Mercury perihelion problem

We can embed the Mercury perihelion problem fully within the general framework given in on theories, symmetries and gauge, and see how to reduce it, naturally, to the usual formulation: Schwarzschild geometry plus test-particle worldline.

1. Formulate the full theory in the general framework

A theory is a mechanism to select some fields:

S = Γ (M, E),

with

E = Lor (M) \oplus {(Λ^{0} (M) \oplus T M)}_{⊙} \oplus {(Λ^{0} (M) \oplus T M)}_{Merc} \to M .

Here:

$Lor (M)$ : bundle of Lorentzian metrics,
$(Λ^{0} (M) \oplus T M)_{⊙}$ : Sun's dust fields: scalar density $ρ_{⊙}$ and 4-velocity $u_{⊙}$ ,
$(Λ^{0} (M) \oplus T M)_{Merc}$ : Mercury's dust fields: scalar density $ρ_{Merc}$ , 4-velocity $u_{Merc}$ .

We write an action as a sum of the Einstein-Hilbert term and two dust terms (one for each body):

\begin{aligned} E [g, ρ_{⊙}, u_{⊙}, ρ_{Merc}, u_{Merc}] & = \frac{1}{16 π} \int_{M} R (g) {vol}_{g} \\ + \int_{M} ρ_{⊙} g_{a b} u_{⊙}^{a} u_{⊙}^{b} {vol}_{g} \\ + \int_{M} ρ_{Merc} g_{a b} u_{Merc}^{a} u_{Merc}^{b} {vol}_{g} \end{aligned}

We must impose the constraints:

$g_{a b} u_{⊙}^{a} u_{⊙}^{b} = - 1$ , (normalization of 4-velocity),
$g_{a b} u_{Merc}^{a} u_{Merc}^{b} = - 1$ ,
$\nabla_{a} (ρ_{⊙} u_{⊙}^{a}) = 0$ , (mass conservation),
$\nabla_{a} (ρ_{Merc} u_{Merc}^{a}) = 0$ .
These can either be imposed via Lagrange multipliers, so they could be introduced in $E$ .

A solution is a pair $(g, ϕ) \in S$ such that:

δ E [g, ϕ] = 0,

yielding:

Einstein's equations: $G_{μ ν} (g) = T_{μ ν} [ϕ]$ ,
Matter field equations: Euler–Lagrange equations for $ϕ$ .

This is the fully coupled gravitational–matter theory.

2. Simplify for the particular situation of Mercury + Sun

We now model the actual setup:

The Sun is a massive, spherically symmetric body. It dominates the stress–energy content. We also assume its particles are not self-interacting.
Mercury is a small planet, much lighter than the Sun, and does not significantly affect the spacetime geometry.
In the full theory, both the Sun and Mercury are encoded inside $ϕ$ as matter fields. But this is analytically intractable, and mathematically unnecessary since:
One source (Sun) dominates the gravitational field, and it will remain constant along time.
The other (Mercury) acts as a probe of the geometry.

2.1. Mercury influence is ignored

Since Mercury does not influence the geometry, the variational principle for $g$ gets simplified to

δ_{g} E [g, ϕ_{Sun}] = 0 ⟹ G_{μ ν} = T_{μ ν}^{Sun} .

and we obtain the Schwarzschild solution for the metric:

d s^{2} = - (1 - \frac{2 G M}{r}) d t^{2} + {(1 - \frac{2 G M}{r})}^{- 1} d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d φ^{2}),

and a constant solution for the Sun
This is the field $g$ selected by the theory. We now fix this metric, treating it as given.

2.2. Replace Mercury’s matter field by a worldline

Instead of including $ϕ_{Mercury} \in Γ (M, F)$ , we now model Mercury by a curve. Any localized matter field (e.g., a narrow Gaussian scalar field) can be approximated by a delta-like field supported near a trajectory $X (τ)$ . In the limit where the width of this localization tends to zero, the matter configuration is modeled by a curve, not a field.

Mathematically, this corresponds to:

ϕ (x) ⟶ δ (x - X (τ)) .

Thus, the field $ϕ$ is effectively replaced by the worldline $X$ .

Instead of a matter action $S_{matter} [g, ϕ]$ , you now have:

S_{particle} [g, X] = - m \int \sqrt{- g_{μ ν} (X (τ)) {\dot{X}}^{μ} {\dot{X}}^{ν}} d τ .

This gives the geodesic equation for $X$ , consistent with the field-theoretic description in the test-particle limit.

All of this must be formalized, of course.

3. Mercury perihelion in this model

You now study the geodesic motion of a massive test particle (Mercury) in the fixed Schwarzschild background.

To be done.