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. The observations can be adequately explained by assuming a plausible amount of ordinary matter M and using the correct quantum law of gravity. There is no need for dark matter. As space accelerates away from us, the resulting apparent polarization would enhance the acceleration, and indeed might cause the acceleration, once the process has begun, due perhaps to some disturbance long ago. If space is collapsing in some remote region, the same process would enhance the collapse. So the cosmos may consist of interspersed regions of expansion and collapse. When expansion becomes extreme, a big bang would result as virtual particles are ripped out of the vacuum. A collapsing region would produce a big crunch, where matter is crushed back into the vacuum. The whole process is presumably infinite and eternal.

# Quantum Gravity May Explain Dark Matter

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