Vacuum lagrangian

In pilitron physics, a vacuum Lagrangian, $$\mathcal{L}_V$$, is a Lagrangian that describes the curvature of space-time, which can be thought of as the action performed by vacuum. It is negative, because space-time contains negative energy. It is given by:


 * $$\mathcal{L}_V = -\frac{hc\ell_P^2}{V}$$

where:
 * $$h$$ is the Planck's constant.
 * $$c$$ is the speed of light.
 * $$\ell_P^2$$ is the Planck area (Planck length squared).
 * $$V$$ is the volume of space in question.

Vacuum action
Vacuum action describes the expansion of space. It is given by:


 * $$\mathcal{S}_V = \int_{t=A}^{t=B} -\frac{hc\ell_P^2}{V} dt$$

Curvature of space
The curvature of space can be described using the tension factor or tension function, $$\tau$$, which is given for two equal volumes of space by:


 * $$\tau = \frac{\mathcal{L}_1}{\mathcal{L}_2} = -\frac{hc\ell_P}{V_1} / -\frac{hc\ell_P}{V_2}$$

Which is the same as:


 * $$\tau = \frac{\mathcal{L}_1}{\mathcal{L}_2} = -\frac{hc\ell_P}{V_1} \cdot -\frac{V_2}{hc\ell_P}$$

Which simplifies to:


 * $$\tau = \frac{\mathcal{L}_1}{\mathcal{L}_2} = \frac{V_2}{V_1}$$

Please note, however, that a Lagrangian ratio of $$\frac{V_2}{V_1}$$ is only valid for empty space, because when energy is put in, the Lagrangians become different. Also please note that because both volumes are equal, then for empty space, $$\tau = 1$$, which means no curvature.

Please note that $$\tau \leq 1$$, so when calculating the tension between two volumes, make sure that $$\mathcal{L}_2 > \mathcal{L}_1$$.

Integration
The distance travelled by an object across space-time containing energy, from the perspective of that object, is given by:


 * $$D = \int_{t=A}^{t=B} \frac{v\tau}{\sqrt{1-\frac{v^2}{c^2}}} dt$$

And from the perspective of an outside observer:


 * $$D = \int_{t=A}^{t=B} v\tau dt$$

For an object with mass, we can further rephrase it as:


 * $$D = \int_{t=A}^{t=B} \sqrt{\frac{\mathcal{L}\tau^2}{m}} dt$$

Where $$\mathcal{L}$$ is the Lagrangian without taking curvature into account. This shows that, at least for objects with mass, we have to multiply the normal, flat-space Lagrangian by $$\tau^2$$ to account for gravity and relativity.