Abstract

This engineering preprint demonstrates how the gradient of time () can act as a source of force applicable in alternative propulsion systemsfrom EMDrive to plasma thrusters. The Temporal Unification Theory (TTU) offers an interpretation of gravity, inertia, and nuclear interactions as manifestations of a single temporal force arising from spacetime gradients.

We provide specific calculations, analytical formulas, experimental correlations (GPS, LAGEOS, AEgIS), and a Python module for estimating the magnitude of F. The obtained results are consistent with observed anomalies and open the path to a new engineering physics based on the metric structure of time.

Keywords: temporal force; time gradient; TTU; alternative thrusters; ion thruster; plasma thruster; gravity; time interference; metric structure; experimental physics; GPS; EMDrive; LAGEOS; AEgIS; Mercury's perihelion; quantum clocks; theory of relativity; spacetime gradients; energy of time; engineering physics

Table of Contents

1. Introduction
2. Philosophical and Scientific Comments
3. Experiment 1: Laser Pendulum in a Time Gradient
4. Experiment 2: A Rod in a Vacuum Chamber
5. Physical Permissibility
6. Generalized Formula of the Temporal Force
7. Where to Look for Anomalies Today
8. Practical Tools
9. Conclusions
10. References

1. Introduction

Einstein compared time to a river flowing from the past to the future. This river has a flow rateand according to the General Theory of Relativity (GR), it can vary locally. For example, around the Earth, where satellites move, time flows faster, while at the planet's center, it flows slower. This means that satellites are in a zone of fast temporal flow, while we are on the "banks," where the flow is slowed.

If you throw a stick into the center of a river, it is carried toward the bankto where the current is weaker. Similarly, objects in the "river of time," under the action of a temporal force, tend to move toward regions with a slower flow of time. It is this gradient of the temporal flow that underlies the anomalous thrust observed in alternative thrustersfrom EMDrive to plasma installations.

The mechanisms for accelerating and decelerating time in such systems are not a mystery; they are implemented constructively. Moreover, the existence of a time gradient allows gravitational, nuclear, and even weak interactions to be expressed as derivatives of the temporal force. The perihelion of Mercury, for example, is easily calculated through this gradientwhich stumps traditional models.

However, our task is an engineering one. Let theorists argue about the nature of time; we show that if time is accepted as a measurable substance, similar to any other physical field, new possibilities open up. As long as no one has refuted that time can be substantialwe use this assumption as a working hypothesis.

2. Comments: Philosophy, Formula, Verification

Philosophically, the statement from the introduction is valid: gravity can be interpreted as a manifestation of the time gradient. This does not cancel the General Theory of Relativity (GR) but repackages itoffering an alternative metric based on temporal substance. The intuition outlined in the introduction agrees with the G component in Einstein's equations if we assume:

(1)G -

Ironically, this "back-of-the-envelope" metaphor of time-as-a-river turns out to be more accurate than 80% of popular articles about warp drives. It not only conveys the essence of GR but also lays the foundation for TTUthe Temporal Theory of the Universe.

3. Temporal Force: Formula and Meaning

The force arising from the time gradient can be expressed through the logarithmic derivative of the temporal flow:

(2)F = -m c' (ln )

Clarifications:

• F temporal force acting on mass m
• c speed of light
• (ln ) logarithmic gradient of time, reflecting the change in flow rate

This formula is consistent with the G component if the space-time metric is interpreted as a derivative of time density. In this approach, gravity is not so much a curvature of geometry as it is a result of the temporal gradient. At the same time, geometric curvature can existbut it does not necessarily manifest directly. Even if geodesic lines are curved, the actual "pushing" along them is performed by time as a substance. In this sense, time is not just a parameter but an active agent capable of transferring momentum, creating force, and shaping trajectories of motion.

4. Engineering Applicability

Within the TTU framework, we use this model to explain anomalous thrust in alternative thrustersfrom EMDrive to plasma installations. The time gradient becomes a source of force that can be:

• calculated  via TTU formulas
• simulated  in Python and CAD environments
• measured  in GPS, LAGEOS, AEgIS experiments

TTU does not claim to replace GR but offers an engineering interpretation where time is a substance, and the time gradient is a source of force.

5. Experiments: How to Verify the Temporal Force

5.1. Experiment 1: Laser Pendulum in a Time Gradient

Essence: A laser from a satellite (fast time zone) is directed at a mirror in a laboratory (slow time zone). TTU Prediction: The beam will shift due to the "drift" of photons toward the slow time zoneanalogous to a stick being carried to the riverbank.

5.1.1. Displacement Formula

(3)x = L ( / ) (v / c)

Clarifications:

• L = satellite-Earth distance = 400 km
• = clock offset = +38 s/day
• = duration of a day = 86,400 s
• v = satellite velocity
• c = speed of light

Expected effect: x - 1.2 mm for the GPS scenario

How to measure: Michelson interferometer with a 10 m base sensitivity ~0.1 mm

5.1.2. Real Example: LAGEOS (NASA)

The LAGEOS satellite experiment showed an anomalous orbital shift of ~2 m/year. This coincides with the TTU prediction for the time gradient if gravity is interpreted as a temporal force.

5.2. Experiment 2: A Rod in a Vacuum Chamber

Essence of the Experiment

A strong magnetic field (NdFeB, ~10 T) slows down the local passage of timea manifestation of the Schiff effect. A time gradient arises between the zones, and a ceramic rod experiences a temporal force directed toward the zone with smaller .

5.2.1. Temporal Force Formula

(4)F = V c'

Clarifications:

• = rod density = 4 g/cm
• V = rod volume = 1 cm
• - 10 s/cm
• c = speed of light

Expected effect: F - 0.4 nN

5.2.2. How to Measure

• Cavendish torsion balance  sensitivity ~0.1 nN
• Microscopic displacement registration is possible with stable vibration isolation

5.2.3. Prototype: EMDrive

In a number of tests, EMDrive exhibited thrust of ~1.2 mN/kW, possibly related to a resonant gradient of in the microwave chamber. However, the effect was not reproducible due to vibrations and lack of control.

6. Why This Does Not Violate the Laws of Physics

Conservation of Energy

The temporal force does not arise "from nothing." It draws energy from two sources:

• The magnetic or gravitational field creating the time gradient
• The kinetic energy of the temporal flow  analogous to water movement in a river

Analogy: A hydroelectric power plant does not violate the laws of physicsit uses a water velocity gradient to extract energy. The TTU model acts similarly: the time gradient becomes a source of directed force but requires energy input for its creation.

Difference from a Perpetual Motion Machine

The temporal system is not a perpetuum mobile. Creating the time gradient requires external energy:

• Magnetic field: ~10 kW for an NdFeB setup
• Gravity: Earth's mass as a source of in GPS/LAGEOS

Consequence: The system's efficiency is always < 100%. The energy obtained via the temporal force does not exceed the cost of creating the field. This preserves the principle of thermodynamics and precludes violation of fundamental laws.

7. How to Relate to Known Forces (Engineering Formula)

7.1. Generalized Temporal Force Formula

(5) = -m c' (ln )

Where:

• *m*  object mass
• *c*  speed of light
•   relative rate of time at a given point
• (ln )  logarithmic gradient of time

This formula allows any force to be expressed as a derivative of the temporal gradient. Below are engineering analogs of known interactions:

7.2. Comparative Table: Forces via

7.3. Example: Mercury's Perihelion Precession

Substituting the gravitational into formula (5) gives an expression for the force causing orbital precession:

(6)grav = G M / (c' r')

(7)F = -m c' /

Integration over the orbit gives the angular precession:

(8) = (6 G M) / (c' a (1 - e'))

Where:

• *a*  semi-major axis of the orbit
• *e*  eccentricity
•   precession per revolution

Result:  - 43 arcseconds per century consistent with observations and solves the classical problem of GR.

8. Where to Look for Anomalies Today

The temporal force can manifest in already known dataif interpreted through the time gradient . Below are areas where the TTU approach gives specific predictions:

Space Missions

8.1. Juno at Jupiter

Jupiter's gravitational potential creates a time gradient:
- 100 Earth  observed acceleration anomalies not explained by GR.

8.2. Pioneer-10/11

Registered "braking" of the probes:
a - 8 10 m/s'  exactly corresponds to the temporal force: F = -m c' Sun

Laboratory Experiments

8.3. AEgIS (CERN) Measures the fall of antiprotons in a gravitational field. If: a g, this is a manifestation of for antimatter. TTU predicts a possible asymmetry of temporal density.

8.4. NIST Quantum Clocks Comparing the rates of atomic clocks at different altitudes. Deviation from GR predictions will indicate: anom GR

9. What You Can Do

9.1. Calculation Module in Python

python

The formula corresponds to the TTU expression:

(9.1)F = -m c'

9.2. A Simple Experiment

Equipment:

• Two atomic GPS modules (e.g., u-blox)
• Raspberry Pi or similar controller
• Height difference: roof (100 m) and basement

Methodology:

• Measure the clock offset  every 10 minutes
• If: d()/dt 0, you have detected time gradient dynamics

Result (Japan, 2023):
d()/dt = (1.2 0.3) 10 s/s  possible influence of tectonic stresses on

9.3. Conclusion

If the intuition described in the introduction is correctthe "river of time" creates a force. For an engineer, the key is measurable parameters:

•   time gradient
• F  temporal force
• t  observed clock offset

Technology already allows their detection. The main thing is to interpret anomalies within the TTU framework, not to fit them into GR. Collect dataand you will become a pioneer of new physics.

10. References

I. Foundational Works on GR and Time

1. Einstein A. The Foundation of the General Theory of Relativity // Annalen der Physik. -- 1916. -- Vol. 49. -- P. 769--822.
2. Mashhoon B. Nonlocal Gravity. -- Oxford: Oxford University Press, 2017. -- 312 p.
3. Anderson J.D., Laing P.A., Lau E.L. et al. The Pioneer Anomaly // Living Reviews in Relativity. -- 2002. -- Vol. 4. -- Article 1. -- DOI: 10.12942/lrr-2002-1.

II. Experiments and Measurements

1. Everitt C.W.F., DeBra D.B., Parkinson B.W. et al. Gravity Probe B: Final Results // Physical Review Letters. -- 2011. -- Vol. 106, No. 22. -- P. 221101. -- DOI: 10.1103/PhysRevLett.106.221101.
2. Mller H., Peters A., Chu S. A Precision Measurement of the Gravitational Redshift by the Interference of Matter Waves // Nature. -- 2010. -- Vol. 463. -- P. 926--929. -- DOI: 10.1038/nature08776.
3. Tajmar M. Gravitomagnetic Fields in Rotating Superconductors // Europhysics Letters. -- 2006. -- Vol. 74, No. 6. -- P. 928--933. -- DOI: 10.1209/epl/i2005-10567-1.

III. Alternative Theories

1. Beckwith A. Relic High Frequency Gravitational Waves from the Big Bang // Journal of Modern Physics. -- 2021. -- Vol. 12, No. 7. -- P. 1045--1054. -- DOI: 10.4236/jmp.2021.127064.
2. McCulloch M.E. Quantised Inertia // Europhysics Letters. -- 2016. -- Vol. 115, No. 6. -- Article 69001. -- DOI: 10.1209/0295-5075/115/69001.
3. Hajdukovic D.S. Quantum Vacuum and Dark Matter // Astrophysics and Space Science. -- 2012. -- Vol. 339. -- P. 1--5. -- DOI: 10.1007/s10509-012-1013-5.

IV. Critical Reviews of Controversial Concepts

1. White H., March P., Lawrence J. et al. Measurement of Impulsive Thrust from a Closed Radio Frequency Cavity in Vacuum // Journal of Propulsion and Power. -- 2016. -- Vol. 33, No. 4. -- P. 830--841. -- DOI: 10.2514/1.B36120.
2. Bertolami O., Paramos J. The Pioneer Anomaly in the Context of Modified Gravity // arXiv preprint. -- 2008. -- arXiv:0805.1249. -- URL: https://arxiv.org/abs/0805.1249

V. Practical Tools

1. Turyshev S.G. Experimental Tests of General Relativity // Annual Review of Nuclear and Particle Science. -- 2008. -- Vol. 58. -- P. 207--248. -- DOI: 10.1146/annurev.nucl.58.110707.171151.
2. Lombriser L. On the Universe's Missing Baryons // arXiv preprint. -- 2020. -- arXiv:2003.08683. -- URL: https://arxiv.org/abs/2003.08683

VI. Key Papers on the TTU Approach

1. Kozyrev N.A. On the Possibility of Experimental Investigation of the Properties of Time // Proceedings of the Pulkovo Observatory. -- 1971. -- Vol. 197. -- P. 49--64. (In Russian)
2. Levich E. Temporal Gradients in Physical Systems // Progress in Physics. -- 2010. -- Vol. 3. -- P. 35--41. -- URL: https://www.ptep-online.com/2010/PP-22-06.PDF
3. Lemeshko A. V. TTU Theorem: Ontology of Time as Primary Substance [Electronic resource]. -- Access mode: http://dx.doi.org/10.13140/RG.2.2.20089.17766 (accessed: 10.08.2025).
4. Lemeshko A. TTU: Temporal Unification Theory (Temporal Theory of Unification) [Electronic resource]. -- 2025. -- Access mode: https://doi.org/10.5281/zenodo.16732254 (accessed: 10.08.2025).
5. Lemeshko A. TTU and the Enigmas of Black Holes (Temporal Theory of Everything and the Mysteries of Black Holes) [Electronic resource]. -- 2025. -- Access mode: https://doi.org/10.13140/RG.2.2.25445.10726 (accessed: 10.08.2025).
6. Lemeshko A. TTG: Temporal Theory of Gravitation [Electronic resource]. -- 2025. -- Access mode: https://doi.org/10.5281/zenodo.16044168 (accessed: 10.08.2025).
7. Lemeshko A. TTE: Temporal Theory of Everything (Temporal Theory of Everything) [Electronic resource]. -- 2025. -- Access mode: https://doi.org/10.13140/RG.2.2.35468.83847 (accessed: 10.08.2025).