Are galaxies really moving away from us or is space just expanding?
This depends on how you measure things, or your choice of coordinates. In one view, the spatial positions of galaxies are changing, and this causes the redshift. In another view, the galaxies are at fixed coordinates, but the distance between fixed points increases with time, and this causes the redshift. General relativity explains how to transform from one view to the other, and the observable effects like the redshift are the same in both views. Part 3 of the tutorial shows space-time diagrams for the Universe drawn in both ways.
In the absence of the cosmological constant, an object released at rest with respect to us does not then fly away from us to join the Hubble flow. Instead, it falls toward us, and then joins the Hubble flow on the other side of the sky, as discussed by Davis, Lineweaver & Webb (2003, AJP, 71, 358). In what are arguably the most reasonable coordinates, the cosmic time t and the distance D(t) measured entirely at the cosmic time t, the acceleration is given by g = -GM(r<D)/D2 where M(r<D) is the mass contained within radius D. This gives g = -(4*pi/3)*G*(rho(t)+3P(t)/c2)*D(t). The 3P/c2 term is a general relativistic correction to the otherwise Newtonian dynamics. Galaxies all move under the influence of this acceleration and their initial position and velocity. In other words, F = ma and gravity provides the force. Nothing extra or weird is needed.
Also see the Relativity FAQ answer to this question.
Why doesn't the Solar System expand if the whole Universe is expanding?
This question is best answered in the coordinate system where the galaxies change their positions. The galaxies are receding from us because they started out receding from us, and the force of gravity just causes an acceleration that causes them to slow down, or speed up in the case of an accelerating expansion. Planets are going around the Sun in fixed size orbits because they are bound to the Sun. Everything is just moving under the influence of Newton's laws (with very slight modifications due to relativity). [Illustration] For the technically minded, Cooperstock et al. computes that the influence of the cosmological expansion on the Earth's orbit around the Sun amounts to a growth by only one part in a septillion over the age of the Solar System. This effect is caused by the cosmological background density within the Solar System going down as the Universe expands, which may or may not happen depending on the nature of the dark matter. The mass loss of the Sun due to its luminosity and the Solar wind leads to a much larger [but still tiny] growth of the Earth's orbit which has nothing to do with the expansion of the Universe. Even on the much larger (million light year) scale of clusters of galaxies, the effect of the expansion of the Universe is 10 million times smaller than the gravitational binding of the cluster.
Also see the Relativity FAQ answer to this question.
