Local effects of cosmic expansion

Question

Suppose we place two galaxies 100 Mpc apart, with zero initial velocity with respect to each other. In other words, they are static with respect to each other initially, with a negligible gravitational attraction, say. They are static is in the physical distance space, not necessarily in the comoving coordinate system.

Now, as the universe expands according to FRW metric, will the distance between the two galaxies increase with time as $d => a(t)d$?

If no, then is the expansion of universe really an expansion of space, or is it just that all matter are moving away from each other due to some initial explosion, like the big bang? Will the light emitted by one galaxy be redshifted to the other galaxy, as there is no relative velocity in the physical distance space, and hence no doppler effect?

If yes, then isn't it a violation of conservation of momentum? Two bodies initially with zero relative velocity and zero interaction suddenly starts to move away from each other? In that case, if the relative motion synchronises with the hubble expansion again with them receding away from each other, then what is the nature of the force/energy which is supplying the potential well to "glue" particles to the expanding comoving coordinate system?

Answer 1

Short answer: the premise of your question is flawed, and therefore your actual questions don't really have answers beyond "that's not how it works."

Longer answer:

The issue with the premise of your question is that you cannot compare the velocities of two galaxies separated by 100 Mpc. The velocity is a quantity that is defined locally (formally, the velocity is a vector in the tangent space at the position of the object). On the other hand, 100 Mpc is a large enough distance that spacetime curvature effects due to the expansion are not negligible (we cannot pretend that the two galaxies live in the same tangent space). So the velocities cannot be compared directly. As an analogy, if two ants on different lines of longitude on Earth pointed their arms directly "North", it doesn't mean anything to ask whether the ants are pointing their arms in the "same" direction on the surface of the sphere. The definition of "North" depends on the tangent space of the sphere you find yourself in, and can't be compared between tangent spaces.

What you can say, is that the distance between galaxies increases with time as the Universe expands, and that there is a gravitational redshift that occurs as light travels from one galaxy to another.

Note that gravitational redshift is like the Doppler effect if you picture the galaxies as having a relative velocity, but it's not really the same thing (since you can't directly compare velocities over such large distances). You can give different physical interpretations to where this redshift comes from (personally I like saying that the expansion of space stretches the photon's wavelength as it travels from one galaxy to the next), but a mathematical derivation of this redshift effect that everyone will agree on is to project the tangent vector of the light's path into the observers' tangent spaces (attached to the locations of each galaxy), and using this projection show that each observer will assign a different frequency to the light.

A related issue with the way your question is framed is that the Big Bang did not happen at a single point; it happened everywhere in space. It's just that space itself was very small, in the sense that the distances between points were small compared to the distances we observe today, or perhaps the distances were actually zero.

Essentially, the issue is that you are trying to apply intuition from your experience with uncurved spaces, to a curved spacetime, and finding (correctly) that your intuition leads to contradictions. What this is telling you is that your intuition is wrong, and the correct explanations require knowledge of how curved spacetimes work. This is a very common issue for people learning GR. On one level, an answer to your questions is for me just say "trust me, the math works" (which is sort of what I'm doing). On another level, if you really want to understand what's going on, the full story is too long for me to fit it into an answer on this site, so I recommend (a) taking a course if that is an option for you, (b) working through a book (the ones by Schutz and Hartle are aimed at beginners), and/or (c) watching some videos (such as the ones by Susskind).

Answer 2

First of all, let us address the argument from the answer by Andrew, that one “cannot compare the velocities of two galaxies separated by 100 Mpc”. While, for a general spacetime statements like this can have merit, FRW spatially flat cosmology is an interesting example of a curved spacetime, that is well described by Newtonian gravitation globally, including on scales exceeding the Hubble radius (provided, of course, that gravitational effects from relativistic sources, such as cosmological constant, radiation etc. are recast in the Newtonian language). Homogeneity of FRW provides us with a natural splitting of spacetime into “slices” of constant cosmic time, which are basically flat Euclidean 3D spaces. So if there are vectors defined at different points of space at the same time, these vectors could be parallel transported to a common tangent space and thus compared. In practical terms, this means that for large scale numerical N-body simulations of cosmological structure formation we can use Newtonian gravity over cosmological distance instead of Einstein equations.

Now, as the universe expands according to FRW metric, will the distance between the two galaxies increase with time as $d=>a(t)d$?

No. The evolution of the distance between the galaxies could be determined form the second Friedmann equation: $$ \ddot{d} = \left(-\frac{4 \pi G \rho_\text{m}}{3} + \frac{\Lambda c^2}{3}\right) d . $$ This is the equation for relative acceleration for particles in the Hubble flow but it would continue to be valid also for particles not moving with it, provided that the relative velocities remain small compared to $c$, otherwise one has to include relativistic corrections to the acceleration.

Note, that this is a 2-nd order ODE and we need to specify not only the initial separation $d_0$ but also the initial velocity, which, as specified in the question, is zero $\dot{d}(0)=0$. In our universe the expression in parenthesis is positive at present, so the distance would be increasing but slowly at first, and only after sufficiently long time (about Hubble time) the growth of distance could be approximated as $d=C a(t)$ (note, that $C\ne d_0$). Also note, that if we imagine the universe without cosmological constant (pure dust cosmology) the acceleration $\ddot{d}$ would be negative, and the distance between these two galaxies would be decreasing until they collide (or approximation of uniform matter distribution stops working).

If no, then is the expansion of universe really an expansion of space, or is it just that all matter are moving away from each other due to some initial explosion, like the big bang?

This is not an either/or situation. Space is expanding and thus bits of matter are moving away from each other. But unlike explosion there is no external reference which allows us to define center or absolute state of rest.

Will the light emitted by one galaxy be redshifted to the other galaxy, as there is no relative velocity in the physical distance space, and hence no doppler effect?

If the light is emitted from the first galaxy when both are static relative to each other, by the time it reaches the second galaxy, this galaxy would be moving, so the light would appear redshifted to the observers of the second galaxy, but this redshift would be smaller than that from the Hubble law.

If yes, then isn't it a violation of conservation of momentum? Two bodies initially with zero relative velocity and zero interaction suddenly starts to move away from each other?

This is no more violation of conservation of momentum than an apple initially at rest relative to Earth and starting falling toward it. Relative acceleration of those two galaxies is determined by the cosmological constant and by the density of matter between them. We can apply the shell theorem of Newtonian gravity and ignore the influence of anything outside the sphere, then the total momentum inside the sphere is conserved.

… then what is the nature of the force/energy which is supplying the potential well to "glue" particles to the expanding comoving coordinate system?

Particles are not “glued” to the comoving system, in Newtonian language they are moving only under the force of gravity which has two components: gravitational attraction toward matter and gravitational repulsion from cosmological constant.

Answer 3

Expansion of space means that distance can increase between two particles with zero velocity. Because $d = d_{0}a(t)$, the particles themselves need not move. Imagine points, drawn on a balloon. As you inflate the balloon, distances increase, yet the points do not move relative to the balloon. This is what it means for the metric to evolve: the very definition of lengths is changing.

Let's suppose a length scale D for which the Universe is homogeneous, and two tracer particles at distance D with 0 co-moving velocity. As the Universe expands, the distance between the two will increase. Moreover, any photons in transit between the two galaxies will be redshifted--this is not a Doppler effect, this is an effect due to the spacetime metric changing. Neither particle moves relative to the co-moving coordinate system and neither gains momentum.

You must abandon the sense of cosmic expansion as a velocity--it is a change in distance without velocity. For $v = H_{0}d$ and $H_{0} \sim 70$ km/s/Mpc, what will you think when $H_{0}d = v > c$ at a measly distance of 4-5 Gpc? You know nothing can move faster than the speed of light. This is why General Relativity is the best way to describe the expansion of the Universe, with the understanding that there are no velocities greater than the speed of light, yet there are galaxies whose distances increase faster than 300,000,000 m/s.