# Distinction between metric expansion and objects just moving apart from each other?

I see explanations likes this:

Galaxies and other sources, then, are not strictly expanding away from each other but rather are attached to the fixed grid on the expanding fabric of spacetime. Thus, the galaxies give us the impression of moving away from each other. (Scientific American)

[...] space is expanding [...] the most correct way to think about it would be that the grid points are getting farther apart. (Physics.se)

How have we determined that it is some underlying grid or space itself that is expanding rather than the objects just moving apart on a static grid of spacetime?

How have we determined that it is some underlying grid or space itself that is expanding rather than the objects just moving apart on a static grid of spacetime?

General relativity tells us that the geometry of spacetime is dynamic, being affected by the matter content and its motion. So a fixed, static background spacetime is by its nature some sort of idealized edge case at best.

Well, let's try it. Imagine that in flat, static Minwkoski spacetime with spherical coordinates $(T,R,\theta,\phi)$, there is an expanding spherically symmetric cloud of galaxies expanding from a center at $T = 0$, each with some velocity $v$, such that the galaxies have a negligible effect on the background geometry or each others' velocity. Thus the radial coordinate of each galaxy is $R = vT$.

Parametrizing by the rapidity $\eta$ at the given radial coordinate ($v = \tanh \eta$) and the time measured by the galaxy $t$ as given by special-relativistic time dilation ($t = T/\gamma = T/\cosh\eta$), in those coordinates, the Minwkoski metric becomes $$\begin{eqnarray*} \mathrm{d}s^2 &=& -\mathrm{d}T^2 + \mathrm{d}R^2 + R^2\mathrm{d}\Omega^2\\ &=& -\mathrm{d}t^2 + t^2\left(\mathrm{d}\eta^2 + \sinh^2 \eta\,\mathrm{d}\Omega^2\right)\text{,} \end{eqnarray*}$$ which is a linearly expanding spatially hyperbolic universe where $\eta$ plays the role of the radial coordinate (up to some dimensionful factor, i.e., $r = r_0\eta$).

We sure didn't get far! A spherically symmetric explosion of galaxies in a fixed, static Minkowski spacetime is equivalent to a spatially hyperbolic universe that in which space itself is expanding, with a scale factor $a\propto t$ in terms of cosmological time $t$.

For a homogeneous and isotropic universe, the time-projection of the Einstein field equation is the first Friedmann equation, $$\frac{\dot{a}^2+k}{a^2} = \frac{8\pi G\rho + \Lambda}{3}\text{,}$$ so in the case of vanishing cosmological constant ($\Lambda = 0$) and negligible energy density ($\rho = 0$), its solution $a(t) = a_0\pm it\sqrt{k}$ is real only if $k<0$, implying an open, spatially hyperbolic universe; conventionally, $k\in\{-1,0,+1\}$, as only the sign of $k$ is important.

Why propose an expanding underlying grid instead of some other mystery force that accelerates objects away faster the farther away they are from us? How would we distinguish between those two scenarios?

If one introduces an extra force and fine-tunes things well enough, you probably can't. However, this is an incredibly silly thing to do: you would need to postulate that (1) gravity simply does not work on the cosmological scale, and that (2) that its effects are exactly mimicked by an extra force that we have absolutely no evidence for. This doesn't achieve anything, not even philosophical pandering, because we would still know that general relativity is correct on smaller scales, so spacetime geometry would still be dynamic on those scales.

There's no difference between relative motion and "space expanding" between objects. There is a very common misconception that there is a difference, which has even made it into textbooks, but there isn't actually a difference.

Metrics in general relativity just describe the shape/geometry of spacetime. The coordinates are a means to an end. Only the shape matters as far as the physics is concerned.

The FLRW family of metrics defines a family of spacetime geometries, using particular coordinates. If you hear somebody talking about a "grid", they're mistaking the coordinates used in the FLRW metric for an aspect of physical reality. The FLRW geometry is coordinate independent and has no grid.

Any spacetime geometry is approximately flat if you zoom in enough. If you zoom in on a FLRW universe, you'll see galaxies moving away from each other in a special-relativistic sense. They don't move apart with respect to FLRW coordinates even at that small scale, but that's because the FLRW coordinates are noninertial even in flat regions. Many people don't understand this and conclude that the galaxies really aren't moving.

The Scientific American quote is wronger than usual, as it talks about expanding spacetime (instead of expanding space), and it says not only that there's a grid but that galaxies are "attached" to it, as though there are no peculiar velocities. I'm actually surprised that SA published that.

For one, the expansion of space allows objects to move away from us faster than light. Einstein said nothing can move faster than light through space; however, space itself can expand between objects faster than light.

That's another common misconception.

In special relativity, nothing can move faster than $$c$$ with respect to inertial coordinates. FLRW coordinates aren't inertial, and neither are the rescaled coordinates $$(t, a(t)x)$$ that are used to define recessional velocity.

In a flat region of spacetime where special-relativistic relative velocities make sense, they are different from cosmological recessional velocities. If the flat region is large enough (possible only if the matter density is very low, and there's no dark energy), there will be galaxy pairs in the flat region whose recessional velocity is larger than $$c$$, but their special-relativistic relative velocity is of course less than $$c$$.

Also, nothing can move faster than light in any case. Things may move faster than $$c=299{,}792{,}458\text{ m/s}$$, but so does light (in those coordinates).

# Time will show

...although that is a very long time.

If the grid of the universe was static, objects moving away from us should continue to move away from us with a continues velocity. That is not the case from observed data: Things farther away from us seems to move away faster. This is known as Hubble's law, and is one of the basic concepts of modern cosmology. The model of an expanding metrics fit that data, in that when there is more space fabric between us and the far object, there are more space that stretches, and therefore the velocity is higher.

That things move away from us, and have an accelerating velocity, should suggest a universe that expands faster and faster. There are however other forces limiting this growth, most notably gravity. The fate of the growth of the universe is determined by that unknown balance, that must also consider dark matter and energy.

The magnitude, or even the validity of those models is the current frontier of astronomy, and cosmology in general. Nothing is settled yet.

• Why propose an expanding underlying grid instead of some other mystery force that accelerates objects away faster the farther away they are from us? How would we distinguish between those two scenarios?
– user10250
Feb 28 '16 at 3:00
• @Dawn I understand your confusion, but it can be quickly explained with relativity and field theory. Feb 28 '16 at 9:46

For one, the expansion of space allows objects to move away from us faster than light. Einstein said nothing can move faster than light through space; however, space itself can expand between objects faster than light.

If it were actually a force pushing these objects away, they would not recede from us faster than light.

Second, according to QFT, there does not seem to be a universally (always) repulsive force. Whether forces can be repulsive or not depends on the spin of their mediating field.

A scalar (spin-0) force is universally attractive, as is a spin-2 force, while a spin-1 is attractive for different charges and repulsive for like charges. So the electromagnetic, the weak and the strong force can be repulsive, while gravity cannot.

Because of this pattern we can assume that there are no universally repulsive forces that can be currently described by fields in QFT.