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Gravitational constant (G): definition, measurement and significance

The gravitational constant G quantifies the strength of Newtonian gravity. This article explains its meaning, units, measurement history, roles in physics, and common confusions such as G vs g.

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Overview

The gravitational constant, commonly written as G and sometimes called Newton's constant or "Big G," is a fundamental empirical parameter that sets the strength of the gravitational attraction between masses. In Newton's law of universal gravitation it appears as a proportionality constant: the magnitude of the attractive force between two point masses is G times the product of their masses divided by the square of their separation. The constant is central to classical gravity and also appears in Einstein's field equations of general relativity as the coupling between matter and spacetime curvature. Physics context and mathematical forms describe how G is used in equations.

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Definition, units and commonly quoted value

In the International System of Units (SI), G has units of newton square-meters per kilogram squared (N·m2/kg2). Numerically, its accepted value from modern measurements is about 6.67430×10−11 N·m2/kg2, often quoted with an uncertainty determined by laboratory experiments. Because it connects force, mass and distance at the scale of point masses, G provides an absolute scale for gravitational interactions in the same way that other constants set scales in electromagnetism or quantum mechanics. For further background on constants as measured quantities, see empirical constants and metrology resources.

Measurement and experimental challenges

Measuring G is technically demanding. Typical experiments compare tiny gravitational torques or deflections produced by known masses at known separations, often using torsion balances, beam balances, or atom interferometry. Sources of uncertainty include mechanical alignment, environmental vibrations, thermal effects and the difficulty of isolating extremely small forces from background influences. Because of these challenges, different high-precision laboratories historically reported slightly different values; community efforts and improved techniques have reduced discrepancies but measurement remains less precise than for many other fundamental constants. See experimental approaches at torsion-balance methods and modern techniques.

History and development

The concept of a universal gravitational law and a proportionality constant emerged with Sir Isaac Newton in the 17th century. Newton's formulation related the force between masses to an inverse square of distance and introduced the idea of a universal constant, though he did not determine its numerical value. Direct laboratory determinations began in the 18th century with the Cavendish experiment and evolved through the 19th and 20th centuries as instrumentation improved. The constant's role was later incorporated into general relativity, where it links mass-energy to spacetime curvature; historical overviews are available at historical review and scientific biographies.

Uses, examples and significance

  • Newtonian gravity: compute forces between bodies, planetary motion and orbital mechanics.
  • Astrophysics and cosmology: combined with other parameters, G helps determine masses of stars, black holes and the dynamics of galaxies.
  • Fundamental physics: G sets the Planck scale when combined with the speed of light and Planck's constant, linking gravity to quantum units.

Applications range from engineering orbital trajectories to estimating the mass of distant objects from their gravitational influence. Additional reading about practical uses and numerical examples is available at applied gravitation.

Distinctions and notable points

It is important not to confuse the universal gravitational constant G with the local gravitational acceleration g (sometimes called "small g"), which denotes the acceleration experienced in a particular gravitational field (for example, ≈9.8 m/s2 near Earth's surface). While G is a fixed constant of nature (to the extent measured), g varies with location and the distribution of nearby mass. For quick comparisons and historical experiments see related physics resources and measurement summaries.

Value and units

In the International System of Units (SI), the value according to the current CODATA 2018 recommendation is:

{\displaystyle G=6{,}674\,30(15)\cdot 10^{-11}\,\mathrm {\frac {m^{3}}{kg\cdot s^{2}}} }

(i.e., with an estimated standard uncertainty of {\displaystyle 0{,}000\,15\cdot 10^{-11}\,\mathrm {m} ^{3}/(\mathrm {kg} \cdot \mathrm {s} ^{2})}).

In the CGS system of units Ghas the value

{\displaystyle G=6{,}674\,30(15)\cdot 10^{-8}\,\mathrm {\frac {cm^{3}}{g\cdot s^{2}}} .}

The gravitational constant can also be expressed using other natural constants, for example Planck's reduced quantum of action \hbar and the speed of light c{"natural units"). According to CODATA 2018, the value is:

{\displaystyle G=6{,}708\,83(15)\cdot 10^{-39}\,{\frac {\hbar c}{(\mathrm {GeV} /c^{2})^{2}}}}

Compared to other fundamental forces of physics, gravity is a very weak interaction, which is expressed in the small value of the gravitational constant. For example, calculating the amount of the ratio between the gravitational force and the electrostatic force between two protons, we get regardless of the distance:

{\displaystyle \left|{\frac {F_{\text{Gravitation}}}{F_{\text{elektrisch}}}}\right|={\frac {Gm_{\text{Proton}}^{2}}{e^{2}/(4\pi \varepsilon _{0})}}\approx 10^{-36}}

Measurements

The gravitational force between the earth and another object, i.e. its weight, can be measured very accurately, but in order to determine the gravitational constant with the same accuracy, one would have to know the mass of the earth or better the whole mass distribution in the earth reliably. But this is not given, so that in order to measure , Gthe extremely small gravitational attraction between bodies of known mass must be determined in the laboratory. For example, the gravitational attraction between two bodies of mass 100 kg each at a distance of 1 m is less than 10-9 (one billionth) of their weight, and all other matter in the laboratory or outside it also exerts gravitation on the test bodies. These measurements are therefore difficult. Even the smallest differences in temperature, air currents, irregularities in the material or creep of the material, even the number of vehicles in the parking lot in front of the institute building, falsify the results.

Current status

A value for G with eight-digit accuracy, as achieved for other natural constants long ago, would thus require here a reduction of such possible perturbations to 10-17 (one hundred billionth) of the weight force of the bodies involved. This has not yet been achieved. Five-digit accuracy is thus the highest, it was given for a measurement of G from the year 2000. However, from the last three decades alone, there are a total of 13 other measurement results from laboratories around the world with various apparatuses, some of which state similarly high accuracy, but which nevertheless differ by up to almost ten times the uncertainty ranges stated in each case. It is assumed that the individual apparatuses still have undetected weak points.

As a result, the relative uncertainty in the value of G cannot be pushed below 2.2 - 10-5 at present. Thus, G is currently the one with the lowest measurement accuracy among the fundamental constants of nature. For comparison, the Rydberg constant is known in SI units with a relative uncertainty of 1.9 - 10-12, which is more than a million times more accurate.

The - in comparison - low accuracy of Gand the too large scattering range of the single results are regarded as deficiencies. Apart from unrecognized weaknesses of the measuring apparatuses, the scattering width could also point to an aspect of gravitation which is not yet understood. The inaccuracy limits the possibility of determining the mass of a celestial body from its gravity. To do this, the celestial body must be orbited by a companion whose orbital radius rand orbital angular frequency ω {\displaystyle\omega are known, so that the gravitational parameter μ \mu = r^3 \omega^2determined. This is often possible with high accuracy, for the Earth, for example, with up to 10-digit accuracy (see WGS 84). Then the mass of the celestial body is given by M=\mu /G(see Kepler's laws). Despite the uncertainty in Gmuch more accurate than estimating the mass from the diameter and the density profile inside the celestial body.

In recent experiments, the gravitational constant is measured using two different methods by varying the experimental setup of the pendulum balance:

  • Vibration time method: a gold-plated quartz block on a fiberglass and two steel balls, each weighing 778 grams, change torsional vibrations, resulting in gravitational constants of {\displaystyle 6{,}674\,184\cdot 10^{-11}\,\mathrm {{m^{3}}\,{kg^{-1}\,s^{-2}}} }±11.64 ppm.
  • Method for measuring angular acceleration: In the second experimental setup, both masses (quartz block and steel balls) are allowed to rotate independently and the turntable is tracked to compensate for torsion, resulting in no rotation angle, but angular acceleration ("angular-acceleration-feedback method") is measured to compensate for the angle of rotation, from which a value of {\displaystyle 6{,}674\,484\cdot 10^{-11}\,\mathrm {{m^{3}}\,{kg^{-1}\,s^{-2}}} }±11.61 ppm was derived.

In previous experiments, the standard deviation was ±47 ppm, so it was improved by ±36 ppm.

The Cavendish Experiment

The first measurement of the gravitational force between two masses of known size was made by Henry Cavendish in 1798 with the help of a specially invented gravitational balance. The balance consisted of two spherical test masses with a combined mass (in today's units) of m= 1.46 kg, which were connected to form a dumbbell and suspended from a torsion wire so that they could perform free horizontal torsional oscillations. Two large spheres of total mass {\displaystyle M_{c}=316\,\mathrm {kg} }, equally spaced r_mclose to each of the test masses, produced the attractive force that deflected the test masses about 1° from rest. From the angle of deflection, the torsional force F_cdetermined, which balanced the attractive force of the large and small balls at this distance. The necessary knowledge of the torsional stiffness of the wire was obtained from the period of the torsional vibration.

From Cavendish's measured values, the following formula is obtained

{\frac {F_{c}\,r_{m}^{2}}{m\,M_{c}}}=G

a value for the constant

{\displaystyle G_{\mathrm {Cavendish} }=6{,}754\cdot 10^{-11}\,\mathrm {{m^{3}}\,{kg^{-1}\,s^{-2}}} .}

This falls short of today's figure by only 1.2 percent.

However, the concept of a gravitational constant was not yet common in Cavendish's time, rather Newton's law of gravitation was used exclusively in the form of proportionalities. Accordingly, he considered the ratio of the two forces F_{c}and F_E , by which the small spheres are attracted to the large ones and to the earth, respectively. According to Newton:

\frac {F_c} {F_E} = \frac { M_c} {M_E}\ \frac {r_E^2} {r_m^2}

F_E is nothing else than the (total) weight of the small spheres, so that the earth mass M_{E}the only unknown in this. Cavendish was able to determine the mass of the Earth from his measurement data. The physically incorrect and, strictly speaking, meaningless formulation that Cavendish "weighed the earth" became popular.

After the mass of the earth, implicitly the value of the gravitational constant, was known, the masses of other celestial bodies of the solar system could be determined.

Cavendish Experiment (1798)
Cavendish Experiment (1798)

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