In the quantum vacuum there are many transient acceleration vectors of mean magnitude a randomly oriented. If the vacuum is viewed from an accelerated frame, the vectors going with the frame appear diminished, and the vectors going against the frame appear enhanced, resulting in a net polarization of the vacuum. If the frame's acceleration g is small, the effect is linear, and if the vacuum is filled with vectors the coefficient of the polarization will be unity. The standard exponential term for suppressing high-energy fluctuations must also be applied. Hence the vacuum polarization is g exp (g/a). The terms of the exponent when multiplied by the dipole moment have the dimensions of energy.
The rest frame of the galaxy, for example, is accelerated with respect to local inertial frames that fall into the center. In this rest frame the vacuum appears polarized and enhances the galaxy's gravitational field g. So we have
g= -GM/r2 + g exp (g/a)
where g is understood to be negative. For g much greater than a, the exponential is negligible and Newton's law results. But for g less than a, the exponential can be expanded to 1 + g/a and we get
g2 = aGM/r2
This is precisely the formula found empirically by Milgrom to explain the motion of stars and galaxies in the weak-field region, except the law of gravity is altered, not the law of motion (Scientific American, August 2002). He finds that a is about one Angstrom per second squared, which is near the "surface gravity" of an electron, the field of a one-kilogram mass at one meter, or the field of a galaxy in its outer parts. Also, the square of a is not far from the value of the cosmological constant, in units where c=1. In this model, a may be viewed as the saturated field strength of the quantum vacuum.
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