61–70 of 86 posts
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"at most one". As you get further away zero protons are reaching you on most cases, and zero photons times infinite stars is zero, which would appear to be darkness
Zero multiplied by infinity does not have to be zero, it can be anything between zero and infinity. Refresh your https://en.wikipedia.org/wiki/Calculus .
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Probably better not to use the term "steady state" here (even if pretty appropriate) in that the "steady state" cosmological model is/was one that is exponentially expanding, with all physical observables statistically time-independent. It solves Olber's paradox due to the radiation redshift. That model was observationally incorrect, but actually has pretty much been reborn in "eternal inflation" in which the Univers…
How does the universe expand? What is it expanding into? And why isn't that thing considered the universe?
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It does, but at the time, we didn't know about that.
Let's assume the universe is not expanding but is infinite. At some point in time stars started lighting up. Maybe they even all lit up at once, long time ago. But some stars are so far away that the light from them has not yet reached us. 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 sim…
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.
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Zero multiplied by infinity does not have to be zero, it can be anything between zero and infinity. Refresh your https://en.wikipedia.org/wiki/Calculus .
How does calculus relate to the context of this discussion though? Photons are discrete, not continuous, so I don't see how calculus applies.
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.
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How does the universe expand? What is it expanding into? And why isn't that thing considered the universe?
Quoting Wikipedia: It is an intrinsic expansion whereby the scale of space itself changes. The universe does not expand "into" anything and does not require space to exist "outside" it. Technically, neither space nor objects in space move. Instead it is the metric governing the size and geometry of spacetime itself that changes in scale https://en.m.wikipedia.org/wiki/Expansion_of_the_universe
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It depends on the way you look at things: 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 t…
I have a related question, borne out of my ignorance as I am not a physicist nor a mathematician. > 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 Noet…
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...
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The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger. My argument above is semi-classical, but it shouldn't change with a full quantum mechanical approach.
> The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger. 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.
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I have a related question, borne out of my ignorance as I am not a physicist nor a mathematician. > 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 Noet…
Are not symmetries necessarily the transformations which conserve quantities 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?
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> The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger. 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 point is not that the probability needs to hit zero, it's that it's not correct to say that you receive a quarter of the power as you move twice as far from the source. It's still true that the expected number of photons per second drops by a factor of 4, but it can drop so far as to render the source dark for an appreciable amount of time. The paradox claims that the sky should appear bright, which I take to mea…
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).
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Quoting Wikipedia: It is an intrinsic expansion whereby the scale of space itself changes. The universe does not expand "into" anything and does not require space to exist "outside" it. Technically, neither space nor objects in space move. Instead it is the metric governing the size and geometry of spacetime itself that changes in scale https://en.m.wikipedia.org/wiki/Expansion_of_the_universe
What is the metric governing the size and geometry of spacetime? Gravity??