True story: I found and completed Rixis long before I knew about Machu Picchu. To me, Rixis will always matter more.

### What’s so “Hyperbolic” About the Hyperbolic Time Chamber?

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Simple concept: in the Hyperbolic Time Chamber one experiences more time than is passing on earth. Plus the name is totally bitchin’. But is the name more than just totally bitchin’? Yes.

This is Albert Einstein.

After experimental physicists found light had a constant speed (http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment) Einstein introduced the topics of special and general relativity to physics, the consequences of which largely boil down to an equivalence of space and time. This called for a graphical representation, so Hermann Minkowski gave us his spacetime diagram.

With a spatial axes of x (displacement) and ct (speed of light times elapsed time; maximum possible displacement), the Minkowski diagram allows us to track movement through space and time and even compare the way different reference frames (here shown as x’ and ct’) experience time and space. To appropriately compare, however, we need a common measurement, which comes in the form of the spacetime interval.

Here *t* represents time displacement, *c* represents the constant speed of light, *r* represents spatial displacement, and *s* represents the total displacement in spacetime. So if I want to make sure any combination of space *r* and time *t* cover an equal interval in spacetime, I define that *s* is equal to an arbitrary constant *k*. Now we have this:

which is the same thing in a time vs. space plot as

is in a *y* vs. *x* plot. Do you recognize what this is?**It’s the equation for a hyperbola**.

That means that any two points on a hyperbola in a spacetime diagram have traversed the same interval of spacetime. So let’s plot a few of these and check it out:

The Hyperbolic Time Chamber (as its name tells us) must experience hyperbolic time, so its time axis, shown in purple, will be one of the hyperbolae while Earth’s time axis, shown in yellow, remains vertical. One can observe that at the end of any interval of purple hyperbolic time, less yellow Earth time has passed. This shows that the hyperbolic nature of time in the chamber is the direct reason why users experience more time than passes on Earth.

It does not end there. The Hyperbolic Time Chamber is said to have a gravity greater than that of Earth. General relativity explains how gravity can be understood as the acceleration of a reference frame. In the above Minkowski diagram, two linear axes, *x*' and *t*’, are shown in blue representing time dilated and length contracted by constant velocity. At rest, the angle between the axes is 90° and at higher velocities the angle gets smaller until both axes approach the 45° angle of the light cone (shown in red). Because of this, if the frame were accelerating then the angle between its two axes would get smaller and smaller until it came to the angle of its ultimate velocity: the “universal speed limit” of the speed of light *c*, at which the axes will asymptotically approach the 45-degree angle of the light cone. When this process starts from rest and has a peak velocity, the result obtained will be a hyperbola for both axes.

Then why is it not also a Hyperbolic *Space* Chamber?

Answer: infinity.

The phenomena which motivate the Minkowski diagram are time dilation and length contraction (or “space contraction”). Our universe, the universe which contains Earth, is comprised of an enormous, constantly expanding, but finite amount of space. As one can see in the anime or notice in *Budokai*, the Hyperbolic Time Chamber contains an infinite amount of space. The phenomenon of length contraction shrinks space by a scaling coefficient by which a length is multiplied, and since the physical universe contains a finite amount of space, the total length of the universe can by contracted by a scaling factor. Since the Hyperbolic Time Chamber contains an infinite amount of space, it cannot be contracted at all; any non-zero constant scaling factor multiplied by infinity yields infinity, therefore the total length of the chamber (infinity) is equal to the total length of the contracted chamber (infinity × constant = infinity). If the total length is the same then all of the length is the same and the contraction of the space does not change the space at all, therefore the spacial axis of the diagram remains straight even as the time axis asymptotically approaches 45° to form a hyperbola.

Let’s wrap this up:

The concept of “hyperbolic time” implicitly, by the characteristics of its geometry, causes more time to be experienced than linear time otherwise provides. In addition, the fact that greater gravity is present in the Hyperbolic Time Chamber than is present on Earth confirms, by general relativity, that it is accelerating with respect to Earth and dilation of time will cause its time axis to approach 45°, slowly shaping itself into a hyperbola — becoming hyperbolic. Though time will undergo relativistic effects due to this acceleration, the space in the Hyperbolic Time Chamber is infinite and can therefore not be contracted with any consequence, causing the spatial axis or axes which the chamber follows to remain linear, solidifying its definition as the Hyperbolic Time Chamber rather than the “Hyperbolic Space and Time Chamber”. Only one worthwhile conclusion to draw from this: Kami deserves a Nobel Prize.

Dwarven Vow #4: Don’t depend on others, walk on your own two legs!