The Infinity Paradox

In Astronomy by Brian Koberlein19 Comments

No matter what direction you look in the night sky, it looks basically the same. In astronomy terms we say the Universe is homogeneous and isotropic. Sure there are areas where galaxies cluster together, and other areas where galaxies are rare, but on average the distribution of stars is pretty even. Because of this, an early idea for the cosmos is that it is the same everywhere forever. It seems both ageless and infinite in expanse. But if that’s the case it raises a few troubling paradoxes. 

Olber argued the sky should be bright as the Sun. Credit: Wikipedia user Htkym

Olber argued the sky should be bright as the Sun. Credit: Wikipedia user Htkym

The first paradox is perhaps the most famous. Known as Olber’s paradox, it questions how an infinite ageless universe could be mostly dark. At first glance it might seem obvious. The more distant a star, the dimmer it appears, so stars very far away are simply too dim to be seen. But the apparent brightness of a star follows a specific relationship known as the inverse square law. A single star some distance away is as bright as four similar stars twice as distant, or nine three times farther away. But if stars are distributed fairly evenly, then there are four times the number of stars twice as far away, and nine times more that are three times away. So while stars appear dimmer with distance, there are more stars at greater distances. So an infinite ageless universe should have a sky as bright as the Sun.

Thermodynamics requires that your coffee and the Universe are getting cold.

Thermodynamics requires that your coffee and the Universe are getting cold.

On the other hand, Clausius’ paradox argues that the sky should be completely dark, with no stars in the sky at all. First postulated by Rudolf Clausius, the paradox is based upon thermodynamics. One of the basic laws of thermodynamics is that heat flows from hotter regions to colder regions until they equalize in temperature. In other words, your morning coffee will always cool down until it reaches room temperature. It will never spontaneously heat up by cooling the surrounding room slightly. According to thermodynamics, even the stars will eventually cool. In an ageless universe the stars should have faded long ago, and the vast cosmos should be a sea of completely uniform temperature. So why is the universe not cold and dark?

Even Einstein thought the Universe was static.

Even Einstein thought the Universe was static.

Of course you might argue that stars still shine because gravity causes clouds of gas and dust to collapse in on themselves. New stars are being formed all the time, so naturally the Universe won’t be completely dark. But this raises another paradox: why does gravity work at all? As with light, gravity obeys the inverse square law. An object some distance away pulls upon you gravitationally with a force four times larger than an object of the same mass twice as far away. With distance a gravitational force gets ever weaker, but it never completely goes away. In an infinite universe the amount of mass at a particular distance also follows the square law. For every gravitational pull in one direction, there will always be enough mass in the other direction to balance it out. This is known as Seeliger’s paradox, and it means that gravity shouldn’t be able to act on anything. Gravitational forces should always balance out, so stars shouldn’t form and planets shouldn’t orbit stars. And yet they do.

The solution to these paradoxes is pretty clear. The Universe is not ageless, nor is it stationary. We now know it is only about 13.8 billion years old, and ever expanding. Because of expansion and a finite age, the observable universe doesn’t extend to infinity, so Olber’s and Seeliger’s arguments don’t apply. Since the Universe is finite in age, Clausius’ argument is also invalid. It seems an obvious solution to us, but it’s an excellent example of how incorrect assumptions are difficult to overcome. Before Hubble’s observation of cosmic expansion, it seemed obvious that the Universe must be ageless and stationary. The idea that it might begin with a primordial fireball seems downright creationist in comparison. But in the end, evidence for the big bang became overwhelming, and the paradoxes of an infinite cosmos were finally solved.

Next time: Nothing can be colder than absolute zero, or can it? Consider an ancient cold white dwarf. It’s temperature is near absolute zero, but it’s matter is tightly squeezed by gravity. If you took a chuck of the white dwarf away, would that chunk expand and cool even further? Arthur Eddington wrestles with stellar thermodynamics in tomorrow’s post.


  1. Another interesting paradox in a both ageless and infinite universe is that everything that can happen will/has happened which means different versions of yourself and and other stuff like Boltzmann brains…

  2. It does seem clear that the Universe as we know it is 13.8 billion or so years old, and it is expanding at an accelerating rate. Now. Do we regard it as a closed system in the thermodynamic sense. That is, does the Universe contain all matter and all energy? If so, then how can it expand at an accelerating rate? The short answer is “Dark Energy”. But you mean to tell me that the quantity of dark energy is constant? To me, it seems another paradox, that makes me question the underlying assumptions.

    1. What a thermodynamic closed system is, in an infinite universe, isn’t so easy to get your head around. And it’s not helped by the fact that, if it’s infinite, we can see only a part of it that takes some hairy mathematics to show is not 0.000000000…% of the whole. The laws of thermodynamics – except perhaps the 0th law, which is more of a definition – can be shown to be what the ‘laws of physics’ (i.e. QM) produce when acting on a bazillion tiny pieces (i.e. atoms, electrons, etc). Add GR to the laws, and make the scope the whole universe, and the textbook laws need some, um, modification (or at least some caveats). If Dark Energy is indeed a static scalar (completely represented by lambda, the cosmological constant), then there’ll need to be some more tweaking.

      Thermodynamics, in other words, is a “less fundamental” law of physics than QM and GR.

      1. Many thanks for that. I’ve never seen the subject presented in quite that way.

      2. Thanks. That’s a much better explanation than my woolly “Add GR to the laws, and make the scope the whole universe, and the textbook laws need some, um, modification (or at least some caveats)”!

  3. The universe expanding is
    constantly ACCELERATING.
    They say that universe is expanding even faster than previously thought. That basically put the Big Bang theory totally unacceptable.
    Do you know that Astronomers face an embarrassing conundrum: they don’t know what 95% of the universe is made of. Atoms, which form everything we see around us, only account for a measly 5%.
    The percentage of our current knowledge of universe, even if we accepted all that is currently believed, the scientists agree that is less than 1%.
    The scientists don’t know nothing about dark matter…the same goes with antimatter, dark energy and so on…

    If you take all these facts, it’s really unprofessional or seriously stupid to say:
    “But in the end, evidence for the big bang became overwhelming, and the paradoxes of an infinite cosmos were finally solved”.

    1. Author

      Feel free to follow the links in the post that outline evidence for the big bang, or just search “big bang” on the website. You’ll find lots of posts outlining the evidence. I realize it may be easier to hide behind the pseudonym “Truth Hurts” and troll however.

    2. “The scientists don’t know nothing about dark matter…the same goes with antimatter, dark energy and so on…”

      Interesting sentence construction. Do you mean that the scientists (who, by the way?) know a great deal about dark matter (etc)? Or at least, they know a lot more than nothing?

      That’s true, of course, but is rather inconsistent with the rest of what you wrote. 😉

    3. Indeed, scientists don’t know what dark matter is. If they did, they’d stop.

  4. Well a closed thermodynamic system simply means there isn’t some other source adding energy, thereby decreasing entropy. Whether a system is finite or infinite doesn’t actually change how entropy works, other than perhaps the rate of change.

    N.B. Technically our universe would be an isolated system, since, as far as our observations tell us, neither matter nor energy can leave or enter our universe. Whether this remains true will obviously depend on our discoveries in frontier physics.

    1. This may seem like a nitpick, but I think it’s not.

      The parts of the universe which are ‘causally disconnected’ cannot influence each other, by definition. So at any particular location, it’s only parts of the universe causally connected to you that can enter any considerations of entropy, thermodynamics, etc. Given this, how is it possible to say anything meaningful about the entropy of the universe as a whole, or its thermodynamic state?

      The universal horizon – the edge of what’s causally connected – changes; sometimes it grows, sometimes it shrinks. How does thermodynamics work in such times?

      Of course, adding an assumption such as “homogeneous and isotropic” helps …

  5. The universe is both finite and infinite depending on how you look at it. Finite in that we live inside of a nuclear explosion, but infinite if we can break out of the vacuum of that explosion into the world that explosion is taking place. Billions of our years would be but a nano second in that universe. Our universe is expanding, but will eventually collapse. If you look at our universe from the outside it is a blinding flash, not the dark night sky we see from in here

  6. Here is another paradox. If our universe is infinite long then it must be infinite old because a star that is 1 000 000 light years away from us must exist at least 1 000 000 years.

  7. If the big bang did take place, and the universe is constantly expanding as some quote, observe, then by definition why are we not capable of observing, defining the central point or área from which everything is expanding away from??

    1. Author

      Because the Universe is expanding at all points of the cosmos, not expanding from a single point. There is no center of the Universe.

  8. One paradox of infinity I never see expressed is that between two finite points, there are an infinite number of divisions that can be made, similar to the elementary principle of calculus. Between two finite points, 0 and 1, there are an infinite number of even divisions, 0.01, 0.001, 0.0001, 1×10^-100….1000….one millionth…trillionth, etc, etc. The paradox questions how can something, whether it be distance, time, or whatever, be divided evenly an infinite number of times. And this can be applied to the real world, an atom can be divided into electrons, protons, neutrons, sub atomic particles, quarks, plancks, and theoretically strings. If mathematics suggests an infinite number of divisions, why does physical matter have a finite end point? or does it? This is the only true paradox of infinity in my opinion.

  9. Okay so the universe might of had a beginning but what came before the beginning of the universe? The universe might have an end but what comes after it?

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