Newton's law of Gravity

Prerequisites: Newton's Second Law. Full path (2 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• Definitions

Intuitive

Newton's law of Gravity tells us that the $\color{blue}{\text{gravitational force}}$ between $\color{red}{\text{two}}$ $\color{orange}{\text{masses}}$ gets smaller as the masses are removed $\color{magenta}{\text{further away from each other}}$.

In addition, it tells us that the exact strength of the $\color{blue}{\text{gravitational force}}$ is determined by the $\color{olive}{\text{gravitational constant}},$ the $\color{red}{\text{two}}$ $\color{orange}{\text{masses}}$ of the objects in the system and the $\color{magenta}{\text{distance between them}}$.

So given an object with some known $\color{red}{\text{mass}}$, we can immediately calculate the $\color{blue}{\text{force}}$ it exerts onto another $\color{orange}{\text{mass}}$.

Concrete

The formula at the top gives the magnitude of the force. As a vector, the force on mass $m_1$ at $\vec r_1$ due to mass $m_2$ at $\vec r_2$ is

$$\vec F_{12} = -G\frac{m_1 m_2}{|\vec r_1 - \vec r_2|^2}\,\frac{\vec r_1 - \vec r_2}{|\vec r_1 - \vec r_2|}, \qquad G \approx 6.674\times 10^{-11}\,\frac{\text{N}\,\text{m}^2}{\text{kg}^2}.$$

The minus sign makes the force point from $m_1$ towards $m_2$: gravity always attracts. As for electric charges, the forces from several masses add as vectors, so we can describe the effect of all sources by a gravitational field $\vec g$ and a potential $\Phi$,

$$\vec F = m\vec g, \qquad \vec g = -\nabla\Phi, \qquad \Phi(\vec r) = -G\sum_i \frac{m_i}{|\vec r - \vec r_i|} \;\to\; -G\int d^3r'\,\frac{\rho(\vec r')}{|\vec r - \vec r'|},$$

and $\Phi$ satisfies the Poisson equation $\nabla^2\Phi = 4\pi G\rho$. A consequence (Newton's shell theorem) is that outside a spherically symmetric body the field is the same as if all its mass sat at the centre.

Example: surface gravity. With the mass of the Earth $M \approx 5.97\times 10^{24}\,\text{kg}$ and its radius $R \approx 6.37\times 10^6\,\text{m}$,

$$g = \frac{GM}{R^2} \approx \frac{(6.674\times 10^{-11})(5.97\times 10^{24})}{(6.37\times 10^{6})^2}\,\frac{\text{m}}{\text{s}^2} \approx 9.8\,\frac{\text{m}}{\text{s}^2}.$$

The mass of the falling object cancels between $F = GMm/R^2$ and Newton's second law $F = ma$, so all bodies fall with the same acceleration. This works because the same mass appears in both laws (equality of inertial and gravitational mass).

Example: circular orbits. For a small body on a circular orbit of radius $r$ around a mass $M$, gravity supplies the centripetal force:

$$\frac{GMm}{r^2} = \frac{mv^2}{r} \quad\Rightarrow\quad v = \sqrt{\frac{GM}{r}}, \qquad T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r^3}{GM}}.$$

This is Kepler's third law, $T^2 \propto r^3$. For the Moon, $r \approx 3.84\times 10^8\,\text{m}$ gives $T \approx 2.4\times 10^6\,\text{s} \approx 27\,\text{days}$, close to the observed sidereal month. (Here we neglected the Moon's mass compared to the Earth's; for two bodies, $M$ is replaced by $M + m$.)

Abstract

As a field theory, Newtonian gravity is described by a scalar potential $\Phi$ with the Lagrangian density

$$\mathcal{L} = -\frac{1}{8\pi G}(\nabla\Phi)^2 - \rho\,\Phi,$$

whose Euler-Lagrange equation is the Poisson equation $\nabla^2\Phi = 4\pi G\rho$. The point-mass potential $\Phi = -Gm/r$ follows from the Green's function identity $\nabla^2(1/r) = -4\pi\delta^3(\vec x)$, exactly as for Coulomb's law, with the opposite overall sign: like "charges" (masses) attract. There is no time derivative in the field equation, so a change in the source is felt everywhere instantly. The theory is invariant under the Galilean group, and the equality of inertial and gravitational mass means that all test bodies follow the same trajectories $\ddot{\vec x} = -\nabla\Phi$.

Weak-field limit of general relativity. In general relativity, with signature $(-,+,+,+)$, a static weak field is described by the metric

$$ds^2 = -\left(1 + \frac{2\Phi}{c^2}\right)c^2 dt^2 + \left(1 - \frac{2\Phi}{c^2}\right)d\vec x^{\,2}, \qquad |\Phi| \ll c^2.$$

For a slowly moving particle ($|\dot{\vec x}| \ll c$) the geodesic equation reduces to $\ddot{\vec x} = -\nabla\Phi$, which only needs $g_{00}$. The $00$ component of the Einstein equation for non-relativistic matter, $T_{00} \approx \rho c^2$, reduces to $\nabla^2\Phi = 4\pi G\rho$. Newton's law is thus the limit of general relativity for weak, static fields and slow sources and test bodies.

Geometric formulation. The universality of free fall allows Newtonian gravity itself to be written geometrically, in Newton-Cartan theory. Spacetime is a four-manifold with an absolute time one-form $dt$, a degenerate spatial metric, and a torsion-free connection compatible with both. In Galilean coordinates $(t, x^i)$ (a non-rotating frame in which $\Phi$ is the usual Newtonian potential) the only non-vanishing connection coefficients are $\Gamma^i{}_{00} = \partial_i\Phi$, so that the geodesic equation $\ddot x^i + \Gamma^i{}_{00} = 0$ is the Newtonian equation of motion. The field equation becomes $R_{00} = 4\pi G\rho$ for the Ricci tensor of this connection. In this formulation gravity is curvature of spacetime with no force, as in general relativity, and $\Phi$ is no longer uniquely defined: the potentials $\Phi$ and $\Phi + \vec a(t)\cdot\vec x$ give the same connection, written in two coordinate systems that accelerate relative to each other with $\vec a(t)$. In a freely falling frame the connection coefficients vanish at the origin, as in general relativity.

Newton's law is the static limit of the Einstein equation.

Why is it interesting?

Newton's law of gravity is a universal law allows us to understand how two planets attract each other, but also why an apple falls onto the earth. It tells us that any two massive objects attract each other.

In addition, it allows us to accurately predict the movements of planets in the solar system. For example, we can use Newton's law to predict the motion of the moon.

It is still used nowadays, for example, by NASA and ESA to design orbits or to work out spacecraft trajectories for space missions.

Famous examples are the Apollo missions in the 60s and 70s. Their routes were calculated using Newton's law to take the attraction of the earth and moon into account. Another example, are the many satellites that orbit the earth nowadays. Thus, in some sense, Newton's law makes satellite television, GPS systems and also the Mars rover possible.

Definitions

• $\color{blue}{\vec F}$ is the force between the two masses.
• $\color{olive}{ G}$ is the gravitational constant.
• $\color{red}{m_1}$ and $\color{orange}{m_2}$ are the masses of the two objects in questions.
• $\color{magenta}{r}$ is the distance between the two objects.

Path to this page

1. Calculus
2. Newton's Second Law

Leads to: Einstein Equation.