Celestial Mechanics in Spherical Space

Scientific Paper
Title	Celestial Mechanics in Spherical Space
Read in full	Link to paper
Author(s)	Tuomo Suntola
Keywords	Cosmology, zero-energy principle, Dynamic
Published	2001
Journal	Apeiron
Volume	8
Number	3
No. of pages	22
Pages	65-86

Read the full paper here

Abstract

The Dynamic Universe model describes space as the surface of a four-dimensional sphere expanding in the direction of the 4-radius. Instead of being defined as a physical constant the velocity of light becomes determined by the velocity of space in the fourth dimension. The changing velocity of light and the dynamics of space allow time to be defined as a universal scalar. Local mass centres modify space in the fourth dimension, giving a space geometry with features that are closely related to those of the Schwarzschild metrics based on four-dimensional space-time. In the modified space geometry the local velocity of light is a function of the local tilting of space in the fourth dimension. The precise geometry of space makes it possible to solve the effect of the 4-D topology on Kepler's laws and the orbital equation. The perihelion shift of planetary orbits can be derived in closed mathematical form as the rotation of the main axis of Kepler's orbit relative to the reference coordinate system. For a full revolution the rotation is D_j = 6pGM/c²a(1 - e²) like the corresponding prediction in the general theory of relativity.

Overview

This 2001 Apeiron paper applies Tuomo Suntola's Dynamic Universe (DU) model to the two-body problem, and derives planetary orbits — including the perihelion advance — without using the field equations of general relativity. In the DU picture, three-dimensional space is the surface of a four-dimensional sphere expanding along its 4-radius. The velocity of that expansion, rather than a postulated constant, is the velocity of light; because the velocity of space is a dynamic quantity and light is tied to it, time can be kept as a universal scalar rather than being made frame-dependent. Near a mass centre the surface is tilted in the fourth dimension, and the local velocity of light is reduced by the cosine of the tilting angle. The fourth dimension in DU is "purely geometrical in nature" — a real direction of space, not a time-like coordinate.

Suntola's method is Newtonian in form and geometrical in content: he computes the flat-space component of the free-fall acceleration in locally tilted space, substitutes it into the classical equation of motion, and then works through the standard Kepler machinery — eccentricity vector, energy integral, orbital equation — to a closed-form result. He obtains a perihelion advance of 6πGM/c²a(1 − e²) per revolution, numerically identical to the general-relativistic first approximation. The paper's distinctive claim is not this agreement but a difference in what happens afterwards: he argues that the standard Schwarzschild-based orbital solution, iterated over many revolutions, degrades — the eccentricity grows, the perihelion shift decays, and for extreme parameters the orbit escapes — whereas the DU orbit conserves its shape, size and perihelion advance indefinitely.

The argument

The instability Suntola identifies in the standard solution

The paper opens with the Schwarzschild line element and the familiar orbital solution for 1/r, in which the perihelion advance arises from a term in sinφ·φ inside the square brackets. Setting φ = 0 to 2π gives the classical first-revolution result

Δφ = 6πGM / [c²a(1 − e²)]

which Suntola acknowledges "is consistent with the observed perihelion shifts of Mercury and several binary pulsars".

His objection concerns multiple revolutions. Evaluating the same expression at φ = n·2π, he finds a decreasing perihelion advance accompanied by a cumulative increase in eccentricity, so that the orbit becomes progressively more asymmetric. His figure, computed for a gravitational factor δ = GM/ac² = 4 × 10⁻³ and e = 0.6, shows the orbit extending to infinity beyond twelve revolutions. He notes for scale that the gravitational factor of Mercury is about 2.6 × 10⁻⁸ and that of the binary pulsar PSR 1913+16 about 4.6 × 10⁻⁶, and states that the calculation does not include gravitational radiation. His conclusion is that "the instability of the orbit predicted by equation (3) is inconsistent with observation, which indicates a problem either in the Schwarzschild metrics or in approximations made in solving the geodesic equations."

The tilted-space line element

The DU line element near a mass centre is written

ds² = −c²_δdt² + dr²_0δ/(1−δ)² + r²_0δ(dθ² + sin²θ dφ²)

with δ = 2GM/rc² the local gravitational factor and cosφ_tilt = (1 − δ). The first term carries the reduction of the local velocity of light through the tilting, c_δ = c_0δcosφ_tilt; the second carries the lengthening of the radial line element, dr_δ = dr_0δ/cosφ_tilt. Suntola stresses two contrasts with Schwarzschild: the fourth dimension here is geometrical rather than time-like, and both the radial line element in the second term and the distance in the last term are measured in the direction of non-tilted, apparent homogeneous space rather than in the local tilted frame.

Free-fall acceleration from energy conservation

Rather than solving geodesic equations, Suntola derives the acceleration from the conservation of total energy. The free-fall velocity and the local imaginary velocity of space sum orthogonally to the imaginary velocity of apparent homogeneous space, giving

v_ff = c_0δ√[1 − (1−δ)²]

Differentiating and projecting onto the flat-space direction yields the acceleration in local space, a_ff = −(GM/r²_0δ)(1−δ)², and its flat-space component

a_ff(0δ) = −(GM/r²_0δ)(1−δ)³

Substituting this for the Newtonian acceleration in d²r/dt² = −μr/r³, and introducing the critical radius r_c = μ'/c²_0δ, gives the DU equation of motion on the base plane:

d²r_0δ/dt² = −μ'(1 − r_c/r_0δ)³ r_0δ/r³_0δ ≈ −μ'(1 − 3r_c/r_0δ) r_0δ/r³_0δ

The whole of the relativistic correction is thus carried by a single factor (1 − 3r_c/r), the cube of the tilting factor linearised for weak fields. The paper is explicit that the result holds where δ ≪ 1 and the approximation (1−δ)³ ≈ (1−3δ) is accurate.

The eccentricity vector and the rotation of the main axis

Angular momentum per unit mass, k_0δ = r_0δ × ṙ_0δ, is shown to be conserved, since the modified force remains central. The eccentricity vector, however, is not. Forming ṙ̇ × k and reducing, Suntola arrives at

de_0δ/dt = −(3r_c/r_0δ) d(r_0δ/r_0δ)/dt

which is zero in Newtonian mechanics. Expanding dr_0δ in polar components, the radial parts cancel exactly and only the perpendicular component survives:

de_0δ/dt = −(3r_c/r_0δ)(dφ/dt) r̂_⊥

This is the paper's cleanest structural result. Because the change in the eccentricity vector is purely rotational, "the orbit conserves its eccentricity but is subject to a rotation of the main axis." Kepler's condition de/dt = 0 is recovered exactly in a coordinate system rotating at

dψ/dt = (3r_c/r_0δ) dφ/dt

Substituting Kepler's r_0δ = a(1−e²)/(1 + ecosφ) and integrating gives

Δψ(φ) = 3r_c(φ + esinφ) / [a(1 − e²)]

so that the orbital equation is Kepler's equation with argument (φ − Δψ). At φ = 2π, with r_c = μ'/c²,

Δψ(2π) = 6πμ' / [c²a(1 − e²)]

identical to the general-relativistic first approximation. Crucially, in the DU derivation this expression is not a first term of a series that decays with successive orbits: the rotation rate is a property of the rotating frame in which the orbit is exactly Keplerian, so the shift is the same on every revolution.

The energy integral and the radial perturbation

Suntola then examines Kepler's energy integral h = v²_r(0δ)/2 − μ'/r_0δ, which is conserved in Newtonian mechanics. Under the modified acceleration its derivative is

dh/dt = 3μ'r_cṙ_0δ/r³_0δ

Tracing this through the relation between h, k and the orbital parameters, he obtains a differential radial perturbation dr_0δ = 6r_cesinφ dφ/(1−e²) and, integrating,

Δr_0δ(φ) = 6r_ce(1 − cosφ)/(1 − e²)

This vanishes at perihelion and reaches its maximum, 12r_ce/(1−e²), at aphelion. The complete flat-space orbital equation is then Kepler's ellipse with both the axis rotation Δψ and the radial perturbation Δr_0δ included. Suntola's comparison figure, again for δ = 4 × 10⁻³ and e = 0.6 over ten revolutions, shows the DU orbit as a slightly deformed ellipse that conserves its shape while its axis rotates uniformly, against the standard solution's progressive radius reduction in the first two quarters, increase in the last two, and degrading perihelion shift.

Restoring the fourth dimension

The planar solution is a projection. To recover the actual path on the curved surface, Suntola adds a z coordinate measuring distance from a reference plane in the direction of apparent homogeneous space:

z_0δ = 2√(r_0δr_0δc) − 2√[a(1−e²)r_0δc]

with dz_0δ = B dr_0δ, B = tanφ_tilt = √[1−(1−δ)²]/(1−δ). In cylindrical coordinates the squared line element becomes ds² = r²_0δ(1 + A² + A²B²)dφ², with A = dr_0δ/dφ, and path length follows by integration. The paper closes with a plot of the z–x profile of Mercury's orbit in the solar gravitational frame, showing an excursion of order 10⁴ km in the fourth dimension against an orbital scale of 10⁸ km. Suntola thanks Ari Lehto, Heikki Sipila, Raimo Lehti and Matts Roos.

Assessment

The paper is carefully executed and, within its own framework, honest about its scope. Its most attractive feature is the transparency of the derivation. Where the general-relativistic result emerges from geodesic equations that most readers take on trust, Suntola's route runs entirely through Newtonian vector mechanics, with the whole of the relativistic effect residing in one geometrically motivated factor, (1−δ)³. The demonstration that the change in the eccentricity vector is purely perpendicular — so that eccentricity is conserved and only the apsidal line rotates — is a genuinely elegant piece of work, and it delivers the perihelion advance as an exact rotation rate rather than as a secular term. That is a real structural difference from the usual textbook presentation, and it is the paper's strongest card: it gives a Kepler orbit that is exactly Keplerian in a slowly rotating frame, with the shift constant on every revolution by construction rather than by approximation. The additional radial perturbation, zero at perihelion and maximal at aphelion, is a concrete and in principle checkable prediction, as is the fourth-dimensional profile computed for Mercury.

The paper's central critical claim, however, does not survive scrutiny, and it is the claim on which the case for preferring DU rests. Suntola's equation (3) — the source of the alleged instability — is a first-order perturbative solution of the relativistic orbit equation, valid for small GM/ac² and for φ not large. Terms of the form φsinφ are secular terms, an artefact of naive perturbation theory that grow without bound and destroy the approximation once φ becomes comparable to the inverse of the small parameter. They are removed by standard resummation methods, and the exact relativistic orbit equation, u″ + u = GM/h² + 3GMu²/c², has bounded, precessing solutions with conserved energy and angular momentum for the parameters in question. So the "instability" Suntola displays in his figure is a property of the truncated series, not of the Schwarzschild solution. His own diagnosis — "a problem either in the Schwarzschild metrics or in approximations made in solving the geodesic equations" — names the correct alternative and then does not pursue it. He also chooses δ = 4 × 10⁻³ for the demonstration, some five orders of magnitude larger than Mercury's and nearly three larger than the Hulse–Taylor pulsar's, precisely the regime where a first-order expansion is least trustworthy. The general-relativistic orbit is stable; the case for DU therefore cannot rest on rescuing it.

That said, the paper's positive result is not damaged by this. Reproducing 6πGM/c²a(1−e²) from a wholly different geometry is a real achievement, and the DU orbit is stable for the right reason rather than by accident. But agreement at this order is also where the test of a gravitation theory begins. The paper does not compute light deflection, Shapiro delay, gravitational redshift, geodetic or frame-dragging precession, or the Nordtvedt effect — the observables that distinguish metric theories from one another at post-Newtonian order and that are now constrained at the 10⁻⁵ level by Cassini ranging and lunar laser ranging. Nor does it engage with binary-pulsar timing beyond a passing mention: the orbital decay of PSR 1913+16 matches the gravitational-radiation prediction to a fraction of a percent, and a theory in which gravity is a geometric tilt of an expanding surface owes an account of that decay. Suntola notes explicitly that his comparison figure ignores gravitational radiation, but the DU model's own prediction for it is not given.

Two further gaps deserve mention. First, the additional radial perturbation Δr_0δ is a prediction not shared with general relativity, and it is quantitative: for Mercury it should be of order 12r_ce/(1−e²), a small but definite aphelion displacement. The paper does not evaluate it numerically for any real system, and does not check it against the planetary ephemerides, which constrain such residuals tightly. This is the paper's most obvious missed opportunity, since it is the one place where DU and general relativity are said to differ observationally. Second, the variable light velocity on which the whole construction rests is used only geometrically here; its consequences for atomic spectra, for the interpretation of the very timing measurements against which the perihelion shift is checked, and for laboratory bounds on c anisotropy are not addressed in this paper, though they are presumably treated in the 2001 monograph to which everything is referred.

Taken on its own terms, then, the work is sound where it computes and overreaches where it criticises. It shows that a curved-surface, universal-time geometry can reproduce the classical test of the perihelion advance in closed form, which is worth knowing. It does not show that the standard solution is unstable, and it does not yet subject the DU model to the range of measurements that a competitor to general relativity must face.

See also

• Tuomo Suntola
• General Relativity
• Gravity
• Speed of Light
• Variable Speed of Light
• Cosmology
• Expanding Universe
• Equivalence Principle
• Gravitomagnetism
• Gravitational Waves
• Apeiron
• Ari Lehto