### Proof of the Impossibility of Time Reversal

Using an example, it will be demonstrated that the assumption of time reversibility leads to a contra­diction. The argument, however, applies to any possible system.

We look at a planetary system between two instants T1 and T2. Suppose we film the process from a point that lies on the straight line through the center of gravity of the system normal to its plane and is located so far away that the differences of the times, which light needs to cover the distances between the various objects of the system and our camera, nearly disappear and can thus be neglected in the following considerations.

We then play the movie reversed in time. The question is: Does the film now represent the real time-reversed process?

The answer is no. As follows:

Let us assume, the process was time-reversible. This means: the backward running film shows the actual backward running process. All objects move along the same paths as before, but in the opposite direction. This assumption is based on two conditions:

1.In the time-reversed process, the velocity of each object is at any point of its path the negative of the velocity that this object possessed at the same point in the original process.

2.In the time-reversed process, the acceleration of each object is at any point of its path equal to the acceleration of this object at the same point in the original process.

Now we proceed as follows: We pause the forward running film at some instant, which corresponds to an instant T between T1 and T2 in the real process. We look at an arbitrary object A. At the time T, it is at the position O. Let us call the totality of the positions, where all other objects are at this instant T, the constellation C(O).

However, since gravitation needs a certain time to cover the distance from any of the objects to the object A, the gravitational effect, which the object A experiences in O at the time T, is not determined by the constellation C(O), but by the hypothetical constellation C'(O), which can be obtained in the following way: One starts with C(O) and moves each object on its path backward – precisely by that amount of time which gravitation, starting from there, needs to reach the object A in O at the time T.

Now we look at the backward running film. Again we stop it, when the constellation C(O) appears. However, also here applies that the gravitational effect which the object A experiences at that instant, is not determined by the constellation C(O) but by a hypothetical constellation C"(O). This constellation C"(O), however, is obviously not identical with the constellation C'(O): In order to construct C"(O), the objects must indeed be displaced – again starting from C(O) – on their paths into the opposite direction than before in the construction of C'(O).

From this follows that, in the time-reversed process, the acceleration of the object A at the position O is not identical with the acceleration of A at the same position in the original process.

Therefore the following applies:

If the process is time-reversible, then all objects move backward in time along the same paths as forward. This is only possible if the acceleration of any object along its path is – at any point of this path – identical for both time-directions. As has just been shown, however, in none of the points this condition is met. Thus the assumption of time-reversibility leads to a contradiction; the backward running film shows no real possible procedure.

As regards our example, the difference between the actual time-reversed process that starts at T2 and the backward running film can be downright dramatic:

Suppose our system has more than two heavy gas planets. So its stability-level is low. Then it is possible that in the original process between the time points T1 and T2 all planets stay in their orbits, whereas in the actual time-reversed process – in contrast to the backward running film – several planets are thrown out of the system.

Thus it can be stated: Since gravitation is active in any system, our argument proves that time reversal is generally impossible. There are no "reversed trajectories".

(It could be argued that including gravity would – in many physical systems – lead only to such absurdly small changes that they would be completely negligible. However, this is not a valid objection, because the question of time-direction is not a practical but a fundamental question.)

Notes

1.The proof is only valid for relative movements of the objects of a system. Due to special relativity, however, it does not apply to the (uniform) motion of the whole system, i.e. to the movement of the center of gravity of the system: whether the system as a whole is moving or not is physically indistinguishable.

2.Of course, the thermodynamic irreversibility arguments still apply. But they are only suited to specify – under certain conditions – one time-direction as the more probable one, while the argument presented here applies to all systems, such that irreversibility appears as ontological necessity.