black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
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Newton's gravitational constant provides a unit conversion between units of spacetime and units of mass/energy.
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:
or (with tensor indices)
However, conventional units of measurement don't allow this equality, and so we write
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 $8 \pi$:
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
in general relativity, it gives the simplest formula for the fictional force? of gravity in the nonrelativistic limit:
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.)
There is good stuff to say here about how we only know this constant to about 6 significant digits.
fundamental scales (fundamental physical units)
speed of light$\,$ $c$
Planck's constant$\,$ $\hbar$
gravitational constant$\,$ $G_N = \kappa^2/8\pi$
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 $e$
string tension$\,$ $T = 1/(2\pi \alpha^\prime)$
string length scale$\,$ $\ell_s = \sqrt{\alpha'}$
string coupling constant$\,$ $g_s = e^\lambda$
Last revised on March 30, 2020 at 05:32:06. See the history of this page for a list of all contributions to it.