Balloon theory

The balloon theory is a theory in pilitron physics, which explains what happens to objects as the space they occupy expands through time.

3D and 4D
First, imagine a 3D balloon. The space is the surface of the balloon, and we can describe the position of an object in that space using spherical coordinates $$r$$, $$\theta$$, $$\varphi$$. The radius of the sphere, $$r$$, is proportional to time, as the sphere inflates, but the angular coordinates remain the same. If we measure the angle in radians, then the distance between two objects in that space is given by:


 * $$D = r\sqrt{|\theta_1 - \theta_2|^2 + |\varphi_1 - \varphi_2|^2}$$

Now, let's add a fourth dimension to the sphere. This means that the space is now 3D, and we have an extra angular coordinate $$\omega$$. Now the distance between two objects is given by:


 * $$D = r\sqrt{|\theta_1 - \theta_2|^2 + |\varphi_1 - \varphi_2|^2 + |\omega_1 - \omega_2|^2}$$

Accelerated expansion
We know that the Universe is not expanding linearly, but instead the expansion accelerates through time. If we know the acceleration of the Universe, $$a$$, then at time $$t$$, the radius can be given by:


 * $$r = at^2$$

Therefore the distance between two objects becomes:


 * $$D = at^2\sqrt{|\theta_1 - \theta_2|^2 + |\varphi_1 - \varphi_2|^2 + |\omega_1 - \omega_2|^2}$$

Force & Work
Since the distance between objects increases as time passes, then also the passage of time exerts a force on them. When we know the rate of expansion, and the mass of an object, $$m$$, then we can tell that the force exerted is:


 * $$|\mathbf{F}| = ma$$

Because the objects are moving as the space expands, work is exerted on them. The work exerted (and therefore the energy pumped into the Universe) between $$t_0$$ and $$t_1$$ is:


 * $$W = ma \cdot \int_{t_0}^{t_1} at$$

Since the acceleration is constant, the shape on the graph of the function $$at$$ is a triangle, therefore the integral $$\int_{t_0}^{t_1} at$$ can be evaluated by calculating the area of that triangle. The base of the triangle is the difference in time (call it $$t$$) and the height is $$at$$ ($$t$$ being equal to the latest time), so we derive:


 * $$W = ma \cdot \frac{1}{2} at^2$$

Which simplifies to:


 * $$W = \frac{1}{2} ma^2t^2$$

Since $$a^2t^2 = v^2$$, then if we probe the relative velocity due to expansion, $$v$$, the formula is exactly the same as kinetic energy in classical mechanics:


 * $$E = \frac{1}{2}mv^2$$