As you may recall, in my last Project post, I outlined my results concerning the creation of a temporal wormhole in a simple electrical circuit. I also mentioned that these results came with certain assumptions, one of which being that the flow of time was the same on either side of the wormhole. I have since been working to figure out what the results would be if this was not the case.
First, I attempted using differential equations and a similar approach as last time, only instead of having a parallel circuit I had two circuits that influenced each other, but each effect was reversed in time.
Something like this.
After struggling with this for a while, I determined that the math was, in fact, actively malevolent. I therefore decided to try a numerical method. While less exact than an algebraic solution, this would in theory be easier to implement. I chose to use Eulers's method instead of Runge-Kutta, again because it was simpler and fit very nicely with the fact that the equations for the current through a capacitor gave me the slope of the voltage. I also knew that the overall calculations would involve some self-referencing iteration, and I wanted to keep from muddying things up as much as possible.
Since numerical solutions sometimes have problems with accuracy (and also since it was so easy to implement changes) I tested my calculations first with a non-wormhole setup to check for accuracy, then with a common-flow wormhole (the one from last time) to check that the whole self-referencing thing worked out. I had to make several revisions as I apparently made some mistakes in my math somewhere, but I eventually got a working version and was able to run it on a reverse-flow wormhole.
When I was thinking about it before-hand, I pictured that the results from a reverse-flow wormhole would be much "smoother;" that the voltage would always be decreasing, just at different rates. I figured this because a way of thinking about the reverse-flow setup is that the two instances of the wormhole both start (or end) and the middle, then spread out from there. This means that the midpoint where they meet should be decently discontinuous. I also viewed the second instance of the wormhole as a capacitor charging in reverse: a somewhat straight line dropping off faster and faster. These two conceptions formed a mental image something like the tangent of negative pi.
Then I actually ran the numbers.
The result is not a smooth, continually decreasing curve, but rather it forms an even deeper bowl than a common-flow wormhole, with steep sides at either end. The middle area is still smooth, but my idea of the wormhole's second instance looking like a capacitor charging in reverse was wrong; from the perspective of the capacitor, it's still charging in normal, forward time, it's just the voltage source that it charging from is in reverse.
While these results are kind of neat, I am kind of disappointed that they didn't come out how I was hoping. If the voltage was continually decreasing, I was planning on making a graph with the time axis stretched and compressed until the curved looked like a typical capacitor discharge in order to give a better idea of what the wormhole did to the time-line of the event. I had also hoped to make an experiment with an event-dependent wormhole, where the second instance wouldn't start until the voltage reached a certain level or something, but if the voltage is not continually decreasing, there may be issues with opening the wormhole at the "wrong" place. Or maybe not. I'll have to think about it.