Gravitational constant

The gravitational constant G appears in Newton's law of universal gravitation: F = G m₁m₂/r², where F is the force between two masses, m_{1 and m2, separated by distance r. It also appears in Einstein's field equations of general relativity, where it sets the coupling between spacetime curvature and energy-momentum.²}

Unlike the speed of light or the elementary charge, G is not exactly known; it is a measured quantity with a relative uncertainty of about 2.2×10⁻⁵ (0.0022%). This limits the precision of many astrophysical calculations, such as the masses of celestial bodies and the age of the universe.¹

The first precise measurement of G was performed by Henry Cavendish in 1798 using a torsion balance, an experiment often described as "weighing the Earth." Cavendish's result was within about 1% of the modern value.³

Subsequent measurements have used various techniques, including torsion balances with larger masses, beam balances, and, more recently, atom interferometry. Despite over two centuries of effort, the value of G remains the least precisely known fundamental constant.¹

The Committee on Data for Science and Technology (CODATA) periodically compiles a recommended value of G based on all available measurements. The 2018 CODATA value is 6.67430(15)×10⁻¹¹ m³ kg⁻¹ s⁻², with a relative standard uncertainty of 2.2×10⁻⁵.¹

Notably, different high-precision experiments have produced values that disagree by more than their reported uncertainties, a puzzle that remains unresolved. For example, measurements at the International Bureau of Weights and Measures (BIPM) and at other laboratories have shown discrepancies of several parts per million.⁴

One lesser-known fact is that the gravitational constant is not directly measurable in the laboratory with high precision because gravity is extremely weak compared to other forces. The torsion balance experiment is delicate and sensitive to environmental disturbances.

Another intriguing aspect is the possibility that G might vary over cosmological timescales, as proposed in some extensions of general relativity, such as Brans–Dicke theory. However, current observations of the cosmic microwave background and lunar laser ranging place stringent limits on any such variation.

In the context of quantum gravity, G is expected to be related to the Planck mass, but a complete quantum theory of gravity remains elusive.

G is essential for determining the masses of planets, stars, and galaxies from their gravitational effects. For example, the mass of the Earth is derived from G and the acceleration due to gravity at the surface.

In geodesy and space navigation, precise knowledge of G is required for accurate orbit determination. The uncertainty in G also affects the calculation of the Sun's mass and the gravitational binding energy of celestial bodies.

Torsion balance

An instrument that measures extremely weak forces by the twisting of a fiber, used by Cavendish to measure G.

CODATA

Committee on Data for Science and Technology, which provides internationally recommended values for fundamental constants.

Planck mass

The mass scale at which quantum effects of gravity become important, defined in terms of G, ħ, and c.