# Is spacetime hyperbolic?

I have always seen answers to this point to measurements of how "flat" local geometry is, but my problem is, thats space not spacetime. I would say its safe to say that the geometry of space in our universe is indeed euclidean, but space isn't over time. The further I look, space should be less, the farther I go, space should be more. To me this seems as if spacetime is hyperbolic along the temporal axis, and that every "instance" cross-section is euclidean. So that ultimately is the question, is spacetime hyperbolic?

• Jan 10 '18 at 19:37
• Eh, they are on the same topic, but I think my question is prodding more at the time element of it, stressing that aspect. They probably have similar answers or foundations though. Jan 10 '18 at 19:48
• I think Pela's answer in the first question I linked is what you want. It says (in summary) that spacetime is, to our ability to measure, flat on the large scale. That applies to the time as well as the space. Jan 10 '18 at 20:33
• It seems you are saying that expansion should affect the geometry of space-time. It would be interesting to see an explanation for why that is not the case (if that is so). Jan 10 '18 at 20:42
• You seem to have a misunderstanding about the geometry of space-time. When astronomers/physicists say our universe is flat, they're talking about space-time, not just space. When discussing the curvature of the universe, space and time are inextricable. It only makes sense to talk about curvature within a given space-time metric. You can't have a space metric alone and have physics (i.e., GR) make sense. Jan 12 '18 at 14:11

There are three kinds of different hyperbolic in this question and the previous answer by AtmosphericPrisonEscape: the behavior of Lorentz transformations, the spatial hypersurfaces in cosmology (which seem to be flat), and the hyperbolic appearance of spacetime when tracing world lines in a universe with accelerated expansion.

TL;DR: Spacetime is expanding exponentially and if you choose an unconventional hypersurface, that hypersurface is hyperbolic.

Lorentz transformations: Lorentz covariance is the mathematical expression of Einstein's principle of the constancy of the speed of light. Changing the frame of reference of an observer is done by a Lorentz transformation which preserves the light cone and is closely related to the concept of causality. The answer to this question explains this for the Minkowski space, where a Lorentz transform is a hyperbolic rotation.

In this flat (Minkowskian) spacetime:

$$ds^2 = c\, dt^2 - dx^2$$

you can use a Lorentz transform to switch to different coordinates $dt$ and $dx$ while the line element $ds$ is invariant, analogous to the hyperbola

$$1 = a^2 - b^2.$$

This kind of hyperbolic is not the point of the question.

Cosmology: If we assume large scale homogeneity and isotropy (Copernican principle), we can choose isochronous coordinates. There you have a synchronous time $t$ and spatial hypersurfaces for given $t$ ("time slices"). Freely falling observers are comoving, i.e. have fixed spatial coordinates $\mathbf x$. Their world lines are perpendicular to the spatial hypersurfaces and their proper time matches the synchronous time $\tau = t$. Galaxy clusters are such freely falling observers.

If you take these assumptions and allow time dependence of spatial coordinates on time, you end up with the FLRW metric:

$$ds^2 = c\, dt^2 - a(t)^2 d\Sigma^2$$

where $d\Sigma^2$ describes the spatial hypersurface. Now additionally to this being a hyperbolic equation, the hypersurfaces $d\Sigma^2$ can have curvature, especially constant positive (closed sphere), zero (Euclidean), or constant negative (open hyperboloid) curvature.

A flat universe is, for example, given by

$$d\Sigma^2 = dr^2 + r^2 d\Omega,\\ \text{where}\quad d\Omega = d\theta^2 + \sin(\theta)^2\, d\phi^2,$$

which is the metric of Euclidean 3-space in spherical coordinates. Here the only relative acceleration between freely falling observers comes from the scale factor $a(t)$.

A hyperbolic spacetime could be

$$d\Sigma^2 = dr^2 + \sinh(r)^2 d\Omega$$

which is a hyperbolic plane in spherical coordinates. In such a spacetime, freely falling observers do not only measure a relative acceleration due to the scale factor, but an additional contribution due to the curvature of space.

Expansion: In the FLRW metric you see the scale factor $a(t)$. The Friedmann equations describe the evolution of the scale factor. It does not influence the qualitative shape of the spatial hypersurfaces, but in an expanding space their curvature becomes more and more flat (just like the surface of a sphere locally becomes more and more flat as the sphere expands).

A flat $d\Sigma$ and a constant $a$ gives you no acceleration between freely falling observers whatsoever. An $a$ with constant change manifests as all observers flying apart, but no relative acceleration. Finally, an $a$ with constant acceleration makes everything fly apart ever faster (currently observed). The Friedmann equations tell us, that the behaviour of $a$ changed during the lifetime of the universe as different effects where dominating (radiation, matter, cosmological constant).

Currently we observe an accelerating universe, so $a$ is exponentially increasing (just at the right end of the illustration). If you look at the world line of two Galaxies, their physical distance increases exponentially. Therefore the circumference of a chunk of space increases exponentially, so the hypersurface spanned by a line of freely falling observers is actually hyperbolic (white grid in the illustration). However, I struggle to put that into an equation that clearly shows the hyperbolicity, so don't trust that too much, yet.

• I would not want to create a new question for this, as it is mostly a continuation off of this, but has it been determined if spacetime is accelerating exponentially or asymptotically? Since I would assume that an asymptotic acceleration would not only mean that it isn't hyperbolic (I don't know exactly what that would be) but also that the universe is finite along the temporal axis? Mar 23 '18 at 0:26
• 1. The expansion doesn't seem to be bounded, i.e. it's not asymptotic. If it had another composition of radiation/matter/Λ, our ΛCDM "standard model" could predict a non-accelerating universe at some stage. Yes, then the white grid is not a hyperbolic surface anymore. 2. Be careful about the word "finite": The expansion has nothing to do with the finiteness of the universe itself :-) Mar 31 '18 at 11:24
• I kinda was referring to if the universe was expanding asymptotically that at some point time would essentially disappear after a finite amount of time, because it would change at a rate of infinity after 0 seconds... Effectively isolating time from 0 to this asymptote. Interesting that it isn't asymptotic, meaning we are dealing with a consistent rate of expansion? Like a force or a "force" like gravity? Apr 27 '18 at 2:22
• I still don't understand what you mean by "time disappearing", because in the model described above it is actually ticking away at the same constant rate everywhere. You might want to read more about the cosmological constant Λ which drives the expansion in this model. Apr 27 '18 at 10:03

As OP still didn't react to the posted answers, I'll try my shot at this.

First of all, I'm not sure what you mean by hyperbolic. A simple Hyperbola $f(x,y) \in R^2$ can be defined by a relation

$$x^2 - y^2 = a^2$$ with some constant $a^2$. One can plot this.

Now the simple Minkowski spacetime metrics from special relativity states that there is a quantity $ds^2$ which stays invariant if an object is observed from different points in spacetime. This invariant quantity can be locally computed from
$$ds^2 = c^2 dt^2 - dx^2$$ which is a hyperbolic relation and also results from general relativity (the more general variant of special relativity) if one imposes a flat universe.

Now Planck-observations show our universe to be approximately flat, expect for the ripples here and there. By ripples I mean sources of gravity, near which the relation between $dt$ and $dx$ becomes more complicated, and in particular non-hyperbolic.

So if you want to make any statement about 'spacetime', you can only do that about the 'spacetime of our current idea of the universe on a global scale'.

• The spacial measurements might be euclidean but if there is more space over more time then isn't that a hyperbolic relation and thus non-euclidean? (looking at it in a 4 dimensional perspective rather than just a cross section of it) Jan 16 '18 at 18:21
• @Terran: "but if there is more space over more time" is not a statement that makes any sense. "(looking at it in a 4 dimensional perspective rather than just a cross section of it" is exactly what we're doing in relativity, and this post is a simplification of it, but as written, correct on the grand scale of the universe. Thus there can't be 'more space than time'. Relativity is btw. the '4-dimensional perspective' you're looking for. Maybe you should start there, if you're not sure what my post means. Jan 16 '18 at 23:21
• If time expanded at the same rate as space, they would counter-act (more distance to cover but you can do it in less time, canceling itself). But the universe is certainly expanding, space is expanding, that would mean that space has a hyperbolic relationship over time. I am aware of relativity, maybe not completely fluent in minkowski space and lorentz transformations though. Unless time expanding would cause the global time rate to slow down? Expanding time is something I am unsure of regarding implications (does that mean the matter timeline function also expands, inside or outside, idk)... Jan 17 '18 at 1:24
• @Terran: Look, expansion and flatness are two different things. Please take a look at en.wikipedia.org/wiki/Friedmann_equations , and particularly the part 'assumptions'. The Friedmann model poses the basis for modern cosmology, and the increase of the parameter $a(t)$ is usually interpreted as universal expansion. However, the solution for $a(t)$ remains finite and positive, so the global metric $ds^2$ remains hyperbolic in nature, at all times, for all solutions of this model. Outside of this model the words 'spacetime' or 'expansion' have no meaning. Jan 17 '18 at 1:48
• Ah... So the Friedmann equations are referencing spacial geometry, which is nearly flat, but not spacetime geometry, which we are apparently hyperbolic? But it seems the equations are using parameters for determining spacial geometry over time. There also seems to be an Omega density variable, is this density a three dimensional density, or four dimensional density? (Not placing time in with space, as mass stays constant over time with time) Because spatial geometry is heavily dependent on time, the scale and topology of space can readily be affected by the topology of time :/ Jan 17 '18 at 19:04