Skip to content

Newtonian Mechanics and Gravity

Scrollable formula. Use Left and Right arrow keys to reveal hidden content.

Scrollable table. Use Left and Right arrow keys to reveal additional columns.

Scrollable figure. Use Left and Right arrow keys to inspect hidden content.

Newtonian theory is the indispensable control problem for relativity. It supplies a kinematic stage, laws for matter on that stage, a gravitational field equation, and observable limits against which later relativistic results must be checked. This chapter builds that structure from inertial frames through tidal gravity and the virial theorem; it excludes relativistic spacetime geometry and post-Newtonian corrections.

Helpful background. Calculus and Linear Algebra for Relativity is used throughout; Differential Equations for Relativity is especially useful for boundary-value and stability reasoning.

As a quick diagnostic, you are ready to begin if you can differentiate a vector trajectory twice, take the gradient and Hessian of a scalar field, and explain why initial data differ from boundary data. The linked preparation pages provide the shortest repair route for any missing step.

Newtonian kinematics, dynamics, and gravity

Section titled “Newtonian kinematics, dynamics, and gravity”

Three layers organize the chapter. Kinematics supplies absolute time, Euclidean spatial geometry, and inertial frames related by Galilean transformations. Dynamics supplies Newton’s equations and their variational and Hamiltonian equivalents. Gravity supplies a scalar potential Φ\Phi, acceleration g=−∇Φ\mathbf g=-\nabla\Phi, and the field equation

∇2Φ=4πGρ.\nabla^2\Phi=4\pi G\rho.

The potential itself contains conventional freedom; potential differences determine work, and its Hessian determines relative acceleration. Boundary conditions are part of the physical problem. For an isolated source the usual normalization is Φ→0\Phi\to0 at infinity, whereas a bounded laboratory or numerical domain may require different data.

Two distinctions recur. First, changing coordinates inside one inertial frame is not the same as choosing an accelerating frame. Second, a position-independent gravitational acceleration can be canceled by frame acceleration, but the spatially varying tidal field cannot. The latter is the decisive Newtonian bridge toward curvature, without claiming that a global Newtonian potential is already spacetime geometry.

The kinematic assumptions, force laws, and inverse-square field used in this route are developed together in Tong 2013, §§1.1–2.3, pp. 2–26, PDF.

For the two field-theory steps not developed there, Tong, Vector Calculus, §§5.1–5.2.3, pp. 92–106, PDF treats Poisson boundary problems and Green functions, while Golwala 2007, §1.3.2, pp. 47–51, PDF derives the bounded-system virial theorem and its inverse-square specialization.

Routes through the chapter

Section titled “Routes through the chapter”

For the shortest mechanics route, begin with Frames, Configurations, and Newton’s Laws and Variational Mechanics and Symmetries. These pages identify the assumed frame structure, derive the Euler–Lagrange equations, and connect continuous symmetries to conserved quantities; Tong 2005, §§2.1 and 2.4, pp. 10–12 and 23–26; §4.1, pp. 80–88, PDF supplies the corresponding action, symmetry, and Hamiltonian development.

For gravitational field theory, continue through The Newtonian Gravitational Field and Potential and Poisson’s Equation and Gravitational Boundary Data. The first separates invariant field content from normalization and frame freedom; the second shows how sources and boundary conditions jointly determine a solution.

For preparation closest to general relativity, add Tidal Gravity and Relative Acceleration, then Conservation Laws, Scaling, and the Virial Theorem. This route turns field gradients into measurable relative acceleration and derives the scales of bound self-gravitating systems. Readers focused on orbital applications can then proceed directly to Orbits and Celestial Dynamics.

What the chapter establishes

Section titled “What the chapter establishes”

After an appropriate route, a reader can state the structures that make Newtonian acceleration absolute; translate among force, action, and Hamiltonian descriptions when the Legendre map is regular; distinguish a potential value from its observable derivatives; solve a Poisson problem only after specifying boundary data; diagnose vacuum tides from a traceless Hessian; and use the virial theorem without mistaking it for a stability proof.

The approximation has firm limits. Newtonian gravity responds instantaneously to ρ(t,x)\rho(t,\mathbf x), contains no gravitational radiation, and assumes a common absolute time. Weak fields and slow motion are necessary for its use as a relativistic limit. Extended bodies, dissipation, external pressure, and finite boundaries may add multipoles, fluxes, or surface terms that the simplest point-particle formulas omit.

Review the chapter

Section titled “Review the chapter”
  1. Why can two inertial observers disagree about velocity but agree on acceleration?
  2. What data, in addition to ρ\rho, make Poisson’s equation well posed?
  3. A vacuum tidal tensor has eigenvalues (−2a,a,a)(-2a,a,a). What relative motion does it produce along each principal axis?
  4. Why does 2T+U=02T+U=0 at one instant not prove virial equilibrium?
Solution
  1. A Galilean boost adds a constant velocity, whose time derivative vanishes; a fixed rotation only rotates the acceleration vector.
  2. One must give boundary behavior—such as Dirichlet values, compatible Neumann derivatives, or decay at infinity. Neumann data leave an additive constant.
  3. For a>0a\gt0, ξ¨=−Eξ\ddot{\boldsymbol\xi}=-E\boldsymbol\xi gives relative acceleration per unit separation +2a+2a in the radial direction and −a-a in each transverse direction. The positive sign stretches a pair initially at relative rest; the negative signs compress it. These coefficients have units of inverse time squared, and their signed sum vanishes.
  4. The virial theorem is a long-time statement requiring a vanishing averaged boundary term in I˙\dot I. A transient system can pass through 2T+U=02T+U=0 while collapsing or expanding.

References

Section titled “References”