Pais PhysicsLaboratory

Equation Visualizers

Interactive tools for exploring the essential physics equations derived from Pais's work. Adjust parameters in real-time to understand the relationships between electromagnetic fields, energy densities, and theoretical temporal effects.

10
Equations
6
Categories
25
Parameters
∞
Combinations

Categories

Time Manipulation

2 equations

Fundamental Relations

1 equations

Schwinger Limit

2 equations

Energy Density

1 equations

Torsion Physics

2 equations

Quantum Vacuum

1 equations

Quantum Mechanics

1 equations

Time Manipulation

2 equations

Reverse Temporal Excursion Equation

Primary

The primary time manipulation formula presented by Salvatore Pais. Describes how breaking the Schwinger limit can produce negative changes in time (reverse temporal excursion).

−Δt=τ0×Δωω2-\Delta t = \tau_0 \times \frac{\Delta\omega}{\omega^2}

Interactive Parameters

Negative change in time
−Δt-\Delta t
0.100000
seconds
Reverse temporal excursion - movement backward in time
Dimensionless constant (tau of s)
τ0\tau_0
0.100000
dimensionless
Critical constant involving energy density, permittivity, charge density, and characteristic radii
Change in angular frequency
Δω\Delta\omega
1.000e+6
rad/s
The change in angular frequency of the system
Angular frequency
ω\omega
1.000e+12
rad/s
The angular frequency of vibration or spin
Physical Meaning

Demonstrates that manipulating frequency at extreme energy densities can theoretically reverse the arrow of time

Required Conditions
  • •Breaking the Schwinger limit (E ≈ 1.32 × 10^18 V/m)
  • •Energy density of 10^25 J/m³
  • •Quantum vacuum breakdown
  • •Pair production threshold

Dimensionless Constant Tau

Primary

The dimensionless constant that encodes the critical energy density and geometric factors required for temporal manipulation.

τ0=1025×ε0σ2×rsrω\tau_0 = 10^{25} \times \frac{\varepsilon_0}{\sigma^2} \times \frac{r_s}{r_\omega}

Interactive Parameters

Critical energy density
102510^{25}
0.100000
J/m³
Energy density in joules per cubic meter required to break the Schwinger limit
Permittivity of free space
ε0\varepsilon_0
0.100000
F/m
The electric constant, a fundamental constant in electromagnetism
Surface charge density
σ\sigma
1.000e-15
C/m²
The distribution of electric charge per unit area
Ratio of characteristic radii
rs/rωr_s / r_\omega
0.100000
Geometric factor relating characteristic length scales in the system
Physical Meaning

Encodes the relationship between electromagnetic field properties, energy density, and geometric factors necessary for breaking the arrow of time

Fundamental Relations

1 equations

Time-Frequency Inverse Relationship

Primary

The foundational principle that time is inversely proportional to frequency. This is the basis for understanding how frequency manipulation can affect time.

t∝1ωt \propto \frac{1}{\omega}
Alternative form:t∝1ft \propto \frac{1}{f}
Physical Meaning

Higher frequencies correspond to shorter time intervals; by manipulating frequency at quantum levels, one can theoretically affect the flow of time itself

Schwinger Limit

2 equations

Schwinger Critical Electric Field

Essential

The critical electric field strength at which the quantum vacuum undergoes pair production (electron-positron pairs spontaneously created from vacuum).

Ec=me2c3qeℏ≈1.32×1018 V/mE_c = \frac{m_e^2 c^3}{q_e \hbar} \approx 1.32 \times 10^{18} \text{ V/m}

Interactive Parameters

Electron mass
mem_e
0.100000
kg
Speed of light
cc
0.100000
m/s
Elementary charge
qeq_e
0.100000
C
Reduced Planck constant
ℏ\hbar
0.100000
J·s
Physical Meaning

At this field strength, the vacuum becomes nonlinear, virtual particles become real, and spacetime structure is fundamentally altered

Schwinger Critical Magnetic Field

Essential

The critical magnetic field strength at which quantum vacuum breakdown occurs.

Bc=me2c2qeℏ≈4.41×109 TB_c = \frac{m_e^2 c^2}{q_e \hbar} \approx 4.41 \times 10^9 \text{ T}

Interactive Parameters

Electron mass
mem_e
0.100000
kg
Speed of light
cc
0.100000
m/s
Elementary charge
qeq_e
0.100000
C
Reduced Planck constant
ℏ\hbar
0.100000
J·s
Physical Meaning

The threshold at which magnetic fields become strong enough to create particles from the vacuum

Energy Density

1 equations

Schwinger Field Energy Density

Essential

The energy density of electromagnetic fields at the Schwinger limit. For E ≈ 10^18 V/m, this yields approximately 10^25 J/m³.

u=ε02E2+12μ0B2u = \frac{\varepsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2

Interactive Parameters

Permittivity of free space
ε0\varepsilon_0
0.100000
F/m
Electric field
EE
1.000e+12
V/m
Permeability of free space
μ0\mu_0
0.100000
H/m
Magnetic field
BB
1000.000000
T

Torsion Physics

2 equations

Torsion Tensor Connection

Supporting

The connection coefficient in Einstein-Cartan theory, showing how spacetime has both curvature and torsion.

Γμνλ={λμν}+Kμνλ\Gamma^\lambda_{\mu\nu} = \{\lambda \mu \nu\} + K^\lambda_{\mu\nu}

Interactive Parameters

Connection coefficient
Γμνλ\Gamma^\lambda_{\mu\nu}
0.100000
Describes how vectors change as they are parallel transported
Christoffel symbols
{λμν}\{\lambda \mu \nu\}
0.100000
Represents curvature (standard General Relativity)
Contorsion tensor
KμνλK^\lambda_{\mu\nu}
0.100000
Related to torsion (Einstein-Cartan addition)
Physical Meaning

Torsion is sourced by spin angular momentum and can propagate as waves, potentially allowing energy extraction from vacuum

Spin-Torsion Coupling

Supporting

The coupling between the spin density tensor and torsion in Einstein-Cartan theory.

Tμνλ=κSμνλT^\lambda_{\mu\nu} = \kappa S^\lambda_{\mu\nu}

Interactive Parameters

Torsion tensor
TμνλT^\lambda_{\mu\nu}
0.100000
Describes the torsion of spacetime
Spin angular momentum tensor
SμνλS^\lambda_{\mu\nu}
0.100000
Describes the spin density of matter
Coupling constant
κ\kappa
0.100000
Determines the strength of spin-torsion coupling
Physical Meaning

Spinning masses create torsion fields that couple to spacetime geometry

Quantum Vacuum

1 equations

Casimir Force (Standard Form)

Supporting

The attractive force between two parallel plates due to quantum vacuum fluctuations.

F=πℏcA480L4F = \frac{\pi \hbar c A}{480 L^4}

Interactive Parameters

Reduced Planck constant
ℏ\hbar
0.100000
J·s
Speed of light
cc
0.100000
m/s
Area of plates
AA
1.000e-6
m²
Distance between plates
LL
1.000e-9
m
Physical Meaning

Demonstrates that vacuum has real, measurable energy that can produce mechanical forces

Quantum Mechanics

1 equations

Energy-Time Uncertainty Relation

Supporting

Heisenberg's uncertainty principle for energy and time.

ΔE×Δt≥ℏ2\Delta E \times \Delta t \geq \frac{\hbar}{2}

Interactive Parameters

Energy uncertainty
ΔE\Delta E
0.100000
J
Time uncertainty
Δt\Delta t
0.100000
s
Reduced Planck constant
ℏ\hbar
0.100000
J·s
Physical Meaning

Provides quantum mechanical foundation for the time-frequency inverse relationship

Important Notes

Experimental Status

Proven: Schwinger field values, Casimir effect, Energy-time uncertainty
Observed: Light-by-light scattering at ATLAS (2019), Gyroscopic weight anomalies
Theoretical: Reverse temporal excursion, Practical time manipulation
Speculative: Breaking arrow of time, Controlled retrocausality

Usage Guidelines

  • Use high-precision arithmetic for calculations involving very large (10²⁵) and very small (10⁻³⁴) numbers
  • Maintain dimensional consistency - all equations are dimensionally correct
  • Interactive sliders help visualize parameter relationships and extreme values
  • Results approaching or exceeding Schwinger limits indicate theoretical thresholds