- Bohr Model Adjustments for High-ZZZ Elements (Eq. 1--2)
- Wave Equation (Eq. 3--4)
- Schrödinger Equation (Eq. 5--6)
- Time-Dependent Schrödinger Equation (Eq. 7--8)
- Wheeler--DeWitt Equation (Eq. 9--10)
- ADM Formalism: Hamiltonian and Momentum Constraints (Eq. 11--14)
- Klein--Gordon Equation (Eq. 15--16)
- Dirac Equation (Eq. 17--18)
- Einstein Field Equations (Eq. 19--20)
- Maxwell's Equations (Eq. 21--22)
- Planck--Einstein Relation (Eq. 23--24)
- Mass--Frequency--Time Relation (Eq. 25--26)
- Energy--Mass Relation (Eq. 27--28)
- Photon Energy Relation (Eq. 29--30)
- Christoffel Symbols (Eq. 31--32)
- Metric Tensor (Eq. 33--34)
- Quantum Mechanics Compatibility (Eq. 35)
- Tensor Rank (Eq. 36)
- Friedmann Equations (Eq. 37--38)
- Raychaudhuri Equation (Eq. 39--40)
- Yang--Mills Equations (Eq. 41--42)
- Schrödinger--Newton Equation (Eq. 43--44)
- Hawking Radiation Equations (Eq. 45--46)
- Feynman Path Integral Extensions (Eq. 47--48)
Important: In the original QGTCD/SDT theory, the same symbols αρt\alpha,\rho_tαρt and kρt\tfrac{k}{\rho_t}ρtk were inserted across different types of equations (energy, time, force, etc.). To maintain dimensional consistency, we now introduce distinct coupling constants αi\alpha_iαi and kik_iki for each equation's domain.
For example, in energy equations we use αE,kE\alpha_E, k_EαE,kE, in time equations we use αT,kT\alpha_T, k_TαT,kT, and so on. Thus, wherever you see
⋯-α(domain)ρtand⋯+k(domain)ρt,\dots - \alpha_{(\mathrm{domain})},\rho_t \quad\text{and}\quad \dots + \frac{k_{(\mathrm{domain})}}{\rho_t},⋯-α(domain)ρtand⋯+ρtk(domain),
they are meant to have the correct units for the equation at hand (energy, wavefunction amplitude, curvature, etc.).
In a fully rigorous version, you would:
- Define the unit of ρt\rho_tρt (the "time density" or "dark time" variable),
- Specify each αi\alpha_iαi and kik_iki with exact physical units,
- Provide physical motivation for each added term.
Standard Bohr Model:
(1)En=-13.6eVn2.(1)\quad E_n ;=; - \frac{13.6,\text{eV}}{n^2}.(1)En=-n213.6eV.
QGTCD-Modified:
(2)En=-13.6eVn2-αEρt+kEρt.(2)\quad E_n ;=; - \frac{13.6,\text{eV}}{n^2} ;-;\alpha_{E},\rho_t ;+;\frac{k_{E}}{\rho_t}.(2)En=-n213.6eV-αEρt+ρtkE.
(αE,kE\alpha_E, k_EαE,kE carry energy dimension.)
Standard Wave Equation:
(3)∂2ψ∂t2-c2∇2ψ=0.(3)\quad \frac{\partial^2 \psi}{\partial t^2} ;-;c^2,\nabla^2,\psi ;=;0.(3)∂t2∂2ψ-c2∇2ψ=0.
QGTCD-Modified:
(4)∂2ψ∂t2-c2∇2ψ-αψρt+kψρtψ=0.(4)\quad \frac{\partial^2 \psi}{\partial t^2} ;-;c^2,\nabla^2,\psi ;-;\alpha_{\psi},\rho_t ;+;\frac{k_{\psi}}{\rho_t},\psi ;=;0.(4)∂t2∂2ψ-c2∇2ψ-αψρt+ρtkψψ=0.
(αψ,kψ\alpha_\psi, k_\psiαψ,kψ must match the dimension of ∂2/∂t2\partial^2/\partial t^2∂2/∂t2.)
Standard:
(5)iℏ∂ψ∂t=-ℏ22m∇2ψ+V(x)ψ.(5)\quad i\hbar ,\frac{\partial\psi}{\partial t} ;=; - \frac{\hbar^2}{2m},\nabla^2 ,\psi ;+;V(x),\psi.(5)iℏ∂t∂ψ=-2mℏ2∇2ψ+V(x)ψ.
QGTCD-Modified:
(6)iℏ∂ψ∂t=-ℏ22m∇2ψ+V(x)ψ-αSρt+kSρtψ.(6)\quad i\hbar ,\frac{\partial\psi}{\partial t} ;=; - \frac{\hbar^2}{2m},\nabla^2,\psi ;+;V(x),\psi ;-;\alpha_{S},\rho_t ;+;\frac{k_{S}}{\rho_t},\psi.(6)iℏ∂t∂ψ=-2mℏ2∇2ψ+V(x)ψ-αSρt+ρtkSψ.
(αS,kS\alpha_S, k_SαS,kS have energy dimension.)
Standard:
(7)iℏ∂ψ∂t=-ℏ22m∇2ψ+V(r,t)ψ.(7)\quad i\hbar,\frac{\partial \psi}{\partial t} ;=; - \frac{\hbar^2}{2m},\nabla^2,\psi ;+;V(r,t),\psi.(7)iℏ∂t∂ψ=-2mℏ2∇2ψ+V(r,t)ψ.
QGTCD-Modified:
(8)iℏ∂ψ∂t=-ℏ22m∇2ψ+V(r,t)ψ-αS′ρt+kS′ρtψ.(8)\quad i\hbar,\frac{\partial \psi}{\partial t} ;=; - \frac{\hbar^2}{2m},\nabla^2,\psi ;+;V(r,t),\psi ;-;\alpha_{S}',\rho_t ;+;\frac{k_{S}'}{\rho_t},\psi.(8)iℏ∂t∂ψ=-2mℏ2∇2ψ+V(r,t)ψ-αS′ρt+ρtkS′ψ.
Standard:
(9)(-ℏ2Gijklδ2δγijδγkl+γ\prescript3R)Ψ[γij]=0.(9)\quad \Bigl(, -,\hbar^2 ,G^{ijkl} ,\frac{\delta^2}{\delta\gamma_{ij},\delta\gamma_{kl}} ;+;\sqrt{\gamma},\prescript{3}{}{R}\Bigr) ,\Psi[\gamma_{ij}] ;=;0.(9)(-ℏ2Gijklδγijδγklδ2+γ\prescript3R)Ψ[γij]=0.
QGTCD-Modified:
(10)(-ℏ2Gijklδ2δγijδγkl+γ\prescript3R-αWρt+kWρt)Ψ[γij]=0.(10)\quad \Bigl(, -,\hbar^2 ,G^{ijkl},\frac{\delta^2}{\delta\gamma_{ij},\delta\gamma_{kl}} ;+;\sqrt{\gamma},\prescript{3}{}{R} ;-;\alpha_{W},\rho_t ;+;\frac{k_{W}}{\rho_t} \Bigr),\Psi[\gamma_{ij}] ;=;0.(10)(-ℏ2Gijklδγijδγklδ2+γ\prescript3R-αWρt+ρtkW)Ψ[γij]=0.
Standard Hamiltonian Constraint:
(11)H=1γ(πijπij-12(πii)2)-γ\prescript3R≈0.(11)\quad H ;=; \frac{1}{\sqrt{\gamma}} \Bigl(\pi^{ij}\pi_{ij} - \tfrac12(\pi^i_i)^2\Bigr) ;-;\sqrt{\gamma},\prescript{3}{}{R} ;\approx;0.(11)H=γ1(πijπij-21(πii)2)-γ\prescript3R≈0.
QGTCD-Modified:
(12)H=1γ(πijπij-12(πii)2)-γ\prescript3R+αAρt+kAρt≈0.(12)\quad H ;=; \frac{1}{\sqrt{\gamma}} \Bigl(\pi^{ij}\pi_{ij} - \tfrac12(\pi^i_i)^2\Bigr) ;-;\sqrt{\gamma},\prescript{3}{}{R} ;+;\alpha_{A},\rho_t ;+;\frac{k_{A}}{\rho_t} ;\approx;0.(12)H=γ1(πijπij-21(πii)2)-γ\prescript3R+αAρt+ρtkA≈0.
Standard Momentum Constraint:
(13)Hi=-2∇j(πij)≈0.(13)\quad H_i ;=; -,2,\nabla_j,(\pi^j_i) ;\approx;0.(13)Hi=-2∇j(πij)≈0.
QGTCD-Modified:
(14)Hi=-2∇j(πij)+g(ρt)+h(ρt)ρt≈0.(14)\quad H_i ;=; -,2,\nabla_j,(\pi^j_i) ;+;g(\rho_t) ;+;\frac{h(\rho_t)}{\rho_t} ;\approx;0.(14)Hi=-2∇j(πij)+g(ρt)+ρth(ρt)≈0.
Standard:
(15)□ϕ+m2c2ℏ2ϕ=0.(15)\quad \Box ,\phi ;+;\frac{m^2 c^2}{\hbar^2},\phi ;=;0.(15)□ϕ+ℏ2m2c2ϕ=0.
QGTCD-Modified:
(16)(□+m2c2ℏ2-αKρt+kKρt)ϕ=0.(16)\quad \Bigl(, \Box ;+;\frac{m^2 c^2}{\hbar^2} ;-;\alpha_{K},\rho_t ;+;\frac{k_{K}}{\rho_t} \Bigr),\phi ;=;0.(16)(□+ℏ2m2c2-αKρt+ρtkK)ϕ=0.
Standard:
(17)(iℏγμ∂μ-mc)ψ=0.(17)\quad \bigl(,i\hbar,\gamma^\mu,\partial_\mu ;-;m,c,\bigr),\psi ;=;0.(17)(iℏγμ∂μ-mc)ψ=0.
QGTCD-Modified:
(18)(iℏγμ∂μ-mc-αDρt+kDρt)ψ=0.(18)\quad \Bigl(, i\hbar,\gamma^\mu,\partial_\mu ;-; m,c ;-;\alpha_{D},\rho_t ;+;\tfrac{k_{D}}{\rho_t} \Bigr),\psi ;=;0.(18)(iℏγμ∂μ-mc-αDρt+ρtkD)ψ=0.
Standard:
(19)Gμν+Λgμν=8πGc4Tμν.(19)\quad G_{\mu\nu} ;+;\Lambda,g_{\mu\nu} ;=; \frac{8\pi,G}{c^4};T_{\mu\nu}.(19)Gμν+Λgμν=c48πGTμν.
QGTCD-Modified:
(20)Gμν+Λgμν+f(ρt)gμν=8πGc4(Tμν+kE Fρtgμν).(20)\quad G_{\mu\nu} ;+;\Lambda,g_{\mu\nu} ;+;f(\rho_t),g_{\mu\nu} ;=; \frac{8\pi,G}{c^4};\Bigl(,T_{\mu\nu} + \tfrac{k_{E!F}}{\rho_t},g_{\mu\nu}\Bigr).(20)Gμν+Λgμν+f(ρt)gμν=c48πG(Tμν+ρtkEFgμν).
Standard:
(21)∇⋅E=ρε0,∇⋅B=0,(21)\quad \nabla\cdot\mathbf{E} ;=;\frac{\rho}{\varepsilon_0}, \quad \nabla\cdot\mathbf{B} ;=;0,(21)∇⋅E=ε0ρ,∇⋅B=0, ∇×E=-∂B∂t,∇×B=μ0J+μ0ε0∂E∂t.\nabla\times\mathbf{E} ;=; - ,\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla\times\mathbf{B} ;=;\mu_0 ,\mathbf{J} ;+;\mu_0,\varepsilon_0,\frac{\partial \mathbf{E}}{\partial t}.∇×E=-∂t∂B,∇×B=μ0J+μ0ε0∂t∂E.
QGTCD-Modified:
(22)∇⋅E=ρε0+αMρt,∇⋅B=0,(22)\quad \nabla\cdot\mathbf{E} ;=;\frac{\rho}{\varepsilon_0} + \alpha_{M},\rho_t, \quad \nabla\cdot\mathbf{B} ;=;0,(22)∇⋅E=ε0ρ+αMρt,∇⋅B=0, ∇×E=-∂B∂t+kMρt,∇×B=μ0J+μ0ε0∂E∂t+kMρt.\nabla\times\mathbf{E} ;=; -,\frac{\partial \mathbf{B}}{\partial t} ;+;\frac{k_{M}}{\rho_t}, \quad \nabla\times\mathbf{B} ;=;\mu_0 ,\mathbf{J} ;+;\mu_0,\varepsilon_0,\frac{\partial \mathbf{E}}{\partial t} ;+;\frac{k_{M}}{\rho_t}.∇×E=-∂t∂B+ρtkM,∇×B=μ0J+μ0ε0∂t∂E+ρtkM.
Standard:
(23)E=hf.(23)\quad E ;=; h,f.(23)E=hf.
QGTCD-Modified:
(24)E=hfN+kP Eρt.(24)\quad E ;=;\frac{h,f}{N} ;+;\frac{k_{P!E}}{\rho_t}.(24)E=Nhf+ρtkPE.
Standard (speculative formula):
(25)m=hNf.(25)\quad m ;=;\frac{h}{N,f}.(25)m=Nfh.
QGTCD-Modified:
(26)m=hNf+kM Fρt.(26)\quad m ;=;\frac{h}{N,f} ;+;\frac{k_{M!F}}{\rho_t}.(26)m=Nfh+ρtkMF.
Standard:
(27)E=mc2.(27)\quad E ;=;m,c^2.(27)E=mc2.
QGTCD-Modified:
(28)E=mc2+kE Mρt.(28)\quad E ;=;m,c^2 ;+;\frac{k_{E!M}}{\rho_t}.(28)E=mc2+ρtkEM.
Standard (again E=hfE = h,fE=hf):
(29)E=hf.(29)\quad E ;=;h,f.(29)E=hf.
QGTCD-Modified:
(30)E=hfN+kPhρt.(30)\quad E ;=;\frac{h,f}{N} ;+;\frac{k_{Ph}}{\rho_t}.(30)E=Nhf+ρtkPh.
Standard:
(31)Γμνλ=12gλσ(∂μgνσ+∂νgμσ-∂σgμν).(31)\quad \Gamma^\lambda_{\mu\nu} ;=; \tfrac12,g^{\lambda\sigma} \bigl( \partial_\mu,g_{\nu\sigma} ;+;\partial_\nu,g_{\mu\sigma} ;-;\partial_\sigma,g_{\mu\nu} \bigr).(31)Γμνλ=21gλσ(∂μgνσ+∂νgμσ-∂σgμν).
QGTCD-Modified:
(32)Γμνλ→Γμνλ+fΓ(ρt)-kΓρt.(32)\quad \Gamma^\lambda_{\mu\nu} ;\to; \Gamma^\lambda_{\mu\nu} ;+; f_{\Gamma}(\rho_t) ;-;\frac{k_{\Gamma}}{\rho_t}.(32)Γμνλ→Γμνλ+fΓ(ρt)-ρtkΓ.
Standard Metric Tensor:
(33)gμν.(33)\quad g_{\mu\nu}.(33)gμν.
QGTCD-Modified:
(34)gμνλ=gμν+δgμνλ(m,time frames)+kgρt.(34)\quad \tilde{g}{\mu\nu\lambda} ;=; g{\mu\nu} ;+;\delta g_{\mu\nu\lambda}(m,\text{time frames}) ;+;\frac{k_{g}}{\rho_t}.(34)gμνλ=gμν+δgμνλ(m,time frames)+ρtkg.
(Or similarly αgρt\alpha_g,\rho_tαgρt.)
(35)Hψ=(p22m+V+αQρt-kQρt)ψ.(35)\quad H,\psi ;=; \Bigl( \tfrac{p^2}{2m} ;+;V ;+;\alpha_{Q},\rho_t ;-;\tfrac{k_{Q}}{\rho_t} \Bigr),\psi.(35)Hψ=(2mp2+V+αQρt-ρtkQ)ψ.
(36)gμνλ.(36)\quad g_{\mu\nu\lambda}.(36)gμνλ.
(Indicating higher-rank metric modifications from ρt\rho_tρt.)
Standard:
(37)(a˙a)2=8πGρ3-ka2+Λ3,a¨a=-4πG3(ρ+3p)+Λ3.(37)\quad \bigl(\tfrac{\dot{a}}{a}\bigr)^2 ;=; \frac{8\pi G,\rho}{3} ;-;\frac{k}{a^2} ;+;\frac{\Lambda}{3}, \quad \frac{\ddot{a}}{a} ;=; -\frac{4\pi G}{3},(\rho + 3p) ;+;\frac{\Lambda}{3}.(37)(aa˙)2=38πGρ-a2k+3Λ,aa¨=-34πG(ρ+3p)+3Λ.
QGTCD-Modified:
(38)(a˙a)2=8πGρ3-ka2+Λ3+kFρt,a¨a=-4πG3(ρ+3p)+Λ3+αFρt.(38)\quad \bigl(\tfrac{\dot{a}}{a}\bigr)^2 ;=; \frac{8\pi G,\rho}{3} ;-;\frac{k}{a^2} ;+;\frac{\Lambda}{3} ;+;\frac{k_{F}}{\rho_t}, \quad \frac{\ddot{a}}{a} ;=; -\frac{4\pi G}{3},(\rho + 3p) ;+;\frac{\Lambda}{3} ;+;\alpha_{F},\rho_t.(38)(aa˙)2=38πGρ-a2k+3Λ+ρtkF,aa¨=-34πG(ρ+3p)+3Λ+αFρt.
Standard:
(39)dθdτ=-13θ2-σμνσμν+ωμνωμν-Rμνuμuν.(39)\quad \frac{d\theta}{d\tau} ;=; -\tfrac{1}{3},\theta^2 ;-;\sigma_{\mu\nu},\sigma^{\mu\nu} ;+;\omega_{\mu\nu},\omega^{\mu\nu} ;-;R_{\mu\nu},u^\mu,u^\nu.(39)dτdθ=-31θ2-σμνσμν+ωμνωμν-Rμνuμuν.
QGTCD-Modified:
(40)dθdτ=-13θ2-σμνσμν+ωμνωμν-Rμνuμuν+kRρt.(40)\quad \frac{d\theta}{d\tau} ;=; -\tfrac{1}{3},\theta^2 ;-;\sigma_{\mu\nu},\sigma^{\mu\nu} ;+;\omega_{\mu\nu},\omega^{\mu\nu} ;-;R_{\mu\nu},u^\mu,u^\nu ;+;\frac{k_{R}}{\rho_t}.(40)dτdθ=-31θ2-σμνσμν+ωμνωμν-Rμνuμuν+ρtkR.
Standard:
(41)DμFμν=Jν.(41)\quad D_\mu,F^{\mu\nu} ;=;J^\nu.(41)DμFμν=Jν.
QGTCD-Modified:
(42)DμFμν+kY MρtFμν=Jν.(42)\quad D_\mu,F^{\mu\nu} ;+;\frac{k_{Y!M}}{\rho_t},F^{\mu\nu} ;=;J^\nu.(42)DμFμν+ρtkYMFμν=Jν.
Standard:
(43)iℏ∂ψ∂t=-ℏ22m∇2ψ+mΦψ,∇2Φ=4πGm∣ψ∣2.(43)\quad i\hbar,\frac{\partial \psi}{\partial t} ;=; -\frac{\hbar^2}{2m},\nabla^2 \psi ;+;m,\Phi,\psi, \quad \nabla^2 \Phi ;=; 4\pi G,m,|\psi|^2.(43)iℏ∂t∂ψ=-2mℏ2∇2ψ+mΦψ,∇2Φ=4πGm∣ψ∣2.
QGTCD-Modified:
(44)iℏ∂ψ∂t=-ℏ22m∇2ψ+mΦψ-αNρtψ+kNρtψ,(44)\quad i\hbar,\frac{\partial \psi}{\partial t} ;=; -\frac{\hbar^2}{2m},\nabla^2 \psi ;+;m,\Phi,\psi ;-;\alpha_{N},\rho_t,\psi ;+;\frac{k_{N}}{\rho_t},\psi,(44)iℏ∂t∂ψ=-2mℏ2∇2ψ+mΦψ-αNρtψ+ρtkNψ, ∇2Φ=4πGm∣ψ∣2+f(ρt).\nabla^2 ,\Phi ;=; 4\pi G,m,|\psi|^2 ;+;f(\rho_t).∇2Φ=4πGm∣ψ∣2+f(ρt).
Standard:
(45)TH=ℏc38πGMkB.(45)\quad T_H ;=; \frac{\hbar,c^3}{8\pi,G,M,k_B}.(45)TH=8πGMkBℏc3.
QGTCD-Modified:
(46)TH=ℏc38πGMkB+αHρt+kHρt.(46)\quad T_H ;=; \frac{\hbar,c^3}{8\pi,G,M,k_B} ;+;\alpha_{H},\rho_t ;+;\frac{k_{H}}{\rho_t}.(46)TH=8πGMkBℏc3+αHρt+ρtkH.
(αH,kH\alpha_{H}, k_{H}αH,kH have temperature dimension.)
Standard Feynman Path Integral:
(47)Z=∫exp (iS[x(t)])Dx(t).(47)\quad Z ;=; \int \exp!\bigl(,i,S[x(t)]\bigr),\mathcal{D}x(t).(47)Z=∫exp(iS[x(t)])Dx(t).
QGTCD-Modified (Dark Time) Path Integral:
(48)Z=∫exp (iS[x(t)]+αPIρt-kPIρt)Dx(t).(48)\quad Z ;=; \int \exp!\Bigl(, i,S[x(t)] ;+;\alpha_{\mathrm{PI}},\rho_t ;-;\frac{k_{\mathrm{PI}}}{\rho_t} \Bigr),\mathcal{D}x(t).(48)Z=∫exp(iS[x(t)]+αPIρt-ρtkPI)Dx(t).
In every equation above, the original pair
±αρtor±kρt\pm,\alpha,\rho_t \quad\text{or}\quad \pm,\frac{k}{\rho_t}±αρtor±ρtk
is replaced by
±αiρtor±kiρt,\pm,\alpha_{i},\rho_t \quad\text{or}\quad \pm,\frac{k_{i}}{\rho_t},±αiρtor±ρtki,
where αi\alpha_{i}αi and kik_{i}ki are chosen so each new term has the correct dimension for the physical quantity (energy, time interval, field strength, curvature, etc.). In principle, each αi\alpha_iαi or kik_iki is a distinct constant (with separate units), to be determined experimentally or theoretically.