Distant Worlds Equations

"ACROSS SPACE & TIME TOWARDS DISTANT WORLDS"

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

Dedicated page to Distant Worlds Equations. Note: I am not qualified Mathematician or Physicist, Almost all equations are either based on my notes or brainstormed through LLM's, Equations may contain Mistakes or make no sense at all, Done with love.

MATHEMATICS

Refer to respective articles for detailed rundown, D-Particles, for AdS Subspace Anti-De-Sitter Drive, for Bridges The Bridges, for Library The Library, for Grav Well Gravitational Well, Angelic Metal.

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)

---

FRACTON LAGRANGIAN

\mathcal{L}_{\text{Fracton}} = \overline{\Psi}_f \left( i \gamma^\mu D_\mu - m_f - \frac{S_b}{2} \Phi_{\text{string}} \right) \Psi_f - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2} \left( \nabla^2 \Phi_{\text{string}} \right)^2

BOREON LAGRANGIAN

\mathcal{L}_{\text{Boreon}} = \frac{1}{2} (\partial_\mu \Phi_B)(\partial^\mu \Phi_B) - \frac{1}{2} m_B^2 \Phi_B^2 - \frac{\lambda_B}{4} \Phi_B^4 + g_B \Phi_B \overline{\Psi} \gamma^\mu \Psi

Mashtakov Metric

A = \frac{1}{1 + \alpha_B \Phi_B^2} \left( 1 + \frac{g_B \Phi_B}{M} \right)

AURORA LAGRANGIAN

\mathcal{L}_{\text{Aurora}} = \frac{1}{2} (\partial_\mu \Phi_A)(\partial^\mu \Phi_A) - \frac{1}{2} m_A^2 \Phi_A^2 - \frac{\kappa}{4} \Phi_A^4 - \xi \Phi_A^2 \Phi_B^2 + \lambda_S S_A^2

GRAVITON FULL LAGRANGIAN

\mathcal{L}_{\text{Graviton}} = -\frac{1}{2} h^{\mu\nu} \mathcal{E}_{\mu\nu}^{\alpha\beta} h_{\alpha\beta} + \frac{\lambda}{2} \left( \partial_\lambda h_{\mu\nu} \partial^\lambda h^{\mu\nu} - \partial_\lambda h \partial^\lambda h \right) + \frac{\xi}{2} \Phi_p^2 h_{\mu\nu} h^{\mu\nu} + \lambda_S S_G^2

DUAL GRAVITON

\mathcal{L}_{\text{Dual-Graviton}} = \lambda_D (h_{\mu\nu} h^{\mu\nu})^2

GRAVITINO LAGRANGIAN

\mathcal{L}_{\text{Gravitino}} = \bar{\psi}_\mu \left( i \gamma^{\mu\nu\lambda} \partial_\lambda - m \gamma^{\mu\nu} \right) \psi_\nu + \frac{1}{2} \kappa \, \bar{\psi}_\mu \gamma^{\mu\nu} \psi_\nu h_{\alpha\beta} + \frac{\lambda_S}{2} S_G^2 \, \bar{\psi}_\mu \gamma^\mu \psi^\mu + \mathcal{L}_{\text{int}}^{\text{dim}}

---

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

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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

EQUATION OF TRANSITION (ADS)

t_{\text{transition}} = \frac{2d}{\sqrt{\frac{2P}{m}} + \sqrt{\frac{2Q}{m}}}

---

Subspace Projection from 4D to 3D

V_{3D} = P_{4 \to 3} \cdot V_{4D}

V_{4D} = \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}, \quad V_{3D} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}

P_{4 \to 3} = \begin{bmatrix} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \end{bmatrix}

Tensorial Pressure & Anti-de Sitter Tensors

T^{\mu\nu}_{AdS} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial^\nu \phi - g^{\mu\nu} \mathcal{L}

Electroweak Leakage Representation

\Psi_{Bino}(x,t) = A \cdot e^{-(\gamma + i\omega)t} \cdot \psi(x)

(AdS) Energy Requirement per Dimension:

E_d = E_3 \cdot k^{(d - 3)} \quad \text{with } k \in [2, 5]

(AdS) Estimated Travel Time Scaling:

T_d = \frac{T_3}{(d - 2)^{1.5}}

(AdS) Symmetry Preservation Function:

S_d = \begin{cases} 1 & \text{if } d \leq 4 \\\\ e^{-(d - 4)} & \text{if } d > 4 \end{cases}

---

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},

---

GRAVITATIONAL WELL

G ⅁(r) = 1 - \frac{2G}{c^2 r} \left( M_c + \int_0^r 4\pi r'^2 \left( \rho_{g0} e^{-\beta r'} + \rho_{q}(r') \right) \, dr' \right) - \int_0^r \frac{G \left( \rho_{g0} e^{-\beta r'} + \rho_{q}(r') \right)}{r'^2} e^{-\alpha r'} \, dr' - \frac{k_B Φ_k}{\lambda_s^2} \int_0^r ρ_s(r') 4\pi r'^2 \, dr'.

---

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)

\ce{^{{314}}_{{126}}{AM}\ }

\ce{^{{318}}_{{126}}\text{AM}\ }

\ce{^{{325}}_{{140}}\text{AM}^{+}\ }

\ce{^{{314}}_{{126}}\text{AM}^{+}\ }

\ce{^{{314}}_{{126}}\text{AM}^{-}\ }

\ce{^{{318}}_{{126}}\text{AM} + \tilde{\gamma} -> \text{AM}^* + \phi_b + E_{\text{burst}} + \tilde{\gamma}_{\text{secondary}}}

\ce{\text{AM}^* \rightarrow \text{AM} + \phi_b + E_{\text{burst}} + \tilde{\gamma}_{\text{secondary}} + \tilde{e}^-}

\ce{ \tilde{e}^- \xrightarrow{\text{Absorption}} E_{\text{usable}} }

\ce{E_{\text{reaction}} = \phi_b + E_{\text{burst}}}

\ce{E_{\text{SSR}} = 500 \, \text{MeV/reaction}}