We human beings have difficulties with understanding infinity in a variety of contexts, not only in pure mathematics. One such context is the development of the universe on big scales.
Many more people are interested in questions about the universe than the number of people exercising mathematics. So, it is no wonder that there are periodic waves of popular science series on TV, covering topics in astrophysics and cosmology. The problem is an adequate translation of related mathematical ideas into a kind of daily human language. And when things get complex, we use analogies to explain processes in the cosmos. The analogies employ simplifying models in math and physics we are better used to from our standard perception of reality.
One science series which people follow e.g. in the US, Australia, Norway and Germany is “How the universe works”. During episodes on a huge variety of topics active astrophysicists really try their best to provide an insight into cosmological concepts on a level that non-physicists can understand.
However, their statements often are presented together with illustrating movies. And some of these movies visualize analogies in ways which, unfortunately, can lead to misunderstandings. One example is the presentation of some exploding clouds to illustrate the Big Bang. In particular these little movies during talks about the universe trigger thoughts and analogies which sometimes obscure more than they clarify.
Editorial Note: The post was revised on 08/29/25 and supplemented with two versions of the cosmological principle, which are more carefully formulated regarding the assumptions on isotropy and homogeneity.
Misunderstandings
During a limited period in my life, I had the privilege to work in astrophysics (multi-dimensional gas dynamics, stellar core collapse, supernovae type II, gravitational waves, neutrino emission). So, it is in a way natural that friends sometimes ask me about topics in astrophysics and cosmology they got interested in. Some months ago, I had such a discussion with an interested friend, an engineer, about the cosmos and its development.
After some glasses of whine during a late dinner, my friend suddenly confronted me with a statement saying that the universe could not be infinite because it came into existence by an explosion-like event, namely the “Big Bang”. And as everything was concentrated within a point-like volume in space at the beginning of cosmic time, and because nothing can move faster than light, the universe must be finite. Even more general: Everything that starts limited must remain limited. During our discussion my friend often referred to the short movies about the Big Bang mentioned above and to other contents of the named popular science series.
Educated physicists would in such a situation probably try to point out multiple erroneous assumptions in the statements and the argumentation of my friend. I tried, too. However, I failed to convince him within the limited time of our discussion. He simply could not accept that an infinite universe actually is possible and that such a universe would not contradict an ongoing evolution from a Big Bang.
He again and again expressed his belief that all physical things are finite. Only, when I told him that his assumption about “everything having been concentrated in a point-like volume” would actually mean an infinite density of a singularity, he got a bit calmer. But directly jumped to yet another “conclusion”, namely that the initial volume of energy concentration must have been finite, then, due to whatever reasons.
Ok, sometimes one cannot break through a wall of “natural”, so called common sense ideas. At least not in a first attempt.
The whole event left me baffled for a while. While the question whether the universe is infinite or not can well be discussed controversially, it is quite a different thing to categorically exclude an infinite universe based on the above assumptions. Although the arguments of my friend have an intrinsic logic, they are based on some misunderstandings resulting, at least in parts, also from a too simple interpretation of popular science movies about the evolution of the cosmos.
An attempt to remove some of typical, but misleading ideas
Despite my failure to explain some complicated things in a discussion late at night and after a consumption of probably too much wine, I still feel an obligation to correct some of the misunderstandings of my friend in a more structured way. The reason is that I am deeply convinced that some small new insights can open up a wide window for a different view on our world and some new fruitful thoughts about our existence in it.
I think the whole topic is not only interesting for educated and mathematically interested students who have passed a basic course in special and general relativity. The problem is that we do need some basic math. But my friend is an engineer – and therefore I will give math a chance. But as soon as we have derived two equations – the so called Friedmann equations – which describe a symmetric overall evolution of the universe, some further aspects of cosmology like scaling factors, different types of horizons, red-shift dependencies and growing distances in the cosmic flow become relatively handsome.
By looking closely into the meaning and interpretation of some variables governing the equations, a reader – in my opinion – can already correct some of the misleading concepts that may have formed the ideas of an allegedly necessarily finite cosmos which my friend had got in his mind. So, I have just decided to write a post series on this topic – hoping that my friend and other interested people may find the time to read it. But: You should not be afraid of some math.
I will try to keep the required math at the lowest possible level. I will also refer to some articles on the Internet, books and movies, which I find instructive. Otherwise, I will try to trigger some insights of my readers by asking them some questions, which may help to find a new perspective.
Note that in the end I will not dare to say whether the universe is finite or not. I just want to make clear that the assumptions of my friend are partially wrong and/or insufficient to decide the question of a finite or infinite cosmos.
A starting point – the cosmological principle
We have to start somewhere. One basic assumption in cosmology is that the part of the cosmos we are able to observe provides us with a representative sample for the general conditions throughout the whole cosmos.
This idea includes two sub-assumptions, for which we indeed have some (though limited) evidence from observations:
- The universe appears to be homogeneous on large scales – with respect to the distribution of matter and energy in space. Meaning: On large scales we can work with averaged quantities and disregard the impact of local fluctuations on the overall conditions and the development of the cosmos. We assume that the universe indeed is homogeneous on large scales.
- The universe appears to be isotropic from our point of view regarding its structure and matter/energy distributions on large scales. The variation of the general relative motion of objects along radial trajectories indicates that this isotropy also holds for other distant observers. We therefore assume that the universe indeed is uniformly isotropic – at least if the observers choose a proper frame of reference and a respective coordinate system to describe their observations.
The term “uniformly” indicates that the view and measurements of any observer within the universe regarding its large scale structure should reproduce the same results everywhere and in every direction of space. The combination of the two generalizations above is a certain version of the called “cosmological principle”.
Cosmological principle – two major versions
Actually, there are different versions of the cosmological principle [CP]. The differences are subtle. One reason to be careful is the point with isotropy. E.g., a given homogeneous vector field is not isotropic in a general (moving) frame of reference. Also complex radially symmetric vector fields may not guarantee isotropy for distant observers. However, there may exist certain coordinate systems for which we get isotropy for certain vector fields.
The other point is the scale on which we discuss the matter and energy distribution in space. The scales have to be big enough to be able to work with averaged quantities ad forget about local fluctuations.
So you could rightfully feel that we should use more careful formulations – even if we have strong observational indications for the two points named above:
- CP – careful version I: On big enough spatial scales the properties of the universe look the same for all observers – under certain equal conditions.
- CP – careful version II: There exist coordinate systems in which – on big enough spatial scales – the averaged properties of the universe are seen and measured to be spatially homogeneous and isotropic.
Version I implies some restrictions which we may have to make. A very intuitive one is that our observers look at the universe at the same stage of its development. This implies the use of some common cosmic “time scale”. Regarding version II, we will see in future posts that in particular for a cosmology based on the Theory of General Relativity, the question of choosing a proper coordinate system to fulfill the conditions of homogeneity and isotropy is a very relevant one. Assumptions about a maximal symmetry of matter/energy distributions in space imply observer coordinate systems which are co-moving with the cosmic substrate.
In my opinion, both versions of the cosmological principle are based on reasonable assertions which, however, are difficult to prove, as we, as a matter of fact, can not look into all parts of the universe. You may therefore not like them. Despite the observational data for a part of the universe’s space-time whose extensions are measured in billions of light years.
On the other side: Why should our observable corner in a vast universe be something special?
Anyway, the whole point for our subject is to show that one can develop a consistent theory of a dynamic evolution of the cosmos – based on some version of the cosmological principle and taking into account cosmological observations – without necessarily having to exclude the possibility that the universe is infinite.
By the way: At some point in our discussion we must define more precisely what the word “infinite” refers to.
Some aspects of the cosmological principle to dwell upon
Some words of caution and some hints for your personal reception of and reflections upon the cosmological principle until the next post:
(1) The cosmological principle does not explicitly exclude a dynamic development of the matter/energy distribution in space with some (global = cosmic?) “time” coordinate. Or does it? Indirectly?
(2) Take an elastic plain (= flat) table cloth. Mark positions on this cloth by coordinate lines of a 2-dimensional Cartesian coordinate system. Assume objects sitting at the crossings of equidistant coordinate lines in x- and y-direction. Assume in addition an observer sitting at a specific crossing of two selected x- and y-coordinate lines – let us say at a point (x,y) with x=3, y=3. Now stretch the cloth uniformly, i.e. both in x- and y-direction by the same factor of 2. By what amount does the distance between observers sitting at points (3,3) and (4,4) change? How does the distance between observers sitting at points (3,3) and (9,9) change?
Stretching a 2-dimensional table cloth “universe”
(3) In what sense did the stretching of the cloth lead to a relative flow of objects versus each other? And versus one of our observers?
(4) Regarding the position of an observer in our scenario of point (2) moving together with the marked coordinate points on our table cloth. Did the coordinates of the observer change during the stretching? If not, what did actually change?
(5) If the table cloth were a part of a 2-dimensional (infinite) universe – would a continuous stretching with time be compatible with a 2-dimensional cosmological principle?
(6) In what sense could space itself be something like an elastic substance which changes its its fabric, its internal properties, its geometrical nature dynamically – without violating the cosmological principle?
(7) Taking your answers to questions (2) up to (6) into account: What would one mean by “passively following the overall flow of objects” through the dynamic evolution of our 2-dimensional table cloth cosmos if it got stretched continuously?
(8) The cosmological principle refers to an homogeneous and isotropic matter/energy distribution in space. So assuming some, yet unknown dynamics, the cosmological principle must hold at any given point in “time” during such a potential dynamic evolution. The cosmological principle refers to the cosmos’ structural appearance in space, but in the given form it only excludes an asymmetric development on large scales. However, the densities could continuously and uniformly sink by a time-dependent, but location independent factor everywhere. True?
(9) If we accept the possibility of an overall dynamic evolution, the cosmological principle already implies a kind of universal time like parameter controlling the evolution of the matter/energy distribution in space (1) in the same way, (2) everywhere. As a consequence the evolution of the universe must look the same everywhere for observers passively following the evolution of matter/energy and a resulting relative flow of objects.
(10) If you are familiar with special relativity, you may rightfully ask: The “time” of which observer? How could the cosmological principle help regarding this question? What about a kind of universal time scale experienced locally in the same way by every observer passively participating in the general cosmic flow of energy and matter – with this flow following an evolution being the same everywhere in space.
(11) You may have heard that matter/energy-distribution determines a so called metric of spacetime, i.e. a defined way of how infinitesimal coordinate differences determine the distance between two infinitesimal close points. So, if the cosmological principle is true, then must we not, at the same time, postulate a universal metric following the same evolution everywhere and only depending on a global time-like parameter? There can not be any dependence of the metric on space coordinates. True?
(12) When we look at large distances we automatically look back in time as the velocity of light is finite. Effects of an assumed or observed dynamic cosmological evolution must therefore be shown to be consistent with the cosmological principle at any point on the assumed global “time” coordinate axis.
(13) The cosmological principle restricts the structure and coordinate dependence of dynamic changes. You may try to get this insight by experimenting with certain assumed laws for changing distances between objects with a universal time coordinate and a dependency on present distances on our table cloth universe.
(14) By assuming that the cosmological principle is true at every time of a potential cosmological evolution we are actually invited to the simple idea that such an evolution may be described by equations which just re-scale the uniform matter and energy density distribution everywhere by the same scaling factor evolving according to a dependence on a universal time-like parameter, only.
(15) You may rightfully ask whether we did not introduce infinity already via a backdoor with our assumptions of homogeneity and isotropy “everywhere” in the universe.
Would a finite universe not automatically imply a kind of “border” that breaks homogeneity and/or isotropy in space?
Well this is a Newtonian view, which assumes a flat universe at any point in “time” with a standard metric and a finite amount of matter concentrated at a specific corner of the otherwise large and matter free universe. A preliminary answer, which we will come back to later on, is: The assumptions of the cosmological principle do not contradict a finite universe, if space-time and the 3-dimensional space itself have a curvature. The standard analogy is that of a 2-dimensional observer on the surface of an expanding sphere: The matter distribution on such a sphere may look the same (homogeneous and isotropic in any direction along “straight” lines on the surface) from every point – although the surface is finite and may nevertheless be evolving.
However, on the other side, the cosmological principle does e.g. not exclude a homogeneous matter/energy distribution throughout an infinitely large flat space remapping its geometrical structure all the time. True?
(16) Would you describe the dynamics of our table cloth experiment as an explosion or an expansion? How would you characterize the difference?
Summary
The cosmological principle alone raises as series of interesting questions. It does not seem to exclude a dynamic evolution, if such an evolution occurred in the same way everywhere in the cosmos. It excludes neither a finite nor an infinite space. It also appears that a universe which continuously re-scales its matter/energy density distribution depending on a “time”-parameter would be compatible with the cosmological principle.
In the next post of this series
I will try to motivate the structure of the so called Friedmann equations by purely (and thus questionable) Newtonian arguments.