Are we becoming more dense?
1. More of what?
So I managed to give my hypothetical spacetime substance a possible flow. Naturally, the next thing I wanted to do was compress it. If spacetime behaves like a fluid, surely some regions can be denser than others? That seems like a fairly reasonable property for a substance to have.
And then I thought about it, and realized, I had no idea what the hell density even means.
Usually, this is not a particularly difficult question. Take some amount of stuff, divide it by the volume it occupies, and there you go:
\[ \rho=\frac{m}{V}. \]
Here \(m\) is the mass inside a region of volume \(V\). Or I could count particles instead and get a number density. Either way, I have something to put in the numerator. If I squeeze the same amount into a smaller volume, its density increases. But what am I counting when the substance is spacetime itself? Events? Some amount of space? I cannot just say “volume per volume”, because that gives me one, which is mathematically correct and completely unhelpful.
Counting events does not immediately work either. In an ordinary smooth spacetime, even a tiny open region contains infinitely many points. A larger region does not simply contain more points in a way that gives me a useful density. I would need a rule assigning an amount to a region instead of just counting the elements in it. Then there's also the denominator to worry about. Do I mean spatial volume, measured by an observer at a particular time, or four-dimensional volume, including time itself? Those are different things. Since my proposed substance is spacetime, I can't just choose whichever one makes the analogy look nicest.
At first I thought gravity might tell me what to do. Clocks run differently near a mass, and the geometry changes how distances are measured. Maybe that is what a denser region of spacetime looks like? But that is just intuition, and I still need to turn it into a definition. A different clock rate does not come with a little badge saying how many units of spacetime have been squeezed into the region.
So I need to ask a more annoying question: if spacetime gets denser, what am I saying there is more of? And if I follow the same piece of this substance while it contracts, what amount is supposed to stay the same?
2. My first attempts
The first thing I tried was comparing volumes. If some region of spacetime occupies less volume than it did before, maybe I can call it denser. That sounds pretty harmless. But less volume than what? I need to identify the same region before and after, decide how to measure its volume, and say what has stayed inside it. Otherwise I have just measured a smaller region and have declared myself victorious. The labels from the previous note help with the first part. I can follow the same bundle of histories instead of comparing unrelated regions. But the labels don't really tell me how much substance belongs to that bundle. Naming every parcel of water does not tell me how much water there is. Unfortunately, giving spacetime coordinates does not either.
So I tried making the measurements local. Instead of worrying about the entire universe at once, why not use an observer moving with the proposed flow? At each event, I can use a momentarily comoving reference frame (MCRF). This gives me a way to talk about small spatial distances and volumes as measured by that observer. Fantastic, I have a ruler. Still no idea what I am measuring the density of. There is another little nuance here. These local rest frames do not need to fit together into one global “space at this instant”. If the flow twists, for example, I can't necessarily assemble all the observers' notions of simultaneity into a single spatial slice. For now, I can keep the volume measurements local, but I can't just turn them into a universal volume.
Then I wondered if my day 1, \(c\), could help (get it? because \(c=1\) and day 1 is like best friend. I can put these shitty jokes here because i'm the only one that reads these). Unfortunately, setting \(c=1\) did not make it my number one source of answers (i could go on forever with these). Anyways, \(c\) is already built into the relationship between space and time, so maybe it could give me some reference for the substance itself. An observer can measure a radar distance by sending out a light signal and timing its return:
\[ L=\frac{c\,\Delta\tau}{2}. \]
Here \(\Delta\tau\) is the proper time between sending and receiving the signal. In a sufficiently small region, this gives a useful local distance measurement. For a distant target in curved spacetime, I have to be more careful about what distance this procedure defines. But \(c\) doesn't give me an uncompressed reference state. It converts a measured time into a distance; I still need the clock, the measurement procedure, and something to compare the result with, and it certainly doesn't tell me how many units of substance occupy that distance.
But these attempts were not useless! I had ways to follow a region, measure its local volume, and compare distances using clocks and light. What I still did not have was an amount of spacetime that those measurements were supposed to contain. That left me with two possible directions. I could introduce a conserved amount and see what follows. Or I could abandon the idea of counting substance, and define “density” as compression relative to some reference geometry. Both sound like something I could work with. Both also require me to fit something into the model that I have not yet justified.
3. A density needs something to count
At this point I decided to try giving the substance something conserved. I still do not know whether spacetime actually has such a quantity, but I can at least ask what would happen if it did. Suppose each labelled piece carries some amount of substance. I am deliberately leaving the units unspecified, because calling it mass or counting particles would start introducing unjustified assumptions. Let's call its local density \(n\): the amount per unit spatial volume, measured in the frame moving with the flow. For a tiny chunk with volume \(V\), its amount is roughly \(nV\). Now I can say what compression is supposed to do. If the chunk shrinks while keeping the same amount, \(n\) increases. This is the behavior I wanted, and we have something in the numerator. The easiest (possibly only? without going into symmetries and whatnot) way of introducing conservation is through the continuity equation, so we should first introduce a current
\[ J^\mu=nu^\mu. \]
Here \(u^\mu\) is the flow from the previous note, so \(J^\mu\) describes how our hypothetical amount is transported. If none is created or destroyed, then
\[ \nabla_\mu J^\mu = \nabla_\mu(nu^\mu)=0. \]
a.k.a the covariant continuity equation. Expanding it gives
\[ u^\mu\nabla_\mu n+n\nabla_\mu u^\mu=0. \]
The first term is the change in density as I follow the flow, and the second has our expansion \(\theta\). So
\[ \frac{\mathrm{d}n}{\mathrm{d}\tau}+n\theta=0. \]
Finally, an equation that does what I expected! If \(\theta<0\), the flow contracts and a positive \(n\) increases. If \(\theta>0\), it expands and \(n\) decreases. Using the local volume relation from before, I can write this as
\[ \frac{\mathrm{d}}{\mathrm{d}\tau}(nV)=0. \]
The amount stays the same while its volume changes. Nothing particularly exotic has happened; I have given my spacetime substance the same conservation law I would give an ordinary fluid. Of course, that is also the problem! I gave it this rule. I have not found a conserved amount of spacetime hiding inside Einstein's equations. There is a conservation equation already present in GR,
\[ \nabla_\mu T^{\mu\nu}=0, \]
but this describes stress-energy conservation. It does not automatically provide a separate conserved substance current \(nu^\mu\). In particular, energy inside a moving parcel doesn't necessarily need to stay constant: pressure can do work as the parcel changes volume. So I now have a consistent way to describe the density of something conserved, as long as I supply the flow and an initial density. What I don't yet have is a reason to identify that something with spacetime itself. The equation tells me how my proposed density behaves, but not that nature owes me such a density.
Still, this gives me a useful distinction. “Density increases under compression” can be a consequence of a defined conservation law. Without that law, it was just a sentence I liked the sound of.
4. Or perhaps I mean relative compression?
Then there was the other route. Maybe I was just getting stuck because I kept trying to count some amount of substance. What if “denser” just meant that the same labelled region occupied less volume compared with a reference? Imagine drawing a grid on a piece of rubber before squeezing it. I do not need to count rubber particles to see that a particular square now occupies less area. I just compare it with its original area. Perhaps I could do something similar with spacetime, although the rubber has once again managed to make this sound easier than it is. The metric gives me a physical spacetime volume element:
\[ \varepsilon_g = \sqrt{|\det g_{\mu\nu}|}\, \mathrm{d}x^0\wedge\mathrm{d}x^1 \wedge\mathrm{d}x^2\wedge\mathrm{d}x^3. \]
For what I am doing here, the main point is that integrating \(\varepsilon_g\) over a region gives its spacetime four-volume. I am using \(x^0=ct\), so all four coordinates have units of length. Now suppose I also assign that region a reference volume element \(\Omega_0\). I can compare the two by writing
\[ \Omega_0=\chi\,\varepsilon_g. \]
Assuming they have the same orientation and units, \(\chi\) is a positive, dimensionless ratio. If \(\chi=2\), a sufficiently small region has twice as much reference four-volume as physical four-volume. That gives me a possible meaning of “compressed by a factor of two”. But here I've just moved the difficult question into \(\Omega_0\). Where did this reference come from? An earlier geometry? A hypothetical undeformed state? Some volume assigned to the labels? Those are possible choices, but GR does not generally hand me a preferred one. For this ratio to mean anything, both volume elements must refer to the same events, or I need an explicit rule identifying the reference events with the physical ones. The reference doesn't have to be an outside container, it could be another structure assigned to the same manifold. Once I have specified both, their ratio is independent of how I write the coordinates. But that does not make it independent of my choice of reference. I could choose \(\Omega_0=\varepsilon_g\) and get \(\chi=1\) everywhere. I could choose another reference and get another answer. So writing down a scalar has not magically removed the assumption.
There is also an important difference from the previous section. There, \(n\) measured an amount per spatial volume in the moving observer's frame. Here, \(\chi\) compares four-volumes, including time. A spatial volume becoming smaller does not, by itself, tell me how the corresponding spacetime four-volume compares with a reference. I also have to say which portion of the histories I am comparing. So this is a second possible interpretation of density: relative compression rather than conserved stuff per volume. It might be closer to what I originally imagined. But I still need to give it a reference, and a reason why that reference deserves to mean “uncompressed spacetime”.
5. Testing it out
Before getting too attached to either definition, I wanted to test the ideas that made me think about density in the first place. The obvious one was length contraction. If a moving object looks shorter, isn't that the same amount of stuff in less space? For an ordinary object, there's kind of something to that. Its length along the direction of motion is
\[ L=\frac{L_0}{\gamma}, \qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}, \]
where \(L_0\) is its rest length and \(v\) is its speed relative to the observer measuring \(L\). If it contains a conserved number of particles, an observer who sees its volume contracted can measure a greater particle number per spatial volume. But that's a density of the object. I still haven't shown that spacetime itself has become denser. The observers measure the endpoints at different sets of simultaneous events. In fact, a Lorentz transformation preserves spacetime four-volume. So a contracted spatial length doesn't automatically give me compression in the four-dimensional sense of the previous section. Time has a say in this too, which is inconvenient but probably fair considering I included it in the substance. Then there's gravity. For the exterior of a spherical, non-rotating mass, the Schwarzschild metric is
\[ \mathrm{d}s^2 = -f(r)c^2\mathrm{d}t^2 +\frac{\mathrm{d}r^2}{f(r)} +r^2\bigl(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2\bigr), \qquad f(r)=1-\frac{2GM}{c^2r}. \]
Here \(r\) is defined so that a sphere has area \(4\pi r^2\), and \(t\) is the time coordinate normalized to clocks far away. I'm considering stationary observers outside the horizon. Their clocks and local radial rulers satisfy
\[ \mathrm{d}\tau=\sqrt{f}\,\mathrm{d}t, \qquad \mathrm{d}\ell=\frac{\mathrm{d}r}{\sqrt{f}}. \]
So the clocks accumulate less proper time per unit coordinate time closer to the mass. But a fixed radial coordinate interval corresponds to more proper distance, not less. Alternatively, a fixed proper ruler spans a smaller coordinate interval. Those are different comparisons, and I can't just call both “space shrinking”. There's another, particularly annoying detail. With \(x^0=ct\), the temporal and radial factors cancel in the determinant:
\[ \det g_{\mu\nu}=-r^4\sin^2\theta. \]
Consequently,
\[ \varepsilon_g = r^2\sin\theta\, \mathrm{d}(ct)\wedge\mathrm{d}r \wedge\mathrm{d}\theta\wedge\mathrm{d}\phi. \]
That's also the expression for Minkowski four-volume in spherical coordinates. If I choose that flat-space volume element as my reference, identifying the same coordinate values, I get \(\chi=1\) throughout this exterior coordinate patch. This doesn't mean Schwarzschild spacetime is flat. It means the volume element alone doesn't describe its curvature. Nor does this coordinate comparison give me a uniquely preferred physical reference. But it does close the doors to the simple idea that slower clocks must mean more spacetime packed into each region.
So neither test lets me go straight from “a length or clock rate changed” to “the spacetime substance got denser”. I have to specify what I'm measuring, who's measuring it, and what I'm comparing it with. Density is annoying.
6. What I can say so far
Have I ended up further than I started? Maybe. I now have 3 weird, unfinished definitions for density.
First, there's \(n\): a conserved amount per unit spatial volume in the frame moving with the flow. This behaves like an ordinary substance density. Compression increases it because I've imposed a conservation law. The missing part is finding what that conserved amount actually is.
Second, there's \(\chi\): physical four-volume compared with an assigned reference four-volume. This doesn't require me to count anything, but it does require a reference. Without a reason to choose that reference, “denser” means denser relative to something I've supplied.
Third, there's energy density. For an observer with four-velocity \(u^\mu\), it's
\[ \varepsilon = \frac{1}{c^2}T_{\mu\nu}u^\mu u^\nu. \]
This one already has a physical meaning in GR. But it depends on the observer, and it isn't automatically a conserved amount per moving chunk. It also isn't the same as my reference-volume ratio. That brings me back to Stebbins' interpretation [1]. Where the stress-energy tensor selects a unique energy-rest frame, its energy density can be related through Einstein's equation to the Ricci part of the geometry. That's a nice way to give the fluid language a density without inventing another substance count. But it doesn't settle everything I was asking. Vacuum can have nonzero curvature even when the stress-energy tensor vanishes. So this energy density can't, by itself, measure every feature I might want to describe as a deformation of the spacetime substance. For now, I think the useful result is that I've stopped using one word for three different ideas. What I'm still trying to figure out is whether a particular choice helps me understand or calculate something, and whether GR chooses the structure it needs, or whether I'm forcing it in myself.
References
- Albert Stebbins. *A Space-Time Fluid (Unabridged)* (2026). https://arxiv.org/abs/2601.16996