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 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 by examples we are better used to from our standard perception of reality.
One such 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.
Misunderstandings
I had a limited period in my life when 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. 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 in 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 claimed that he was in general convinced 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 mean an infinite density of a singularity, he got a bit calmer. But directly jumped to yet another “conclusion”, namely that the initial volume 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 with an accompanying 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. However, the problem is that we 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 aspects as scaling factors and deriving 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, one – 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 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. We assume that the universe indeed is homogeneous on large scales.
- The universe appears to be uniformly isotropic regarding its structure and matter/energy distributions on large scales. We assume that it is indeed uniformly isotropic.
The word 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 called “cosmological principle”.
In my opinion, this principle is based on reasonable assertions which, however, are difficult to prove as we, as a matter of fact, cannot look into all parts of the universe. You may therefore not like them. But on the other side: Why should our observable corner in a vast universe be something special?
The whole point for our subject is anyway to show that one can develop a consistent theory of a dynamic evolution of the cosmos – based on the cosmological principle and reflecting 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.