Dmitry Ponomarev V.
The description of antigravity as a special case of gravitational interaction and the mechanism of its occurrence (generation) are presented in the works of the relativistic model of antigravity interaction of bodies [1]. The general theory of relativity (GTR) is the generally accepted fundamental theory of gravity, and the classical laws of Newtonian mechanics are derived from its equations in the Newtonian limit. The complete derivation of the general equation of the gravitational force acting on a extended test object (an element of matter) with a velocity gradient from another material object and the determination of the conditions for changing its direction vector (antigravity) according to GTR and its mathematical apparatus are presented in the work «Antigravity as a consequence of the principle of extremality of proper time for a long object with a velocity gradient in general relativity» [2]. This work lays the fundamental theoretical basis for the relativistic model of antigravity interaction between bodies, relying on GTR as the only consistent modern theory of gravity. The articles «The basic equation of antigravity» [3], «The antigravity point» [4], and «The antigravity force» [5] describe antigravity in the Newtonian limit with relativistic corrections.
In this article, we will focus only on the general logical sequence of deriving the equations describing the antigravitational interaction of bodies and present the basic equations of [2].
Note: Since the "Dzen" platform does not allow for the correct placement and display of mathematical formulas and designations of physical quantities, we recommend that you refer to the original source of this article: Ponomarev Dmitry V. Antigravity in the framework of the general theory of relativity // COGNITIO RERUM. 2026. № 3. С. 7-13.
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The derivation of the equations describing the possibility of antigravity is based on the consistent identification of precise relationships between the geometry of space-time, the kinematics of an object in its stationary state, and the resulting force. The fundamental basis for this derivation is one of the fundamental principles of general relativity, which is the principle of extremality of proper time. According to this principle, a freely moving test particle follows a geodesic, a world line that makes its proper time extremal. In other words, the test particle moves towards a region where time flows more slowly. This corresponds to the idea that everything in the world tends to move towards the future. In regions where time flows faster, the future has already passed, while in regions where time flows slower, the future has not yet arrived compared to the first region. Therefore, the region with slower time flow relative to the region with faster time flow represents the future. This is where the test particle moves towards, occupying a more «comfortable» position in space-time.
Thus, the derivation of the necessary equations begins with the definition of a physical system and its state: an infinitesimal extended test element of matter in a stable dynamic configuration is considered, radially oriented in a static spherically symmetric mass M field (shown in Figure 1 as a blue line), described by the Schwarzschild metric.
For further work, we introduce the following notation:
M – the mass of the central gravitating body;
dl – the intrinsic length of an infinitesimal extended test element of matter;
1 and 2 – the first and second equipotential surfaces of the gravitational field of a body with mass M;
G – the gravitational constant;
c – the speed of light in a vacuum;
r_s = 2‧G‧M/c^2 – the Schwarzschild radius;
A(r) = 1 — r_s/r – the component of the metric tensor responsible for the temporal part of the interval and directly related to gravitational time dilation;
ω – the coordinate angular velocity with which an extended test element of matter moves in a curvilinear motion relative to a body with mass M;
t, r, θ, φ – the Schwarzschild coordinates (coordinate time, radial coordinate, polar and azimuthal angles);
τ – the proper time.
It should be noted that Figure 1 does not depict a body with a mass of M, so the space is not represented as curved. It is assumed that a spherically symmetric body with a mass of M will be located in the center of the depicted coordinate systems, and the space-time will be curved. It is important to note that Figure 1 is a schematic representation and reflects the situation when ω = 0. This is done to provide a visual and schematic representation of how the element of matter is positioned in space (in this case, in a flat space-time), and the direction of ω is indicated to demonstrate the future movement of the element. However, when a body with a mass of M and ω > 0 is present, the space-time will be curved, which will be described mathematically (without a visual graphical representation) in this paper.
Let us add that the word «extended» (when applied to an element of material matter) does not mean any significant value of size, but rather that the presence of any material matter implies at least dl → 0, i.e., dl ≠ 0, otherwise (when dl = 0) there is simply no material matter. Let us also agree that for brevity, we will refer to such an extended radial test element of material matter as «element» («extended radial test element of material matter in a stable dynamic state» = «element»).
It is necessary to understand that the element under consideration will actually be a component part of a larger object, namely a complex technical structure designed to create a gravitational lift force. This structure is known as an «antigravity wing» and is designed to provide directional movement of matter along a curved, closed trajectory at high speeds, while also maintaining structural integrity under calculated mechanical loads.
The key is to describe the dynamic state: the element is performing a curvilinear motion (in this case, a uniform circular motion, i.e., rotation) relative to the body M, and in its own (comoving) reference frame, the distance between the ends, i.e., the intrinsic length L_0 = dl, remains constant. This state does not describe a hypothetical perfectly rigid body, but rather a physically realizable stationary configuration within the framework of general relativity, where all internal stresses and relativistic deformations are balanced.
Therefore, based on the above, it follows that an element in a moving state is not the same object as an element in a stationary state.
This physical state, defined in the accompanying system, must be correctly expressed in the global coordinates used to describe the entire system. In the Schwarzschild coordinates, the condition of constant intrinsic length of the element manifests itself as the constancy of the coordinate angular velocity ω(t) = dφ/dt for all points of the element (i.e., ω(t) = const). This correspondence is not an obvious identity, but rather a precise consequence of the metric: in order for the ends of the element, which are located at points with different gravitational potentials, to maintain a constant distance in their rest system, their motion in the coordinates must be synchronized in a specific way. If ω were to depend on t, it would imply the presence of angular acceleration, which would require external moments of force or changes in internal stresses, violating the very property of a stable dynamic state. Therefore, the condition of ω = const is not an independent postulate; it is derived as a mathematical consequence of the physical properties that describe the state of the object.
The next and most important step is to calculate the local, physically measurable velocities of the element’s ends. The physical velocity is determined relative to the momentarily comoving reference frame (MCRF) according to the following rule:
For a point at a distance r_d from the axis of rotation, the proper azimuthal displacement is r_d ∙ dφ, and the proper time of a stationary observer there is dτ = (A(r))^(1/2) ‧ dt. If we project points 1 and 2 on the z-axis in Figure 1, then in the Schwarzschild metric, r_d(r) = r ∙ cos(α), where: r_d – is the radius of rotation of the element point relative to the z-axis, and the angle α = (π/2) — θ. Thus, r_d = r · cos(α) – is true both in the coordinate and physical sense (for the radius of rotation), and it does not depend on the speed of motion. The physical radius of rotation (locally measured) is equal to r · cos(α) because in the azimuthal direction, the Schwarzschild metric gives:
Therefore, the physical velocity of the point of the element is:
Therefore, the local speed is obtained as:
This formula is the core of the derivation, as it shows that in general relativity, the local velocity is determined not only by the geometric factor r, but also by the relativistic factor 1/(A(r))^(1/2), which reflects the gravitational time dilation at a given point in the field.
A direct analysis of this formula for the ends of the element with radii r_1 = r + dl/2 (far) and r_2 = r — dl/2 (near) allows us to establish the exact nature of their motion. Since:
and the A(r) function increases with radius, the denominator for υ_1 is greater than for υ_2. The numerator for υ_1 is also greater. Quantitative analysis of the ratio:
shows that, while remaining greater than one (i.e., υ_1 > υ_2), it is also less than the purely Newtonian ratio r_1/r_2. Thus, the gravitational time dilation does not negate the fact that the far end is moving faster than the near end, but it reduces the difference in their velocities compared to flat space. It is this precisely calculated difference, rather than the simple relationship υ = ω ∙ r ∙ cos(α), and more importantly, the gradient of Lorentz factors it generates:
along the element, they become the primary source of asymmetry, which, within the framework of the Einstein field equations, leads to a qualitatively new effect: antigravity.
Having the exact expressions for υ(r) and, therefore, for the 4-velocity U^μ, we can proceed to a rigorous calculation of the force as a measure of deviation from free (geodesic) fall, which is determined by the principle of extremality of proper time. To do this, we use the fundamental equation of geodesic deviation, which is the main tool of general relativity for analyzing the relative acceleration of test masses. This equation involves the radial separation vector ξ^μ, which fixes the geometry of the element, the calculated components of the 4-velocity, which contain υ(r) and γ(r), and the specific non-zero components of the Riemann curvature tensor for the Schwarzschild metric. Solving this tensor equation yields an expression for the relative 4-acceleration of the ends of the element.
The resulting relative acceleration is then transformed, through the geodesic equation with Christoffel symbols, into the 4-acceleration a^μ of the mass element dm itself. This transformation is a direct application of the principle of extremality of proper time: the calculated 4-acceleration a^μ quantifies how much the actual motion of the element differs from the motion along a local geodesic (free fall). According to the principle of extremality of proper time and its consequence, the geodesic equation of motion, the presence of such a non-zero 4-acceleration means that a force is acting on the element. After algebraic transformations, the radial component a^r takes a form that explicitly depends on the square of the velocity υ^2 and the square of the Lorentz factor γ^2. The elementary force acting on the element is given by dF^r = dm ⋅ a^r, where dm is the mass of the element.
Since the quantities υ and γ in the expression for a^r are themselves functions of the radial coordinate (υ(r’), γ(r’)), the elementary force dF^r also varies along the element. Therefore, to find the total force acting on an element of finite length dl, it is necessary to integrate dF^r from r_2 to r_1. It is crucial to use the exact dependence υ(r’) = (r’ ‧ ω ‧ cos(α))/(A(r’))^(1/2), which was derived earlier, at this stage.
After integrating and expanding the result in a series with respect to the small parameter dl/r, we obtain the final formula for the gravitational force F in the Schwarzschild metric (see formula (8)).
Equations (4) – (7) indicate a very important fact: antigravity is only possible when the interacting material bodies move in a curved path relative to each other. When they move in a straight line relative to each other, antigravity is impossible, because in this case, υ_1 = υ_2.
Thus, the physical mechanism of gravity or antigravity is based on the constant competition between two relativistic corrections to the rate of flow of one’s own time:
- Gravitational time dilation ((A(r))^(1/2)), which tends to accelerate an extended object towards the center of the field;
- Kinematic time dilation (1/γ = (1 — υ^2/c^2)^(1/2)), which is associated with the curved motion of an extended object.
At high speeds, the kinematic correction becomes dominant. For an element of a given object with a velocity gradient (υ_1 > υ_2), the kinematic deceleration is stronger at the far end. This creates an effective gradient of «total time rate» (due to the gravitational and kinematic corrections) along the element, which manifests itself as a resultant force in the geodesic deviation equation. When the critical speed υ > υ_crit is exceeded (see equations (8) and (9)), the contribution of the kinematic deceleration becomes dominant, resulting in a configuration of relative accelerations of the extended object’s parts where the resultant force is directed away from the center of the mass M, indicating antigravity.
The final logic of deriving the formula for the gravitational force with the possibility of changing its sign (i.e., with the possibility of antigravity):
- The principle of extremality of proper time (geodesics) →
- The definition of a well-established dynamic state with a radial orientation of the element →
- The condition of constant proper length →
- The coordinate condition ω = const in the Schwarzschild metric →
- The exact formula for the local velocity in MCRF:
υ(r) = (r ‧ ω ‧ cos(α))/(A(r))^(1/2) →
- The establishment of the exact gradient of velocities and Lorentz factors:
(υ_1 > υ_2, γ_1 ≠ γ_2) →
- Substitution into the equation of geodesic deviation and calculation of Christoffel symbols →
- Calculation of the 4-acceleration a^μ as a measure of deviation from the geodesic (a direct consequence of the principle of extremality of proper time) →
- Integration of the elementary force dF^r = dm ⋅ a^r →
- The final formula for the gravitational force F, which demonstrates the possibility of changing the sign (direction) when υ > υ_crit.
Thus, the entire chain of reasoning is a consistent and necessary progression from the fundamental principle and the definition of a specific dynamic configuration to its precise coordinate description in curved spacetime, from there to the calculation of local dynamic quantities taking into account relativistic effects, and finally to the substitution of these quantities into the fundamental equations of general relativity to obtain a quantitative result. The calculated force is the force acting specifically on an element of matter in its stationary moving state with a radial orientation, and it is fundamentally different from the force that would act on the same element of matter in a different configuration. Each step of the reasoning serves to establish the precise quantitative nature of the relationships under the conditions of general relativity, leading to the profound conclusion that the sign of the gravitational force can change when the critical velocity υcrit is reached, as a direct consequence of Einstein’s equations.
A rigorous derivation of the gravitational force F equation in full general relativity and a simplified derivation of the gravitational force F equation in linearized general relativity based on the above logic are presented in [2], and their detailed examination is beyond the scope of this article. However, we will present the final formulas for the gravitational force F from this work below.
The full formula for the gravitational force F in the Schwarzschild metric is as follows:
The formula for the gravitational force F in the linearized approximation (weak field):
This expression is a limiting case of the full formula when r_s/r → 0, i.e. A(r) = 1 — r_s/r ≈ 1.
The critical speed is the speed at which the gravitational force F changes sign (direction):
- In full GTR:
- In the linearized approximation:
The paper [2] also provides a description of the key role of a material body that is both a source of gravity and a «point of support» for antigravity (including analogies with the lift of an airplane wing and the movement of a sailboat against the wind), examines all the energy conditions that determine the possibility of effective repulsion (antigravity) with a positive energy-momentum density of bodies and without considering negative mass or exotic forms of matter, and presents other evidence of the reality of antigravity (invariant quantities, local measurability, observer consistency, equations in covariant form, and correspondence with the Dixon equations, and analogies with electromagnetism).
A clear proof of the principles discussed above is the technology used by humanity on a daily basis in GPS/GLONASS satellites, namely the gravitational and kinematic corrections that are incorporated into the algorithms of these systems. The speed of a satellite is something that we can control, and the speed of an object determines the rate of time flow, which means that we can also control this. The gravitational and kinematic corrections that are incorporated into the GPS/GLONASS system (which depend on the speed of the satellite) lead to accurate results for humans, and therefore humans are already using the results of their control over time. Similarly, in the definition of gravity and antigravity, the same principle of the sum of gravitational and kinematic time dilation applies, but the only difference is the scale. To change the direction of the gravitational force vector (i.e., to achieve antigravity), the object must be accelerated to more than 70.7% of the speed of light.
The analogy between an airplane wing and an antigravity wing is also illustrative:
- Airplane wing: The geometry of an airplane wing provides for the appearance of a lift force, which is caused by the velocity gradient of the air flow above and below the wing (the difference in air pressure), creating a resultant upward force.
- Antigravity wing: The curvilinear motion of a material matter element as part of an antigravity wing provides for the appearance of an antigravity force, which is caused by the velocity gradient between the far and near ends of this element (the difference in the flow of time), creating a resultant upward force.
If the airplane wing is flat, there is no lift force. If the motion of matter is straight, there is no antigravity force. It’s very simple: the curved space of time, like air, is a real physical entity. In both cases, we don’t just «spin/accelerate the material matter elements in an antigravity wing» or «spin the blades of a helicopter» or «accelerate an airplane,» but we provide the system with additional energy, which, due to the geometry of the airplane wing, results in different air flow velocities at the top and bottom of the wing, and in the case of an antigravity wing, results in different time flow velocities at the top and bottom of the antigravity wing element, and in both cases, this different flow determines the force vector.
In addition to the above, it should be noted that from a technical point of view, rotation seems to be the most convenient way to achieve curvilinear motion. However, this does not mean that an antigravity wing, for example, will be represented by a rotating disk; it will be much more complex. No one is planning to rotate iron blocks at insane speeds in reality, and no one is planning to rotate disks; this will not happen in reality. Several articles on the resource [1] have already been dedicated to the experimental proof and technical implementation of antigravity.
Let’s summarize the final results. The derivation within the framework of general relativity allowed us to obtain a formula for the gravitational force acting on a material element moving in the gravitational field of a massive body. We established the existence of a critical speed, exceeding which leads to a change in the sign of the force from attraction to repulsion.
Let us note the key points:
- The antigravity effect is caused by the difference in conditions at the ends of a long object;
- It is possible with a positive mass and without involving exotic forms of matter, and it does not violate fundamental physical principles;
- The effect is invariant and measurable, which confirms its physical reality.
Thus, a new mechanism of gravitational interaction in general relativity has been discovered, expanding our understanding of the dynamics of long objects and opening up prospects for both theoretical and experimental research in the future.
References:
- Relativistic model of antigravity interaction of bodies. – URL: https://antigravity-theory.ru (date of request: 03.03.2026).
- Ponomarev Dmitry V. Antigravity as a consequence of the principle of extremality of proper time for a long object with a velocity gradient in general relativity. – URL: https://antigravity-theory.ru/antigravity-gtr (date of request: 03.03.2026).
- Ponomarev Dmitry V. The basic equation of antigravity // Internauka: Electronic Scientific Journal. 2025. No. 26(390). Part 3. Pp. 17-23. – URL: https://internauka.org/journal/science/internauka/390 (date of request: 03.03.2026).
- Ponomarev Dmitry V. The antigravity point // Internauka: Electronic Scientific Journal. 2025. No. 27(391). Part 2. Pp. 34-44. – URL: https://internauka.org/journal/science/internauka/391 (date of request: 03.03.2026).
- Ponomarev Dmitry V. The antigravity force // Internauka: Electronic Scientific Journal. 2025. No. 28(392). Part 2. Pp. 64-68. – URL: https://internauka.org/journal/science/internauka/392 (date of request: 03.03.2026).
Date of publication:
March 16, 2026, St. Petersburg
The original article was published in the journal "COGNITIO RERUM":
The official website on the relativistic model of antigravity interaction of bodies: