Gravitational coupling constant

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The gravitational coupling constant is a fundamental physical constant and a coupling constant characterizing the strength of gravitation between typical elementary particles. Because it is a dimensionless quantity, its numerical value does not vary with the choice of units of measurement.

The defining expression and the currently known value is

$\alpha_G = \frac{G m_e^2}{\hbar c} = \left( \frac{m_e}{m_P} \right)^2 \approx 1.752 \times 10^{-45}$

where:

G, the Newtonian constant of gravitation;
m_e, the mass of the electron;
c, the speed of light in a vacuum;
$\hbar$ , Dirac's constant or the "reduced" Planck's constant.
m_P, the Planck mass;

1 Measurement and uncertainty
2 Related definitions
3 Physical interpretation
4 See also
5 References
6 External links

[edit] Measurement and uncertainty

Theory does not yet provide a prescription for a direct measurement of this constant, as the Quantum Hall Effect does for its electromagnetic analogon. Therefore, our knowledge of the value depends on measurements of G, $\hbar$ and m_e.

α_G is normally expressed to only four or five significant digits, because G is known to only about one part in 7000. Comparatively, the meter and second are defined so as to make c and ε₀ exact, and e, m_e, and h are known to better than one part in 5,000,000.

[edit] Related definitions

The definition above is referred to in Barrow and Tipler (1986) except that they, following Eddington, replace m_e with the mass of the proton m_p = βm_e, in which case α_G = 1.752x10^-45xβ² ≈ 10^-39. Barrow and Tipler invoke α_G freely, but never give it a name.

A third possible definition of α_G involves the mass of one proton and one electron, in which case α_G = βx1.752x10^-45 = 3.217x10^-42, and α/α_G ≈ 10³⁹. α/α_G defined in this manner is (C) in Eddington (1935: 232) (except that he employs Planck's constant in place of Dirac's) and (4.5) in Barrow and Tipler (1986).

These three sensible definitions of α_G differ merely by a factor of β or its square (β = m_p/m_e = 1836.15267261 is the dimensionless ratio of the rest mass of the proton to that of the electron) The arbitrariness of choice made among which particle mass, resulting in the three definitions of α_G proposed here (whereas there is one unique choice of particle charge for α), and the relatively low precision with which it can be measured, may explain why the physics literature seldom mentions α_G.

[edit] Physical interpretation

α_G is to gravitation what the fine-structure constant is to electromagnetism and quantum electrodynamics.

α_G can be thought of as the square of the electron's mass (in units of Planck mass) and is therefore related to the question of masses of subatomic particles.

$\alpha_G = G m_e^2 / \hbar c = t_P^2 \omega_C^2$ , where $t P$ is the Planck time, relates it to $ω C$ , the Compton angular frequency of the electron.

α_G may be defined in terms of the masses of any two elementary particles. However, if the two masses are those of the proton or the electron, the two stable elementary particles with charge e and nonzero mass, then the ratio α/α_G measures the relative strengths of gravitation and electromagnetism between these fundamental particles. Assuming Planck units (so that $G=c=\hbar=4\pi\epsilon_0=1$ ) and defining α_G in terms of a pair of electrons, α_G = m_e², α = e², and α/α_G = (e/m_e)². Thus the ratio of the electron's mass to its charge, when both are measured in Planck units, grounds the relative strengths of gravitation and electromagnetism. The reason why gravitation is ostensibly such a weak force is because the charge of subatomic particles (if non-zero) is approximately the natural unit of charge (Planck charge) where the mass of the particles is far, far less than the natural unit of mass (Planck mass).

Because α is 39 orders of magnitude greater than α_G, the electrostatic force between the subatomic particles having charge is vastly stronger than the corresponding gravitational attraction. In fact, the gravitational attraction among subatomic particles can be ignored. That gravitation is relevant for macroscopic objects proves that they are electrostatically neutral to a very high degree.

[edit] See also

[edit] References

John D. Barrow and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford University Press. Invokes α_G freely.
John D. Barrow, 2002. The Constants of Nature. Pantheon Books.
Arthur Eddington, 1935. New Pathways in Science. Cambridge Univ. Press.