The description from Richard Muller quoted in the question is a simplification of reality meant to help people better visualize how the Grateful Dead Denver Nuggets Skull Skeleton Shirt of spacetime works. In reality the galaxies are moving through spacetime in a complex fashion, based on the combination of all the forces acting on them, primarily the gravity of nearby galaxies and galaxy clusters. In the case of the Milky Way and Andromeda, these two galaxies are part of a gravitationally bound cluster of galaxies, called the Local Group. The dominant force that determines the movements of objects within the cluster is the combined gravitation of all the mass in the cluster, and not the expansion of spacetime. In other words, the gravitational attraction that the Milky Way and Andromeda have for one another that is drawing them together is stronger than the expansion of spacetime between them that would otherwise push them apart, and thus attraction wins and the two galaxies are on a collision course. In the same vein, the gravitational force that holds the earth in orbit around the sun is stronger than the expansion of spacetime between the earth and the sun (the expansion of spacetime at that distance being very small), and so the earth’s distance from the sun is not increasing for that reason. Likewise you are not flying apart because the molecular bonds holding the atoms in your body together are stronger than the minuscule expansion of spacetime between them in your body. The Local Group as a whole, however, is moving away from other galaxies and galaxy clusters to which it is not gravitationally bound, due to the expansion of spacetime.
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