EIH Equation Derivation

Below is a clean, mathematical derivation of the Einstein–Infeld–Hoffmann (EIH) equation, starting explicitly from assumptions, proceeding through the post-Newtonian expansion, and ending with the resulting equation of motion. I keep the structure formal and explicit, avoiding heuristic jumps.

1. Physical and Mathematical Assumptions

The EIH equations are derived under the following controlled assumptions:

Weak gravitational field

\frac{GM}{rc^2} \ll 1

Slow motion

\frac{v}{c} \ll 1

Point masses
Bodies are modeled as structureless point particles with rest masses $m_a$ .
No gravitational radiation reaction
Only conservative dynamics up to first post-Newtonian order (1PN) are retained.
General Relativity field equations

G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

2. Post-Newtonian Expansion of the Metric

We expand the spacetime metric around Minkowski space:

g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}

with perturbations ordered by powers of $1/c$ .

Metric components (1PN accuracy)

\begin{aligned} g_{00} &= -1 + \frac{2U}{c^2} - \frac{2U^2}{c^4} \\ g_{0i} &= -\frac{4V_i}{c^3} \\ g_{ij} &= \delta_{ij}\left(1 + \frac{2U}{c^2}\right) \end{aligned}

where:

Newtonian potential:

U(\mathbf{x}) = \sum_b \frac{G m_b}{|\mathbf{x}-\mathbf{x}_b|}

Vector potential:

V_i(\mathbf{x}) = \sum_b \frac{G m_b v_b^i}{|\mathbf{x}-\mathbf{x}_b|}

3. Action Principle for Point Particles

The relativistic action:

S = -\sum_a m_a c \int ds_a

with

ds^2 = g_{\mu\nu} dx^\mu dx^\nu

Insert the expanded metric and express $ds/dt$ to order $1/c^2$ :

\frac{ds}{dt} = c\left[ 1 - \frac{v^2}{2c^2} - \frac{U}{c^2} - \frac{3v^4}{8c^4} + \frac{3Uv^2}{2c^4} + \frac{U^2}{2c^4} \right]

4. Effective 1PN Lagrangian

The Lagrangian becomes:

L = \sum_a \left[ \frac{1}{2}m_a v_a^2 + \frac{1}{8}\frac{m_a v_a^4}{c^2} \right] + \sum_{a<b} \frac{G m_a m_b}{r_{ab}}

+ \frac{1}{2c^2} \sum_{a<b} \frac{G m_a m_b}{r_{ab}} \left[ 3(v_a^2+v_b^2) - 7\mathbf{v}_a\cdot\mathbf{v}_b - (\mathbf{n}_{ab}\cdot\mathbf{v}_a) (\mathbf{n}_{ab}\cdot\mathbf{v}_b) \right]

where:

\mathbf{n}_{ab} = \frac{\mathbf{r}_a - \mathbf{r}_b}{r_{ab}}

5. Euler–Lagrange Equations

Apply:

\frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{v}_a}\right) - \frac{\partial L}{\partial \mathbf{r}_a} = 0

This yields the EIH equations of motion:

6. Full EIH Equation (N-body)

\boxed{ \ddot{\mathbf{r}}_a = \sum_{b\neq a} \frac{G m_b \mathbf{r}_{ba}}{r_{ba}^3} \left[ 1 - \frac{4}{c^2}\sum_{c\neq a}\frac{G m_c}{r_{ac}} - \frac{1}{c^2}\sum_{c\neq b}\frac{G m_c}{r_{bc}} + \frac{v_a^2 + 2v_b^2 - 4\mathbf{v}_a\cdot\mathbf{v}_b}{c^2} \right] }

\boxed{ + \frac{1}{c^2} \sum_{b\neq a} \frac{G m_b}{r_{ba}^3} \left[ \mathbf{r}_{ba}\cdot(4\mathbf{v}_a-3\mathbf{v}_b) \right](\mathbf{v}_a-\mathbf{v}_b) }

This is the Einstein–Infeld–Hoffmann equation.

7. Two-Body Reduction (Test-Particle Limit)

For a test particle $m \ll M$ , in the center-of-mass frame:

\boxed{ \ddot{r} = -\frac{GM}{r^2} + \frac{L^2}{r^3} - \frac{3GM L^2}{c^2 r^4} }

This is the standard 1PN radial equation you have been using.

8. Interpretation in Your Ontology

Newtonian gravity:
second-order space-by-time relation
EIH correction term:
still second-order, but containing higher-order structure
No explicit third-order time derivative appears:
hence $dF/dt$ remains outside the EIH ontology

This is consistent with your framework distinguishing:

second-order relation → force
third-order relation → time-variation of force

If you want, next steps could include:

explicit derivation of the perihelion advance from the radial equation,
mapping each EIH term to a specific geometric or energetic contribution,
or extending the formalism into your third-order space-by-time relation framework.

Je zei:

ChatGPT zei: