# Local effects of cosmic expansion

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?

• Nov 14 '21 at 18:47
• I also don't understand what "glues" stuff to the comoving frame (though I am not sure it is necessary). Opposite, regarding your concerns with conservation in general, I think that is not clear if there should be any at cosmological scale, and there is dark energy... Nov 15 '21 at 9:20
• In some way that I haven't yet been able to fully grasp, Davis & Lineweaver (the physicists who designed the widely-used diagrams of cosmic & particle horizons in 3 different coordinate systems) have claimed that astrophysical bodies (stars etc.) are stationary with respect to the expansion of space: Davis has specified that it is not a "force or drag" carrying those bodies with it. Nov 16 '21 at 15:34
• I believe it relates to the Heisenberg Uncertainty Principle's requirement that energy and time cannot be understood simultaneously, which would leave the spatial expansion (that would not require energy, but would involve time) coincidental with regard to the movement of astrophysical bodies (that would necessarily involve energy), so that an understanding of Heisenberg's math might be a prerequisite for grasping it. Nov 16 '21 at 15:45

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."

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).

• @A.V.S. I think what you are saying is that you can define a comoving velocity. I agree with that of course, but then the velocity is zero for galaxies that are at rest in the cosmic frame. That's also fine, but I thought that this would be confusing for the OP. You shouldn't include any part of the expansion in this "velocity", otherwise you can get particles moving at faster than $c$ if they are separated by more than a Hubble length. I don't think what either of us is saying is wrong, but we're focused on different parts of the story. Nov 14 '21 at 23:43
• @A.V.S. I don't think this answer is wrong. I think you actually make the same point in your answer when you say you need to do parallel transport to compare three vectors at different points in the manifold. I welcome different ways of looking at things, but I'd appreciate it if you rephrased your comment or provided more justification. The fact that you can't directly compare vectors in different tangent spaces is an absolutely central concept in differential geometry. And the comoving velocity is zero so it doesn't match the OP's notion of velocity in their last paragraph. Nov 16 '21 at 15:57
• What I disagree with, is your “short answer“ which critiques the premise of Q for being based on Galilean kinematics. My viewpoint is that spatially flat FRW universe (and derived models such as ΛCDM) has a global and natural structure of a Galilean world: it has absolute time and flat space, meaning that concepts of Galilean kinematics, like relative velocities are actually well defined (this is independent of the underlying dynamical theory). Your longer answer has a lot of good points with which I agree (maybe with addendum that FRW is an almost unique exception for some of them). Nov 16 '21 at 16:45
• The fact that you can't directly compare vectors in different tangent spaces is an absolutely central concept in differential geometry but if the underlying manifold is a Euclidean space, parallel transport required to move vectors between different tangent spaces is quite transparent, so much so that the ability to compare (and also add etc.) vectors initially defined at different points is at the core of machinery of Newtonian mechanics. Nov 16 '21 at 16:55

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.

• Overall, this is a good answer, but two comments. (1) "FRW spatially flat cosmology... is well described by Newtonian gravitation globally, including on scales exceeding the Hubble radius." -- I think this is only true in a matter-dominated Universe, where you can derive the Friedman equation from an expanding gas of matter moving under gravity (and btw the Friedman equation will be the same for spatially curved Universes too). But not for radiation or Lambda. (2) If you need to parallel transport vectors to the same tangent space, you also agree you can't directly compare two vectors. Nov 14 '21 at 23:54
• Also this is not correct: "the acceleration 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 the acceleration is negative, the relative velocity is decreasing. But they don't need to collide (ie end up with $d=0$). In Newtonian language, the matter can be expanding out at a velocity greater than or equal to the escape velocity. In GR language, a matter dominated Universe doesn't need to have a Big Crunch. Nov 14 '21 at 23:59
• @Andrew: I meant that equations also have to be modified to include relativistic sources in the Newtonian language (I made corrections to clarify it). I have to go now, but I'll think on whether the galaxies would collide or not some more. Nov 15 '21 at 4:34
• For a spatially-flat matter-dominated universe, the scale factor grows as a power law $d \propto t^{2/3}$, with a positive proportionality constant, which is always increasing. The first derivative $\dot d \propto \frac{2}{3} t^{-1/3}$ is positive, while the second derivative $\ddot{d} \propto - \frac{2}{9} t^{-4/3}$ is negative. Incidentally I'm not saying galaxies won't collide (that can happen even in Newtonian gravity just due to random motion and gravitational attraction), but that the scale factor won't become zero (or even decrease) in a spatially flat matter dominated Universe. Nov 15 '21 at 5:58
• @Edouard: First of all my answer is in no way some radical viewpoint on cosmology, I think it is compatible with the treatments of cosmology in introductory GR textbooks. As for “subdividing of space” I do not know what this means. Do you have some formalization of that idea? Can you write it up as a full question (without triggering non-mainstream physics policy)? Nov 16 '21 at 18:54

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.