So about a week ago, I watched this video on YouTube, and it inspired me to think some more on the topic of time travel. The scope of my current views on the subject is very large and underdeveloped, and I'm trying to get this post up in reasonable time, so I won't go into them here. Suffice it to say that my various thought processes led me to the idea of time travel within a simple RC circuit.
This is because much of the talk surrounding potential time travel deals with wormholes in space-time and the possibility of sending something through one. However, all of the scenarios that I have heard described concerning such wormholes only speak of sending something through in one direction. My understanding of wormholes, on the other hand, is that they would operate more like gates, allowing for bi-directional travel. The model of an electrical circuit works very well for this, since charge will flow from higher to lower potentials whichever side of the wormhole they're on, the quantities being discussed are continuous rather than discrete, and there is the potential for both positive and negative voltages. My thought experiment took the following form:
Under normal conditions, the voltage across the capacitor would decay exponentially according to the time constant tau.
However, I would alter the conditions on this particular circuit by introducing a wormhole between the resistors connecting two points in the discharge time: t1 and t2. The wormhole had to be between the resistors because opening it at ground would give me nothing (always at 0 volts) and opening it at the capacitor would effectively and instantly place the capacitor in parallel with itself at a time when it has a lower voltage, and you can't instantly change the voltage across a capacitor. Also, since an instantaneous spike or drop in voltage within the circuit would have little lasting effect (capacitors are often used specifically to get rid of such spikes), I would have to leave the wormhole open for some duration td.
I assumed that the flow of time on either side of the wormhole to be the same, so that t1 corresponded to t2 and t1+td corresponded to t2+td. Therefore, for the period of time(s) the wormhole was open, the circuit would look like this:
where C1 is the first instance of the capacitor, starting at t1, and C2 is the second instance of the capacitor, starting at t2. There is no need to make a distinction between the instances of the resistors, since their voltages and currents can change instantaneously. The starting voltage of C1 would be the typical RC voltage after decaying for a time of t1, and the starting voltage of C2 would be the voltage of C1 after it had decayed according to the parallel "wormhole" circuit for td then decayed naturally for t2-(t1+td).
My first calculations were performed using the values of R1=R2=1kohm, C=1uF, t1=0.5ms, t2=2ms, and td=0.1ms. Solving the parallel circuit required using some differential equations, but overall it wasn't too difficult, and the result came out looking like this:
As you can see, the voltage across the capacitor dropped slightly once the wormhole was opened at t1, then rose back to about normal when the wormhole was opened at t2. There appeared to be a slight difference between the final voltages of the normal circuit and the wormhole circuit, but this difference was small enough that it could be attributed to rounding errors.
To find out whether there was a net effect to adding a wormhole or if it was indeed just a rounding error, I decided to work through the equations again in general form. This is where things started to get ugly. None of the equations simplified, but instead got more and more complex the further I got. I therefore present the result in the form of many constants that require their own definition.
The discharge cycle of the capacitor is divided into five distinct sections:
With constants defined as follows:
(If you're able to simplify things more, please let me know.)
While the complexity of the equations made it difficult to verify if there was a net effect of the wormhole by algebraic means, I was able to plug them into Excel and tweak parameters until the answer became more obvious. First, I lengthened the duration of the wormhole to 0.5ms:
Then I moved the wormhole "closer" to the capacitor by increasing R1 and decreasing R2. This kept the time constant the same while increasing the influence of the wormhole on the capacitor.
At this point it became obvious that there was a net effect to the creation of the wormhole: the final voltage of the capacitor was greater than it otherwise would have been. Since the rate of decay at the end of both the typical case and the wormhole case was the same, this effectively meant that the voltage decay had been delayed by some amount of time equal to
In order to guard against paradoxes, I calculated the cumulative energy output of the capacitor, just to see if the postponed decay was due to the wormhole creating energy or something. This was relatively easy to calculate by finding the voltage across the resistors, using that to find the instantaneous power, then integrating over time.
The result is that the wormhole capacitor has expended less energy than usual. Since the graph of energy is cumulative, this means that the energy expenditure of the capacitor has been delayed, and quick investigation shows that this delay is the same as that for the voltage decay. This means that the total effect of the wormhole is that, without introducing energy into the system, it delayed the event of capacitor discharge.
In other words, I effectively (and temporarily) slowed the passage of time for the circuit. Pretty cool, huh?
Now, these results are hardly comprehensive. For example, I mentioned earlier that I assumed that time flowed in the same direction on either side of the wormhole, and this is not necessarily valid. The typical picture of a wormhole is fold in space-time, and, at least with the 2-D analogy, if you're traveling in one direction on one side of the wormhole, the fold will reverse your direction by the time you reach the other side.
Applying this would mean recalculating the parallel circuit while one of the capacitors is experiencing time in reverse. This is mind-boggling (though I think the results would fit nicer with the whole slowing-down-time thing), but I hope to give it a try. Whether I get around to it, though, is uncertain, as I have a completely different, physics-based project planned for this weekend.