
Olbers’ Paradox: Why Is the Sky Dark at Night? (2000) - fauria
http://www.amnh.org/education/resources/rfl/web/essaybooks/cosmic/cs_paradox.html
======
snowwrestler
There is one more complimentary answer, which is that the night sky is not as
dark as it seems, but our eyes cannot see all the wavelengths of light that
populate the night sky.

If we could see deep into the infrared, we might see more faraway stars and
galaxies, whose light has been red-shifted by the expansion of space.

If we could see wavelengths in the single cm range, we might perceive a dim
uniform glow in the night sky: the cosmic background radiation.

If we could see radio waves we might see all sorts of light sources in the
night sky.

~~~
paulddraper
Yep, explained here: [http://www.physics.org/facts/sand-
dark.asp](http://www.physics.org/facts/sand-dark.asp)

Of course, if there were an infinite amount of light (visible or not) reaching
Earth, you'd be dead.

~~~
archgoon
Suppose that the Energy distribution for a certain amount of light (as seen by
earth) was given by

    
    
              /- 1 if f < f0  
       E(f) = >
              \- 1/f^2 if f > f0
    

Then we would have light "at all frequencies" which could be said to be "an
infiinite amount of light" but the total amount of energy would still be
finite.

Most of these explanations don't seem to first figure out whether if you had a
uniform distribution of stars (at discrete points) in an infinite (Euclidean)
space, if the amount of radiated energy seen at an arbitrary point would
diverge if you summed over all available frequencies. If it doesn't, it seems
like lack of sensitivity is the main issue here, and the cosmological
arguments are simply additional reasons for why it doesn't happen.

~~~
wfn
Pity someone down-voted you.

From what I take it, you were basically simply asking, "what if the 'infinite
amount' at 'all frequencies' were reaching you in a non-uniform distribution
of frequencies? - would the original arguments and premises still stand?"

------
dantillberg
A mathematical explanation would involve a comparison of the density of stars
throughout the universe with the inverse-square drop-off in light intensity
from each.

If you assume for the sake of argument that all stars are as bright as the
sun, and that stars have a uniform density throughout the universe (either a
finite-but-huge or infinite universe, it makes little difference here), then
you could solve for the mean intensity of light throughout, or in our case,
"how bright the sky is at night."

The article makes a classic (and really hard to escape! I've seen this sort of
thing called a "paradox" on many occasions.) mistake of examining two infinite
quantities (infinite stars, up to an infinite distance away) and assuming that
they must cancel each other out, that infinity/infinity=1. But not all
infinities are the same, and infinity/infinity can work out to 0, or to 0.1,
or to pi, or to infinity (as in the case of "# real numbers"/"# integers"), or
to anything else; you have to look at the details -- what kind of infinity is
it? -- and figure out how they compare.

~~~
jakobegger
The article makes no mistake, it just doesn't go into enough detail.

To determine the amount of energy reaching us, we need to integrate over the
energy density divided by the square of the distance over the whole 3d space.
As any physicist can tell you, an integral over the whole space will diverge
unless the kernel falls off faster than 1/r^2.

This means that one of the following must be true

a) either the universe is finite, ie. the density falls off

b) we don't integrate over the whole space (we can't see far enough due to age
of universe)

Which is exactly the argument this article is making

Edit: in other words, infinite or finite-but-huge is an important difference
when integrating over constant density; the former integral will diverge, the
latter will not

------
goodside
"Astronomers have concluded that the universe began some 12 to 15 billion
years ago. That means we can only see the part of it that lies within 12 to 15
billion light-years from us."

This is false, because it ignores the metric expansion of spacetime. We can
see radiation from objects that are now ~46.5 billion light-years from Earth.

This isn't picking nits. The explanation given here makes it sound like the
Universe is a Euclidean volume that got instantaneously filled with stars, at
roughly their present density, and that this density is low enough that light
from distant stars hasn't reached us yet. If this were true, it wouldn't only
explain why the sky is dark now, but predict that it has always been dark. We
know this is false: for the first 380,000 years after the Big Bang, the
Universe was filled with hydrogen plasma and completely opaque to radiation,
which we can still see in the form of CMBR. As spacetime expands, energy
density decreases, space becomes less light-emitting and more transparent, and
light already in transit becomes red-shifted. Our field of vision is still
technically flooded with light, as it always has been, but by now the red-
shift is so extreme that the light is in the radio spectrum.

~~~
ank_the_elder
But how can something that is further away than the maximum travel distance
allowed by the time elapsed since the big bang exist? Is there a non-locality
of the time dimension that makes it so? Was the expansion rate of the universe
faster than the speed of light? Or... what?

~~~
goodside
[https://en.wikipedia.org/wiki/Metric_expansion_of_space](https://en.wikipedia.org/wiki/Metric_expansion_of_space)

The short version: The distant object wasn't traveling through space, it sat
stationary while more space came into existence in the gap between us and it,
resulting in it being farther away. This happens everywhere, all the time --
it's what cosmologists are talking about when they say the Universe is
"expanding". The laws that prohibit faster-than-light travel concern movement
within space ("peculiar motion"), not the expansion of space itself.

------
rdiddly
Subsequent edit: Assuming the universe is infinite, is where the problem
begins. There is no reason to assume that.

Original comment: Is this really not obvious to people? It's about subtended
angles - simple geometry. As the Earth's distance from a star increases, it
subtends a decreasing angle in that star's "sky." Assuming radiation does
indeed proceed radially and in straight lines, at increasing distance the
Earth is capable of catching a decreasing portion of that star's radiation. As
the distance approaches infinity, the subtended angle approaches zero and the
amount of light reaching us approaches zero. The star is just as bright, but
most of its radiation goes somewhere else besides Earth.

Imagine building a giant spherical shell around a star. Its radius equals the
distance between earth and the star. It catches 100% of that star's radiated
energy. Now paint the Earth on the inside of the shell, actual size. It looks,
to someone standing on the surface of that star, like the real earth, i.e. a
tiny dot. Now divide the surface area of the dot, by the surface area of the
sphere (4)(pi)(r^2). That's how much of that star's energy our planet catches.
Assuming zero degradation across space.

So imagine you're in a forest of trees the closest of which are 800 miles away
from you. You'd see the goddamned sky just fine.

~~~
pdonis
_> As the Earth's distance from a star increases, it subtends a decreasing
angle in that star's "sky." _

But the number of stars at a given distance increases as the distance
increases, and this increase exactly cancels out the decrease in the amount of
light from a given star that reaches the Earth. So your argument does not
explain why the night sky is dark.

 _> Now divide the surface area of the dot, by the surface area of the sphere
(4)(pi)(r^2). That's how much of that star's energy our planet catches._

Now do it the other way: draw a sphere around the Earth with some radius r.
Calculate how many stars are on that sphere. The number increases as r^2; and,
as you have shown, the amount of light reaching Earth from a given star
decreases as r^2. So, as I said above, the increase in the number of stars
exactly cancels out the decrease in the amount of light from each star that
reaches the Earth; thus, the total starlight reaching Earth from a given
radius r is a constant, independent of r.

~~~
rdiddly
There's no reason to believe the universe is infinite nor that stars are
evenly distributed throughout its volume. So "the number of stars increases as
r^2" is the part that doesn't hold and is causing the paradox.

~~~
mannykannot
This is Kepler's theory, and it gets into philosophical trouble, as
cosmologists generally take it as axiomatic that the laws of physics are the
same everywhere, and so don't have much time for theories that depend on us
being in a special place in the universe. A universe having a lower density of
stars everywhere above some radius from Earth does put us in a special place,
is hard to reconcile with universal laws, and there is no empirical evidence
for it. An explanation that derives from the universe evolving over time,
however, has none of the philosophical problems and does have supporting
empirical evidence.

------
VeilEm
This didn't mention the expansion of space at all. The universe may be
infinite in size but we'll never see the light of stars that exist so far from
us that they are beyond the distance at which space expands faster than light
can travel. That's why we have a dark sky and it will just get darker as more
things we see now eventually expand beyond that barrier. One suggested end to
the universe is the Big Rip in which all matter eventually separates in this
way such that the distance between all matter is infinite:

[https://en.wikipedia.org/wiki/Big_Rip](https://en.wikipedia.org/wiki/Big_Rip)

~~~
paulddraper
Yep. Currently, some galaxies are traveling away from us faster than light.

[http://curious.astro.cornell.edu/about-us/104-the-
universe/c...](http://curious.astro.cornell.edu/about-us/104-the-
universe/cosmology-and-the-big-bang/expansion-of-the-universe/616-is-the-
universe-expanding-faster-than-the-speed-of-light-intermediate)

~~~
joering2
That blew my mind! So now it turns out Einstein was wrong??

We can't get particle to seriously travel faster than light and there is
entire galaxy going faster??

~~~
joehilton
I've also had a long-time question around this point. I've asked various
university physics professors, and they all stutter and disagree (which means
either we really don't know or I'm going to all the wrong universities). The
underlying question is: When we say in general relativity that particles can't
travel faster than light, what is that speed measured relative to?

If it's photons emitting from flashlights pointed in opposite directions
relative to each other, then are they compressing space as they travel so as
not to exceed the speed of light? Or even massive particles accelerated to
over half the speed of light at different times around an ellipse so there is
a moment when their opposite directions make one traveling at greater than the
speed of light relative to the other? (Or even particle accelerators on two
different planets already expanding away from each other - does this just mean
that space is really twisting and turning all the time to make sure no
particle ever exceeds the speed of light relative to any other particle?)

It has been my understanding - and appreciation at Einstein's unbelievable
insight and brilliance - that relativity is truly relative because there
really isn't any such thing as an arbitrary particle or space that all other
motions are measured from. But if this is true, does that really mean that
space is really so constantly twisting and turning that SOL can't be exceeded?

I'm sure there are great answers to this, but I've always wanted a definitive
one. Anyone have any comments on this?

~~~
pdonis
_> When we say in general relativity that particles can't travel faster than
light, what is that speed measured relative to?_

Relative to a local inertial frame, i.e., relative to an inertial observer
(i.e., an observer in free fall, feeling no force) who is at the same spatial
location as the particle whose speed is being measured. That is the only
context in which the concept of "relative velocity", as it appears in the
"can't travel faster than light" condition, has any physical meaning.

All of your suggested examples attempt to compare "relative velocity" between
objects that are not at the same spatial location. That has no physical
meaning in GR.

------
jcoffland
One answer to this problem is that the night sky isn't dark at all. It's just
less bright than during the day. Hear me out. As a species that evolved on
this planet under it's particular conditions we perceive day to be bright and
night to be dark because of the particular range of light sensitivity of our
eyes. Too often science leaves the human observer out of the equation. It lets
us feel like gods presiding over our universe but is a big mistake. All that
said it's not unreasonable to question why the night sky is not brighter than
it is.

~~~
alanwatts
>Too often science leaves the human observer out of the equation.

It seems this problem may have come full circle with the Measurement Problem.
But yes, human beings themselves emit light, however that light resonates on a
frequency that our eyes have not evolved to see, so we invented infrared
optics.

------
paulddraper
This is a great explanation, up until the last part, which is slightly
misguided.

"So even if the universe were infinitely old as well as infinitely large, it
would not contain enough fuel to keep the stars shining forever and to fill up
all of space with starlight."

If the universe were infinitely old, fuel was consumed, and fuel had finite
density, then the sky would be completely dark. Because there would be no
starts.

The author is really just thinking "It would require infinite time to receive
light from infinitely far away." Which is the relevant answer to the paradox.

------
grondilu
I've always been puzzled by this paradox. More particularly, I'm baffled that
apparently the mainstream explanation involves cosmological arguments such
that the expansion of the Universe. Is it really that complicated?

Take Hubble's deep field image, for instance. It was shot in a tiny area of
the sky that looks completely black. But with enough exposure, Hubble found
that it's actually full of galaxies. Can't I just conclude that if this part
of the sky if completely black, it's not because space is expending, but just
because our eyes are not sensitive enough?

Olber's Paradox points out that wherever we look at, at some distance our line
of sight will meet the surface of a star. But that does not mean that this
point will register as a light for our eyes. Our eyes would have to be
sensitive enough.

~~~
nostromo
Imagine Hubble's deep field image again, but this time imagine you can see
towards infinity (beyond the observable universe). If the universe is
infinitely big (and old), then there would be so many stars in the image that
there would be an infinite amount of light in the image.

In this hypothetical universe, the Sun would actually be a dark spot in the
sky -- because it would be blocking the infinite amount of light coming from
an infinite amount of stars behind it.

That's the paradox -- that we do not see that. No matter how sensitive our
eyes are, we would be able to see infinite light. (Actually, our eyes would be
melted... and all current forms of life impossible... but you get the
picture.)

~~~
grondilu
But stars are getting fainter and fainter as they are further and further
away. I get that there would be an infinite amount of them, but then it's an
infinite times a zero. The limit is not obvious and I've never seen anyone
discussing the actual maths of this. I don't know those maths, but it seems
reasonable to me that since the sky is indeed black, then it has to be that
it's because the dimness of stars prevails over their number and/or density.

~~~
nostromo
> But stars are getting fainter and fainter as they are further and further
> away. I get that there would be an infinite amount of them, but then it's an
> infinite times a zero.

No, it never goes to zero, it just gets fainter. An infinite number of small
numbers adds up to infinity.

> since the sky is indeed black, then it has to be that it's because the
> dimness of stars prevails over their number and/or density.

As the article mentions, the sky is black because we cannot see infinity. We
can only see the observable universe.

~~~
czinck
An infinite number of small numbers does not always add up to infinity! Since
the apparent brightness of a star is an inverse-square law, if you assume
there are an infinite number of stars, each 1 light year further away from you
in a line and all the same brightness, you end up with that area being only
64%[1] brighter than it would seem with only the first star. Seems like a
pretty simple resolution to this paradox to me.

[1]
[https://en.wikipedia.org/wiki/Particular_values_of_Riemann_z...](https://en.wikipedia.org/wiki/Particular_values_of_Riemann_zeta_function)

~~~
tamana
I am going to guess that physicists have considered your resolution and it is
incorrect in the face of current measurements and models.

~~~
zuminator
The part that grandparent is overlooking is that while luminosity decreases as
to the square of the distance, the number of stars in a given angular diameter
of space increases as to the square of the distance. The two effects cancel
each other out and so Olbers Paradox can't be resolved by that method.

------
gusmd
I'm missing something here and would so happy if someone explained to me. I am
sorry if I'm making false assumptions as my knowledge on the subject is very
limited.

If the universe started from a single point in space and time (i.e., Big
Bang), and no matter moves faster than the speed of light, how could ANY star
be more distant from us than light could have traveled in the same time frame?

(I'm assuming no matter is travelling at v > c/2, otherwise my argument would
only be valid for things traveling in the same hemisphere of the expansion as
Earth)

~~~
Practicality
The matter isn't moving. Space is expanding, ie, space is being inserted in-
between all bits of matter.

Edit: To elaborate further: Since all the newly expanded space also expands,
the distance to travel continuously increases as the light travels, to the
point where over a certain distance the light never completes the journey.

So, the net effect is that the two objects become further apart faster than c,
even though neither of them are moving.

------
EGreg
Actually this was the argument that bothered Newton, but about gravity. In an
infitely old but finitely large universe, why wouldn't gravity pull everything
together? Unless in all directions gravity canceled out.

Similarly, we can ask why the universe isn't filled with light. If the
universe was infinitely old then indeed the light would eventually heat up all
the matter and penetrate everywhere. But if it's expanding, then the light
didn't have time to fill the space-time before it expanded further.

~~~
paulddraper
In an infinitely large universe, the net force of cosmic gravity would be 0.

Kind of like how an real number divides the number line into "same size"
infinite intervals.

Granted, entropy in the system would cause matter to clump into larger and
larger bunches. But if the matter were perfectly arranged by an outside force
(e.g. God) and didn't deteriorate, the universe could remain static.

~~~
Strilanc
Gravity actually acts kind of... weird... in an infinitely large uniform
universe. You end up with gravity acting like a time-dependent position-
independent scaling factor on position. Leonard Susskind goes over the
derivation (for Newtonian gravity) in a Cosmology lecture available on youtube
[1].

1:
[https://youtu.be/ERjkSbdn6-4?t=53m50s](https://youtu.be/ERjkSbdn6-4?t=53m50s)

------
cristianpascu
Or the Universe is neither infinite or as old as it's assumed to be. Although
there's strong evidence for it being as old as we (me included as a figure of
speech) currently believe it, our belief is based on some fundamental
assumptions that can only be strengthen but not proven. At least for the sake
of honesty, one must be aware of the nature of this kind of opinions. Science
is part of the larger discipline called epistemology, so all the rules valid
there, are valid in science too.

------
3pt14159
Isn't the issue one of lumens? The light from a star is inversely
quadratically related to its distance. Assuming a random distribution of stars
across a ring or plate in 3D space we should expect the quadratically growing
number of stars for the incremental space. The two cancel and it becomes a
ratio of the average brightness of a star (bright) versus the density of stars
in space (super sparse).

~~~
paulddraper
Imagine an infinite series of nested hollow shells of equal thickness around
Earth.

The number of stars in each shell is quadratic relative to the radius. The
brightness of each star is inversely quadratic relative to the radius of the
shell. Thus, each shell has a constant total brightness, as perceived from
Earth. An infinite number of these shells add up to infinite brightness.

~~~
3pt14159
I'm not quite following. The light in each "shell" continues to dissipate in
an inversely quadratic method from the shells distance from earth, no? It
would help if I had this drawn out.

~~~
paulddraper
No, the total light in each spherical "shell" is the same as every other
shell.

Stars that are 20 light-years away are 4x dimmer than stars 10 light-years
away. But there are 4x more stars 20 light-years away than 10 light-years
away. As you go outward from Earth, the stars get dimmer and more numerous,
but these balance each other, and the infinite sum is 1 + 1 + 1 + ...

(This is no chance happening; regardless of the # of spatial dimensions the
universe has, this relationship holds true.)

------
jonmccreery
This looks like a question that actually looks like two questions; namely, 'is
the universe infinite' (and as a side question, what is the actual dimensional
and geometric configuration of the universe), and if the first question is
true, 'why do we not observe an infinite flux of photons, or really any other
type or radiation, bathing our chunk of the universe'.

This one's easy. The first, foundational, assumtion is way easy to knock down,
via all the accumulated evidence that we erupted from a singularity 13-some
billion years ago. So.. no infinite time.

But we could probably get by with, even with an infinite universe, if we just
assume a universal speed, which is most definitely supported by current
academic consensus. This is because of the concept of the light cone, which
says that you can't be illuminated by things that lie outside your temporal
and spacial vicinity, as defined by how long it takes light in a vacuum to
reach you. That's a stronger statement than just how bright the sky is... it
applies to any conceivable mechanism of information transfer, and it's
universally supported by any experiments that we've tried (note: wormholes...
possible, super crazy loop hole here... maybe an infinite universe has
infinitely many wormholes to pump radiation into our light cone, so basicly
multiverse, but we've already talked about the universe having an age).

I see no paradox here, but then again the Big Bang and Special Relativity are
pretty new as ideas and we all benefit from the shoulders of the giants we
stand on.

Edit: On further reflection, a truly infinite universe probably would park an
infinite number of unstable wormholes on top of our neighborhood, but in that
case the the fact that our sky is in fact dark just adds more evidence to the
pile that we already have that it's not.

------
DINKDINK
>The entire sky would be about as bright, and as hot, as the surface of the
Sun.

Not so according to “The Inverse Square Law”.[1] If one measures the light
flux through a hypothetical sphere drawn around a light source and then
doubles that spheres size, the light density must decline or you've broken
conservation of energy. [2]

[1]
[http://bccp.berkeley.edu/o/Academy/workshop08/08%20PDFs/Inv_...](http://bccp.berkeley.edu/o/Academy/workshop08/08%20PDFs/Inv_Square_Law.pdf)

[2]
[http://www.compuphase.com/electronics/candela_lumen.htm](http://www.compuphase.com/electronics/candela_lumen.htm)

~~~
nostromo
If you reduce the power of infinite stars, you still have infinite stars
producing infinite light.

x * infinity = infinity (so long as x > 0; even if x is infinitesimally small)

~~~
jwally
But you can't have infinite stars in your field of view since they'd get in
each other's way; blocking radiation. Using the tree example from the article,
you wouldn't see infinite trees, you'd only see the pieces that are directly
in front of you.

------
maskedinvader
isnt the answer that most of the stars are inside galaxies that are for the
most part moving away from us and for that matter from each other (due to
space itself expanding) with the speed of expansion or their apparent motion
away from us proportional to their distance from us. This means most of the
star light reaching us is red shifted (due to doppler effect of light) and
since we humans cannot see this red shifted light, the night sky appears dark
to us. anybody with a physics background can point out where I got this wrong,
but I thought this is why the sky is dark.

------
mnw21cam
I am quite amazed that noone has pointed out that in terms of the cosmic
microwave background, the sky is _not_ black. The radiation that travels the
furthest before hitting us is this background radiation. It only appears to be
dim because it has been redshifted a huge amount due to the expansion of the
universe since then.

But yes - everywhere you look in the sky (unless there is something in the
way), you are looking directly at the intense fire of the birth of our
universe. Red-shifted until you can't even see it any more.

------
jdimov10
I don't get the "running out of fuel" argument... If the universe (as assumed
in the text) is infinitely old and infinitely large, why should there be any
scarcity of nuclear fuel?

~~~
randlet
Stars fuse light elements it into heavier elements. Eventually all the lighter
elements will be used up.

~~~
pc86
I think they were saying if we're conceding infinite size and an infinite
number of stars, why not an infinite amount of those lighter elements as well?

~~~
kyberias
Imagine an infinite space divided in infinite number of finite volumes each
containing a finite number of "lighter elements". All the light elements form
a star in their local volume and burn up all the light elements locally. The
result: all the infinite number of finite volumes are void of the light
elements. The infinite space is dark.

------
amelius
An easy explanation is here:
[https://www.youtube.com/watch?v=gxJ4M7tyLRE](https://www.youtube.com/watch?v=gxJ4M7tyLRE)

------
jonmccreery
This looks like a question that actually looks like two questions; namely, 'is
the universe finite' (and as a side question, what is the actual dimensional
and geometric configuration of the universe), and if the first question is
true, 'why do we not observe an infinite flux of photons, or really any other
type or radiation, bathing our chunk of the universe'.

------
mjn
I can recommend the book _Darkness at Night_ by Edward Harrison [1] for a
detailed look at the history of this paradox. An enjoyable read, and ties it
in with several closely related historical debates in cosmology.

[1]
[http://www.hup.harvard.edu/catalog.php?isbn=9780674192713](http://www.hup.harvard.edu/catalog.php?isbn=9780674192713)

------
lutusp
As is often the case, this explanation fails to take cosmological expansion
into account. An expanding universe is a cooling one. The explanation is
correct as far as it goes, but failing to take expansion into account either
dates the explanation or fails to evaluate a factor that's a bit more complex
than relying on the finite age of the universe.

------
eutropia
I find it odd that so many people seem to have trouble accepting this
explanation. Light travels at a finite speed, ergo anything far enough
away(adjusted for expansion of space) is invisible, because we've only existed
for a dozen billion years. If the universe were infinitely old, then we'd have
something interesting to talk about.

------
frogpelt
If there are an infinite number of stars in an infinitely large universe that
is infinitely old, how is the nuclear fuel finite?

Help me here.

------
kjell
[https://twitter.com/5point9billion](https://twitter.com/5point9billion) calls
out the inverse: light traveling from earth out towards the nearest stars.

It's a twitter bot that takes a twitter user's birthday, and tweets to them as
their "light cone" expands across the universe.

------
mrdufrain
Isn't it also the case that, because of the accelerated expansion of the
universe, some stars are moving away from us at more than the speed of light,
and their light will therefore never reach us?

~~~
paulddraper
Yes, this is currently true, and will continue to be, unless we end in the Big
Crunch.

------
mrnobody4
So should the earth heat up over time as more light reaches us or does the
expansion of the universe negate that?

------
jwally
Stab in the dark: Inverse Square Law + The number of stars in your field of
vision can't be infinite.

------
mixmastamyk
Hmm, didn't seem to mention the Inverse-square law, which I was waiting
patiently for.

------
vorotato
what about all the light that doesn't hit us.

------
pc86
Title fix: "Olbers'" not "Olber's," named after Heinrich Olbers. It is wrong
in the linked article's title as well.

~~~
dang
Thanks, fixed.

------
dschiptsov
This is mere ray-tracing problem.

What we call bright sky is reflections of Sun rays from particles in
atmosphere. What we see is reflected light.

When Sun is shadowed by Earth we see only light reflected by distant object,
such as Moon.

The darkness of the night sky is relative - some rays are here, but our visual
system is conditioned for days (reflected light), not nights.

The idea that everything should be filled with starts is just wrong. Rays from
distant stars were deflected, blocked by particles, etc.

