Space DOES NOT Expand Everywhere (Script) (Patreon)

Space is big, and it’s getting bigger. But where does all that new space actually come from… And is it popping into existence all around you right now? Is that why the remote control is always further away than I thought?

Nearly a hundred years ago, a collection of observations and theoretical ideas came together to reveal the universe is expanding on the largest scales. The distant galaxies are all racing away from us, and interpreted through the lens of Einstein’s general theory of relativity, this only makes sense if all distances everywhere are increasing evenly. This revelation led us to the discovering of the beginning of the universe at the big bang, and lets us predict its future - it’ll probably expand forever, FYI.

We know the effect of cosmic expansion on the most enormous scales, but what about the here and now? If space is expanding everywhere, is it expanding inside the solar system, inside you? And what does it even mean for space to be expanding? Does that stretching stretch space thin? Will it eventually snap like overdrawn taffy? Or does new space get created on the fly - in which case, where does it come from?

Objects in space tend to move around due to nearby gravitational influences - planets orbit stars, stars orbit in the mutual gravity of their galaxies, galaxies whirl and collide in local groups and clusters. But if you expand your view to large enough scales, that local motion or “peculiar velocity” is small compared to the fact that distant points are moving apart - receding - and the further distant the faster the recession. We call this the Hubble flow, after Edwin Hubble, the guy who first measured the rate of expansion.

The Hubble flow could be interpreted as everything moving THROUGH space, but there’s a cleaner picture. Soon after Einstein finished general relativity, four scientists independently solved its equation for the entire universe, giving us the Friedman-Lemaitre-Robertson-Walker or FLRW metric. You can think of a metric as the coordinate system of a patch of spacetime. Similar to how longitude and latitude and the coordinate system of the 2-D surface of the Earth, a metric in GR is the coordinate system of a chunk of spacetime with  3 spatial and 1 temporal dimensions.

And just as latitude and longitude describe the spherical geometry of the surface of the Earth, the FLRW metric gives the geometry or shape of the universe. The spatial part of that geometry could be the 3-D analog of our surface of a  sphere, which would be a space that loops back on itself in all directions. Or it could be an infinite plane, either flat or hyperbolic. These shapes can be thought of as 2-D slices of 3-D spaces. But those 3-D spaces are also just slices out of 4-D spacetimes - representing single instants in time. If you use general relativity to track how the FLRW metric changes over time, you find that it has to change - either growing or shrinking in size everywhere evenly.

The FLRW metric makes some pretty big assumptions - that the matter in the universe is perfectly evenly spread out - homogeneous, and looks the same in all directions - isotropic. That seems to be a good representation of the universe on its largest scales. But keep this assumption in mind. It’ll be important to notice when it’s wrong.

OK, just for laughs let’s look at the math. This is the FLRW metric - it’s basically Pythagorus for 4-D spacetime - the squared proper distance between two points is the sum of the squares of x, y & z, but here adding the dimension of time. And this thing is the scale factor, which represents the overall size of the universe, and this is the thing that has to change. In our universe it’s getting bigger. When we say “the universe” or “space” is expanding, we mean the scale factor is getting bigger. But what does that mean beyond the math? What’s physically happening?

There’s a classic classroom representation of the expanding universe that’s actually pretty useful here. Glue galaxies to a balloon and inflate. The galaxies recede from each other with apparent speeds that are proportional to distance. But the balloon analogy isn’t even really an analogy - at any one instant it’s a fair representation of a 2-D slice out of a closed universe, and its growth represents the increase in the scale factor over time. The FLRW metric - the coordinate grid of this slice could be represented in spherical polar coordinates. In that case it looks a lot like latitude and longitude. Those lines are smooth because of our founding assumptions of homogeneity and isotropy. But if we zoom in we can see that the shape of the gridlines change. They pull towards massive objects like galaxies. In fact the shape of spacetime around massive objects is NOT the FLRW metric because the matter isn’t spread out evenly. For example, near a compact massive object the correct description of the spacetime is the Schwarzschild metric. The space in the Schwarzschild metric by itself doesn’t expand - in fact space is pulled inwards - pinched - and it’s completely static over time. But what about if it's embedded in a larger FLRW spacetime? Does the underlying expanding sphere fight against the inward pull of gravity? Is the space inside, say, a galaxy growing but overcome by the gravitational attraction between the stars?

The answer is … no. Space within any gravitationally bound system is unaffected by the surrounding expansion. In the balloon analogy it’s tempting to think of the galaxy as being held together despite the expanding material that it's attached to. But that’s not what’s happening. The gravitational field isn’t somethin,g that lies on top of the fabric of spacetime. The gravitational field IS the fabric of spacetime. If the gravitational field of the galaxy is this static, inwardly-pinched grid of the Schwarzschild metric, then that’s it. That’s the spacetime in that area. There isn’t another underlying expanding grid that this field has to fight against. In fact, the spacetime inside the Milky Way doesn’t even know that the universe is expanding.

The other way the balloon analogy fails is that the fabric of spacetime doesn’t get stretched in the way the rubber gets stretched - it doesn’t thin out and it doesn’t build up tension in a way that can pull against these embedded static fields. It’s as though the balloon adds more rubber as it inflates, always maintaining the same level of stretch.

Let's try something else. Lose another dimension of space so the universe becomes an expanding ring instead of a ,sphere, with grid points instead of gridlines. But then we can add the dimension of time into our picture so that the expanding ring traces a sort of cone, and the points trace lines. This is nice because you can show the changing rate of expansion - perhaps a period of rapid accelerating growth during inflation, then the expansion slows under the influence of gravity, then dark energy takes over and expansion accelerates again.

The expansion of space is seen in the divergence of our FLRW coordinate gridlines. Now the distance between those lines represents the growing scale factor. Let’s zoom in again. What does an individual galaxy look like? If the galaxy causes gridlines to be pulled together on a sphere, then on the ring it causes points to be pulled together, and they maintain that separation as the ring expands. So on our expanding cone, we see distant gridlines diverging, but nearby lines in a gravitational field remain parallel.

So there is no constant tug of war between the expanding universe and the gravitationally bound systems it contains. That tug of war did happen, but it happened a long time ago. In certain regions expansion won, and threw apart objects and the spacetime grids they trace. But in sufficiently dense regions gravity won and there the only evidence of the greater expansion comes from the receding view of distant galaxies.

I told you that space doesn’t get stretched out like rubber. In a sense new space gets created as it expands. What does that actually mean? This is a tough one - in fact an impossible one because we don’t have an accepted theory for what that the fabric of space is actually made OF, if anything. That’s the province of a long-sought theory of quantum gravity. But let’s see what we can say.

In general relativity, space can be infinitely divided. That means we can start with a universe that’s small and grid it up and watch it expand. The grid lines diverge, but we can choose to re-grid it at any point. What was once a millimeter might become a meter and the meter becomes a kilometer, but we can just redefine everything and start again. We’ll never run out of subdivisions. Similarly, we can take the universe of the present and define a grid of space - or a grid of points on our 1-D ring universe. Then we rewind the universe and every single one of those points traces a path back to the big bang. We can make the starting grid as fine as we like and get the same result. Every point in the modern universe can be traced back to a point at the big bang.

This is related to the idea of geodesic completeness, which we’ve talked about before. All spacetime paths can be traced to the infinite future or past until they hit a singula,rity - the big bang or a black hole. As the universe expands, we don’t have new patches of space appearing between the old ones to fill out the universe. Geodesics don’t just pop out of nowhere.

This all works if space is infinitely divisible. But we know that at the smallest scales, general relativity comes into conflict with quantum mechanics. There IS a smallest measurable length called the Planck length. When space expands, what happens to this smallest length? Does it expand too? Well the Planck length stays the same - it’s just defined as a combination of the Gravitational constant, the Planck constant, and the speed of light - so if those aren’t changing - and there’s no evidence that they change -then the Planck length remains the same. But as the universe expands it adds more and more of these “Planck lengths” which must expand out of the old Planck lengths. That implies an infinitely divisible space emerging from within each Planck length.

This apparent conundrum is related to something called the transplanckian problem, which is to do with how Hawking radiation seems to emerge from sub-Planck-wavelength radiation at the event horizon of a black hole. Its solution takes us to the realm of the holographic principle, and so we can’t go there today. But long story short - even with a quantized fabric of space, it’s still possible to expand it OR condense it infinitely without changing its fundamental nature. Empty space can be rescaled infinitely.

And that’s even true if empty space contains something. Like dark energy. Empty space has a very weak energy density, even in the absence of particles. As space expands, that density doesn't change - remember, the balloon skin doesn’t thin out. The result of this is that the total dark energy content of the universe depends on the amount of space in the universe, which means dark energy increases as the universe expands. But again, that will never have any effect inside bound gravitational systems. Its effect only manifests when there’s an enormous amount of empty space compared to the amount of matter. But the ratio of empty space to matter inside galaxies doesn’t change. Now if the level of dark energy per unit volume were to increase, that would be a different matter - and we talked about the resulting cataclysm in our episode on the big rip.

So there’s your answer. The infinite scalability of space means the universe can and probably will expand forever with no effect on this little bubble of relatively static space that we call the Milky Way. Well, except for permanently isolating our bubble from all of the others. Let’s just be grateful that we came into being soon enough to catch a glimpse of those receding galaxies - they’ll spend most of cosmic time far beyond the horizon, in which case we never would have even known that we live in an infinitely expanding spacetime.