nLab
gravitational constant

Newtons gravitational constant

Context

Gravity

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Newton's gravitational constant

Idea

Newton's gravitational constant provides a unit conversion between units of spacetime and units of mass/energy.

Motivation

According to Einstein's theory of general relativity, the Einstein tensor GG of the pseudoRiemannian metric gg on spacetime is proportional to the stress-energy tensor TT of the matter and radiation in spacetime. (For simplicity, let any cosmological constant or other dark energy be included in TT.) The simplest way to state this proportion is as an equality:

G=T G = T

or (with tensor indices)

G a,b=T a,b. G_{a,b} = T_{a,b} .

However, conventional units of measurement don't allow this equality, and so we write

G a,b=kT a,b, G_{a,b} = k T_{a,b} ,

where the constant kk is about 1.67732×10 9m 3kg 1s 21.67732 \times 10^{-9} \m^3 \kg^{-1} \s^{-2} in SI unit?s. (Depending on how one handles the conversion between the space and time components of GG and TT, there could be extra factors of the speed of light constant 2.99792458×10 8ms 12.99792458 \times 10^8 \m \s^{-1}.)

This factor is essentially the gravitational constant; however, the gravitational constant is traditionally taken to be smaller by a factor of 8π8 \pi:

G=6.67408(31)×10 11m 3kg 1s 2, G = 6.67408(31) \times 10^{-11} \m^3 \kg^{-1} \s^{-2},

where the standard uncertainty (about 46ppm) is in parentheses. Yes, the same letter is used for this constant as for the Einstein tensor! Although this yields

G a,b=8πGT a,b G_{a,b} = 8 \pi G T_{a,b}

in general relativity, it gives the simplest formula for the fictional force? of gravity in the nonrelativistic limit:

F=Gm 1m 2r 2 F = G \frac{m_1 m_2}{r^2}

for the force exerted by either mass m im_i on the other at a distance rr. (It was in this context that Newton used the constant.)

Measurement

There is good stuff to say here about how we only know this constant to about 6 significant digits.

References

Last revised on April 2, 2019 at 19:40:49. See the history of this page for a list of all contributions to it.