7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity

Newton's law of universal gravitation describes how every object with mass attracts every other object with mass. It's a simple yet powerful formula that explains everything from falling apples to planetary orbits.

While Newton's law works for most everyday situations, Einstein's theory of general relativity provides a more accurate description of gravity in extreme conditions. Instead of treating gravity as a force, Einstein described it as a warping of spacetime caused by mass and energy.

Newton's Law of Universal Gravitation

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Newton's Gravity vs Einstein's Relativity

Newton and Einstein offer two different frameworks for understanding gravity. Newton's version is simpler and perfectly accurate for most situations you'll encounter. Einstein's version is more complete but only becomes necessary under extreme conditions.

Newton's approach:

• Describes gravity as a force proportional to the masses of two objects and inversely proportional to the square of the distance between them (the inverse square law)
• Assumes gravity acts instantaneously between objects, a concept called "action at a distance"
• Treats gravity as a distinct force, separate from electromagnetism, the strong nuclear force, and the weak nuclear force
• Works extremely well for practical applications like planetary orbits and projectile motion, but breaks down near black holes or at speeds approaching the speed of light

Einstein's approach:

• Describes gravity not as a force but as a consequence of the curvature of spacetime caused by mass and energy
• Predicts that gravitational effects propagate at the speed of light (as gravitational waves), not instantaneously
• Treats gravity as a geometric property of spacetime rather than a force pulling objects together
• Provides more accurate predictions in strong gravitational fields (near black holes) and at high velocities (approaching the speed of light)
• Reduces to Newton's law in weak gravitational fields and at low velocities, which is why Newton's formula still works perfectly for everyday physics on Earth

Calculations with Newton's Gravity

The gravitational force between any two objects is given by:

$F = G \frac{m_1 m_2}{r^2}$

• $F$ = gravitational force between the two objects (in Newtons)
• $G$ = gravitational constant = $6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2 / \text{kg}^2$
• $m_1$ and $m_2$ = masses of the two objects (in kg)
• $r$ = distance between the centers of the two objects (in meters)

Notice that $r$ is measured center-to-center. For an object on Earth's surface, $r$ equals Earth's radius, not zero.

Gravitational acceleration at a distance $r$ from a massive object can be found by dividing the gravitational force by the smaller object's mass:

$a = \frac{F}{m} = G \frac{M}{r^2}$

• $M$ is the mass of the larger body (Earth, the Sun, etc.)
• $r$ is the distance from the center of that body

This is where the familiar $g = 9.8 \text{ m/s}^2$ on Earth's surface comes from. Plug in Earth's mass ($5.97 \times 10^{24} \text{ kg}$) and Earth's radius ($6.37 \times 10^{6} \text{ m}$), and you get approximately $9.8 \text{ m/s}^2$.

Gravity's Effects on Celestial Objects

Orbits result from the balance between gravitational pull and an object's tangential velocity. Gravity causes celestial bodies to follow elliptical orbits around more massive objects (planets around the Sun, moons around planets). The shape and size of each orbit depend on the masses involved and the object's initial velocity, which determine properties like eccentricity and semi-major axis length. General relativity also predicts a small precession of orbits. This is most famously observed in the perihelion precession of Mercury, which Newton's law alone could not fully explain.

Tides on Earth are caused by the gravitational pull of the Moon and, to a lesser extent, the Sun. Because gravity weakens with distance, the side of Earth closest to the Moon feels a stronger pull than the far side. This difference in force stretches Earth's oceans, creating tidal bulges (high tides) on opposite sides of the planet. When the Sun, Moon, and Earth align, their combined pull produces especially strong spring tides. When the Sun and Moon pull at right angles to each other, the result is weaker neap tides.

Gravitational lensing is a prediction of general relativity. Massive objects curve spacetime around them, causing light passing nearby to follow a curved path. This can produce multiple images of a single distant quasar, magnify faraway galaxies, or distort objects into arcs and rings. Astronomers use this effect to study objects that would otherwise be too faint to observe.

Gravitational time dilation is another consequence of general relativity: clocks in stronger gravitational fields tick slower compared to clocks in weaker fields. This isn't just theoretical. GPS satellites orbit where gravity is weaker than at Earth's surface, so their onboard clocks run slightly faster. Without corrections for this time dilation, GPS positioning would drift by roughly 10 km per day.

General Relativity Concepts

The equivalence principle is the foundation of general relativity. It states that the effects of gravity are indistinguishable from the effects of acceleration in a small region of spacetime. If you were in a sealed elevator accelerating upward at $9.8 \text{ m/s}^2$ in deep space, you'd feel exactly the same as you do standing on Earth's surface. There's no experiment you could perform inside that elevator to tell the difference.

In general relativity, free-falling objects follow geodesics, which are the straightest possible paths through curved spacetime. A planet orbiting the Sun isn't being "pulled" by a force in Einstein's view. It's following the straightest available path through spacetime that the Sun's mass has curved.

The Schwarzschild radius defines the boundary of a black hole's event horizon. It's the distance from the center of a sufficiently compact mass at which the escape velocity equals the speed of light. Anything crossing inside this radius cannot escape, not even light.