Gravitation – Advanced University‑Level Study Notes
(Prepared for AIIMS Physics Examination – MIT‑style, first principles and rigorous reasoning)
Table of contents
- [Newtonian Gravitation from First Principles]
- [Gravitational Field Equations]
- [Potential Theory & Poisson’s Equation]
- [Conservative Nature & Energy Conservation]
- [Central‑Force Dynamics & Orbital Mechanics]
- [Effective Potential & Stability]
- [Kepler’s Laws from Newtonian Dynamics]
- [Perturbation Theory & Secular Effects]
- [General Relativity – Field Equations & Weak‑Field Limit]
- [Perihelion Precession & Light Deflection]
- [Gravitational Waves – Linearised Theory]
- [Edge Cases & Exotic Scenarios]
- [Challenging Problem – Perihelion Precession of Mercury]
1. Newtonian Gravitation from First Principles
1.1 Empirical Law (Inverse‑Square)
In 1687 Newton postulated that the gravitational force between two point masses (m_1, m_2) separated by distance (r) is
[ \boxed{ \mathbf{F}_{12} = -,G, \frac{m_1 m_2}{r^{2}}, \hat{\mathbf{r}} } ]
where (G) is the universal gravitational constant and (\hat{\mathbf{r}}) is the unit vector from (m_1) to (m_2).
1.2 Derivation from Symmetry & Conservation
- Isotropy & Homogeneity: The force depends only on relative separation, not on orientation.
- Action–Reaction & Newton’s Third Law: Forces are equal and opposite.
- Superposition: For a system of point masses, total force on a mass (m) is the vector sum of pairwise forces.
Using these principles, the most general central force that satisfies conservation of angular momentum and energy and is consistent with the inverse‑square law must be of the form above.
2. Gravitational Field Equations
2.1 Gravitational Field (\mathbf{g})
Define the gravitational field produced by a mass density (\rho(\mathbf{r})):
[ \mathbf{g}(\mathbf{r}) \equiv \frac{\mathbf{F}}{m} = - G \int \frac{\rho(\mathbf{r'}) (\mathbf{r}-\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^{3}} , d^3r'. ]
This is a vector field that satisfies the following differential equations:
[ \nabla \cdot \mathbf{g} = -4\pi G \rho, \qquad \nabla \times \mathbf{g} = \mathbf{0}. ]
The curl‑free condition implies that (\mathbf{g}) is a gradient of a scalar potential.
2.2 Poisson’s Equation
Since (\mathbf{g} = -\nabla \Phi), substituting into the divergence equation yields
[ \boxed{ \nabla^{2}\Phi(\mathbf{r}) = 4\pi G,\rho(\mathbf{r}) }. ]
This is the cornerstone of Newtonian gravitation: the mass density is the source of the gravitational potential.
3. Potential Theory & Poisson’s Equation
3.1 Green’s Function Solution
For a localized source, the solution of Poisson’s equation in free space is
[ \Phi(\mathbf{r}) = - G \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}, d^3r'. ]
The kernel (1/|\mathbf{r}-\mathbf{r'}|) is the Green’s function of the Laplacian in three dimensions.