Sunday, July 31, 2011

Dnoces a Em Evig

Greetings, Internet.

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.

Monday, July 18, 2011

Perils of Procrastination

I have learned an important lesson this week: NEVER put off booking a flight.

A good friend of mine is getting married toward the beginning of next month, and both Hopps and I were invited to the wedding. I eagerly responded that I would attend, and proceeded to get the necessary time off from work. I did not, however, book my flight. The reason I did this is because I wanted to coordinate my travel with Hopps, if possible. The actual location of the wedding is about an hour away from the airport, and if we both arrived at the same time it would save on a taxi or be more convenient if someone was picking us up. For his part, Hopps, having just started work, was trying to work out if he could get time off and then waiting for confirmation.

We checked ticket prices about a week ago, and while they were a bit pricey (the airport we would be flying into is not frequented by airlines as often as others) we decided that we could manage to fit them into our respective budgets. Some more time went by without Hopps knowing for sure if he had the day off, and eventually I decided to go ahead and book my flight anyway. So this last weekend I got on the internet to book my flight. I wasn't too concerned about the price; the wedding was still three weeks away, and prices couldn't have gone up that much, right?

WRONG.

In the several days since we had last checked, the price for a round trip had increased to almost 150% of its previous value, putting it way beyond what we had decided was pushing it for our wallets.

And now, I have just gotten of the phone with my friend after explaining to him why I won't be coming even though I had told him many times that I would. Uhhhg.


I have another friend getting married at the end of August. I will be booking that flight tomorrow.

Thursday, July 7, 2011

Give Me a Second

Greetings.

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.

Saturday, July 2, 2011

A Benign Crazy

I don't think there are very many people in the world who would spend their Friday evenings/early Saturday mornings calculating the effects of a temporal wormhole on an RC circuit, but I am one of them.

I'll re-run my calculations and get a "project" post up about it soon, but for now I'll just say that I've always tended to think of time travel in terms of an op-amp feedback circuit, so it just felt natural to run some initial thought experiments with electrical components. Although the circuit just had two resistors and a charged capacitor, the complications introduced by the wormhole were enough that I had to review my old notes from Differential Equations to get a handle on it. I was also aided in my endeavors by Grimholt and Sam, the Homework Gremlins (two whiteboards that I inherited from the Powersuite), who were happy to have something to do.

In other news, Hopps has been moved in for about two weeks now, and things are going fine. We're both somewhat solitary, but it's nice to have another person around. Another one of my friends going to be visiting over the weekend, so we got a grill (and lighter fluid!) and hopefully we won't kill ourselves with it, any July 4 activities, or just the sheer heat outside.

Or rather, hopefully Hopps won't kill himself. I am invincible!