Olbers' Paradox
en.wikipedia.org
en.wikipedia.org
Were the succession of stars endless, then the background of the sky would present us a uniform luminosity, like that displayed by the Galaxy – since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all.”
Take this together with the fact that intensity falls off as the square of the distance and it seems like the sky should be dark.
More formally, each star emits some amount of power in each frequency band: P(f) so that \int_0^\infty P(f) df = P_total.
For each frequency then, we have a total of P(f)/(hf) photon emitted per second. The total number of photons emitted per second by the star is then \int_0^\infty df P(f)/hf which is a finite number.
The total number of photons received per unit area a distance r away from the star would then be
\frac{1}{4\pi r^2} \int_0^\infty df P(f)/hf
If your detector has an area A (e.g. your retina or some other device), you'd expect to see
\frac{A}{4\pi r^2} \int_0^\infty df P(f)/hf
photons per second from the star. As r gets really large, you'd see this drop arbitrarily low. Conversely, the amount of time you'd need to wait to see a single photon from that star then grows, making the star dark.
No, it won't. If you're going to use a quantum model of light (which you have to to use the concept of "photon"), then you have to use the quantum interpretation of "intensity". The quantum interpretation of "intensity" is the probability of detecting a photon; and this is a continuous quantity which can get smaller and smaller indefinitely without ever dropping to zero.
My argument above is semi-classical, but it shouldn't change with a full quantum mechanical approach.
Yes, but the probability is never zero, and the expected time is never infinite. So saying "the intensity drops to zero" is never correct.
The paradox claims that the sky should appear bright, which I take to mean that a detector should be receiving light from each point in the sky at each moment in time. It does not say that the detector will receive light from each part of the sky at some point, but that you may need to wait a million years before a particular point flickers and that, even then, there's nothing that guarantees that all points will flicker at the same time.
Ah, I see what you mean: yes, the intensity will be 1/4, but because of quantization, you now have to draw a distinction between the time-averaged power (which behaves like the power does in the classical case--more precisely, this would be the expectation value of the power in the quantum case) and the actual power at a given time, which can vary from the average (even to the point of being zero).
I don't think this is required for the paradox. All that is required is that the average flux of radiation received from the sky as a whole should be constant, and equal, roughly speaking, to the flux corresponding to the surface brightness of a star. That will still be true, under the specified conditions of the paradox (a universe in steady state and infinitely old) even if quantization is taken into account.
I agree that countability vs. uncountability seems like it should come into play.
No, it doesn't. See my other post in response to you upthread.
What I feel is the core of the resolution of the paradox is (global or local) conservation of energy. Even in an infinitely large eternal steady state universe, if we assume the total energy of the universe is conserved one cannot have an increase in energy density everywhere at once.
If in this universe stars live forever, you'd have eternal "sources" of energy, and for energy to be conserved you'd need compensatory "sinks" draining energy out of the universe, like black holes which don't increase in size. In which the resolution of the Olbers' paradox would be that most of your lines of sight would end in such a black hole.
If, like is usual in physics, you assume local conservation of energy, stars cannot live forever, so in an eternal steady state universe there must be a mechanism recycling the radiation back into a star. In this case, again every line of your sight would eventually hit a star, but most of the radiation would never reach you, being used underway to make a new star. (This is a blatant violation of the second law of thermodynamics of course, which is the actually issue with eternal steady state universes).
In either classic steady-state or eternal inflation case, energy conservation is not necessarily a problem: you can have vacuum energy that converts steadily into radiation, while being generated by the expansion.
The future, literally. To grossly oversimplify: if all of space is east-west, and time is north-south, the Big Bang is the north pole. Only the universe is the map rather than the globe, and the globe doesn’t have to exist for the map to exist and to have the same expansion in space (/longitude) with respect to time (/latitude).
Also there may or may not be a Big Crunch/south pole, this is all just a way to get into the nature of the geometry by way of a convenient frame of reference.
When you have a space that isn't just normal globally euclidean space (such as: on the surface of a sphere), the idea of a vector as in like "a direction you can go in and an amount of how much or how fast or whatever" isn't something that makes sense as something independent of a base location. Instead, each point in the space has associated with it a space of "tangent vectors" at that point, and these spaces are related to each other.
The metric tensor associates to each point a "bilinear form" with some properties, essentially a way of doing something like a dot product of two tangent vectors at that same point.
This in turn allows for defining the notion of the length of some curve through the manifold.
Noether's second theorem works fine in General Relativity (in fact, this was the historic context of her paper). So for any given time-like vector field, you'll get an energy conservation law. In case of Friedmann cosmology and chosing cosmological time as said vector field, you'll get a term proportional to H² which picks up the change in energy.
However, you won't be able to make this into a covariant expression: Gravitational energy-momentum can be expressed in terms of pseudo-tensors at best...
> Noether's first theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
Why is this not obvious? Are not symmetries necessarily the transformations which conserve quantities, and the types of symmetry equivalent to the type of conservation?
Because Noether got a lot of respect for her work despite working (in a time of great sexism, no less!), I know that either I have totally misunderstood or this is a Columbus’ Egg [0] — but I am curious which, and if it isn’t a Columbus’ Egg, what I’ve misunderstood (assuming that sort of thing can even fit into a HN-sized reply and I’m not asking something that would normally be the conclusion of a final year degree level physics module).
A priori, a symmetry is a transformation that, when applied to any valid trajectory, yields another valid trajectory. It is not obvious to me why (certain types of) symmetries necessarily yield conserved quantities...
If I have a motion vector and a force vector, and if work done is ∫ force • displacement dx, and my transformation is one which conserves the scale of and relationship between both vectors (e.g. displacement), then is it not automatically true that work done is also conserved under that transformation?
To get a feel for it (and why it may not necessarily be intuitive) consider the following pairs of symmetries and conservation laws:
- The laws of physics are invariant under time translation (i.e. repeating an experiment at different times gives you the same result ceteris paribus). The corresponding conserved quantity is energy.
- The laws of physics are invariant under spatial translation (i.e. repeating an experiment at two different places gives you the same result ceteris paribus). The corresponding conserved quantity is momentum (this is a vector: one component of momentum for each possible direction of translation).
- The laws of physics are invariant under rotation (i.e. repeating an experiment under different orientations gives you the same result ceteris paribus). The corresponding conserved quantity is angular momentum (again a vector, since the rotation group SO(3) has three generators).
- Electromagnetism, constructed as a classical field, exhibits an internal symmetry in the field quantities. The corresponding conserved quantity is the electrostatic charge (actually a 4-vector current).
It is not a priori obvious from from the kind of arguments you've supplied why these should be the case. Demonstrating these would require access to the Lagrangian formalism (i.e. action principles) and how it behaves under these symmetry operations. I would say that, as you suspect, you are fundamentally misunderstanding the situation.
I’ve heard of the Lagrangian before, but my knowledge of physics is probably somewhere around the level of someone who has only just finished the first year of their degree, at least judging by the modules my alma mater lists for their physics degree (I got Software Engineering from them 14 years ago, all my physics knowledge since then is personal geekery).
So at one degree of perception, we have an empty void, and at another, a bright flush of light and activity.
I think a more fitting example of "an empty void yet a bright flush of light" would be the microwave background. With eyes sensitive to longer wavelengths the entire sky is indeed bright.
Suppose the experiment is repeated on a black pixel from the Deep Field image, and another swell of stars are observed, hinting at a kind of fractal distribution.
Were the universe eternal and static, why could this pattern not repeat indefinitely in infinite time, space and matter? The paradox seems to assume a kind of infinite level of sensitivity of the observer.
The figure explains it visually - the further away you go from the observer, the more stars you capture in your camera's field of view and the apparent brightness stays the same. The 1/r^2 term for light intensity is cancelled by the r^2 for the number of stars.
It's interesting think what an experimental result you describe would imply. It either contradicts the nature of light or that we're in the center of a cloud of stars where the density of stars falls with distance from us.
With eyes sensitive to the CMB, the CMB is dimming and reddening. Such eyes witnessing an essentially isotropic and homogeneous CMB would be Eulerian observers of the CMB. (In contrast to observers who see a dipole anisotropy because of acceleration along one spatial axis, for example, or observers immersed in the gravitational field of a massive system like a galaxy cluster or a planet).
Such Eulerian observers see a clearly peaked spectrum, essentially identical to a that of a blackbody radiator that is cooling. If such observers are in deep inter-galaxy-cluster space and equipped with instruments to augment their CMB-sensitive eyes, they'd detect plenty of bright spots with frequencies much much lower than the peak in the CMB. As an example an https://en.wikipedia.org/wiki/Radio_galaxy will be much brighter in wavelengths longer than that of the CMB (spectral radiance peak of CMB is ~ 160 GHz and falls off quickly away from the peak).
That -- correcting for proper motions and atmospheric effects -- our view of the sky is pretty uniform in the CMB (cf. https://en.wikipedia.org/wiki/BOOMERanG_experiment and beyond) but far from uniform in VLF, radio, IR, UV, X-Rays, gamma rays, and so on (as known since roughly the 1930s thanks to https://en.wikipedia.org/wiki/Karl_Guthe_Jansky#Radio_astron... ) just like it is in visible light, and that the bright spots at different frequencies aren't coincident in our sky, are important pieces of evidence which must be dealt with by any prospective model of physical cosmology.
I'm not a physicist so presumably I'm wrong. But why is this wrong? As far as I understand it, this paradox is the reason we have the theory that space is expanding at a rate which makes objects in effect move away from each other faster than the light they emit.
If it did, the same thing would happen not just for stars, but for all sorts of things. If you position a computer screen on a hiltop far enough away that, on a dark night, you can just make out whether it's on or off, then your eyes are getting about 6 photons per second. It doesn't matter whether these come from a million separate pixels, or from one light-bulb of similar total brightness.
seconded, but i think youre misunderstanding the paradox's axioms. part of the assumptions are infinite homogenous distribution, which i think addresses your point (the "the paradox" section).
i think the salient point is why youd assume stars are both infinite (both in existence and lifespan?) but homogenous on an arbitrary scale.
edit: additionally, as long as orbital motion is included in this contrived model, blockage would certainly produce a less than perfectly bright sky.
If you are going to use a "photon" interpretation, you are using quantum mechanics, and in QM "brightness" involves the probability of detecting a photon and does not require there to be an exact integral "number" of photons. So brightness is still continuous in QM; the probability of detecting a photon can keep getting smaller and smaller indefinitely, without ever having to discontinously jump to zero.
However, think of it another way ... if the stars had infinite time to pump out these "photon" particles you speak of (harmph! highly dubious), then there ought to be an infinite number of them in any given space, as they have had an infinity of time to reach you.
In case that renders as a link by Hacker News, I've added a space here and removed the https part:
en.wikipedia.org/wiki/Olbers%27_paradox #:~:text=In%20astrophysics%20and%20physical%20cosmology,infinite%20and%20eternal%20static%20universe.
What it does is that it highlights that text in the page, with a yellow background color. Making the rest of the text difficult to read (your eyes are drawn to the very yellow highlighted part).
Introducing any tiny amount of absorption in space would fix this, even in an infinite size, infinitely old universe.
The Universe full of matter/anti-matter bubbles popping in and out of existence. Most of them don’t straddle an event horizon but they still occur.
How much would such a fog contribute to the darkness at night?
Edit: Aha, clouds are irrelevant. Any dark clouds would heat up until they are the same brightness as the rest of the sky. Thanks, article!
Even though there are an infinite number of stars we can only see the light from a subset of them because the light from the rest of them has not reached us yet.
So I think the simple resolution of the paradox is that the speed of light is not infinite. No?
What I also never understood was what about matter that is not lit up? There could be many more unlit heavenly bodies than stars, which would block the light from the stars.
Having expansion just-about implies there must be such a time. Because if you run it backwards, eventually the stars will all be touching each other, and clearly can't then behave exactly like stars do -- something else must have been going on.
Our modern answer does have such a time, and expansion. And the answer for why no stars shine before some time is that they took a while to condense into dense clumps from the initially quite smooth hot gas. What you see in the gaps between them is precisely this hot gas, at the moment it first became transparent to light, this is the microwave background radiation.
In their scenario of infinite time, matter which isn't lit up doesn't help. It would, like your eyeball, get the light of the stars from all directions, and would soon equilibrate to the same temperature as these surfaces.
I see, that may be the crux of the paradox.
But if we assume that stars were born at some time then dark planets could be too. And if stars became to existence at some time it is not too crazy to think that new stars might be continually become into existence and planet too so new planets would get created continually to block the light.
But if the stars have been there forever, the comet is effectively in an oven of uniform temperature. The equilibrium state at which it radiates heat as fast as it gets it has the comet's surface the same temperature as the rest of the oven. So it glows.
And now imagine every other room is uccupied. Still an infinite number of rooms are occupied, but this line, from a distance, will be half as dim.
Now imagine ever millionth room is occupied. Still, we have an infinite number of occupied rooms. And now, depending on the distance from which you are viewing this hotel, you may notice points of light instead of a dim continuum.
In your example, the number of doors is linear with distance but light falls of quadratically.