In fact this conclusion does not rest merely on the plausibility of the argument I have given, but can be proven mathematically, as follows:
The behaviour of a classical mechanical system (a system composed of masses and forces) is governed by Newton's three laws of motion. The most important law here is the second, which says that the acceleration a of a body at time t is proportional to the force F exerted on it, thus:
| a = | --- m | F |
The proportionality constant is one over the mass m. Recall that the acceleration is the time rate of change (in calculus: ``derivative with respect to time'') of the velocity. Similarly the velocity is the time rate of change of the position x of the body. Thus in calculus we say the acceleration is the ``second derivative with respect to time'' of the position, thus:
| a = | --- d t2 | = | --- m | F |
If we solve this differential equation, we can find the trajectory of the body, that is the position of the body as a function of time x(t).
Aficionados of differential equations will recall that a ``second-order'' equation (one that has a second derivative in it, as above) requires two initial conditions for the complete solution. That is, it is not enough to have some function x(t) that satisfies the equation above. One must also specify two ``initial conditions'' to give a unique solution to Newton's second law. These two initial conditions are, unsurprisingly, the initial location and initial velocity of the body.
Now let's try an interesting thought experiment. Suppose we have one trajectory x(t) that we know satisfies Newton's second law (because, for example, we film some system following it). What can we then say about x(-t) ?
First of all, let us consider the meaning of x(-t). Remember x(t) is just a shorthand way of saying that if we are given t we can find x. We could represent this knowledge most easily with a giant table in which every possible value of t has next to it the corresponding value of x. This is the same way an income-tax table, which gives the value of the tax for every possible value of the income, could be said to represent a function TAX(INCOME).
Suppose the function x(t) is this rather simple one:
| at time t = | body is at x = |
|---|---|
| Alioto's in San Francisco | |
| Yosemite National Park | |
| U.C. Irvine |
then the function x(-t) would have to be (just reversing the sign of t everywhere):
| at time t = | body is at x = |
|---|---|
| Alioto's in San Francisco | |
| Yosemite National Park | |
| U.C. Irvine |
So having a body follow x(-t) instead of x(t) is exactly what we mean mathematically by the English statement ``we film the body moving normally and then run the movie backwards.''
Now what happens if we try x(-t) in Newton's second law, like so:
---- d t2 |
? = |
--- m | F |
---- d t2 |
= | --- d t |
---- d t |
= | --- d t |
---- d t |
---- d (-t) |
= | --- d t |
(-1) | ---- d (-t) |
= | (-1) | --- d t |
---- d (-t) |
| = | (-1) | ---- d t |
--- d (-t) |
---- d (-t) |
= | (-1)×(-1) | --- d (-t) |
---- d (-t) |
= (+1) | ---- d (-t)2 |
---- d (-t)2 |
? = |
--- m | F |
---- d s2 |
? = |
--- m | F |
---- d t2 |
= | --- m | F |
Hence we must conclude in general that if x(t) is a solution to Newton's second law, then the ``time-reversed'' trajectory x(-t) is also a solution. Therefore if we pick the right initial conditions, we can get the system to follow x(-t) just as ``naturally'' as it would follow x(t).
In particular, suppose we have been observing the system following x(t). If we suddenly put the system into the right initial conditions, we should be able to get it to start following x(-t) instead, and hence retrace its steps perfectly. That is, if we do the right thing to a body following the simple trajectory above, then when it gets to U.C. Irvine tomorrow we should be able to get it to reverse its course and arrive back in San Francisco two days after that. Cool! What a great party trick, eh?
But what is it we need to do? This can be determined by considering the simplest possible system: a body in free motion, where the force is zero. In this case the body travels in a straight line with a constant velocity equal to its initial velocity. To cause the particle to retrace its steps it is only necessary to reverse its velocity.
Interestingly, this prescription can be shown to work in general: to get any system to retrace its trajectory exactly it is only necessary to reverse the velocities of each body in it.
Even more astonishingly, this can be shown to be true no matter how large and complex the system may be, so long as it obeys Newton's Laws. For example, you are a system following Newton's Laws. You have followed a (very complex) trajectory from your birth until now. To get you to follow the time-reversed trajectory leading right back to your birth, it is only necessary to instantly reverse the velocity of every single atom of which you are composed.
Pretty much. There are only two minor clarifications to be made: First of all, you interact with your surroundings, in general. It turns out the theorem above only holds for isolated systems. Hence we must actually reverse the velocity of every atom in you and every atom that ever came into contact with you -- i.e. each atom in all your past boyfriends/girlfriends, many of the atoms in the Earth's atmosphere, the atoms in various objects now languishing in garbage dumps, etc. Second, atoms don't actually obey Newton's second law, but the Schroedinger equation of quantum mechanics. Luckily the theorem above can be shown to hold equally well for quantum mechanical systems.
In fact, this theorem can be shown to hold for any law of mechanical motion the form of which does not change with time (that is, where the equations are the same today as they were yesterday). In this form the theorem is called the time symmetry or symmetry under time reversal of mechanical systems.
It is the bald fact that the mathematically provable time symmetry of an isolated mechanical system conflicts so dramatically with the observed time asymmetry of our experience (nobody is observed traveling back to their youth, having accidentally or deliberately reversed the velocity of their atoms) which led leading philosophers of the 19th century -- who were no dummies --- to doubt that complex everyday phenomena, e.g. life, could be purely mechanical system.