Distant Worlds Equations

"ACROSS SPACE & TIME TOWARDS DISTANT WORLDS"

HOME PAGE
CATALOGS

EQUATIONS
DICTIONARY

TIMELINE

This article takes place in the 24 & 26 centuries of Distant Worlds.

Dedicated page to Distant Worlds Equations

MATHEMATICS

Refer to respective articles for detailed rundown, D-Particles, for AdS Anti-De-Sitter Drive, for Bridges The Bridges, for Library The Library

M THEORY ACTION

S_{\text{M-Theory}} = \int d^{11}x \, \sqrt{-g} \left( -\frac{1}{2\kappa^2} R + \frac{1}{12} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) - \frac{1}{6} \int d^{11}x \, \epsilon^{\mu_1 \dots \mu_{11}} C_{\mu_1 \mu_2 \mu_3} F_{\mu_4 \dots \mu_7} F_{\mu_8 \dots \mu_{11}} + \mathcal{L}_{\text{M2}} + \mathcal{L}_{\text{M5}}

11D SURGA ACTION

S_{\text{SUGRA}} = \frac{1}{2 \kappa^2} \int d^{11}x \, \sqrt{-g} \left( R - \frac{1}{48} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) - \frac{1}{6} \int d^{11}x \, \epsilon^{\mu_1 \dots \mu_{11}} C_{\mu_1 \mu_2 \mu_3} F_{\mu_4 \dots \mu_7} F_{\mu_8 \dots \mu_{11}} + \text{fermionic terms}

SUPERSTRINGS ACTION

S_{\text{Superstrings}} = \int d^2 \sigma \left( -\frac{1}{2} \sqrt{-h} h^{ab} \partial_a X^\mu \partial_b X^\nu g_{\mu\nu} - i \bar{\psi} \gamma^a \partial_a \psi \right)

MSSM ACTION

S_{\text{MSSM}} = \int d^4x \left( \mathcal{L}_{\text{SM}} + \sum_{\text{superpartners}} \left( \bar{\psi} i \gamma^\mu D_\mu \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right) + \mathcal{L}_{\text{Yukawa}} + \mathcal{L}_{\text{Higgs}} \right)

SUSY ACTION

S_{\text{SUSY}} = \int d^4x \left( \bar{\psi} \gamma^\mu D_\mu \psi + \frac{1}{2} F_{\mu\nu} F^{\mu\nu} + \text{superpotential terms} \right)

CDT ACTION

S_{\text{CDT}} = \sum_{\Delta_2} \left( \frac{1}{2 \kappa^2} \text{Vol}(\Delta_2) R(\Delta_2) \right) - \sum_{\Delta_3} \text{Vol}(\Delta_3)

---

M + SURGA + MSSM + CDT

S_{\text{Total}} = \int d^{11}x \, \sqrt{-g} \left( -\frac{1}{2\kappa^2} R + \frac{1}{12} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) - \frac{1}{6} \int d^{11}x \, \epsilon^{\mu_1 \dots \mu_{11}} C_{\mu_1 \mu_2 \mu_3} F_{\mu_4 \dots \mu_7} F_{\mu_8 \dots \mu_{11}}
+ \frac{1}{2 \kappa^2} \int d^{11}x \, \sqrt{-g} \left( R - \frac{1}{48} F_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} \right) + \text{fermionic terms}
+ \int d^2 \sigma \left( -\frac{1}{2} \sqrt{-h} h^{ab} \partial_a X^\mu \partial_b X^\nu g_{\mu\nu} - i \bar{\psi} \gamma^a \partial_a \psi \right)
+ \int d^4x \left( \mathcal{L}_{\text{SM}} + \sum_{\text{superpartners}} \left( \bar{\psi} i \gamma^\mu D_\mu \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right) + \mathcal{L}_{\text{Yukawa}} + \mathcal{L}_{\text{Higgs}} \right)
+ \int d^4x \left( \bar{\psi} \gamma^\mu D_\mu \psi + \frac{1}{2} F_{\mu\nu} F^{\mu\nu} + \text{superpotential terms} \right)
+ \sum_{\Delta_2} \left( \frac{1}{2 \kappa^2} \text{Vol}(\Delta_2) R(\Delta_2) \right) - \sum_{\Delta_3} \text{Vol}(\Delta_3)

---

DUAL-GRAVITON

S_{\text{dual-graviton}} = \int d^{11}x \left[ \frac{1}{2} \partial_{[\mu} h_{\nu]\lambda} \partial^{[\mu} \tilde{h}^{\nu]\lambda\rho\sigma} + \frac{1}{2} \tilde{h}^{\mu\nu\rho\sigma} \Box \tilde{h}_{\mu\nu\rho\sigma} + \text{gauge} + \text{fermion} \right]

---

FRACTON LAGRANGIAN

\mathcal{L}_{\text{Fracton}} = \frac{i}{2} \left( \bar{\psi}_F \gamma^\mu D_\mu \psi_F \right) - m_F \bar{\psi}_F \psi_F + g_{\text{Fracton}} \bar{\psi}_F \psi_F^2 + \Phi_k \cdot D \cdot \bar{\psi}_F \nabla \psi_F

BOREON LAGRANGIAN

\mathcal{L}_{\text{Boreon}} = \frac{1}{2} \partial_\mu \phi_B \partial^\mu \phi_B + g_{\text{Boreon}} \phi_B \bar{\psi}_F \psi_F + \Phi_k \cdot D \cdot \phi_B \nabla \phi_B + \alpha_{\text{B}} \cdot \phi_B \cdot p_0

AURORA LAGRANGIAN

\mathcal{L}_{\text{Ավրորա}} = \frac{1}{2} \partial_\mu \phi_A \partial^\mu \phi_A + g_{\text{AB}} \phi_A \phi_B^2 + \Phi_k \cdot D \cdot \phi_A \cdot \nabla \phi_A - \alpha_{\text{A}} \cdot \partial^\mu \phi_A \cdot p_0

AXION LAGRANGIAN

\mathcal{L}_{\text{Axion}} = \frac{1}{2} (\partial_\mu \varphi)(\partial^\mu \varphi) - \frac{1}{2} m_\varphi^2 \varphi^2 - g_{\varphi\gamma} \varphi F_{\mu\nu} F^{\mu\nu} - \frac{\kappa}{4} \varphi T^{\mu\nu} h_{\mu\nu}

GRAVITON FULL LAGRANGIAN

\mathcal{L} = \frac{1}{2} \left( \partial_{\lambda} h_{\mu\nu} \partial^{\lambda} h^{\mu\nu} - \partial_{\lambda} h^{\lambda}_{\ \mu} \partial_{\nu} h^{\nu\mu} + \partial_{\mu} h^{\nu}_{\ \nu} \partial^{\mu} h^{\lambda}_{\ \lambda} \right) + \frac{1}{M_{\text{Pl}}} h^{\mu\nu} T_{\mu\nu}^{\text{EM}} + g_{\text{Fr}} \, h^{\mu\nu} \, \bar{\psi}_{\mu\nu} \gamma^\rho \psi_{\rho} + g_{\text{Ax}} \, \phi \, \partial_{\mu} h^{\mu\nu} \partial_{\nu} \phi + \mathcal{L}_{\text{Matter}}

DUAL GRAVITON

\mathcal{L}_{\text{Dual Graviton}} = \frac{1}{2} h_{\mu \nu} \left( \partial^\mu \partial^\nu h_{\rho \sigma} \right) + \frac{\alpha}{2} \epsilon^{\mu \nu \rho \sigma} h_{\mu \nu} h_{\rho \sigma} + \lambda \left( h_{\mu \nu} h^{\rho \sigma} \partial_\rho \partial_\sigma \right) + V_{\text{grav}}(h_{\mu \nu}, h_{\rho \sigma})

GRAVITINO LAGRANGIAN

\mathcal{L}_{\psi}^{(5D)} = \bar{\psi}_{\mu} \left( i \gamma^{\mu\nu\rho} D_{\nu} - m_{\psi} \gamma^{\mu\rho} \right) \psi_{\rho}+

---

POTENTIAL INTERACTIONS

The potential interaction between the Fracton and Boreon fields, including the reduced interdimensional coupling constant \Phi_k:

\mathcal{L}_{\text{Fracton-Boreon}} = K_{\text{FB}} \cdot \bar{\psi}_{\mu\nu} \varphi \psi^{\mu\nu} + M \cdot \partial^\mu \varphi \partial_\mu \psi

DUAL GRAVITON INTERACTION WITH MATTER FIELDS

\mathcal{L}_{\text{Dual Graviton Interaction}} = \frac{1}{2} \left( h_{\mu \nu} h^{\mu \nu} \right) \left( \bar{\psi}_\mu h^{\mu \nu} \psi_\nu \right) + \Phi_{11}(h_{\mu \nu}, h_{\rho \sigma})

COMBINED LAGRANGIAN FOR TOPOLOGICAL ENTANGLEMENT

\mathcal{L}_{\text{entanglement}} = \mathcal{L}_{\text{Fracton}} + \mathcal{L}_{\text{Boreon}} + \frac{k}{4\pi} \epsilon^{\mu\nu\alpha} A_{\mu} \partial_\nu \varphi \partial_\alpha \bar{\psi} + g \cdot \bar{\psi} \gamma^\mu (\partial_\mu \varphi) \psi + \xi \cdot |\nabla \varphi|^2 + \lambda \cdot \varphi^4

TOPOLOGICAL ENTANGLEMENT WITHIN BRIDGES

V_{\text{total}} = \sum_{i=1}^{N_B} V_{\text{wormhole}, i} - \beta \sum_{i=1}^{N_B} \sum_{j > i}^{N_B} \left( E_B^{(i)} \cdot E_B^{(j)} \right) \cdot \Phi_{ij} - \gamma \sum_{i=1}^{N_B} \lambda_i \left( \frac{1}{r_{\text{throat}, i}} \right)^{2}

LAGRANGIAN FOR AXION-GRAVITON-GRAVITINO AKA "GRAVIOLI" INTERACTION WITH WORMHOLE POTENTIAL

\mathcal{L}_{\text{Gravioli}} = \frac{1}{2} (\partial_\mu \varphi)(\partial^\mu \varphi) - \frac{1}{2} m_\varphi^2 \varphi^2 - g_{\varphi h} \varphi \, h^{\mu\nu} R_{\mu\nu} - g_{\varphi \psi} \varphi \, \bar{\psi_\mu} \gamma^\mu \psi_\nu - \left( \alpha_1 \varphi^2 R + \alpha_2 (\bar{\psi_\mu} \gamma^\mu \psi_\nu) h^{\mu\nu} + \alpha_3 \varphi^4 - \alpha_4 \frac{1}{M_{\text{Pl}}^2} (\bar{\psi_\mu} \gamma^\mu \psi_\nu)^2 \right)

---

WORMHOLE POTENTIAL

V_{\text{wormhole}}(\varphi, h_{\mu\nu}, \psi_\mu, g_{AB}) = \alpha_1 \varphi^2 R_4 + \alpha_2 (\bar{\psi}_A \Gamma^A \psi_B) h^{AB} + \alpha_3 \varphi^4 + \alpha_5 \, \frac{1}{M_{\text{Pl}}^2} (\bar{\psi}_A \Gamma^A \psi_B)^2 + \alpha_6 \, R_5

5D CYLINDRICAL WORMHOLE METRIC

ds^2 = - c^2 dt^2 + d\rho^2 + r^2(\rho) \, d\Omega_3^2 + \lambda^2(\rho) \, d\psi^2

TRAVERSABILITY OF WORMHOLE

The equation for throat radius stability is: \frac{d^2r(\rho)}{d\rho^2} + 4 \frac{r(\rho)}{\lambda(\rho)} \frac{d\lambda(\rho)}{d\rho} = 0

The effective gravitational force on a 3D object passing through the wormhole is: F_{\text{effective}} = - \frac{d}{d\rho} \left( r(\rho) \right) + \frac{\lambda(\rho)}{r(\rho)^2}

BRIDGE STABILITY LAGRANGIAN

\mathcal{L} = \int d^5x \left( -\frac{1}{2} \left( \partial_\mu h_{\nu \rho} \partial^\mu h^{\nu \rho} - \frac{1}{2} \partial_\mu h \partial^\mu h \right) + \frac{i}{2} \left( \bar{\psi}_\mu \gamma^\mu D_\nu \psi^\nu - \bar{\psi}_\nu \gamma^\mu D^\nu \psi_\mu \right) + \frac{1}{2} \left( \partial_\mu \phi_F \partial^\mu \phi_F - m_F^2 \phi_F^2 \right) + \frac{1}{2} \left( \partial_\mu \phi_A \partial^\mu \phi_A - m_A^2 \phi_A^2 \right) + g_{\text{FA}} \phi_F \phi_A + g_{\text{GrF}} h_{\mu \nu} \partial^\mu \phi_F \partial^\nu \phi_F + \frac{1}{2} \epsilon^{\mu \nu \rho \sigma \tau} \partial_\mu A_\nu \partial_\rho B_{\sigma \tau} \right)

BRIDGE EVAPORATION

\frac{dM_{\text{Bridge}}}{dt} = -\alpha \frac{\hbar c^2}{G} \frac{\Delta D}{r_{\text{Horizon}}^2} \exp\left(-\frac{\Phi_k}{k_B T}\right)

BRIDGE CONFIGURATION

\mathcal{S}_{\text{Bridge}} = \int d^5x \sqrt{-g} \left[ \frac{1}{2} R - \Lambda - \frac{1}{2} \left(\alpha_F |\nabla \phi_F|^2 + \alpha_B |\nabla \phi_B|^2 \right) + \beta \sum_{n} \frac{|\phi_{F,n}|^2}{\left(1 + r^2 \right)^{n}} + \gamma \epsilon^{\mu \nu \rho \sigma \tau} \partial_\mu A_\nu \partial_\rho B_{\sigma \tau} + \delta_{FB} \phi_F \phi_B + \eta_{GG} h_{\mu \nu} h^{\mu \nu} + \zeta \mathcal{A}_{\text{Boreon}} \right]

BRIDGE STABILITY SAFEGUARD

\mathcal{L}_{\text{Stability}} = \lambda_{\text{GF}} \bar{\psi}_\mu \gamma^\mu \Psi

---

ANGELIC METAL RESONANCE

\mathcal{R}_{\text{AM}} = \lambda_\text{res} \int d^4 x \left( \bar{\psi}_\mu h^{\mu \nu} \right))

\mathcal{L}_{\text{res}}^{(5D)} = \alpha \left( \bar{\psi}_{\mu} \gamma^{\nu} \psi^{\mu} \right) \left( h_{\mu\nu}^{(5D)} h^{\mu\nu(5D)} \right)

---

AURORA-BOREALIS ASYMPTOTIC SAFETY

\mathcal{L}_{\text{Aurora-Boreon}} = g_{\text{AB}} \phi_B \partial_\mu \phi_A + \lambda_{\text{AB}} \phi_A \phi_B^2

V_{\text{total}}(d) = \frac{1}{d^\beta} - \frac{1}{d^\gamma}, \quad \gamma > \beta

---

GENERAL ACTION OF ADS DRIVE DYNAMICS

S = \int d\tau \, \mathcal{L}_{\text{AdS}}(x^\mu, \dot{x}^\mu, \phi, g_{\mu\nu}, \epsilon, E)

LAGRANGIAN FOR ADS DRIVE DYNAMICS

\mathcal{L}_{\text{AdS}} = \frac{1}{2} m \left( g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu - 1 \right) - \epsilon \, g_{\mu\nu} \, \phi \, F(\dot{x}^\mu, \partial_\mu \phi) + \frac{\alpha}{2} \dot{x}^\mu \frac{d}{d\tau} \dot{x}_\mu - \frac{\beta}{r^2 + \gamma} + \mathcal{L}_{\text{AdS Transition}}

EQUATION OF MOTION (ADS)

\frac{d}{d\tau} \left( \frac{\partial \mathcal{L}_{\text{AdS}}}{\partial \dot{x}^\mu} \right) - \frac{\partial \mathcal{L}_{\text{AdS}}}{\partial x^\mu} = 0

GALACTIC NUMERICAL SYSTEM

\text{S}_r = \text{hex}\left( \left\lfloor \frac{r}{R_{\text{max}}} \times 0xFFFF \right\rfloor \right), \quad

\text{S}_\theta = \text{hex}\left( \left\lfloor \frac{\theta}{360^\circ} \times 0xFFFF \right\rfloor \right), \quad

\text{S}_\phi = \text{hex}\left( \left\lfloor \frac{\phi + \phi_{\text{offset}}}{\Phi_{\text{max}}} \times 0xFFFF \right\rfloor \right)

LIBRARY'S BERKENSTEIN BOUND

I_{\text{max}} \leq \frac{D R^\prime E^\prime}{\hbar c \ln 2},