Newton's constant appears in the proportionality between the spacetime curvature and the energy density component of the stress–energy tensor. The scaled gravitational constant κ = 8π/c4G ≈ 2.071×10−43 s2⋅m−1⋅kg−1 (depending on the choice of definition of the stress–energy tensor it can also be normalized as κ = 8π/c2G ≈ 1.866×10−26 m⋅kg−1) is also known as Einstein's constant.
Value and dimensions
The gravitational constant is a physical constant that is difficult to measure with high accuracy. This is because the gravitational force is extremely weak compared with other fundamental forces.[a]
In other words, in Planck units, G has the numerical value of 1.
Thus, in Planck units, and other natural units taking G as their basis, the gravitational constant cannot be measured as it is set to its value by definition. Depending on the choice of units, variation in a physical constant in one system of units shows up as variation of another constant in another system of units; only variation in dimensionless physical constants is preserved independently of the choice of units; in the case of the gravitational constant, such a dimensionless value is the gravitational coupling constant,
Diagram of torsion balance used in the Cavendish experiment performed by Henry Cavendish in 1798, to measure Big G, with the help of a pulley, large balls hung from a frame were rotated into position next to the small balls.
The gravitational constant appears in Newton's law of universal gravitation, but it was not measured until seventy-one years after Newton's death by Henry Cavendish with his Cavendish experiment, performed in 1798 (Philosophical Transactions 1798). Cavendish measured G implicitly, using a torsion balance invented by the geologist Rev. John Michell. He used a horizontal torsion beam with lead balls whose inertia (in relation to the torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. Cavendish's aim was not actually to measure the gravitational constant, but rather to measure Earth's density relative to water, through the precise knowledge of the gravitational interaction. In modern units, the density that Cavendish calculated implied a value for G of 6.754×10−11 m3⋅kg−1⋅s−2.
The accuracy of the measured value of G has increased only modestly since the original Cavendish experiment. G is quite difficult to measure, because gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies. Furthermore, gravity has no established relation to other fundamental forces, so it does not appear possible to calculate it indirectly from other constants that can be measured more accurately, as is done in some other areas of physics. Published values of G have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive. This led to the 2010 CODATA value by NIST having 20% increased uncertainty than in 2006. For the 2014 update, CODATA reduced the uncertainty to less than half the 2010 value.
In the January 2007 issue of Science, Fixler et al. described a new measurement of the gravitational constant by atom interferometry, reporting a value of G = 6.693(34)×10−11 m3⋅kg−1⋅s−2. An improved cold atom measurement by Rosi et al. was published in 2014 of G = 6.67191(99)×10−11 m3 kg−1 s−2.
A controversial 2015 study of some previous measurements of G, by Anderson et al., suggested that most of the mutually exclusive values can be explained by a periodic variation. The variation was measured as having a period of 5.9 years, similar to that observed in length-of-day (LOD) measurements, hinting at a common physical cause which is not necessarily a variation in G. A response was produced by some of the original authors of the G measurements used in Anderson et al. This response notes that Anderson et al. not only omitted measurements, they also used the time of publication not the time the experiments were performed. A plot with estimated time of measurement from contacting original authors seriously degrades the length of day correlation. Also taking the data collected over a decade by Karagioz and Izmailov shows no correlation with length of day measurements. As such the variations in G most likely arise from systematic measurement errors which have not properly been accounted for.
Under the assumption that the physics of type Ia supernovae are universal, analysis of observations of 580 type Ia supernovae has shown that the gravitational constant has varied by less than one part in ten billion per year over the last nine billion years.
The GM product
The quantity GM—the product of the gravitational constant and the mass of a given astronomical body such as the Sun or Earth—is known as the standard gravitational parameter and is denoted μ. Depending on the body concerned, it may also be called the geocentric or heliocentric gravitational constant, among other names.
This quantity gives a convenient simplification of various gravity-related formulas. Also, for celestial bodies such as Earth and the Sun, the value of the product GM is known much more accurately than each factor independently. Indeed, the limited accuracy available for G limits the accuracy of determination of such masses in the first place.
For Earth, using M⊕ as the symbol for the mass of Earth, we have
^For example, the gravitational force between an electron and proton one meter apart is approximately 10−67N, whereas the electromagnetic force between the same two particles is approximately 10−28 N. The electromagnetic force in this example is some 39 orders of magnitude (i.e. 1039) greater than the force of gravity—roughly the same ratio as the mass of the Sun to a microgram.
^Gundlach, Jens H.; Merkowitz, Stephen M. (2002-12-23). "University of Washington Big G Measurement". Astrophysics Science Division. Goddard Space Flight Center. Since Cavendish first measured Newton's Gravitational constant 200 years ago, "Big G" remains one of the most elusive constants in physics
Standish., E. Myles (1995). "Report of the IAU WGAS Sub-group on Numerical Standards". In Appenzeller, I. Highlights of Astronomy. Dordrecht: Kluwer Academic Publishers. (Complete report available online: PostScript; PDF. Tables from the report also available: Astrodynamic Constants and Parameters)
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