nLab 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 $G$ of the pseudoRiemannian metric $g$ on spacetime is proportional to the stress-energy tensor $T$ of the matter and radiation in spacetime. (For simplicity, let any cosmological constant or other dark energy be included in $T$.) The simplest way to state this proportion is as an equality:

$$G = T$$

or (with tensor indices)

$$G_{a,b} = T_{a,b} .$$

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

$$G_{a,b} = k T_{a,b} ,$$

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

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

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

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

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

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

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

for the force exerted by either mass $m_i$ on the other at a distance $r$. (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.

Related concepts

fundamental scales (fundamental/natural physical units)

• speed of light$\,$ $c$

• Planck's constant$\,$ $\hbar$

• gravitational constant$\,$ $G_N = \kappa^2/8\pi$

• Planck scale

  • Planck length$\,$ $\ell_p = \sqrt{ \hbar G / c^3 }$

  • Planck mass$\,$ $m_p = \sqrt{\hbar c / G}$

• depending on a given mass $m$

  • Compton wavelength$\,$ $\lambda_m = \hbar / m c$

  • Schwarzschild radius$\,$ $2 m G / c^2$

  • depending also on a given charge $e$

    • Schwinger limit$\,$ $E_{crit} = m^2 c^3 / e \hbar$

• GUT scale

• string scale

  • string tension$\,$ $T = 1/(2\pi \alpha^\prime)$

  • string length scale$\,$ $\ell_s = \sqrt{\alpha'}$

  • string coupling constant$\,$ $g_s = e^\lambda$

References

• Wikipedia, Gravitational constant