropy increase orientation. The causality-violation argument against FTL travel applies to FTL travel of any controllable signal, including sending a person, which could be sent; it doesn't apply to the signal sent by a spin or polarization measurement of one of the pair to the location of the other of the pair for an entangled pair which is in the singlet state used in tests of locality via the Bell inequality, since these signals are uncontrollable by the sender. (Many people deny that a signal is sent in this situation, even though there is a correlation between the results of the distant measurements which cannot be explained by a local theory (either deterministic or probabilistic). They are wrong. What signal is sent is decided by inanimate nature, however, rather than the experimenter. Probably relevant to this is a Japanese conference paper titled approximately "Controllable and uncontrollable signaling", by Abner Shimony, which I haven't been able to obtain, but might yet if I try harder.) All this is gone into in greater detail in my unpublished paper "A possible severe conflict between quantum mechanics and special relativity", available at https://www.logic-physics-settheory-math.com/conflict-between-qm-and-sr.
The causality argument against the possibility of FTL travel doesn't depend on any
judgement of which of 2 spacelike separated s-t points a or b is earlier, except for using a time-oriented region U, or on whether in traveling a s-t path in one path-direction something is going forward or backward in time. According to special relativity, if a & b are 2 s-t points, whether a or b occurs earlier is not physically significant. The causality argument against FTL travel uses only the assumption that for some pair of spacelike separated s-t points, a controllable action at one of them guarantees a specified response at the other. This is what is meant by being able to send a controllable signal FTL. The argument proceeds as follows:
Suppose, for some inertial frame IF, signaling FTL wrt IF from one s-t point to some other
space-like related s-t point were possible. Then, by the principle of special relativity & the Lorentz transformations, it would be possible, for each inertial frame IF, to signal
instantaneously wrt IF from each s-t point to every other space-like related s-t point. In
such a case, signaling from some s-t point (x,t) into the past of (x,t), & next preventing that signaling, could be arranged as follows: Send a signal instantaneously, in some inertial frame IF, from (x,t) to some (y,t), with y =/ x, from (y,t) send, instantaneously in frame IF' moving away from x, a signal to some (z,t') in the past light cone of (x,t), which is possible since a simultaneity slice in IF' thru (y,t) intersects the past light cone of (x,t), next from (z,t'), by slower-than-light signaling, send a signal to a mechanism that will prevent the initial sending of the signal from (x,t) to (y,t).