empty space has a constant amount of energy content, a small but fixed amount for each element of volume. The Dark Energy*. But do take care to note that this really doesn't equate to an everyday concept very well, meaning that's it's not like we have a sample of it to play with. This recycles a concept brought up by Einstein himself, the so-called 'cosmological constant'. Back then Albert was trying to achieve something rather different though. He wanted to make a universe which wouldn't change much in size, a static flavour.
So if you interpret Dark Energy as manifestation of the cosmological constant, then there is another interesting aspect. Is the cosmological constant realy constant?
If you take the results of the WMAP mission, then the ingredients of the universe were changing over time. At the time, where the microwave background emerged (when the universe was 380000 years old - "scource wikipedia/wiki/dunkle_enrgie"), this were 10% neutrinos, 15% photones, 12% atoms and 63% dark matter. No dark energy, and, by definition, a cosmological constant with value 0. Today the ingredients are 4,9% atoms (baryionic matter), 26,8% dark matter and 68,3% dark energy. That means, that the dark energy increast over the the last 13,3 billion years. And therefore the cosmological constant should have changed with the increasing dark energy.
Thanks for the encouragement chaps, let us plow on ... :-)
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If the Hubble flow is measurable on the cosmic scale, are the gravity wave detectors or similar able to measure this closer to home?
IIRC there is a general low level rumble of gravitational waves ( the so called 'stochastic background' ), a sort of combination of lots of different disturbances all about the place. The events/scenarios which initially produced them may have long subsided, but they remain propagating about for us to 'hear'. Though not as separate distinguishable sources. More like the accumulated murmuring of a nearly quiet room of people whom are whispering. Now when I say low level here, I mean on the already diminished scale at which all gravitational waves are detected. So that's even smaller amplitude again than the relatively stronger stuff we attempt to catch here at E@H. I would expect the expansion of the universe ( = Hubble Flow ) to modulate this behaviour in the sense that the echo-chamber is getting bigger!! As for what type of detector would be best at disclosing this I'll have to look that up, and get back if I find something easily explicable. :-)
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Regarding the Hubble flow, 70 Km/s/Mpc is the local figure, but how much does it change far away (long ago)?
About 10 to 15% lower at high distance. In these discussions it's often quoted as a number ( z ) representing the redshift, and left up to whom-so-ever is reading to deduce a flow figure according to some chosen interpretation of that redshift. Ultimately it is a statistical estimate, indeed one 'breakthrough' was recognising that some of the distant supernovae were being dimmed by intervening material and various effects coming into play during the propagation of the photons ( other than inverse square reduction ). By excluding those dimmer supernovae the data curves out at high redshifts looks different, and has rather radically different implications too for both the past and future of the cosmos. I don't fully understand the reasoning behind the selection of some supernovae over others, for that matter I may well not be alone in worrying about that aspect of the analysis.
Note that the Hubble Flow has dimensions of inverse time and in a sense represents a measurement of time since the Point Of Confluence. So a higher flow figure means we haven't been around as long ( ie. it was quicker to reach our current size ) and a lower figure means the universe is older ( as a slower flow means it took longer to get this big ). But to be exact that is, in detail, model dependent and goes back to my very early comments about how astronomical thinking operates.
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So if you interpret Dark Energy as manifestation of the cosmological constant, then there is another interesting aspect. Is the cosmological constant realy constant?
If you take the results of the WMAP mission, then the ingredients of the universe were changing over time. At the time, where the microwave background emerged (when the universe was 380000 years old - "scource wikipedia/wiki/dunkle_enrgie"), this were 10% neutrinos, 15% photones, 12% atoms and 63% dark matter. No dark energy, and, by definition, a cosmological constant with value 0. Today the ingredients are 4,9% atoms (baryionic matter), 26,8% dark matter and 68,3% dark energy. That means, that the dark energy increast over the the last 13,3 billion years. And therefore the cosmological constant should have changed with the increasing dark energy.
The Nobel in physics in 2011 was awarded for this accelerating universe work on distant supernovae. In that regard it's well worth a couple of quotes from the Nobel committee's more involved explanation of that award ( I've highlighted certain words to emphasise the advocate nature of the reasoning, presumably based on Occam's Razor ):
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Within the framework of the standard cosmological model, the acceleration is generally believed to be caused by the vacuum energy (sometimes called â€dark energyâ€) which – based on concordant data from the SNe, the observations of the anisotropies in the CMB and surveys of the clustering of galaxies – accounts for about 73% of the total energy density of the Universe. Of the remainder, about 23% is due to an unknown form of matter (called â€dark matterâ€). Only about 4% of the energy density corresponds to ordinary matter like atoms.
and
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The evolution of the Universe is described by Einstein’s theory of general relativity. In relativistic field theories, the vacuum energy contribution is given by an expression mathematically similar to the famous cosmological constant in Einstein’s theory. The question of whether the vacuum energy term is truly time independent like the cosmological constant, or varies with time, is currently a very hot research topic.
and
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Conclusion
The study of distant supernovae constitutes a crucial contribution to cosmology. Together with galaxy clustering and the CMB anisotropy measurements, it allows precise determination of cosmological parameters. The observations present us with a challenge, however: What is the source of the dark energy that drives the accelerating expansion of the Universe? Or is our understanding of gravity as described by general relativity insufficient? Or was Einstein’s “mistake†of introducing the cosmological constant one more stroke of his genius? Many new experimental efforts are underway to help shed light on these questions.
So indeed does the evolution of the vacuum energy depend upon itself? Mathematically that leads to exponential behaviour. If it is a decaying exponential then would we be here to discuss that? Whereas positive exponentiation certainly yields 'explosions'. But I personally avoid anthropic ( human centred or 'I think therefore I am' ) arguments - for me they are just long looped tautologies where the circularity is hidden behind a pleasant but superfluous tour. And one needs to carefully define when one crosses over from the 'what is happening' to the 'why' in terms of recorded data.
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ASIDE : Which is why I no longer read New Scientist or Scientific American. Even good old Science is suspect now. The writers are/have losing/lost the ability to distinguish the insides of their heads from the outside.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Now here's a real gem that I've just found when looking up stochastic background material. Yes : it's by our fearless leader, Professor Bruce Allen in 1996 ! I'll come back to answering more about the topic when I've read it totally, but I thought I'd quote this part for now :
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The second type of source is pulsars. Once again, the form of the signal is known very precisely: it’s just a sine wave in the solar system barycenter, a coordinate system at rest with respect to the Sun (Niebauer et al. 1993). In this case however, the signal processing problem is far more difficult, because the orbital motion of the earth around the sun and the rotational motion of the earth around its axis modulates the pulsar frequency. This means one must separately analyze the signal for each of ≈ 10^14 separate patches on the celestial sphere, each of which would have a distinctive pattern of frequency modulation. The required processing speeds are currently beyond the limits of even the most powerful computers. Of course, it may be just a question of waiting until a better search algorithm is developed, or faster computers become available!
Or both.
So there you go. And here we are now at E@H in 2013 .... :-) :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Dark Matter has a different pedigree than Dark Energy although they are often mentioned in the same sentences. Like Dark energy it is also an intellectual placeholder or "I don't know" label, IMHO.
Let's go back to Kepler's Laws of planetary motion, a brilliant set of conclusions about what the solar system does and later well validated by Newton's work. As one steps away from the Sun, our local gravitational dominator, we find that the period of a planet's orbit decreases in a very well understood way. The radius of orbit cubed goes like the period of orbit squared, and hence they increase together or decrease together. I could write this relationship as the period being proportional to the three-halves power of the radius. So that's between linear ( two-halves or a single power ) and quadratic ( four-halves or two powers ).
Suppose as an alien I was some large distance from the solar system ( eg. at Alpha Centauri ) and, for some obtuse reason, was only able to see and record the motion of the planets but not the Sun. If we Centaurians had Keplerian and Newtonian equivalent knowledge then we could deduce the existence of a Sun-like central body in Earth's system based upon our study of planetary motions. Or at least we would probably accept such a thing being the simplest explanation for our data.
Forward a bit to Newton. Create a spherical thin shell of uniform matter at some given radius. So the inside is empty space, and here 'thin' implies that the shell's thickness is rather smaller than the overall dimensions of the shell. If you like, make it the same size as the Earth, but only a kilometer thick. 'Uniform matter' here means that : whatever it is made of, the density is the same everywhere. Put yourself inside the shell at the exact centre, and ask the following simple question. What is the force of gravity upon you due to the mass of the shell ?
Without even knowing the exact law of gravity fall-off with distance ( inverse square ) you could confidently give an answer of ZERO. There is no force at the exact centre of this construction. The argument is purely on the grounds of symmetry, meaning that if it wasn't zero then because force is a vector there must be some nett direction of force. If space, and the force of gravity, are isotropic ( no preferred directions ) then this is a contradiction, recalling our insistence upon a uniformly dense shell and that the centre is, by definition, equidistant from all shell points.
OK, now move off the shell's centre, while staying inside the shell. Will the total force from the shell now be zero ? The answer now is still ZERO !! But here we have to also rely upon a useful mathematical happenstance. If I move off centre then I am closer to some side of the shell, for which there will be some single nearest point to me. With co-ordinates centred upon myself consider a slab of solid angle equally surrounding that nearest shell point. This will be cone shaped with the axis of the cone running from myself to the nearest shell point. Evidently there will be another slab of solid angle - of the same size - centred upon another point on the shell which is diametrically opposite. This will be the furthest point on the shell from me. So there is another cone, this time the axis of which is from myself to the farthest shell point. It turns out that while any part of the shell within that nearest solid angle slab/cone has a greater gravitational influence ( varying as inverse square of distance ), that farthest slab of solid angle/cone encloses more of the shell in proportion to area ( or the square of distance ). For a given solid angle choice : moving further away from the centre implies moving closer to some inner shell side. That nearer cone encompasses less actual shell, while the further cone encompasses more actual shell.
Yep, you guessed it, for any given solid angle choice ( with cones emanating from my position, and not the shell's centre ) : the increase in force ( per shell amount ) by moving closer to one side of the sphere is exactly balanced by having more shell mass on the other. Because this analysis is true for any choice of solid angle cone pairs centred upon myself, the upshot is that throughout the interior of the shell I experience no nett gravitational force. The shell may as well not be there !! :-)
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We could go further and claim, correctly, that only an inverse square law achieves this. Again we meet that 'coincidence' of an inverse square force law that reduces in magnitude relating to spherical surface area that increases in magnitude. Rather deeper down this rabbit hole is a number of theorems regarding the stability of orbits in non-inverse-square-law cases. Indeed GR does vary gravity off Newton's inverse-square rule, and thus we deduce that the solar system is ultimately unstable on such grounds. This is thought to be true as planetary motion has quadrupolar components, which thus radiate gravitational waves/radiation, energy is lost, the system contracts and in some long distant future all ought merge together.
I won't labour the derivation for the situation of being outside the shell, you could do an analysis based on solid angle. You probably already know that such a shell will act upon me as if all the mass were at the centre of the shell.
These are key results for what follows. One good question is why does the behaviour of uniform shells matter? How many of those are about? Very few per se, actually. But think a wee bit further along ....
A solid sphere of uniform density can be considered as a set of concentric shells ie. all centred on the sphere's centre with one enclosing the next. Onion like. A ( small ) body embedded at any position within will feel no force from any shell that encloses it ( at greater radius from the sphere's centre ), and all shells that are at radius less than that body will act as if all mass is at the sphere's centre. For this to be more generally true, it turns out the entire solid sphere doesn't have to have uniform density, but a given thin shell at any radius must. Another way to assert that is : the sphere's density is spherically symmetric and may vary radially.
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There's a classic question that recurrently arises in late-high-school/early-tertiary physics courses : imagine a hole from the surface of the Earth ( non-rotating and symmetrically dense ) going through the core ( exact centre ) and out to the other side. What is the mathematical form of the force on a body dropped down the hole, and what is the subsequent evolution of it's motion ?? It will go down/back and up/forth along the tunnel, forever oscillating ( classically ), unless something else comes into play.
We will apply this sort of thinking to objects which have some suitable symmetry, but are not spherical. More like a pizza or a fried/poached egg, that is :
What's Happening In The Galaxies ?
Cheers, Mike.
( edit ) Another related idea is the tendency of sufficiently large bodies ( ie. big enough that gravity has a decent say in the shaping of ) to form layers/shells of uniform density per shell. This is a sort of Archimedes water displacement principle writ large. The attractive/central nature of gravity means that if a lump of stuff has greater density that some lump just closer in then they will tend to swap places. Over the entire volume of the body there will be a natural sorting of material from densest in the middle to least dense on the outer surface. Some reckon that carbon ( in diamond form ? ) lies at Jupiter's core for instance, carbon being the densest material available when that planet formed. Earth is the same story of course too, the light gases form a fine layer on the rind while the heavy metals lie at the core. Fortunately this enables our solar system navigators to view the planets as mass points - vastly simplifying calculations - at least from a good distance away. Which is most of the time.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Addendum : A picture is worth a thousand words ! :-)
This you can see is a 2D slice of the uniform shell, analysed for a test mass ( me ) situated off centre. The mathematical full gore comes out in an integrand where the inverse square behaviour of gravity is offset by the differential surface element, yielding a constant ( = 0 ) regardless of location ( provided it is within the shell ).
If you're on the outside the solid angle on one side has no mass enclosed by it, but the solid angle on the other side has both near-side and far-side shell sections contributing to a non-zero nett in that direction. But similiar geometric cuteness applies as the total force from those two shell sections gives an effective centre of mass at their average distance : the shell's centre !*
Cheers, Mike.
* Of course you know this already. Standing on planet Earth the 'down' direction is to the centre. Naturally the Earth is not quite uniform, indeed variations in the strength and direction of gravity is useful for, say, finding concentrations of minerals. The Apollo missions had variations upon their close-in Moon ( expected ) orbits due to 'mass-cons' or concentrations of mass ( volumes within of higher than average density ) not previously apparent. Until they got there. :-)
( edit ) If it helps, you may want to think of this scenario as involving a flux of gravitons from the shell travelling inwards and focussing onto the test mass. The math & geometry evolve the same .... indeed some of you may be thinking of another well known force with inverse square behaviour. That has two charges ( gravity has one ) and may attract and repel ( gravity only attracts ). Now if you can work out how to plonk all of that ( electromagnetism and gravity ) consistently and accurately under the one umbrella then : you have yourself a Nobel plus more !! :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Back to good old Doppler again. Get your telescope out and look at a galaxy and measure the frequency shifts of spectral lines across the galaxy. Typically you will find that it varies, meaning the velocity of galaxy parts is different depending on which piece of the galaxy that you look at. Many galaxies look like huge whirlpools and indeed the velocity measurements reflect that. A really big swirling bunch of stars orbiting around what obviously looks like a central accumulation of matter ( most galaxies ).
Now assume that the laws of gravity as disclosed to us hereabouts ( solarsystem ) also apply, then one could take the pattern of velocity as disclosed above and by 'reverse engineering' deduce what amount of matter must be there in order to give that appearance. So this is the inverse of our shell analysis : if a given star is orbiting at a certain radius and at a certain velocity then what matter must be contained within the star's orbit? Do this for all the luminous ( light emitting ) material that you can see in the galaxy .....
..... and one finds that by a very big margin there must be far more matter there than we can see. Specifically instead of the velocities reducing with distance from the centre, by some function approximating Kepler's Laws, the velocities are rather higher and so we have a conundrum to solve. Resolution ?
- assume no change to laws of gravity ( as derived from hereabouts ) for distant galaxies and over the much larger scale ie. galaxy widths. That approach gives you the 'where is all the mass ?' and hence the phrase Dark Matter simply meaning that there is stuff there but we can't see it. Now what could that stuff actually be ? Have we looked hard enough ? What about other modes of detection ?
- assume that what we see pretty much represents what mass is actually there. So we reconcile the velocity distribution by claiming that we must apply a law of gravity different than what we currently understand to be true. That is a harder road for several reasons. Not least is having to come up with that different law which would explain the galaxies, but also not forgetting that such a law would also have to apply to our local neighbourhood too. Put another way : we'd have to admit that the form of the law is scale dependent. By that we don't mean simply that the force of gravity changes with distance, we already know that to be the case, but we mean that the formulation of the law changes with larger distances. That's a bit like saying that accounting standards for a business will change depending upon the size of the business : it's not just that more or less money is involved but that the rules of how you calculate depend on the quantity of money involved.
The door is wide open here for new ideas - suitably verified by observation of course - to plug the breach in our understanding.
For instance there was a fairly recent article in Science ( the AAAS magazine ) outlining Spitzer ( the infrared space telescope ) studies on, well, balls of gas intermediate in size/mass between say Jupiter and the Sun. This is a range where nuclear fusion may or may not happen ( eg. deuterium ) depending on whether the object's core temperature is sufficient. This would hence determine whether it would simply softly glow in the heat generated from gradual gravitational contraction, or is there in addition heat production from fusion? Anyway this study suggests there are quite alot more bodies of that type out there, than were previously expected. So these would presumably be lone bodies, not in some star system as 'planets', but drifting about the place as 'heavy Jupiters'. I give this as an example of the sort of thing that might, if the numbers are right, turn out to be appropriate to fit in as Dark Matter candidates ie. are responsible for keeping the galactic stars orbiting as quick as we see.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
It's occurred to me that I haven't especially covered the
Cosmic Microwave Background
which is a relatively recent astronomical discovery. There's considerable subtlety in this topic so I'll start slow.
'Cosmic' and 'background' have expected meanings so I'll define 'microwave' for you. There's this thing called electromagnetic radiation, otherwise known as light or photons. It's what you see with, but that in turn depends on the frequency or 'color' of the light and what you use to 'see' with. 'Microwave' refers to a range of wavelengths corresponding to frequencies of the order of 10^10 per second ( give or take ). The wavelengths are from about a meter down to a millimeter, the term originally/historically meaning 'microscopic' compared to radio wavelengths then used. To coherently produce microwaves one has to have some device that will produce oscillations of about that 10^10 cycles per second. It wasn't until about WWII that gadgets called klystrons* and cavity magnetrons were able to achieve that, for radar and the like technologies. Nowadays you'll find magnetrons performing the role of food and water heaters in kitchens as ovens**, and as I look out my office window today I can see several towers on mountain tops that have microwave relay dishes atop.
One can incoherently produce microwaves rather more easily by having an object of suitable temperature just radiate in that frequency band. Coherence refers to whether there is any phase relationship, or not, b/w various distinct radiation emitters. Phase refers, basically, to the moment of time when one can deem an oscillation to be at a particular part of a cycle. Sort of like 'when do I pass GO ?' in Monopoly, say, as I keep going around the board.
In thermodynamics there's this thing called a black body. This is not anything to do with Dark Matter or Dark Energy. In all likelihood you have never seen one, nor ever will. To be exact it is an ideal/theoretical construct or a scenario which may be closely approximated, but not quite achieved in reality. Indeed the Cosmic Microwave Background ( CMB ) is probably the best 'natural' example to date. This picture illustrates something akin to the idea :
... recognisable as hot ceramic pots within a kiln. If it wasn't for some parts having a cooler profile than others then it would be harder to distinguish 'pot' features here. The bases of the pots are difficult to define, evidently opening the kiln to take the shot has cooled the tops. Of course you've noted that the pots aren't actually black ( I won't be describing the uninteresting & misleading historical reasons for why the word 'black' is used in 'black body' ). As the full technical definition is somewhat horrible, let's stick with a simpler statement : radiation in equilibrium with oscillators. For our kiln above the pots have oscillators - the electrons in the outer shells of the pot atoms - and the radiation is the photons ( mainly of orange color ) flying about in the kiln. Here's an example where it is even harder to define the pot outlines :
.... so those tissue paper looking objects up the back of the oven on the right are actually solid but quite hot ceramics. Again they aren't black. You don't actually have to have solid objects like these examples to have 'black bodies'.
In the late 19th century there was alot of experimenters dealing with insulated hot ovens that had a small hole in the outer wall so that the radiation coming out of the hole could be measured in both frequency/wavelength as well as strength. Bear in mind that back then the concepts of photons, atoms, electrons etc was a very contentious topic indeed. That is : even the existence of such things was no where near as certain or agreed upon as today. So what does 'in equilibrium with' mean ? Next up :
Equilibrium Thermodynamics
( relax, it's not as bad as that sounds )
Cheers, Mike.
* .... and related stuff. When a lad I used to think all scientists hung around these sort of things :
which, on reflection, one ought not get too close to during operation !! Devices with designs descending from these ( though not necessarily known by the klystron tag ) are still used today in, say, particle accelerators. Naturally one can recognise true scientists by their white coats .... :-O
** Curiously early British radar research had a sideline stemming from the question as to whether one could cook/kill a sheep from some distance away. This inquiry was just a subtle way of re-phrasing 'could you kill airmen in enemy aircraft at a distance by using a death-ray' ? I kid you not. :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
.... But for really long distances away ( = long times ago ) the speeds ( redshifts ) are more than expected. Where 'expected' means assuming the expansion rate that we observe 'nearby' ( = recent times ) holds there/then too. Thus because the redshifts are greater for those older objects that means we are moving away faster now than we were then ...
.... which is 180 degrees wrong. Sorry. I think I got the axes mixed up on a graphic somewhere. The older/distant redshifts/speeds are less than expected based upon nearby/recent ones. Hence the recession is accelerating.
Equilibrium Thermodynamics
Let's make a cup of hot coffee, then put it in a magical container that lets no energy* in nor energy out. I say 'magical' as a perfect one of these containers doesn't really exist : if you are in a universe you must interact with it. There are no truly force blocking containers ( even a Faraday cage interacts ). But we can have containers which highly approximate this ideal of isolating a hot cup of coffee. I only use that as an example because I'm drinking one now, you could apply this discussion to literally anything up to and including entire universes.
What happens within is that the components of the coffee and cup are going to interact amongst themselves, mainly by pushing and shoving each other about. Suppose over one corner of the cup the atoms generally have more energy than those elsewhere. It turns out that energy gets distributed overall from the energy 'rich' to the energy 'poor', but for a fairly subtle reason. It turns out that there are many, many more ways to be energy poor than energy rich and physical laws tend to proportion/distribute available energy equally amongst those ways. Since energy poor ways are far more numerous than energy rich ones, a sort of 'democracy of ways' applies as energy becomes distributed. We have names for 'energy richer' and 'energy poorer' states : hotter and colder respectively ie. the concept of temperature or average energy of a group of things. These are relative terms of course, but one absolute does apply - there is a temperature which one cannot go lower than ( or strictly speaking even equal ) and that is absolute zero.
[ We have a name for the 'ways' too : entropy is the counting of occupied ways or states. Apart from saying that entropy is maximised at equilibrium, we'll leave this aspect alone. ]
Now it takes time for such distribution to occur. Exactly how long depends upon messy internal details, but we can be sure that if one waits long enough then all that distributing will occur and the temperature will be the same throughout. Well, not quite. Close, but no cigar. It could happen that by a random sequence of events - at a microscopic level now - a group of atoms in the coffee cup might briefly acquire extra energy than the others. Strictly speaking this depends upon what you mean by 'which group' and 'extra', but nonetheless it might happen. However that wouldn't last even if it did, as interactions would dissipate any energy bonus away from the given group. The overall state of the cup of coffee is now deemed to be in equilibrium**. Or to be most exact thermal equilibrium, there being other sorts of equilibrium about which aren't relevant here.
The assumption of isolation is crucial here. If one is shoving energy into, or is sucking energy out of, the system ( cup of coffee here ) then the nett energy distribution amongst system components may never end. But all perfection is broken in this universe and so there are many methods to approximate isolation.
One obvious way is to keep the leaking of energy in/out low by various barriers that aren't very excitable, so to speak. So the barrier doesn't interact much with the system contents inside the container ( which the barrier defines ), nor with whatever is outside the barrier. Note that the terms 'inside' and 'outside' are what we choose to apply to some scenario, and the barrier in detail is indifferent to both. Inside and outside apply because the barrier is shaped to enclose, that's a global property.
Another 'trick' sometimes used is to pick a time interval. Take the explosion of an air/petrol mixture in the chamber of an engine. Choose a time long enough for combustion to complete and the gaseous components to equilibrate, but short enough so that said combustion products do not significantly transmit heat to the walls, or change the chamber's volume by pushing the cylinder head away or opening a valve. This is a 'cheat' of course and truly one could only rely on modelling from that if measurement of predictions agreed.
Above I used the phrase 'mainly by pushing and shoving'. By that I meant more or less up close and personal contact. This is largely an electromagnetic thing that occurs provided the things are significantly close together. But just as one can shout across a crowded room to attract attention ( rather than by messages passed from one person to the adjacent ), one can throw long. Of course both are really electromagnetic in truth, an electron in one atom is going to be playing throw and catch using ( virtual ) photons with electrons in nearby atoms. So it's a matter of scale/length and language used. Here is where gravity may come in to play - thinking of the cosmos now - as we ignore ( rightly or wrongly ) gravity over short distances and low masses.
Cheers, Mike.
* I mean matter here too. As per Einstein : energy and matter are aliases.
** This doesn't mean exactly the same energy per atom/molecule, it's an average or statistical statement that applies to a large group of things. So one atom is never in equilibrium, that's a mis-application of the concept. Equilibrium is a sort of group summary. Also in a stricter sense we can only really talk of temperature when discussing bodies in equilibrium. I have a thermometer in my office which one gently points inside someone's ear canal and it reads out a temperature value. To be exact the device is an infrared radiometer which upon the assumption that it is pointing to an equilibrated body, is calibrated to display the temperature value of such a body. A 'steady state' is not equilibrium ! Steady state is where the overall measured properties of a system remain constant in time, and says nothing about isolation or waiting or entropy or energy distribution. If you like equilibrium = isolation + waiting which ultimately also has constant measured global properties. There's one especially famous area of modern 'science', that I won't name and shame, that shows little evidence of understanding even these basic definitions .... in particular none of the lessons of equilibrium thermodynamics can be applied to any seriously non-isolated system. It is not even vaguely the same sort of cow !!
( edit ) I have also deliberately blurred the distinction between 'macrostates' and 'microstates' ( you may safely ignore what follows ). This is partly a matter of degree in how one chooses to cognitively lassoo and/or label system components. But a deeper point lies within in that many actually distinct microstates may be present 'underneath' a certain macrostate. So things like temperature, pressure, volume and especially total energy content are macrostate parameters : it's what a system looks like overall. But of course in reality each tiny part of a system ( eg. atom or photon ) has particular qualities, the listing of all such qualities as these over all system components is what defines a microstate. This is called fungibility, which isn't a kind of nasty infection, but rather the ability to substitute low level detail and not notice a change at larger scale. So if I was caught throwing a can of baked beans at police in a riot, then there wouldn't be any concern ( in court later ) with the exact arrangement of the beans in the can. One can would be deemed as good as the next in terms of assaulting police. Of course the aggregate/average values over a given microstate will yield a set of macrostate parameters. You also see this, say, in the drawing of lotto balls - the precise ordering of the balls and exactly when etc is not considered relevant for a payout. One approach to defining entropy is to say that it is a numerical measure ( typically involving logarithms ) of the number of microstates per macrostate. Because the numbers are typically horribly large then using logarithms gives briefer notations.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
So let's take some lump of stuff, isolated and in equilibrium, and ask what is the distribution of energy amongst it's components. Even a system of moderate size is going to contain an enormous number of elements. Rather than listing each element/atom/molecule/whatever and the energy that each has ( tedious ), we will flip the question : for a given energy level how many elements in the system have that energy. To be exact we have to talk of energy bands - between such & such energy and a bit more - because of quantum & other reasons. Thus we cut up the energy spectrum into little narrow slices and discuss how many things have the energy lying within each slice. Economists do this sort of thing with groups of people, finding out what fraction of some large population has an income say, within a given band. One can come up with an overall ranking scheme by 'value'.
The maths has been done and dusted for this. On the assumption of equilibrium one can describe how the total energy of the system may be divided up in terms of energy bands. Maxwell and Boltzmann nailed this first. Theirs was a classical analysis ie. before relativity and quantum mechanics, and usually named Maxwell-Boltzmann ( M-B ) statistics. It has an exact mathematical form. This work has been added to over the years, the significant changes being exactly how one goes about doing the counting of things.
[ You may safely ignore this paragraph : It depends on whether one can truly tell the difference between 'distinct' objects. In a classical world we could ( if only as a Gedankenexperiment ) track and label each tiny bit of a big system without upsetting the system sufficiently to change it's equilibrium state. That's the original Boltzmann approach. But quantum mechanics says no, you can't set any arbitrarily lower limit on your interaction with the system ( as a counting observer ). So to count states and remain in equilibrium one has to accept some ambiguity in the counting procedure, because you can't track particles individually now. Depending upon whether the particles are bosons ( which like to clump together ) or are fermions ( which don't ) then different counting techniques are involved. The former distribution is called Bose-Einstein ( B-E ) statistics, the latter Fermi-Dirac ( F-D ) statistics. At high temperatures ( read : high average energy per particle ) the M-B, B-E and F-D approaches converge and give almost identical answers. Indeed one early suspicion, so to speak, of quantum behaviour arose when 'classical' systems were examined at low temperatures. What was discovered was that the systems lost 'degrees of freedom' - or that there should have been more ways to be energy poor than were actually found. Nowadays we interpret this as quantum objects having well spaced discrete energy levels down low near their base/ground states, but up at high energy the gaps b/w energy levels are much closer and effectively continuous. So high temperatures imply that the small but fixed quantum nudges make no significant alteration, relatively, to the energy level populations. Hence B-E and F-D are well approximated by M-B if it's hot enough. ]
Now if one samples the electromagnetic radiation of an equilibrated system then one obtains a special 'black body curve' that gives the amount of radiation at each frequency/energy. This pattern of emission reflects/reveals the underlying equilibrium. From that one can deduce a temperature for the entire system. Penzias and Wilson in the 1960's were studying radio transmission issues b/w ground stations and satellites ( a new technology then ). They found a low level 'hiss' or static in the signals coming from all over the sky. Day and night it was uniform. Even across the seasons. They took great pains to rule out errors and problems in their own equipment. After a chance meeting with an astronomer on a plane trip they deduced that they had tripped over evidence for a prediction made several times over the previous decades : if the Universe began as a 'Big Bang' then there ought be an 'afterglow' of that. The rest is history ..... :-)
A couple of important/interesting points :
- strictly speaking Penzias and Wilson only measured at a single frequency. So they had to assume the radiation they sampled was from an equilibrium 'body' to deduce it's 'effective' temperature. However subsequent - and for that matter prior - measurements gave several other frequencies. The COBE satellite/experiment nailed the whole curve to a high degree of precision thus satisfying our confidence that we are indeed looking at an equilibrated thing. Later on WMAP gave even better results. The Planck mission is currently in play as we speak.
- or should I say nearly in equilibrium because the pattern across the sky shows slight variation of the order of one part in ten thousand from different directions. Some of that is specific objects and regions - certainly not in equilibrium - that radiate in the frequency band of interest, polluting if you like our studies. Such extras can be studied, understood and accounted for by subtracting away from the whole picture. You've probably seen various graphics on this, usually under the casual/throwaway heading of 'Face of God' or somesuch ( thanks to George Smoot ).
- something in perfect equilibrium will have no structure or contrasts. That's why I emphasised the difficulty of distinguishing the features of the pots and stuff in the kilns earlier. A pot in true equilibrium would have to be a mist of atoms floating around in the kiln ie. no pot structure or shape at all ! :-0. Most of us will never see anything even close to equilibrium, because we live in a world with structure. We ourselves have structure. Our entire planet is well away from equilibrium ( seriously non-isolated ), fortunately for us !! :-)
- why do we speak of 'counting' things ? What's that got to do with their overall group behaviour ? Well, in a sense the elements of each system count each other. They, by mutually interacting, sample each other's characteristics like energy, momemtum, spin etc. Indeed a large group of things at equilibrium may well have some fixed number/fraction within some energy band, but it's not going to be the same actual particles from moment to moment. Like a very highly socially mobile society every particle is perpetually exchanging their characteristics, energy included. Recall that energy, momentum, spin ( angular momentum ) and the like are all conserved quantities provided no 'leaking' in or out occurs. Which is why we insisted earlier upon isolation as a criteria for equilibrium. So 'counting' is an intellectual mnemonic for 'assessment of true interaction'.
Next up :
So Where & When Did The CMB Originate ??
[ I ought add : if any of you are keeping up with this description/narrative, then you are doing exceedingly well. It is quite thick porridge, definitely non-trivial theory, a complex edifice for sure. Conversely, if you are having trouble : don't panic, just skim and grasp the highlights. ]
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: ( edit ) BTW - doesn't
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Thanks Mike for putting up these, this forum is a favourite, i try to read and keep up.
OK i thought of some questions.
If the Hubble flow is measurable on the cosmic scale, are the gravity wave detectors or similar able to measure this closer to home?
Regarding the Hubble flow, 70 Km/s/Mpc is the local figure, but how much does it change far away (long ago)?
RE: empty space has a
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So if you interpret Dark Energy as manifestation of the cosmological constant, then there is another interesting aspect. Is the cosmological constant realy constant?
If you take the results of the WMAP mission, then the ingredients of the universe were changing over time. At the time, where the microwave background emerged (when the universe was 380000 years old - "scource wikipedia/wiki/dunkle_enrgie"), this were 10% neutrinos, 15% photones, 12% atoms and 63% dark matter. No dark energy, and, by definition, a cosmological constant with value 0. Today the ingredients are 4,9% atoms (baryionic matter), 26,8% dark matter and 68,3% dark energy. That means, that the dark energy increast over the the last 13,3 billion years. And therefore the cosmological constant should have changed with the increasing dark energy.
Thanks for the encouragement
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Thanks for the encouragement chaps, let us plow on ... :-)
IIRC there is a general low level rumble of gravitational waves ( the so called 'stochastic background' ), a sort of combination of lots of different disturbances all about the place. The events/scenarios which initially produced them may have long subsided, but they remain propagating about for us to 'hear'. Though not as separate distinguishable sources. More like the accumulated murmuring of a nearly quiet room of people whom are whispering. Now when I say low level here, I mean on the already diminished scale at which all gravitational waves are detected. So that's even smaller amplitude again than the relatively stronger stuff we attempt to catch here at E@H. I would expect the expansion of the universe ( = Hubble Flow ) to modulate this behaviour in the sense that the echo-chamber is getting bigger!! As for what type of detector would be best at disclosing this I'll have to look that up, and get back if I find something easily explicable. :-)
About 10 to 15% lower at high distance. In these discussions it's often quoted as a number ( z ) representing the redshift, and left up to whom-so-ever is reading to deduce a flow figure according to some chosen interpretation of that redshift. Ultimately it is a statistical estimate, indeed one 'breakthrough' was recognising that some of the distant supernovae were being dimmed by intervening material and various effects coming into play during the propagation of the photons ( other than inverse square reduction ). By excluding those dimmer supernovae the data curves out at high redshifts looks different, and has rather radically different implications too for both the past and future of the cosmos. I don't fully understand the reasoning behind the selection of some supernovae over others, for that matter I may well not be alone in worrying about that aspect of the analysis.
Note that the Hubble Flow has dimensions of inverse time and in a sense represents a measurement of time since the Point Of Confluence. So a higher flow figure means we haven't been around as long ( ie. it was quicker to reach our current size ) and a lower figure means the universe is older ( as a slower flow means it took longer to get this big ). But to be exact that is, in detail, model dependent and goes back to my very early comments about how astronomical thinking operates.
The Nobel in physics in 2011 was awarded for this accelerating universe work on distant supernovae. In that regard it's well worth a couple of quotes from the Nobel committee's more involved explanation of that award ( I've highlighted certain words to emphasise the advocate nature of the reasoning, presumably based on Occam's Razor ):
and
and
So indeed does the evolution of the vacuum energy depend upon itself? Mathematically that leads to exponential behaviour. If it is a decaying exponential then would we be here to discuss that? Whereas positive exponentiation certainly yields 'explosions'. But I personally avoid anthropic ( human centred or 'I think therefore I am' ) arguments - for me they are just long looped tautologies where the circularity is hidden behind a pleasant but superfluous tour. And one needs to carefully define when one crosses over from the 'what is happening' to the 'why' in terms of recorded data.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Now here's a real gem that
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Now here's a real gem that I've just found when looking up stochastic background material. Yes : it's by our fearless leader, Professor Bruce Allen in 1996 ! I'll come back to answering more about the topic when I've read it totally, but I thought I'd quote this part for now :
Or both.
So there you go. And here we are now at E@H in 2013 .... :-) :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Dark Matter Dark Matter
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Dark Matter
Dark Matter has a different pedigree than Dark Energy although they are often mentioned in the same sentences. Like Dark energy it is also an intellectual placeholder or "I don't know" label, IMHO.
Let's go back to Kepler's Laws of planetary motion, a brilliant set of conclusions about what the solar system does and later well validated by Newton's work. As one steps away from the Sun, our local gravitational dominator, we find that the period of a planet's orbit decreases in a very well understood way. The radius of orbit cubed goes like the period of orbit squared, and hence they increase together or decrease together. I could write this relationship as the period being proportional to the three-halves power of the radius. So that's between linear ( two-halves or a single power ) and quadratic ( four-halves or two powers ).
Suppose as an alien I was some large distance from the solar system ( eg. at Alpha Centauri ) and, for some obtuse reason, was only able to see and record the motion of the planets but not the Sun. If we Centaurians had Keplerian and Newtonian equivalent knowledge then we could deduce the existence of a Sun-like central body in Earth's system based upon our study of planetary motions. Or at least we would probably accept such a thing being the simplest explanation for our data.
Forward a bit to Newton. Create a spherical thin shell of uniform matter at some given radius. So the inside is empty space, and here 'thin' implies that the shell's thickness is rather smaller than the overall dimensions of the shell. If you like, make it the same size as the Earth, but only a kilometer thick. 'Uniform matter' here means that : whatever it is made of, the density is the same everywhere. Put yourself inside the shell at the exact centre, and ask the following simple question. What is the force of gravity upon you due to the mass of the shell ?
Without even knowing the exact law of gravity fall-off with distance ( inverse square ) you could confidently give an answer of ZERO. There is no force at the exact centre of this construction. The argument is purely on the grounds of symmetry, meaning that if it wasn't zero then because force is a vector there must be some nett direction of force. If space, and the force of gravity, are isotropic ( no preferred directions ) then this is a contradiction, recalling our insistence upon a uniformly dense shell and that the centre is, by definition, equidistant from all shell points.
OK, now move off the shell's centre, while staying inside the shell. Will the total force from the shell now be zero ? The answer now is still ZERO !! But here we have to also rely upon a useful mathematical happenstance. If I move off centre then I am closer to some side of the shell, for which there will be some single nearest point to me. With co-ordinates centred upon myself consider a slab of solid angle equally surrounding that nearest shell point. This will be cone shaped with the axis of the cone running from myself to the nearest shell point. Evidently there will be another slab of solid angle - of the same size - centred upon another point on the shell which is diametrically opposite. This will be the furthest point on the shell from me. So there is another cone, this time the axis of which is from myself to the farthest shell point. It turns out that while any part of the shell within that nearest solid angle slab/cone has a greater gravitational influence ( varying as inverse square of distance ), that farthest slab of solid angle/cone encloses more of the shell in proportion to area ( or the square of distance ). For a given solid angle choice : moving further away from the centre implies moving closer to some inner shell side. That nearer cone encompasses less actual shell, while the further cone encompasses more actual shell.
Yep, you guessed it, for any given solid angle choice ( with cones emanating from my position, and not the shell's centre ) : the increase in force ( per shell amount ) by moving closer to one side of the sphere is exactly balanced by having more shell mass on the other. Because this analysis is true for any choice of solid angle cone pairs centred upon myself, the upshot is that throughout the interior of the shell I experience no nett gravitational force. The shell may as well not be there !! :-)
I won't labour the derivation for the situation of being outside the shell, you could do an analysis based on solid angle. You probably already know that such a shell will act upon me as if all the mass were at the centre of the shell.
These are key results for what follows. One good question is why does the behaviour of uniform shells matter? How many of those are about? Very few per se, actually. But think a wee bit further along ....
A solid sphere of uniform density can be considered as a set of concentric shells ie. all centred on the sphere's centre with one enclosing the next. Onion like. A ( small ) body embedded at any position within will feel no force from any shell that encloses it ( at greater radius from the sphere's centre ), and all shells that are at radius less than that body will act as if all mass is at the sphere's centre. For this to be more generally true, it turns out the entire solid sphere doesn't have to have uniform density, but a given thin shell at any radius must. Another way to assert that is : the sphere's density is spherically symmetric and may vary radially.
We will apply this sort of thinking to objects which have some suitable symmetry, but are not spherical. More like a pizza or a fried/poached egg, that is :
What's Happening In The Galaxies ?
Cheers, Mike.
( edit ) Another related idea is the tendency of sufficiently large bodies ( ie. big enough that gravity has a decent say in the shaping of ) to form layers/shells of uniform density per shell. This is a sort of Archimedes water displacement principle writ large. The attractive/central nature of gravity means that if a lump of stuff has greater density that some lump just closer in then they will tend to swap places. Over the entire volume of the body there will be a natural sorting of material from densest in the middle to least dense on the outer surface. Some reckon that carbon ( in diamond form ? ) lies at Jupiter's core for instance, carbon being the densest material available when that planet formed. Earth is the same story of course too, the light gases form a fine layer on the rind while the heavy metals lie at the core. Fortunately this enables our solar system navigators to view the planets as mass points - vastly simplifying calculations - at least from a good distance away. Which is most of the time.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Addendum : A picture is worth
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Addendum : A picture is worth a thousand words ! :-)
This you can see is a 2D slice of the uniform shell, analysed for a test mass ( me ) situated off centre. The mathematical full gore comes out in an integrand where the inverse square behaviour of gravity is offset by the differential surface element, yielding a constant ( = 0 ) regardless of location ( provided it is within the shell ).
If you're on the outside the solid angle on one side has no mass enclosed by it, but the solid angle on the other side has both near-side and far-side shell sections contributing to a non-zero nett in that direction. But similiar geometric cuteness applies as the total force from those two shell sections gives an effective centre of mass at their average distance : the shell's centre !*
Cheers, Mike.
* Of course you know this already. Standing on planet Earth the 'down' direction is to the centre. Naturally the Earth is not quite uniform, indeed variations in the strength and direction of gravity is useful for, say, finding concentrations of minerals. The Apollo missions had variations upon their close-in Moon ( expected ) orbits due to 'mass-cons' or concentrations of mass ( volumes within of higher than average density ) not previously apparent. Until they got there. :-)
( edit ) If it helps, you may want to think of this scenario as involving a flux of gravitons from the shell travelling inwards and focussing onto the test mass. The math & geometry evolve the same .... indeed some of you may be thinking of another well known force with inverse square behaviour. That has two charges ( gravity has one ) and may attract and repel ( gravity only attracts ). Now if you can work out how to plonk all of that ( electromagnetism and gravity ) consistently and accurately under the one umbrella then : you have yourself a Nobel plus more !! :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
What's Happening In The
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What's Happening In The Galaxies ?
Back to good old Doppler again. Get your telescope out and look at a galaxy and measure the frequency shifts of spectral lines across the galaxy. Typically you will find that it varies, meaning the velocity of galaxy parts is different depending on which piece of the galaxy that you look at. Many galaxies look like huge whirlpools and indeed the velocity measurements reflect that. A really big swirling bunch of stars orbiting around what obviously looks like a central accumulation of matter ( most galaxies ).
Now assume that the laws of gravity as disclosed to us hereabouts ( solarsystem ) also apply, then one could take the pattern of velocity as disclosed above and by 'reverse engineering' deduce what amount of matter must be there in order to give that appearance. So this is the inverse of our shell analysis : if a given star is orbiting at a certain radius and at a certain velocity then what matter must be contained within the star's orbit? Do this for all the luminous ( light emitting ) material that you can see in the galaxy .....
..... and one finds that by a very big margin there must be far more matter there than we can see. Specifically instead of the velocities reducing with distance from the centre, by some function approximating Kepler's Laws, the velocities are rather higher and so we have a conundrum to solve. Resolution ?
- assume no change to laws of gravity ( as derived from hereabouts ) for distant galaxies and over the much larger scale ie. galaxy widths. That approach gives you the 'where is all the mass ?' and hence the phrase Dark Matter simply meaning that there is stuff there but we can't see it. Now what could that stuff actually be ? Have we looked hard enough ? What about other modes of detection ?
- assume that what we see pretty much represents what mass is actually there. So we reconcile the velocity distribution by claiming that we must apply a law of gravity different than what we currently understand to be true. That is a harder road for several reasons. Not least is having to come up with that different law which would explain the galaxies, but also not forgetting that such a law would also have to apply to our local neighbourhood too. Put another way : we'd have to admit that the form of the law is scale dependent. By that we don't mean simply that the force of gravity changes with distance, we already know that to be the case, but we mean that the formulation of the law changes with larger distances. That's a bit like saying that accounting standards for a business will change depending upon the size of the business : it's not just that more or less money is involved but that the rules of how you calculate depend on the quantity of money involved.
The door is wide open here for new ideas - suitably verified by observation of course - to plug the breach in our understanding.
For instance there was a fairly recent article in Science ( the AAAS magazine ) outlining Spitzer ( the infrared space telescope ) studies on, well, balls of gas intermediate in size/mass between say Jupiter and the Sun. This is a range where nuclear fusion may or may not happen ( eg. deuterium ) depending on whether the object's core temperature is sufficient. This would hence determine whether it would simply softly glow in the heat generated from gradual gravitational contraction, or is there in addition heat production from fusion? Anyway this study suggests there are quite alot more bodies of that type out there, than were previously expected. So these would presumably be lone bodies, not in some star system as 'planets', but drifting about the place as 'heavy Jupiters'. I give this as an example of the sort of thing that might, if the numbers are right, turn out to be appropriate to fit in as Dark Matter candidates ie. are responsible for keeping the galactic stars orbiting as quick as we see.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
It's occurred to me that I
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It's occurred to me that I haven't especially covered the
Cosmic Microwave Background
which is a relatively recent astronomical discovery. There's considerable subtlety in this topic so I'll start slow.
'Cosmic' and 'background' have expected meanings so I'll define 'microwave' for you. There's this thing called electromagnetic radiation, otherwise known as light or photons. It's what you see with, but that in turn depends on the frequency or 'color' of the light and what you use to 'see' with. 'Microwave' refers to a range of wavelengths corresponding to frequencies of the order of 10^10 per second ( give or take ). The wavelengths are from about a meter down to a millimeter, the term originally/historically meaning 'microscopic' compared to radio wavelengths then used. To coherently produce microwaves one has to have some device that will produce oscillations of about that 10^10 cycles per second. It wasn't until about WWII that gadgets called klystrons* and cavity magnetrons were able to achieve that, for radar and the like technologies. Nowadays you'll find magnetrons performing the role of food and water heaters in kitchens as ovens**, and as I look out my office window today I can see several towers on mountain tops that have microwave relay dishes atop.
One can incoherently produce microwaves rather more easily by having an object of suitable temperature just radiate in that frequency band. Coherence refers to whether there is any phase relationship, or not, b/w various distinct radiation emitters. Phase refers, basically, to the moment of time when one can deem an oscillation to be at a particular part of a cycle. Sort of like 'when do I pass GO ?' in Monopoly, say, as I keep going around the board.
In thermodynamics there's this thing called a black body. This is not anything to do with Dark Matter or Dark Energy. In all likelihood you have never seen one, nor ever will. To be exact it is an ideal/theoretical construct or a scenario which may be closely approximated, but not quite achieved in reality. Indeed the Cosmic Microwave Background ( CMB ) is probably the best 'natural' example to date. This picture illustrates something akin to the idea :
... recognisable as hot ceramic pots within a kiln. If it wasn't for some parts having a cooler profile than others then it would be harder to distinguish 'pot' features here. The bases of the pots are difficult to define, evidently opening the kiln to take the shot has cooled the tops. Of course you've noted that the pots aren't actually black ( I won't be describing the uninteresting & misleading historical reasons for why the word 'black' is used in 'black body' ). As the full technical definition is somewhat horrible, let's stick with a simpler statement : radiation in equilibrium with oscillators. For our kiln above the pots have oscillators - the electrons in the outer shells of the pot atoms - and the radiation is the photons ( mainly of orange color ) flying about in the kiln. Here's an example where it is even harder to define the pot outlines :
.... so those tissue paper looking objects up the back of the oven on the right are actually solid but quite hot ceramics. Again they aren't black. You don't actually have to have solid objects like these examples to have 'black bodies'.
In the late 19th century there was alot of experimenters dealing with insulated hot ovens that had a small hole in the outer wall so that the radiation coming out of the hole could be measured in both frequency/wavelength as well as strength. Bear in mind that back then the concepts of photons, atoms, electrons etc was a very contentious topic indeed. That is : even the existence of such things was no where near as certain or agreed upon as today. So what does 'in equilibrium with' mean ? Next up :
Equilibrium Thermodynamics
( relax, it's not as bad as that sounds )
Cheers, Mike.
* .... and related stuff. When a lad I used to think all scientists hung around these sort of things :
which, on reflection, one ought not get too close to during operation !! Devices with designs descending from these ( though not necessarily known by the klystron tag ) are still used today in, say, particle accelerators. Naturally one can recognise true scientists by their white coats .... :-O
** Curiously early British radar research had a sideline stemming from the question as to whether one could cook/kill a sheep from some distance away. This inquiry was just a subtle way of re-phrasing 'could you kill airmen in enemy aircraft at a distance by using a death-ray' ? I kid you not. :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Correction Whoops. From
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Correction
Whoops. From this earlier post :
.... which is 180 degrees wrong. Sorry. I think I got the axes mixed up on a graphic somewhere. The older/distant redshifts/speeds are less than expected based upon nearby/recent ones. Hence the recession is accelerating.
Equilibrium Thermodynamics
Let's make a cup of hot coffee, then put it in a magical container that lets no energy* in nor energy out. I say 'magical' as a perfect one of these containers doesn't really exist : if you are in a universe you must interact with it. There are no truly force blocking containers ( even a Faraday cage interacts ). But we can have containers which highly approximate this ideal of isolating a hot cup of coffee. I only use that as an example because I'm drinking one now, you could apply this discussion to literally anything up to and including entire universes.
What happens within is that the components of the coffee and cup are going to interact amongst themselves, mainly by pushing and shoving each other about. Suppose over one corner of the cup the atoms generally have more energy than those elsewhere. It turns out that energy gets distributed overall from the energy 'rich' to the energy 'poor', but for a fairly subtle reason. It turns out that there are many, many more ways to be energy poor than energy rich and physical laws tend to proportion/distribute available energy equally amongst those ways. Since energy poor ways are far more numerous than energy rich ones, a sort of 'democracy of ways' applies as energy becomes distributed. We have names for 'energy richer' and 'energy poorer' states : hotter and colder respectively ie. the concept of temperature or average energy of a group of things. These are relative terms of course, but one absolute does apply - there is a temperature which one cannot go lower than ( or strictly speaking even equal ) and that is absolute zero.
[ We have a name for the 'ways' too : entropy is the counting of occupied ways or states. Apart from saying that entropy is maximised at equilibrium, we'll leave this aspect alone. ]
Now it takes time for such distribution to occur. Exactly how long depends upon messy internal details, but we can be sure that if one waits long enough then all that distributing will occur and the temperature will be the same throughout. Well, not quite. Close, but no cigar. It could happen that by a random sequence of events - at a microscopic level now - a group of atoms in the coffee cup might briefly acquire extra energy than the others. Strictly speaking this depends upon what you mean by 'which group' and 'extra', but nonetheless it might happen. However that wouldn't last even if it did, as interactions would dissipate any energy bonus away from the given group. The overall state of the cup of coffee is now deemed to be in equilibrium**. Or to be most exact thermal equilibrium, there being other sorts of equilibrium about which aren't relevant here.
The assumption of isolation is crucial here. If one is shoving energy into, or is sucking energy out of, the system ( cup of coffee here ) then the nett energy distribution amongst system components may never end. But all perfection is broken in this universe and so there are many methods to approximate isolation.
One obvious way is to keep the leaking of energy in/out low by various barriers that aren't very excitable, so to speak. So the barrier doesn't interact much with the system contents inside the container ( which the barrier defines ), nor with whatever is outside the barrier. Note that the terms 'inside' and 'outside' are what we choose to apply to some scenario, and the barrier in detail is indifferent to both. Inside and outside apply because the barrier is shaped to enclose, that's a global property.
Another 'trick' sometimes used is to pick a time interval. Take the explosion of an air/petrol mixture in the chamber of an engine. Choose a time long enough for combustion to complete and the gaseous components to equilibrate, but short enough so that said combustion products do not significantly transmit heat to the walls, or change the chamber's volume by pushing the cylinder head away or opening a valve. This is a 'cheat' of course and truly one could only rely on modelling from that if measurement of predictions agreed.
Above I used the phrase 'mainly by pushing and shoving'. By that I meant more or less up close and personal contact. This is largely an electromagnetic thing that occurs provided the things are significantly close together. But just as one can shout across a crowded room to attract attention ( rather than by messages passed from one person to the adjacent ), one can throw long. Of course both are really electromagnetic in truth, an electron in one atom is going to be playing throw and catch using ( virtual ) photons with electrons in nearby atoms. So it's a matter of scale/length and language used. Here is where gravity may come in to play - thinking of the cosmos now - as we ignore ( rightly or wrongly ) gravity over short distances and low masses.
Cheers, Mike.
* I mean matter here too. As per Einstein : energy and matter are aliases.
** This doesn't mean exactly the same energy per atom/molecule, it's an average or statistical statement that applies to a large group of things. So one atom is never in equilibrium, that's a mis-application of the concept. Equilibrium is a sort of group summary. Also in a stricter sense we can only really talk of temperature when discussing bodies in equilibrium. I have a thermometer in my office which one gently points inside someone's ear canal and it reads out a temperature value. To be exact the device is an infrared radiometer which upon the assumption that it is pointing to an equilibrated body, is calibrated to display the temperature value of such a body. A 'steady state' is not equilibrium ! Steady state is where the overall measured properties of a system remain constant in time, and says nothing about isolation or waiting or entropy or energy distribution. If you like equilibrium = isolation + waiting which ultimately also has constant measured global properties. There's one especially famous area of modern 'science', that I won't name and shame, that shows little evidence of understanding even these basic definitions .... in particular none of the lessons of equilibrium thermodynamics can be applied to any seriously non-isolated system. It is not even vaguely the same sort of cow !!
( edit ) I have also deliberately blurred the distinction between 'macrostates' and 'microstates' ( you may safely ignore what follows ). This is partly a matter of degree in how one chooses to cognitively lassoo and/or label system components. But a deeper point lies within in that many actually distinct microstates may be present 'underneath' a certain macrostate. So things like temperature, pressure, volume and especially total energy content are macrostate parameters : it's what a system looks like overall. But of course in reality each tiny part of a system ( eg. atom or photon ) has particular qualities, the listing of all such qualities as these over all system components is what defines a microstate. This is called fungibility, which isn't a kind of nasty infection, but rather the ability to substitute low level detail and not notice a change at larger scale. So if I was caught throwing a can of baked beans at police in a riot, then there wouldn't be any concern ( in court later ) with the exact arrangement of the beans in the can. One can would be deemed as good as the next in terms of assaulting police. Of course the aggregate/average values over a given microstate will yield a set of macrostate parameters. You also see this, say, in the drawing of lotto balls - the precise ordering of the balls and exactly when etc is not considered relevant for a payout. One approach to defining entropy is to say that it is a numerical measure ( typically involving logarithms ) of the number of microstates per macrostate. Because the numbers are typically horribly large then using logarithms gives briefer notations.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
More Equilibrium
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More Equilibrium Thermodynamics
So let's take some lump of stuff, isolated and in equilibrium, and ask what is the distribution of energy amongst it's components. Even a system of moderate size is going to contain an enormous number of elements. Rather than listing each element/atom/molecule/whatever and the energy that each has ( tedious ), we will flip the question : for a given energy level how many elements in the system have that energy. To be exact we have to talk of energy bands - between such & such energy and a bit more - because of quantum & other reasons. Thus we cut up the energy spectrum into little narrow slices and discuss how many things have the energy lying within each slice. Economists do this sort of thing with groups of people, finding out what fraction of some large population has an income say, within a given band. One can come up with an overall ranking scheme by 'value'.
The maths has been done and dusted for this. On the assumption of equilibrium one can describe how the total energy of the system may be divided up in terms of energy bands. Maxwell and Boltzmann nailed this first. Theirs was a classical analysis ie. before relativity and quantum mechanics, and usually named Maxwell-Boltzmann ( M-B ) statistics. It has an exact mathematical form. This work has been added to over the years, the significant changes being exactly how one goes about doing the counting of things.
[ You may safely ignore this paragraph : It depends on whether one can truly tell the difference between 'distinct' objects. In a classical world we could ( if only as a Gedankenexperiment ) track and label each tiny bit of a big system without upsetting the system sufficiently to change it's equilibrium state. That's the original Boltzmann approach. But quantum mechanics says no, you can't set any arbitrarily lower limit on your interaction with the system ( as a counting observer ). So to count states and remain in equilibrium one has to accept some ambiguity in the counting procedure, because you can't track particles individually now. Depending upon whether the particles are bosons ( which like to clump together ) or are fermions ( which don't ) then different counting techniques are involved. The former distribution is called Bose-Einstein ( B-E ) statistics, the latter Fermi-Dirac ( F-D ) statistics. At high temperatures ( read : high average energy per particle ) the M-B, B-E and F-D approaches converge and give almost identical answers. Indeed one early suspicion, so to speak, of quantum behaviour arose when 'classical' systems were examined at low temperatures. What was discovered was that the systems lost 'degrees of freedom' - or that there should have been more ways to be energy poor than were actually found. Nowadays we interpret this as quantum objects having well spaced discrete energy levels down low near their base/ground states, but up at high energy the gaps b/w energy levels are much closer and effectively continuous. So high temperatures imply that the small but fixed quantum nudges make no significant alteration, relatively, to the energy level populations. Hence B-E and F-D are well approximated by M-B if it's hot enough. ]
Now if one samples the electromagnetic radiation of an equilibrated system then one obtains a special 'black body curve' that gives the amount of radiation at each frequency/energy. This pattern of emission reflects/reveals the underlying equilibrium. From that one can deduce a temperature for the entire system. Penzias and Wilson in the 1960's were studying radio transmission issues b/w ground stations and satellites ( a new technology then ). They found a low level 'hiss' or static in the signals coming from all over the sky. Day and night it was uniform. Even across the seasons. They took great pains to rule out errors and problems in their own equipment. After a chance meeting with an astronomer on a plane trip they deduced that they had tripped over evidence for a prediction made several times over the previous decades : if the Universe began as a 'Big Bang' then there ought be an 'afterglow' of that. The rest is history ..... :-)
A couple of important/interesting points :
- strictly speaking Penzias and Wilson only measured at a single frequency. So they had to assume the radiation they sampled was from an equilibrium 'body' to deduce it's 'effective' temperature. However subsequent - and for that matter prior - measurements gave several other frequencies. The COBE satellite/experiment nailed the whole curve to a high degree of precision thus satisfying our confidence that we are indeed looking at an equilibrated thing. Later on WMAP gave even better results. The Planck mission is currently in play as we speak.
- or should I say nearly in equilibrium because the pattern across the sky shows slight variation of the order of one part in ten thousand from different directions. Some of that is specific objects and regions - certainly not in equilibrium - that radiate in the frequency band of interest, polluting if you like our studies. Such extras can be studied, understood and accounted for by subtracting away from the whole picture. You've probably seen various graphics on this, usually under the casual/throwaway heading of 'Face of God' or somesuch ( thanks to George Smoot ).
- something in perfect equilibrium will have no structure or contrasts. That's why I emphasised the difficulty of distinguishing the features of the pots and stuff in the kilns earlier. A pot in true equilibrium would have to be a mist of atoms floating around in the kiln ie. no pot structure or shape at all ! :-0. Most of us will never see anything even close to equilibrium, because we live in a world with structure. We ourselves have structure. Our entire planet is well away from equilibrium ( seriously non-isolated ), fortunately for us !! :-)
- why do we speak of 'counting' things ? What's that got to do with their overall group behaviour ? Well, in a sense the elements of each system count each other. They, by mutually interacting, sample each other's characteristics like energy, momemtum, spin etc. Indeed a large group of things at equilibrium may well have some fixed number/fraction within some energy band, but it's not going to be the same actual particles from moment to moment. Like a very highly socially mobile society every particle is perpetually exchanging their characteristics, energy included. Recall that energy, momentum, spin ( angular momentum ) and the like are all conserved quantities provided no 'leaking' in or out occurs. Which is why we insisted earlier upon isolation as a criteria for equilibrium. So 'counting' is an intellectual mnemonic for 'assessment of true interaction'.
Next up :
So Where & When Did The CMB Originate ??
[ I ought add : if any of you are keeping up with this description/narrative, then you are doing exceedingly well. It is quite thick porridge, definitely non-trivial theory, a complex edifice for sure. Conversely, if you are having trouble : don't panic, just skim and grasp the highlights. ]
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal