Consider this thought experiment. Different emitters are placed at different heights from the ground. The earth plays the role of the source of gravity.
They emit light, which from an earth observer, would be blueshifted. According to Einstein’s Equivalence Principle: we can say that the Doppler effect is equal to the gravitational shift ( see equation 2 in The Essential General Relativity , reproduced below)
(1) (Δf/f)gravity = -(Δf/f)doppler = -Δv/c
For emitter 1, we can say,
(2) Δv1 = g(d1) Δt1 (definition of acceleration, where g(d1) is the gravitational potential field at d1.
(3) Define g(d1) = g1
(4) Δv1 = g1 (d1/c) (time = distance/velocity)
(5) However, g1 = (GMsource)/ R21
= (GMsource)/ (Rsource + d1)2
= (GMsource) (Rsource + d1)−2
= (GMsource/ R2source) ( 1 + d1 / Rsource)−2
≈ (GMsource/ R2source) ( 1 − 2d1 / Rsource)For 2d1 << Rsource
(6) g1 = (GMsource)/ R2source
Substitute (6) into (4),
(7) Δv1 = (GMsource)/ cR2source )(d1)
We can get the same result for emitters 2 and 3,
(8) Δv2 = (GMsource)/ cR2source )(d2)
(9) Δv3 = (GMsource)/ cR2source )(d3)
We can generalize equations 7,8 and 9 as,
(10) Δv = Hd ,
where H = (GMsource)/ (cR2source)
In case you haven’t recognized this, it is Hubble’s equation. When he discovered that all galaxies have a redshifted spectrum, Hubble concluded that all the galaxies were moving away. That is the Doppler Effect. However using Einstein’s Equivalence Principle, we can say that galaxies are at rest, and photons are redshifted( they are moving against gravity). Note that Hubble discovered not a change in velocity but just a velocity. In his days, he did not have the technology to observe such a small change in the galaxies' velocities, and it took nearly 70 years before it was discovered that galaxies are actually accelerating.
In our thought experiment, the emitters are at rest, so one can easily say that such emitters would start to accelerate as they cannot be “nailed” in outer space. What about the galaxies, can they be “nailed” so that we can claim they are at rest with respect to each other? Consider one galaxy against all others.
According to Gauss’ theorem (see fig.3 in Newton's Law of Gravity), a galaxy would be attracted as if all the matter inside the sphere were concentrated at the center of that sphere. One can ignore all the other galaxies outside that sphere. At the same time, one can draw an infinite number of spheres, in which the galaxy would be attracted to the center of each sphere. Here’s a diagram with just three spheres drawn.
If the universe is infinite, we can safely say that the total force on a galaxy is zero, and therefore, the galaxies are nearly at rest.
Does it mean that the Big Bang theory is wrong? No. The Big Bang theory says that for every galaxy, all other galaxies are moving away. But Einstein’s Equivalence Principle also says that we can look at every galaxy at rest with their light being redshifted. Both pictures are equivalent.
Equation 10 is true for every galaxy, since each one is emitting light from a source of gravity that has an infinite radius. This argument is only valid if the universe is infinite.