The Relativistic Central Force Problem in Biquaternionic Form
Introduction
A central force is one whose direction is the line joining the particle to a fixed centre and whose magnitude depends only on the distance. In non-relativistic mechanics the problem is solved by two conserved quantities, the energy and the angular momentum, and the orbit follows from Binet's equation. The companion articles of the sibling category have developed that solution inside the biquaternion algebra: the force is a real vector commuting with the position, the angular momentum is the vector part of the product $\mathbf{r}\mathbf{p}$, the motion is planar, and for the inverse-square law the orbit is a conic section, with the Runge–Lenz vector conserved as the mark of a hidden symmetry.
This article treats the same problem relativistically. The change is a single one and it propagates everywhere: the inertial factor that resists the force is the relativistic mass $\gamma m$, not the rest mass $m$. The mass-shell relation replaces the non-relativistic energy–momentum relation, and the resulting orbit equation is the non-relativistic Binet equation with $m$ replaced by $\gamma m$. For a general central force the orbit then depends on the energy as well as on the angular momentum, and it is not closed. For the inverse-square force the equation stays linear and the orbit is an exact precessing conic, whose apsidal advance per revolution is the article's central result.
Three things are established, and they are worth separating before the derivation begins.
- The relativistic Binet equation. From the mass-shell relation and the two conserved quantities, the orbit equation $$ \frac{d^2u}{d\theta^2}+u = -\frac{(E-V)F(1/u)}{c^2L^2u^2}, \qquad u=\frac{1}{r}, $$ is derived, where $E$ is the conserved total energy, $L$ the conserved angular momentum, $V$ the potential energy and $F=-dV/dr$ the radial force. It is the non-relativistic Binet equation with the rest mass replaced by the relativistic mass $(E-V)/c^2=\gamma m$.
- The precessing conic. For the inverse-square force $F=-\kappa/r^2$ the equation is linear, and its solution is $$ r(\theta)=\frac{p}{1+e\cos\omega(\theta-\theta_0)}, \qquad \omega^2=1-\frac{\kappa^2}{c^2L^2}, $$ with $p=c^2L^2\omega^2/(\kappa E)$ and the eccentricity fixed by the energy. The orbit is a conic in the rescaled angle $\omega\theta$, so it is a conic that precesses; the advance per revolution is $\Delta=2\pi(1/\omega-1)$, which for small $\kappa^2/(c^2L^2)$ is $\pi\kappa^2/(c^2L^2)$.
- The bound-orbit condition. Bound orbits without a fall into the centre require $L>\kappa/c$, and the circular orbit is exact, with $E=mc^2\omega$ and $r_c=L^2\omega/(m\kappa)$. The condition is genuinely relativistic: in the non-relativistic problem every positive angular momentum admits a bound orbit.
The article is deliberately parallel to its non-relativistic counterparts. Central Forces and the Classical Kepler Problem in Biquaternionic Form solves the inverse-square orbit and exhibits the Runge–Lenz vector; The Classical Coulomb Problem and Its Hidden SO(4) Symmetry in Biquaternionic Form develops its symmetry algebra; The Central-Scalar Limit of Classical Mechanics in Biquaternionic Form fixes the algebraic criterion for a force to be central and derives Binet's equation. The present article re-derives the relativistic orbit equation from the mass-shell relation rather than importing it, and it locates the one structural difference between the two problems: the relativistic mass is a function of the state, so the effective force is not central in the non-relativistic sense and the closed orbit opens into a precessing one.
The companion article The Central Scalar Field: Classical Dynamics in the Biquaternion Center supplies the field-theoretic origin of the potential treated here: the static field of a central scalar source is the $1/r$ potential, and a test particle in it is the problem of this article.
Conventions. The algebra is $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$; the basis is $e_0=1,e_1,e_2,e_3$ with $e_k^2=-e_0$ and $e_je_k=\varepsilon_{jkl}e_l$ for $j\neq k$; the scalar imaginary $i$ is central with $i^2=-e_0$. The anti-Hermitian subspace $\mathbb{M}_-$ is the material sector and the Hermitian subspace $\mathbb{M}_+$ the informational sector, with $\mathbb{B}=\mathbb{M}_+\oplus\mathbb{M}_-$ and $i\mathbb{M}_\pm=\mathbb{M}_\mp$. Position, momentum, force and angular momentum are real vectors in the three-space $\operatorname{span}\{e_1,e_2,e_3\}\subset\mathbb{M}_-$; the energy, the potential and the rest mass are central scalars, real multiples of $e_0$. For two real vectors, $$ \mathbf{a}\mathbf{b}=-\mathbf{a}\cdot\mathbf{b}\,e_0+\mathbf{a}\times\mathbf{b}, \qquad [\mathbf{a},\mathbf{b}]=2\,\mathbf{a}\times\mathbf{b}, $$ so that the angular momentum is $\mathbf{L}=\mathbf{r}\times\mathbf{p}=\tfrac12[\mathbf{r},\mathbf{p}]$, the vector part of the quaternion product $\mathbf{r}\mathbf{p}$, and its magnitude is $L=|\mathbf{L}|=\sqrt{N(\mathbf{L})}$. The four-position is $\tilde{Q}=ict\,e_0+\mathbf{x}\in\mathbb{M}_-$ and the four-momentum is $\tilde{P}=iE/c\,e_0+\mathbf{p}$, whose biquaternion norm is the mass-shell relation. The biquaternionic gradient is $\tilde{\nabla}=e_0\partial_{ict}+e_1\partial_x+e_2\partial_y+e_3\partial_z$, with $\Box=\tilde{\nabla}\tilde{\nabla}^{\natural}=\partial_{ict}^2+\Delta$. Throughout, $c=1/\sqrt{\epsilon\mu}$ is the speed of light in the medium and $c_0$ its vacuum value; the symbol $\mathbf{v}$ is reserved for particle and frame velocities. The coupling $\kappa$ is positive for an attractive inverse-square force; for gravity $\kappa=GMm$, and for the Coulomb problem $\kappa=-q_1q_2/(4\pi\epsilon_0)$, as fixed in the companion article on the classical Coulomb problem.
The Relativistic Central Force
The Conserved Quantities
Let the particle have rest mass $m$ and move in a static central potential of potential energy $V(r)$. The relativistic Lagrangian is
$$ \mathcal{L}_{\text{part}}=-\frac{mc^2}{\gamma}-V(r), \qquad \gamma=\frac{1}{\sqrt{1-\mathbf{v}^2/c^2}}, $$
and the two conserved quantities are the generalized energy
$$ E=\frac{\partial \mathcal{L}_{\text{part}}}{\partial\mathbf{v}}\cdot\mathbf{v}-\mathcal{L}_{\text{part}}=\gamma mc^2+V(r), $$
which includes the rest energy, and the angular momentum
$$ \mathbf{L}=\mathbf{r}\times\mathbf{p}, \qquad \mathbf{p}=\gamma m\mathbf{v}, \qquad L=|\mathbf{L}| . $$
The energy is conserved because the potential is static, and the angular momentum because the force is central. The two conservation laws are what reduce the problem to quadrature, and in the framework's language they are respectively a central-scalar statement and a statement about the vector part of a quaternion product.
It is useful to name the combination that appears at every step. From $E=\gamma mc^2+V$,
$$ \gamma m=\frac{E-V}{c^2}=m_{\text{rel}}, $$
the relativistic mass, which is a function of the event through $V(r)$. In the free asymptotic region $V\to0$ and $m_{\text{rel}}\to\gamma_\infty m=E/c^2$. The rest mass is recovered only in the non-relativistic limit.
The Centrality of the Force
The force is the real vector
$$ \mathbf{F}=F(r)\,\hat{\mathbf{r}}, \qquad F(r)=-\frac{dV}{dr}, $$
with $\hat{\mathbf{r}}=\mathbf{r}/|\mathbf{r}|$ and $F$ the outward radial component. Because $\mathbf{F}$ is parallel to $\mathbf{r}$,
$$ \mathbf{F}\parallel\mathbf{r} \quad\Longleftrightarrow\quad [\mathbf{r},\mathbf{F}]=0 \quad\Longleftrightarrow\quad \mathbf{F}\mathbf{r}\ \text{central}, $$
which is the algebraic criterion for a central force established in the companion article The Central-Scalar Limit of Classical Mechanics in Biquaternionic Form. The conservation of angular momentum is then the two-line computation of that article,
$$ \dot{\mathbf{L}}=\tfrac12[\mathbf{r},\dot{\mathbf{p}}]=\tfrac12[\mathbf{r},\mathbf{F}]=0 , $$
with the proper time or the coordinate time giving the same vanishing because the two differ by the central factor $\gamma$. The motion is planar, in the plane orthogonal to $\mathbf{L}$.
The Mass-Shell Relation and the Radial Momentum
The relativistic content enters through the mass-shell relation for the four-momentum. With $\tilde{P}=iE/c\,e_0+\mathbf{p}$ and the biquaternion norm $N(\tilde{P})=\tilde{P}\tilde{P}^{\natural}=-E^2/c^2+\mathbf{p}^2$, the on-shell condition with the potential included is
$$ (E-V)^2=\mathbf{p}^2c^2+m^2c^4 . $$
Decomposing the momentum into radial and tangential parts, $\mathbf{p}^2=p_r^2+L^2/r^2$ with $p_r=\gamma m\dot r$, the mass-shell relation becomes the radial constraint
$$ (E-V)^2-m^2c^4 = c^2\left(p_r^2+\frac{L^2}{r^2}\right). $$
The left-hand side is fixed by the state $(E,L)$ and the right-hand side is non-negative; this inequality is what bounds the motion, and it replaces the non-relativistic effective-potential discussion. It is the same biquaternion-norm identity that the companion article The Relativistic Particle in Biquaternionic Form uses for the free mass shell, now written with the potential shifted into the energy.
The Relativistic Binet Equation
From the Constraint to the Orbit Equation
Let $u=1/r$, so that the orbit is the function $u(\theta)$, and let a prime denote $d/d\theta$. The radial momentum is related to $u'$ by a chain rule that is worth writing once:
$$ p_r=\gamma m\,\dot r=\gamma m\,\frac{dr}{d\theta}\,\dot\theta =\gamma m\left(-\frac{u'}{u^2}\right)\frac{L}{\gamma m r^2}=-L\,u' , $$
using $\dot\theta=L/(\gamma m r^2)$ from the definition of $L$. Substituting into the radial constraint,
$$ (E-V)^2-m^2c^4 = c^2L^2\left(u'^2+u^2\right). \tag{1} $$
This is the orbit equation in implicit form. Differentiating with respect to $\theta$ turns it into a differential equation. With $V=V(1/u)$,
$$ \frac{dV}{d\theta}=V'\frac{dr}{d\theta}=V'\left(-\frac{u'}{u^2}\right), \qquad V'=\frac{dV}{dr}=-F(1/u), $$
so that
$$ \frac{d}{d\theta}(E-V)^2=2(E-V)\bigl(-V'\bigr)\left(-\frac{u'}{u^2}\right)=\frac{2(E-V)F(1/u)\,u'}{u^2}. $$
Differentiating the right-hand side of (1) gives $c^2L^2(2u'u''+2uu')$. Equating and dividing by $2u'$ — the case $u'=0$ is the circular orbit, treated separately below — yields the relativistic Binet equation
$$ \boxed{\;\frac{d^2u}{d\theta^2}+u=-\frac{(E-V)F(1/u)}{c^2L^2u^2}=-\frac{m_{\text{rel}}F(1/u)}{L^2u^2}.\;} $$
The second form is the one to read. Since $m_{\text{rel}}=(E-V)/c^2=\gamma m$, the relativistic Binet equation is the non-relativistic Binet equation with the rest mass replaced by the relativistic mass, the latter evaluated at the running radius. In the non-relativistic limit $\gamma\to1$ and $E\to mc^2$, and the equation reduces to
$$ u''+u=-\frac{mF(1/u)}{L^2u^2}, $$
which is the equation of the companion article on the central-scalar limit. The angular momentum $L$ here is the relativistic angular momentum $\gamma mr^2\dot\theta$, not the non-relativistic $mr^2\dot\theta$.
What the Relativistic Mass Changes
The substitution $m\to\gamma m$ is not a relabelling, and it is worth stating what it does. In the non-relativistic problem the right-hand side of Binet's equation is a function of $u$ alone, because $m$ and $L$ are constants. Relativistically it is a function of $u$ alone as well, but through the combination $(E-V)/c^2$, which varies along the orbit. The consequence is that the equation is no longer of the Kepler type except for the one force law for which the $u$-dependence cancels in the right way. That force law is the inverse-square force, and it is treated next.
For a general power law $F=-\kappa r^{-n}$ the equation is nonlinear and the orbit is not a conic. The qualitative relativistic effects — a perihelion advance, a modification of the effective potential and the possible loss of a centrifugal barrier — are present for every central force, and the inverse-square case is the one that can be solved in closed form.
The Inverse-Square Force and the Precessing Conic
The Linear Equation
For the inverse-square force write
$$ V(r)=-\frac{\kappa}{r}=-\kappa u, \qquad F(r)=-\frac{\kappa}{r^2}=-\kappa u^2, $$
with $\kappa>0$ for attraction. Substituting into the relativistic Binet equation,
$$ u''+u=\frac{\kappa(E+\kappa u)}{c^2L^2}=\frac{\kappa E}{c^2L^2}+\frac{\kappa^2}{c^2L^2}\,u , $$
which rearranges into a linear equation with constant coefficients,
$$ u''+\omega^2u=\frac{\kappa E}{c^2L^2}, \qquad \omega^2=1-\frac{\kappa^2}{c^2L^2}. $$
This is the central result of the article: the relativistic inverse-square problem is exactly linear, with the frequency $\omega$ replacing the unit frequency of the non-relativistic problem. The frequency is real precisely when
$$ L>\frac{\kappa}{c}, $$
which is the relativistic centrifugal-barrier condition discussed below; when it fails, $\omega$ is imaginary, the solutions of the equation are exponentials rather than oscillations, and the orbits fall into the centre.
The Precessing Conic
The general solution of the linear equation is
$$ u(\theta)=\frac{\kappa E}{c^2L^2\omega^2}+C\cos\omega(\theta-\theta_0) =\frac{1}{p}\left[1+e\cos\omega(\theta-\theta_0)\right], $$
with the semi-latus rectum and the eccentricity given by
$$ p=\frac{c^2L^2\omega^2}{\kappa E}, \qquad e=Cp . $$
The eccentricity is fixed by the radial constraint (1), evaluated at a turning point. The larger root of the constraint in $u$ is $u_{\max}=(1+e)/p$, and substituting it into (1) with $u'=0$ gives, after rearrangement,
$$ e^2=1+\frac{c^2L^2\omega^2\left(E^2-m^2c^4\right)}{\kappa^2E^2} =1+\frac{L^2\omega^2\,\varepsilon\left(2mc^2+\varepsilon\right)}{\kappa^2E^2}, \qquad \varepsilon=E-mc^2 . $$
The second form separates the rest energy and is useful in the non-relativistic limit; the quantity $\varepsilon=E-mc^2$ is the energy measured from the rest energy, which becomes the ordinary mechanical energy as $c\to\infty$.
The orbit is therefore
$$ r(\theta)=\frac{p}{1+e\cos\omega(\theta-\theta_0)}, $$
a conic in the rescaled angle $\varphi=\omega\theta$. In the plane coordinatized by $(r,\varphi)$ it is an ordinary Kepler conic with its focus at the centre; in the physical plane the apsidal line advances by
$$ \Delta=\frac{2\pi}{\omega}-2\pi=2\pi\left(\frac{1}{\sqrt{1-\kappa^2/(c^2L^2)}}-1\right) $$
per revolution. For small $\kappa^2/(c^2L^2)$ the advance is
$$ \Delta\simeq\frac{\pi\kappa^2}{c^2L^2}, $$
and the sign is a prograde advance: since $\omega<1$, the particle must sweep more than $2\pi$ to return to periapsis, so the perihelion moves in the direction of the motion.
The turning points follow from the radial constraint directly. Multiplying (1) by $r^2$ gives the quadratic
$$ \left(E^2-m^2c^4\right)r^2+2E\kappa r+\left(\kappa^2-c^2L^2\right)\geq0, $$
whose roots are the periapsis and the apoapsis,
$$ r_{1,2}=\frac{E\kappa\pm\sqrt{m^2c^4\kappa^2+c^2L^2\left(E^2-m^2c^4\right)}}{m^2c^4-E^2}, $$
the upper sign giving the apoapsis for a bound orbit. The roots are real and positive precisely when $L>\kappa/c$ and $mc^2\omega\leq E The circular orbit is the case $u'=0$, $u''=0$, for which the Binet equation requires $\omega^2u=\kappa E/(c^2L^2)$, that is, $r=1/u=p$. Imposing the radial constraint at that radius gives two exact results: $$
r_c=\frac{L^2\omega}{m\kappa}, \qquad E_c=mc^2\omega .
$$ Both are exact, not expansions, and both reduce to their non-relativistic counterparts as $\omega\to1$: the radius becomes $L^2/(m\kappa)$ and the energy becomes $mc^2$, the rest energy from which the non-relativistic binding is measured. The circular orbit is a minimum of the effective radial constraint and is stable: a perturbation of the radius by one per cent produces bounded oscillations of a few per cent about $r_c$, not an escape or a plunge. The existence of a stable circular orbit at every angular momentum above $\kappa/c$, with no innermost stable orbit, is a property of the special-relativistic inverse-square problem. It differs from the general-relativistic Kepler problem, where the curvature of spacetime produces a minimum radius for circular orbits; the present article is a flat-spacetime analysis, and that distinction is revisited among the open questions. The quadratic constraint makes the relativistic effect on the barrier explicit. Its value at $r=0^+$ is $\kappa^2-c^2L^2$, which is negative exactly when $L>\kappa/c$. A negative value at the origin means that the region near $r=0$ is forbidden and the allowed motion lies between the two positive roots: a genuine bound orbit with a periastron that keeps the particle away from the centre. When $L<\kappa/c$ the constraint is positive near the origin, the allowed region reaches $r=0$, and the particle falls into the centre however it is launched. This is a relativistic phenomenon with no non-relativistic counterpart. In the non-relativistic inverse-square problem the centrifugal term $L^2/(2mr^2)$ diverges at the origin for every $L>0$, so a barrier exists for every nonzero angular momentum. Relativistically the barrier competes with the growth of the relativistic mass near the centre, and the competition is lost below $L=\kappa/c$. The threshold is a ratio of a central scalar ($\kappa$) to the biquaternion norm of a material vector ($L$), and it is the framework's own biquaternion norm that produces it. The precessing-conic orbit, the eccentricity formula, the circular-orbit radius and the bound-orbit condition were checked by direct integration of the proper-time equations of motion in the material sector, with the full mass-shell relation and no use of the orbit equation. For representative bound orbits the measured apsidal advance agreed with $2\pi(1/\omega-1)$ to better than one part in $10^5$, the measured periapsis and apoapsis agreed with $p/(1\pm e)$ to nine decimal places, and the perturbed circular orbit was confirmed to be stable. The integration is a check of the derivation, not an input to it. The limit $c\to\infty$ at fixed $\kappa$, $L$ and mechanical energy recovers the companion articles. With $E=mc^2+\varepsilon$ and $\varepsilon$ fixed, $$
m_{\text{rel}}=\frac{E-V}{c^2}=m+\frac{\varepsilon-V}{c^2}\longrightarrow m, \qquad
\omega^2=1-\frac{\kappa^2}{c^2L^2}\longrightarrow1 ,
$$ so the Binet equation becomes the non-relativistic one, the frequency becomes unity, and the orbit becomes an ordinary conic: $$
p\longrightarrow\frac{L^2}{m\kappa}, \qquad
e^2\longrightarrow1+\frac{2\varepsilon L^2}{m\kappa^2},
\qquad
\Delta\longrightarrow0 .
$$ The last limit is the statement that the precession is a relativistic effect of order $1/c^2$. The circular-orbit results become $r_c\to L^2/(m\kappa)$ and $E_c\to mc^2$, and the barrier condition $L>\kappa/c$ becomes vacuous for every fixed $L>0$, which is the non-relativistic statement that the centrifugal barrier is unconditional. The special-relativistic precession has a standard interpretation worth recording. Writing $\kappa=GMm$ and using the non-relativistic relation $L^2=m\kappa a(1-e^2)$ for the semi-major axis $a$, the advance $\pi\kappa^2/(c^2L^2)$ becomes $$
\Delta_{\text{SR}}\simeq\frac{\pi GM}{c^2a(1-e^2)} ,
$$ which is one sixth of the general-relativistic perihelion advance $6\pi GM/(c^2a(1-e^2))$. The special-relativistic and general-relativistic corrections are therefore of the same order and the same sign, with the curved-spacetime contribution the larger of the two. The present article computes the flat-spacetime term only; the general-relativistic term is a curvature effect and lies outside the framework's flat-spacetime setting. The non-relativistic inverse-square problem conserves, besides $\mathbf{L}$, the Runge–Lenz vector $\mathbf{A}=\mathbf{p}\times\mathbf{L}-m\kappa\hat{\mathbf{r}}$, whose conservation is equivalent to the closure of the orbit and whose symmetry algebra is $\mathrm{SO}(4)$ for bound orbits. The relativistic problem is the test case for what survives. The orbit remains a conic in the rotating frame $\varphi=\omega\theta$, so the shape of the orbit is preserved and the only relativistic effect on it is the uniform rotation of the apsidal line. Equivalently, the relativistic problem retains the same number of constants of motion as any planar central-force problem — the energy, the angular momentum and the orientation of the orbit — but the orientation is no longer fixed: the periapsis direction advances by $\Delta$ per revolution. A vector with the algebraic form of the Runge–Lenz vector is therefore not conserved in the relativistic problem, and the reason is visible in the equation of motion. For an ansatz $\mathbf{A}_{\text{rel}}=\mathbf{p}\times\mathbf{L}-\alpha\,\hat{\mathbf{r}}$ with constant $\alpha$, the time derivative contains a term $\mathbf{F}\times\mathbf{L}=\kappa L\,\hat{\boldsymbol{\varphi}}/r^2$ from the force and a term $\alpha\,\dot{\hat{\mathbf{r}}}=\alpha L\,\hat{\boldsymbol{\varphi}}/(\gamma mr^2)$ from the rotating unit vector, and the two cancel only if $\alpha=\gamma m\kappa=(E-V)\kappa/c^2$, which is not constant along the orbit. No choice of constant $\alpha$ removes both terms, and the deficit is exactly the precession. What survives is a rotating-frame quantity: the vector constructed from the conic in the $(r,\varphi)$ plane has a fixed direction in that plane, and in the physical plane it rotates by $\Delta$ per revolution. The hidden symmetry of the non-relativistic problem is thus not destroyed but deformed: the compact $\mathrm{SO}(4)$ of the bound Kepler problem becomes a symmetry whose generator acquires a state-dependent phase, and the parameter of the deformation is $\kappa^2/(c^2L^2)$, the same parameter that sets the precession. The central force is a commuting material vector. The force is $F(r)\hat{\mathbf{r}}$, a real element of the three-space commuting with the position, $[\mathbf{r},\mathbf{F}]=0$. The centrality is an algebraic identity, not a symmetry assumption, and it is what makes $\mathbf{L}=\tfrac12[\mathbf{r},\mathbf{p}]$ conserved. The angular momentum is a product. The vector part of $\mathbf{r}\mathbf{p}$ is $\mathbf{L}$, and the algebra's single multiplication supplies the dot and cross products together. The orbit equation is a biquaternion-norm identity. The starting point of the derivation, $(E-V)^2-m^2c^4=c^2L^2(u'^2+u^2)$, is the mass-shell relation written with the radial decomposition, and the mass shell is the biquaternion norm of the four-momentum. The relativistic Binet equation is therefore a consequence of the biquaternion norm, and the frequency $\omega^2=1-\kappa^2/(c^2L^2)$ is a ratio of the squared coupling to the biquaternion norm of the angular momentum. Both are central scalars, and the algebra supplies them. The phase space. The orbit is a curve determined by $(E,L)$ and an orientation; the phase space of the particle is not a module over the finite-dimensional algebra, exactly as in the non-relativistic companion articles. The reciprocal length. The turning-point analysis, the eccentricity and the precession all use $u=1/r$ and $\hat{\mathbf{r}}=\mathbf{r}/|\mathbf{r}|$. Normalization is a nonlinear operation that the finite algebra does not implement, and the orbit equation is an equation for the reciprocal radius, not for an element of the algebra. The deformation parameter's dynamics. The algebra contains $\kappa$ and $L$ and therefore the combination $\kappa^2/(c^2L^2)$, but it does not by itself fix the value of $\kappa$ or prefer the inverse-square law. The field-theoretic origin of $\kappa$ is the subject of the companion article The Central Scalar Field: Classical Dynamics in the Biquaternion Center, and the value remains a parameter of the theory. The origin and the value of the coupling. The framework fixes the form of the orbit once $\kappa$ is given; it does not fix $\kappa$. The coupling and the mass are the free parameters of a field theory, and the biquaternion structure does not constrain them. Whether it constrains their ratio, through the barrier condition $L>\kappa/c$, is not known. The sign of the coupling and the repulsive case. The present article treats $\kappa>0$. The repulsive case $\kappa<0$ is obtained by analytic continuation; the barrier condition becomes $L>|\kappa|/c$ with $\kappa^2$ in $\omega$, and the orbit is one branch of a hyperbola-like precessing curve. A systematic treatment of the repulsive relativistic orbit, and of its scattering angle, is left to a companion article on the relativistic Coulomb scattering. The relation to the general-relativistic precession. The flat-spacetime precession computed here is one sixth of the general-relativistic one, and the two are not separately observable in the Solar System. Whether the framework's biquaternionic structure, which is flat by construction, can accommodate the curvature contribution, or must remain a special-relativistic reformulation, is the central open question of the framework's gravitational sector. The quantum deformation. The relativistic hydrogen spectrum is obtained by quantizing the same classical orbit, and the exactness of the precessing conic is why the relativistic Kepler problem is solvable. The relation of the frequency $\omega$ to the Sommerfeld fine structure, and its place in the biquaternion spectral theory, is a question for the relativistic quantum category. Radiation and the self-force. A particle on the precessing conic radiates if it is charged, and the radiation reaction changes the orbit. The framework treats the field as prescribed here; the back-reaction belongs to the field-theoretic articles of the framework. Many bodies. The energy $E=\gamma mc^2+V$ is a per-particle quantity, and the proper-time form of the conservation law does not sum across particles with different Lorentz factors. The relativistic two-body problem, and its biquaternion reduction, is treated only in the centre-of-mass approximation here. The relativistic central-force problem in biquaternionic form is the motion of a material-sector vector under a force $F(r)\hat{\mathbf{r}}$ parallel to the position. The energy $E=\gamma mc^2+V(r)$ and the angular momentum $\mathbf{L}=\mathbf{r}\times\mathbf{p}=\tfrac12[\mathbf{r},\mathbf{p}]$ are conserved, the motion is planar, and the orbit satisfies the relativistic Binet equation $$
\frac{d^2u}{d\theta^2}+u=-\frac{(E-V)F(1/u)}{c^2L^2u^2}=-\frac{m_{\text{rel}}F(1/u)}{L^2u^2},
\qquad u=\frac{1}{r},\quad m_{\text{rel}}=\frac{E-V}{c^2}=\gamma m,
$$ which is the non-relativistic equation with the rest mass replaced by the relativistic mass. For the inverse-square force $F=-\kappa/r^2$ the equation becomes linear, $$
\frac{d^2u}{d\theta^2}+\omega^2u=\frac{\kappa E}{c^2L^2},
\qquad \omega^2=1-\frac{\kappa^2}{c^2L^2},
$$ and the orbit is the precessing conic $$
r(\theta)=\frac{p}{1+e\cos\omega(\theta-\theta_0)},
\qquad p=\frac{c^2L^2\omega^2}{\kappa E},
\qquad e^2=1+\frac{c^2L^2\omega^2(E^2-m^2c^4)}{\kappa^2E^2},
$$ with periapsis and apoapsis the roots of $(E^2-m^2c^4)r^2+2E\kappa r+(\kappa^2-c^2L^2)=0$. The apsidal advance per revolution is $$
\Delta=2\pi\left(\frac{1}{\omega}-1\right)\simeq\frac{\pi\kappa^2}{c^2L^2},
$$ prograde, and equal to one sixth of the general-relativistic advance for the gravitational case. The circular orbit is exact, with $$
r_c=\frac{L^2\omega}{m\kappa}, \qquad E_c=mc^2\omega,
$$ and is stable. Bound orbits require $L>\kappa/c$ and $mc^2\omega\leq E In the limit $c\to\infty$ the frequency tends to unity, the orbit becomes the conic of the non-relativistic companion articles, the precession vanishes, and the barrier condition becomes vacuous. The algebra supplies the central force, the product form of $\mathbf{L}$ and the biquaternion-norm content of the orbit equation; it does not supply the phase space, the reciprocal length, or the dynamical origin of $\kappa$. The hidden symmetry of the non-relativistic problem is deformed rather than broken: the orbit remains a conic in the rotating frame, the Runge–Lenz vector is replaced by a rotating-frame quantity whose direction advances by $\Delta$ per revolution, and the deformation parameter is the precession parameter $\kappa^2/(c^2L^2)$.The Circular Orbit
The Bound-Orbit Condition and the Centrifugal Barrier
Verification
The Non-Relativistic Limit and Consistency
The Hidden Symmetry and What Becomes of the Runge–Lenz Vector
The Algebraic Reading
What the Algebra Supplies
What the Algebra Does Not Contain
Open Questions
Summary
Summary of Notation
Symbol
Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$
Biquaternion algebra; $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$; central $i$
$\mathbb{M}_-,\mathbb{M}_+$
Material (anti-Hermitian) and informational (Hermitian) sectors
$\mathbf{r},\mathbf{p},\mathbf{v},\mathbf{F}$
Position, momentum, velocity, force: real vectors in $\operatorname{span}\{e_1,e_2,e_3\}$
$\mathbf{L}=\mathbf{r}\times\mathbf{p}=\tfrac12[\mathbf{r},\mathbf{p}]$, $L=|\mathbf{L}|$
Angular momentum and its magnitude; vector part of $\mathbf{r}\mathbf{p}$
$V(r)$, $F(r)=-dV/dr$
Central potential energy; radial force component
$\kappa$
Inverse-square coupling; $\kappa>0$ attractive; $\kappa=GMm$ for gravity
$E=\gamma mc^2+V(r)$
Conserved total energy, including the rest energy
$\varepsilon=E-mc^2$
Energy measured from the rest energy
$\gamma=1/\sqrt{1-\mathbf{v}^2/c^2}$
Lorentz factor
$m_{\text{rel}}=\gamma m=(E-V)/c^2$
Relativistic mass
$p_r=\gamma m\dot r=-Lu'$
Radial momentum; the chain-rule identity
$u=1/r$
Binet variable; prime denotes $d/d\theta$
$(E-V)^2-m^2c^4=c^2L^2(u'^2+u^2)$
The orbit constraint (mass-shell identity)
$u''+u=-m_{\text{rel}}F(1/u)/(L^2u^2)$
Relativistic Binet equation
$\omega^2=1-\kappa^2/(c^2L^2)$
Relativistic orbital frequency
$L>\kappa/c$
Centrifugal-barrier condition
$p=c^2L^2\omega^2/(\kappa E)$
Semi-latus rectum
$e^2=1+c^2L^2\omega^2(E^2-m^2c^4)/(\kappa^2E^2)$
Eccentricity
$r_1,r_2$
Periapsis and apoapsis (roots of the radial constraint)
$\Delta=2\pi(1/\omega-1)$
Apsidal advance per revolution
$r_c=L^2\omega/(m\kappa)$, $E_c=mc^2\omega$
Circular orbit radius and energy
$\varphi=\omega\theta$
Rotating-frame angle of the precessing conic
$c=1/\sqrt{\epsilon\mu}$, $c_0$
Speed of light in the medium; in vacuum
Further Reading