I have a previous high impliciation uncertainty about this (that would be a crux?). ” you can’t accelerate enough to turn around ” seems false to me. The mathematical rotation seems like it ought to exist. The prevoius reasons I thought such a mathematical rotation would be impossible I have signficantly less faith in. If I draw a unit sphere analog in spacetime having a visual observation from the space-time diagram drawn on euclid paper is not sufficient to conclude that the future cone is far from past cone. And thinking that a sphere is “all within r distance” it would seem it should be continuous and simply connected under most instances. I think there also should exist a transformation that when repeated enough times returns to the original configuration. And I find it surprising that a boost like transformation would fail to be like that if it is a rotation analog.
I have started to believe that the standrd reasoning why you can’t go faster than light relies on a kind of faulty logic. With normal euclidean geometry it would go like: there is a maximum angle you can reach by increasing the y-coordinate and slope is just the ratio of x to y so at that maximum y maximum slope is reached so maximum angle that you can have is 90 degrees. So if you try to go at 100 degrees you have lesser y and are actually going slower. And in a way 90 degrees is kind of the maximum amount you can point in another direction. But normally degrees go up to 180 or 360 degrees.
In the relativity side c is the maximum ratio but that is for coordinate time. If somebodys proper time would start pointing in a direction that would project negatively on the coordinate time axis the comparison between x per coordinate time and x per proper time would become significant.
There is also a trajectory which seems to be timelike in all segments. A=(0,0,0,0),(2,1,0,0),B=(4,2,0,0),(2,3,0,0),C=(0,4,0,0),(2,5,0,0),D=(4,6,0,0). It would seem awfully a lot like the “corner” A B C would be of equal magnitude but opposite sign from B C D. Now I get why physcially such a trajectory would be challenging. But from a mathematical point of view it is hard to understand why it would be ill-defined. It would also be very strange if there is no boost you can make at B to go from direction AB to direction BC. I get why you can’t rotate from AB to BD (can’t rotate a timelike distance to spacelike distance if rotation preserves length).
I also kind of get why yo woudl need infninte energy make such “impossibly sharp” turns. But as energy is the conserved charge of time translation, the definition of time might depend on which time you choose to derive it from. If you were to gain energy from an external source it would have to be tachyon or going backwards in time (which are either impossible or hard to produce). But if you had a thruster with you with fuel the “proper time energy” might behave differently. That is if you are going at signficant C and the whole universe is frozen and whissing by you should still be able to fire your rockets according to your time (1 second of your engines might take the entire age of the universe to external observers but does that prevent things happening from your perspective?). If acceleration “turns your time direction” and not “increases displacement per spent second” at some finite amount of acceleration experienced you would come full circle or atleast long enough that you are now going to the negative direction that you started in.
I have a previous high impliciation uncertainty about this (that would be a crux?). ” you can’t accelerate enough to turn around ” seems false to me. The mathematical rotation seems like it ought to exist. The prevoius reasons I thought such a mathematical rotation would be impossible I have signficantly less faith in. If I draw a unit sphere analog in spacetime having a visual observation from the space-time diagram drawn on euclid paper is not sufficient to conclude that the future cone is far from past cone. And thinking that a sphere is “all within r distance” it would seem it should be continuous and simply connected under most instances. I think there also should exist a transformation that when repeated enough times returns to the original configuration. And I find it surprising that a boost like transformation would fail to be like that if it is a rotation analog.
I have started to believe that the standrd reasoning why you can’t go faster than light relies on a kind of faulty logic. With normal euclidean geometry it would go like: there is a maximum angle you can reach by increasing the y-coordinate and slope is just the ratio of x to y so at that maximum y maximum slope is reached so maximum angle that you can have is 90 degrees. So if you try to go at 100 degrees you have lesser y and are actually going slower. And in a way 90 degrees is kind of the maximum amount you can point in another direction. But normally degrees go up to 180 or 360 degrees.
In the relativity side c is the maximum ratio but that is for coordinate time. If somebodys proper time would start pointing in a direction that would project negatively on the coordinate time axis the comparison between x per coordinate time and x per proper time would become significant.
There is also a trajectory which seems to be timelike in all segments. A=(0,0,0,0),(2,1,0,0),B=(4,2,0,0),(2,3,0,0),C=(0,4,0,0),(2,5,0,0),D=(4,6,0,0). It would seem awfully a lot like the “corner” A B C would be of equal magnitude but opposite sign from B C D. Now I get why physcially such a trajectory would be challenging. But from a mathematical point of view it is hard to understand why it would be ill-defined. It would also be very strange if there is no boost you can make at B to go from direction AB to direction BC. I get why you can’t rotate from AB to BD (can’t rotate a timelike distance to spacelike distance if rotation preserves length).
I also kind of get why yo woudl need infninte energy make such “impossibly sharp” turns. But as energy is the conserved charge of time translation, the definition of time might depend on which time you choose to derive it from. If you were to gain energy from an external source it would have to be tachyon or going backwards in time (which are either impossible or hard to produce). But if you had a thruster with you with fuel the “proper time energy” might behave differently. That is if you are going at signficant C and the whole universe is frozen and whissing by you should still be able to fire your rockets according to your time (1 second of your engines might take the entire age of the universe to external observers but does that prevent things happening from your perspective?). If acceleration “turns your time direction” and not “increases displacement per spent second” at some finite amount of acceleration experienced you would come full circle or atleast long enough that you are now going to the negative direction that you started in.