Proper energy density
(1) In the following and in general both signs will be shown except when one is justified by one or the other.
During the movement, when — it is positive and
It is observed that for — the two signs in (4.22) make sense, as discussed in (2.5). Of course, for the trip to the past to be possible, the negative sign in (4.22) has been chosen for —.
The form of the function (4.27) that reaches a maximum in allows the temporary recoil. At the instant of S the temporal inversion occurs.
Inspection (4.32) shows that for both the positive and negative solutions of (2.5) it makes sense. In order to differentiate between both solutions, it is necessary to look at the force density that depends on the sign of since. This means that in the reference system S different values are obtained according to the positive or negative sign of. It is this circumstance, which makes possible the trip to the past, because for the positive and negative solutions in (2.5) they differ physically in S, which allows to choose between one or the other. In addition it can be verified that the solution (4.32) changed sign when also allows the trip to the past. Both solutions differ only in the sign of angular velocity —.
It is observed that in S’the force density in (4.34) does not depend on —. This means that an observer located in S ‘can not force the time backward because the positive and negative solutions of (2.5) lead to the same external force density. For this reason, the force density of (4.34) only provides. On the other hand, an observer located in the reference system S, does physically distinguish both solutions and can choose the solution of (4.32) that causes the temporary recoil.
Again it is observed that the temporary reinvestment is possible because —- depends on the sign of — since —.