I’ll be away surfing in Biarritz next week, so I’ll leave readers with a puzzle to mull over. Nigel Cook’s comments on the post below reminded me of a slight problem I have with Hubble’s Law. The problem is laid out below: the challenge is for anyone to supply a straightforward answer in simple language (damned if I can).
As every schoolgirl knows, Hubble discovered that distant galaxies are moving away from us (or any other point) with a velocity that is proportional to their distance. This is the crux of the evidence for the expanding universe, not to mention a major piece of the evidence for the Big Bang.
The law arises from experimental observation and is usually written as
v = Hd
where v is the recessional velocity of a galaxy, d is the displacement of the galaxy from us and H is the Hubble ‘constant’, or the slope of the graph.
(Note that relativity predicts that it’s really space that’s expanding and the galaxies ride the wave, but this doesn’t affect the question coming. We can also ignore the fact that there is a correction factor for the time it takes light to reach us).
Every physicist reads this law as v1/d1 = v2/d2 =v3/d3 = H and it works fine. However, consider what Hubble’s Law says about any one particular galaxy. The equation clearly implies that the velocity v of a galaxy A (relative to some point) is proportional to its displacement d (relative to that point). But for non-zero velocity, the displacement d must be changing in time – therefore Hubble implies that the galaxy’s velocity is also changing in time – which is another way of saying that galaxy A is accelerating!
So there’s the puzzle: Does Hubble’s Law predict that distant galaxies are not just moving away from us, but accelerating? On the face of it, it does. If so, then a force must be acting. Hmm. Suspect the equation is misleading. After all, why all the fuss/surprise about the recently observed acceleration of the universe expansion? According to the logic above, it must be accelerating..
P.S. The question can be framed in terms of basic mechanics – surely any object that has a velocity that is proportional to its displacement it must be accelerating?
Antimatter 
um, Hubble’s constant depends on time?
Erm, no … think of it as a bomb that explodes in deep space (i.e. vacuum + zero gravity). When the bomb explodes, all the future fragments are in the same place. Afterwards the distance of the fragments from the site of the explosion is proportional to the velocity they set off at, so if you observe at a later time T the distance from the initial point d=vT. So the “Hubble constant” for this is just the inverse time since the explosion. And, because of vector addition of velocities, one gets the same answer in a co-ordinate system comoving with any fragment.
Or am I missing something here?
(so it is Hubble’s “constant”, not the speed of the galaxy that is changing)
Dr Chris Oakley:
Thanks! Taking your last point first, the flat universe cosmology has
t = 1/H
or:
H = 1/t
so it does suggest that the Hubble “constant” is falling as the universe expands. But Hubble’s constant is not an time-independent constant, but merely
v/r = H
= 1/t
in flat cosmology, where t is age of universe.
So v is only a “constant” with respect to r as far as Hubble was concerned. H is not a variable as observed in the Minkowsky flat spacetime metric of the universe we see.
Taking
v/r = H = 1/t,
your argument (‘so it is Hubble’s “constant”, not the speed of the galaxy that is changing’) suggests that v is constant and in
v/r = 1/t
each denominator (r and t) is increasing in proportion.
That’s a nice simple idea. Unfortunately, it’s completely wrong, because the time t in this formula is the age of the universe, whereas r = cT where T is time past.
The age of the universe, t, is not proportional to time past T = r/c. E.g., the closest star, the sun is T = 8.3 light-minutes away, but the universe is not t = 8.3 minutes old. (We’re just seeing the sun as it was 8.3 minutes in the past.) So you can’t set v/r = H = 1/t and try to get rid of a changing v by saying that r is proportional to t!
Now for your earlier comment above about the explosion analogy. This particular explosion analogy has been tried and criticised. In about 1931, when initial attempts were being made to understand the Hubble law (before the Friedmann-Robertson-Walker metric was dogma), people like Lemaitre were suggesting that the universe was like an explosion in a pre-existing space.
A letter appeared in Nature in I believe 1931 (I think you will find the details discussed in Eddington’s book The Expanding Universe) pointing out that the Hubble law is not what you expect from say a bursting bomb.
Just before the bomb explodes, the compressed hot gas of explosion debris will have a Maxwell-Boltzmann distribution of velocities, which is skewed so that most of the molecules have low velocities, and the above the peak there is a long tailing-off to a small number of molecules with very high velocities: see http://www.webchem.net/notes/how_far/kinetics/maxwell_boltzmann.htm or http://theory.ph.man.ac.uk/~judith/stat_therm/node85.html
The letter in Nature pointed out that the Hubble distribution is quite different, it is in fact an anti-Maxwell-Boltzmann distribution. In the universe, the greatest number of galaxies have the greatest recession velocities, which is contrary to what the Maxwell-Boltzmann distribution predicts for molecules from a bursting bomb.
‘Afterwards the distance of the fragments from the site of the explosion is proportional to the velocity they set off at, so if you observe at a later time T the distance from the initial point d=vT. So the “Hubble constant” for this is just the inverse time since the explosion. And, because of vector addition of velocities, one gets the same answer in a co-ordinate system comoving with any fragment.’
There’s a graph showing the Hubble law for distribution of air molecule speeds behind the shock front in an explosion, in a paper by Sir G. I. Taylor, ‘The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945’, Proceedings of the Royal Society of London, Series A, v. 201, pp. 175–186.
However, that’s an air burst detonation, not an explosion in space. The die-hard general relativists who believe in curved space (even though the universe has been shown to be flat, and curvature is anyway just an approximation to a lot of quantum graviton interactions), will tell you (falsely) that spacetime in the universe curves back on itself at great distances, so any effort to model the big bang as some kind of explosion is automatically void.
I’m not sympathetic with those people who want to use the authority of popular speculations to rule out the simplest possible physical model. Within seconds, the big bang universe became mainly compressed, ionized hydrogen gas. As in a H-bomb, fusion occurred in the extremely high temperature and pressure but was not complete; the expansion of the universe quenched the fusion rate by reducing the temperature and pressure before all of the hydrogen could fuse into helium, etc.
If you get away from the curved space of general relativity, and move on to a model of gravitation where accelerations result from the exchange of gravitons in discrete interactions (so spacetime curvature is just an approximation for the effect of a large number of discrete interactions), then it might make sense to try to model the late stages of the universe as a 10^55 megatons H-bomb in a pre-existing space. But this won’t address the very early-time physics (less than 1 second after the big bang), where the energies were initially so high that the binding energy of hadrons was trivial by comparison, so there was a quark soup instead. Also, it will annoy all the pacifist physicists who don’t like the morality of using an explosion as any kind of analogy to the big bang. Finally, general relativity indicates that space was created with the big bang, not pre-existing (this is because spacetime is defined by the gravitational field, so where you don’t have such a field there is no spacetime).
I find these arguments vacuous because general relativity is just a classical continuous-field line model for gravitational fields. Because of the clever way it incorporates conservation of mass-energy for fields (something which Newton’s gravity law ignored), it makes checkable predictions that differ from Newtonian gravity, and which is correct. But the evidence supporting general relativity is just supporting the inclusion of conservation of mass-energy by general relativity, it isn’t specifically supporting the classical curved spacetime model of general relativity. Curved spacetime at best is just a classical approximation to many discrete graviton interactions. Sum the Feynman diagram interaction histories for many graviton interactions, and you get something that approximates to the spacetime curvature of general relativity.
A simple physical way to get the observed cosmological acceleration out of big bang is by spin-1 graviton exchange between masses. All masses have the same gravitational charge (say positive gravitational charge), so they all repel by exchanging gravitons. The repulsion makes masses accelerate away from one another, giving the Hubble law. The same effect predicts gravitation with the correct strength (within observational error bars).
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Hello Cormac: I found you via our mutual link at Tommaso’s blog. As you can see from my writings, some “constants” are not constant. Hubble’s constant decreases in inverse proportion to age of the Universe, H ~ 1/t. As a galaxy recedes, gravitation would cause it to decelerate as seen from our frame.
As for an “accelerating” universe, that observation depends entirely on the redshifts of Type Ia supernovae. Many astrophysicists are not sure their luminosity is constant. For instance, the metallicity of early supernovae is known to be different. My previous guest post at Tommaso’s blog was about Inconstant Constants.
There is a standard metaphore for this: dots (or little pictures of galaxies, if you will) painted on a balloon. Inflate the balloon so its radius grows at a constant rate. Pick some dot A and measure its distance to some other dot B (along the shortest path on the surface of the balloon, not through it). It also grows at a constant rate.
Now pick a third dot C which just happens to be located on the continuation of that shortest path from A to B, at an equal distance from B as B from A. Diagrammatically: A—B—C. Clearly, by the symmetry of the situation, dot C must be receding from B, and at the same rate as B is receding from A.
Now add it all up: B recedes from A at a constant rate; C recedes from B at the same constant rate, and in the same direction. What does that amount to? Yes: C recedes from A at twice the rate that B recedes from A, because it is twice as far away.
So a constant rate of growth of the diameter of the balloon makes the dots on its surface recede from each other at a rate proportional to distance.
In cosmology, the “balloon” is a 3-sphere, i.e. it has one more dimension than the balloon in the example (which is a 2-sphere, i.e. an object with a 2-dimensional surface) but the math works the same way. If the concept of radius of a 3-sphere makes your head spin, just think of it as a scale factor. That’s what everybody does.
The surprise about the purported accelerated expansion, which is far from an established fact (no matter what some or even most cosmologists would have you believe) is about this scale factor. In the standard big bang cosmology of yore, its growth was supposed to be decelerating due to the gravitational pull between galaxies. If you believe the current concordance model, it is actually growing faster as time goes by.
‘As for an “accelerating” universe, that observation depends entirely on the redshifts of Type Ia supernovae. Many astrophysicists are not sure their luminosity is constant.’ – Louise Riofrio
Louise,
In addition to Type Ia supernovae, gamma ray bursters (stars collapsing into black holes) also provide alleged evidence of “acceleration”, albeit an “evolving dark energy” (changing cosmological constant), see the plotted gamma ray burster data at: http://cosmicvariance.com/2006/01/11/evolving-dark-energy/
However, as Nobel Laureate Philip Anderson has argued,
‘… the flat universe is just not decelerating, it isn’t really accelerating …’ – http://cosmicvariance.com/2006/01/03/danger-phil-anderson/
The radial recession of galaxies in the Friedmann-Robertson-Walker metric of general relativity is the Hubble recession law with a gravitational slowing down at large distances due to attraction to the mass centred around us (in our frame of reference, which is the frame of reference we’re observing the universe from).
The 1998 results of Perlmutter on Type Ia supernovae suggested that this metric is wrong because there is no observable gravitational slowing down of the expansion.
So the mainstream knee-jerk response was to say that the inward-directed gravitational acceleration (which is only important at large distances, i.e. immense redshifts, according to general relativity) must be cancelled out by an outward-directed acceleration on the order 10^{-10} metres/second^2. The outward-directed acceleration would be due to a universal repulsion of masses, i.e. a small positive cosmological constant.
However, this explanation makes quite a lot of assumptions. As mentioned, gamma ray burster data indicate an evolution of the cosmological ‘constant’. There is also the idea that gravity gets weaker over cosmological distance scales. If you believe that gravity is due to spin-2 gravitons being exchanged directly between the masses which are attracting, then for redshifted masses the gravitons can be expected to be redshifted and this received in a degraded energy condition, weakening the gravity coupling constant G from that measured in a lab where the masses are not relativistically receding from one another.
But if gravitons have spin-1 rather than spin-2 (and thereby act by pushing masses together instead of pulling them together), the spin-1 graviton exchange actually causes the cosmological acceleration as well as gravity. This makes checkable predictions.
I think one thing to be made clear is that a gravitational field can exist without something actually accelerating. I’m sitting in a gravitational field of 9.81 metres/second^2 and I’m not being accelerated downward, because there’s a normal reaction force from the chair that stopping me.
It’s exactly the same with the cosmological acceleration:
1. The most distant receding galaxies, supernovae and gamma ray bursters etc have an inward-directed gravity-caused acceleration towards us observers on the order 10^{-10} metres/second^2. (This effect is similar in nature to the gravitational slowing down of a bullet fired vertically upward.)
2. Such distant receding matter also has an outward-directed ‘dark energy’ (spin-1 graviton, I argue) caused acceleration away from us on the order of 10^{-10} metres/second^2.
The outward cosmological ‘dark energy’ acceleration cancels out the inward gravitational acceleration, so there is no net acceleration.
This is what Philip Anderson meant when he wrote:
‘… the flat universe is just not decelerating, it isn’t really accelerating …’
In my case, I’m not being accelerated downward by gravity because that acceleration is being cancelled out by an equal upward acceleration due to electromagnetic repulsion of the electrons in me and the electrons in my chair.
In the case of the universe, the cosmological acceleration of the universe is being cancelled out by gravitational attraction.
“In the standard big bang cosmology of yore, its growth was supposed to be decelerating due to the gravitational pull between galaxies. If you believe the current concordance model, it is actually growing faster as time goes by.” – SomeRandomGuy
The Friedmann-Robertson-Walker metric of general relativity up to 1998 predicted that the expansion rate is slowing down, and that this should be observable at extreme redshifts.
Perlmutter simply found that the expansion rate isn’t slowing down in his observations in 1998. The mainstream interpreted the lack of gravitational (inward-directed) deceleration to imply that gravity is being cancelled out by an outward-directed cosmological acceleration due to some unknown dark energy.
The universe isn’t actually accelerating; there is an acceleration field which isn’t causing matter to accelerate because gravitational attraction is cancelling out that outward acceleration (see Nobel Laureate Phil Anderson’s criticism of “acceleration” on mainstream cosmologist Professor Sean Carroll’s blog as I quoted it in the previous comment; Sean didn’t repudiate this point!).
The cosmological acceleration is a small but universal repulsion between masses. Gravitation is a universal attraction between masses. On cosmological distance scales these two opposite accelerations cancel one another out, so there is no net acceleration of matter.
“Does Hubble’s Law predict that distant galaxies are not just moving away from us, but accelerating?”
Yes, where “accelerating” here is taken to be “change in redshift of object with respect to proper time of an inertial observer.”
“On the face of it, it does. If so, then a force must be acting.”
No. Three ways to see it:
1. the definition of acceleration above is rather a strange one, and only reduces to the ordinary definition in flat spacetime.
2. even worse, the definition gives different accelerations for different inertial observers (and the differences are not related by a Lorentz transformation — the relativistic generalization of the Galilean transformations.) But Newton’s laws are invariant for inertial observers.
3. Newton’s law is modified in curved spacetime — it becomes the geodesic equation and thus, for example, an apple “accelerating” towards the Earth is actually not accelerating at all.
Nigel #9, the concordance model definitely features accelerated expansion, in the sense of a FRW scale factor growing as time^k with k > 1.
Whatever Phil Anderson (a great solid state physicist, by all means) had in mind, it was not the concordance model. Maybe he was thinking of the linearly coasting cosmology proposed by Kolb back in the 80s, which also fits the supernova data well.
Sean Carroll is not a professor, he is a “senior research associate”, whatever that means at CalTech.
SomeRandomGuy,
The Lambda-CDM model, which is the mainstream model now (the Cold Dark Matter model with a small positive cosmological constant lambda worked out from the data), involves a repulsive force that increases as a function of distance.
The gravitational deceleration effect decreases with increasing distance.
Therefore at small distances, gravitation predominates, then at a larger distance (on the order 5*10^9 light years) the acceleration of the universe cancels out gravitational deceleration on receding matter, and at still greater distances the cosmological acceleration exceeds gravitation.
The observations of red-shifts made so far from Type Ia supernovae and gamma ray bursters are concentrated in the region where cosmological acceleration (repulsion) is cancelling out gravitational acceleration which is trying to slow the expansion of the universe.
For bigger distances than have currently been well observed, the mainstream Lambda-CDM model suggests an overall net acceleration outward. Whether the model is right is another matter (see the evidence from gamma ray bursters which suggests that the value of Lambda isn’t a constant: http://cosmicvariance.com/2006/01/11/evolving-dark-energy/ ).
This acceleration as an extrapolation from the Lambda-CDM model isn’t a fact or even a scientific prediction, because it’s not really a falsifiable prediction because the small positive Lambda/cosmological constant value is already an ad hoc modification, not a piece of genuine scientific prediction. You can endlessly introduce ad hoc ‘epicycle’ type corrections into a model to make it fit unpredicted effects. Whatever new data comes, they can just find a formula to fit Lambda’s variation (if any) as a function of redshift/distance. But this isn’t real physics.
The real physics concerns the nature of the dark energy. It’s spin-1 gravitons, causing universal repulsion between similar gravitational charge; this causes the cosmological acceleration and also pushes relatively small masses towards one another because they exchange gravitons more forcefully with the large masses in the distant universe than with one another. Both are checkable predictions.
Spin-2 gravitons don’t lead to any checkable predictions. Firstly spin-2 gravitons are based on the bizarre idea that mass A and mass B are solely exchanging gravitons with one another, and not exchanging gravitons with every other mass in the universe, including the immense masses at great distances. Secondly, spin-2 gravitons seem to need some incredibly ugly and extravagant theoretical framework such as string theory with 10 dimensions. (String theory is inherently vague because 6 dimensions are supposed to be too small to probe experimentally, so nobody knows their moduli if they are compactified in a Planck scale Calabi-Yau manifold. Without knowing all the parameters of these extra dimensions, you can’t make falsifiable predictions.)
Nigel #12, you should not rely on blogs for observational data. At the very least, you should check the literature for updates when looking at old posts. Regarding Schaefer’s work on high-z GRBs, he changed his mind about them less than a year after the post you link to. See
http://arxiv.org/abs/astro-ph/0612285
which concluded that “the Dark Energy can be described well as a Cosmological Constant that does not change with time”.
As an aside, your use of the term “acceleration outward” suggests that you do not yet understand FRW. You really seem to be using a mental image of an ordinary explosion in 3-dimensional space.
Your claim of repulsive “spin-1 gravitons” is of course completely unsubstantiated.
SomeRandomGuy,
Thanks for that up to date reference. Normally blog posts are updated or have trackbacks from new posts when updates are made, which makes them more dynamic than the usual literature not less so, but as you point out this is not so for Cosmic Variance. (I’ll avoid linking to posts at Sean’s blog from now on in case they become obsolete and are not updated by a trackback.)
It’s interesting that the latest gamma ray burster data is compatible with an unchanging cosmological constant!
“As an aside, your use of the term “acceleration outward” suggests that you do not yet understand FRW. You really seem to be using a mental image of an ordinary explosion in 3-dimensional space.”
See https://coraifeartaigh.wordpress.com/2008/08/12/cosmological-distance-at-trinity-college/#comments
We’re seeing earlier epochs with increasing distances, which is quite different from a purely 3-dimensional Euclidean space. As I explained there, flat spacetime is Minkowski spacetime, where you time changes with distance. This is why the variation in recession velocity with “distance” that Hubble reported is also a variation of velocity with time (conventionally referred to as acceleration).
“Your claim of repulsive “spin-1 gravitons” is of course completely unsubstantiated.”
There is scientific evidence to back it up: http://nige.wordpress.com/2008/01/30/book/
The cosmological acceleration is a univeral repulsion of masses suggesting a spin-1 mediator, and differentiating Hubble’s law, a = dv/dt = d(Hr)/dt gives you a way of making solid predictions of forces, which predicts that the same spin-1 mediators that cause cosmological acceleration also produce gravitation. This is simple physical calculations ujsing long-established empirical laws and observations. It predicted the cosmological acceleration of the universe in a publication in 1996, two years before observational discovery by Perlmutter!
The same calculation predicts gravity parameter G accurately. It doesn’t contain any speculations, unlike string theory. It’s already made falsifiable predictions and been vindicated. So it does seem scientifically accurate, although string theorists have censored it out!
I’m sorry Nigel, but in terms of the balloon analogy, the comment which you linked to confirms that you are thinking of the “acceleration” of the dots, as measured along lines on the surface of the balloon. Cosmologists who discuss accelerated expansion are thinking of the growth of the balloon’s radius, i.e. of the scale factor.
Let me put it this way: galaxies which are further away from us recede faster because there is more space (on the balloon surface, that would be more rubber) between us and them; since the whole effect is caused by the expansion of space, the rate at which it separates two objects is directly proportional to the amount of space between the two objects.
Again, this has nothing to do with what cosmologists mean by accelerated expansion.
SomeRandomGuy,
No, I’ve never mentioned a balloon analogy!
Your statement that ‘the whole effect is caused by the expansion of space’ is vague enough to be compatible not only with what you’re thinking (the idea that the vacuum is powering cosmological expansion) but is also compatible with the physics I’ve stated: spin-1 gauge boson exchange between masses in the vacuum causes a repulsive force, giving rise to expansion of the universe (recession of masses) over large distances and pushing matter together on small scales.
Cosmologists don’t know what dark energy is so they don’t mean anything by accelerated expansion, apart from what I explained.
I.e., the cosmological acceleration is an outward radial acceleration (acceleration is a vector so it has direction and can be either outward or inward towards us when it is ascertained from redshifts). It’s needed – as far as cosmologists are concerned – to make the observed redshift data compatible with the predicted gravitational deceleration of the universe which is based on the density of the universe.
If the universe had a high enough density, the gravitational deceleration would not merely slow down the expansion but would cause the universe to begin contracting at some point in the future. I think it’s important to understand that the gravitational deceleration is always a vector, represented by arrows pointed radially inwards towards the observer. The acceleration of the universe by ‘dark energy’ is an acceleration outward, a vector represented by arrow pointed radially away from the observer. Hence the two accelerations oppose each other.
This physical explanation makes clear what is going on. Ideally the quantitative magnitude of the acceleration needs to be explained to people, on the order of 10^{-10} metres/second^2. If all this had been done when the acceleration was discovered in 1998, then there would be less confusion today. E.g., differentiate the Hubble law (v = Hr) and you get
a = dv/dt = d(Hr)/dt = H*dr/dt + r*dH/dt = Hv + 0 = rH^2.
This predicts the acceleration of the universe quantitatively. Mainstream cosmologists can’t predict this, so they don’t really ‘mean’ anything about acceleration. This was predicted in 1996, two years before being confirmed, while I was doing a cosmology course at university. Further calculations predicted gravity accurately. This model is observationally confirmed and predicts not only cosmological acceleration but also gravity accurately, using spin-1 gauge bosons. It debunks spin-2 graviton speculations, which are physically vacuous.
Nigel,
You make me think of one of those electronic parrots that listens to conversations and then repeats chunks of them back at random. Reading popular science books, or reading chunks of scientific papers without understanding them does not make you a physicist. You do not even have a bachelor’s degree in physics, and all your copious writings on the internet do is to increase the noise level. Instead of wasting everyone’s – and your own – time getting involved in discussions that are above your head you should spend time studying the subject properly instead. You may even be able to get someone to sponsor you.
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Hi Chris,
I did two years of undergraduate physics before switching disciplines to IT and graduating in programming. I think this is pretty relevant to your comment.
With regard to your advice, I think in comment 4 above I make it clear that your argument that the Hubble constant is varying is misplaced. This isn’t electronic parrot talk.
I’ve read your own page, and while you are justifiably proud of your Oxford PhD in theoretical physics, I don’t see any falsifiable predictions in your papers. Could you point them out please, if there are any, so maybe we can discuss the merits of your research, since you find my argument to be so much noise?
Maybe I can make my key point from comment #4 above clearer to you as follows:
Hubble law: v/r = H
where H = 1/t,
Chris Oakley argument: v/r = H = 1/t,
hence: v/r = 1/t
therefore (according to Chris Oakley in comment #3): v is constant and r is directly proportional to t. I.e., your argument is that both denominators in v/r = 1/t vary the same way.
This is wrong because r is not directly proportional to t (age of the universe).
Instead, r is merely directly proportional to time past T = r/c.
Actually 1/t is constant on the right hand side, because t is the age of the universe for the observer, not for the star being observed.
“Instead of wasting everyone’s – and your own – time getting involved in discussions that are above your head you should spend time studying the subject properly instead.”
I’ve been studying physics in my all my spare time since as far back as I can remember. I don’t think that your comment is addressing the point I made, maybe you can do so? Have you actually studied modern cosmology, or just mathematical quantum field theory?
Dr Chris Oakley’s site includes an interesting partly completed brief textbook on quantum field theory or rather “relativistic quantum mechanics”: http://www.cgoakley.demon.co.uk/qft/RQM.pdf
It’s good to have this kind of mathematical material on the Poincare group and other topics for reference, and the criticisms of the existing treatment of quantum field theory on page 1 begin clearly enough:
“… We are therefore implying the behaviour of the more comprehensive quantum world from the far less general classical world, which clearly is the wrong way round. …”
Quantum field theory should reduce to classical physical for large numbers of quantum interactions. I agree that the traditional approach is the wrong way around. Unfortunately, Chris continues:
“2. The traditional approach makes extensive use of the interaction picture, which does not exist (Haag’s
theorem).
“3. The result of subtracting infinity from infinity is indeterminate. If one ignores this fact one will produce
theories of no scientific value.”
Well, Chris, if you use Haag’s theorem to throw out the quantum interaction picture from quantum field theory, you’re left with just a sea of mathematics unlinked to clear pictorial physical processes. I’ve always found that the less physical the mathematical model (the less tied down to real physical facts like interactions), the more likely people are to end up digging themselves into a hole and not producing any checkable predictions. Equations that aren’t linked directly to hard physical facts end up having a landscape of interpretations, which is surely the problem with string.
Renormalization of charge etc. makes quite a lot of sense physically: in strong fields the vacuum contains pairs of virtual fermions which polarize, absorbing field energy and weakening the field which creates them. This reduces the apparent electric charge of a fermion as seen from a long distance. The naive mathematical model for running couplings based on this variation of charge is a simple logarithmic expression which would make the bare core charge of the electron become infinite at zero distance, so a cutoff must be imposed on the variation of the charge to get a realistic bare core charge to use in predicting the magnetic moment of leptons and the Lamb shift. This is physically justifiable if leptons don’t have zero size. No vacuum pair production can occur within the physical size of a lepton, so it’s quite natural to have a cutoff on the running coupling corresponding to the smallest physically sensible distance.
“Although it covers issues normally associated with quantum field theory, this treatise has the title “relativistic quantum mechanics” on the grounds that the quantum field is not viewed as fundamental here, being derived instead from
annihilation and creation operators, which in turn are defined as operators on Fock space. It will be shown that the presence of interactions does not invalidate this analysis. Interactions will be seen, in effect, to be just interference patterns between free field states.”
That paragraph (again on page 1 of your book) indicates that you’re sure that the quantum field isn’t fundamental, and that quantum interactions are just interference patterns. You need to prove that assertion if you’re really claiming this. Your treatment just seems to show that it’s possible to treat interactions as interferences; i.e. at best you just have an alternative way of representing physics. My argument is the opposite of yours. Rather than moving away from Feynman diagram quantum interactions to greater mathematical abstraction, Feynman diagrams should be taken even more literally than they currently are. E.g., virtual radiations can be treated for some purposes like real radiations (after all they cause all known forces). If the boots fit, wear them. If you can get falsifiable predictions out of simple well established physics, why attack it? Having said that, I’ll read your new book draft carefully and update my domain accordingly ASAP.
The Wikipedia page on Haag’s theorem (which Dr Oakley has edited) links to Haag’s major paper: http://doc.cern.ch/yellowrep/1955/1955-008/p1.pdf
I wil read Dr Haag’s paper when time permits in addition to Dr Oakley’s draft book, because it seems obvious to me that the point Dr Oakley makes about quantum field theories is that Haag’s theorem has negated the physical relevance of the interaction picture (so my work building entirely on quantum interactions sounds is just unwelcome noise to his ears).
I wonder if anyone knows if Feynman made any comments about Haag’s theorem, since it is allegedly a disprove of the physical relavance of the interactions Feynman’s diagrams represented?
Is Haag’s theorem supposed to debunk the quantum field interactions of the weak field gauge bosons discovered at CERN in 1983? Are those gauge bosons just mathematical “interferences in free field states”? Is this really a helpful physical way of looking at the different fundamental interactions?
(Apologies for the typing errors above.)
To restate briefly Dr Oakley’s problem:
Hubble law: v/r = H
where H = 1/t,
Chris Oakley argument: v/r = H = 1/t,
hence: v/r = 1/t
where Dr Oakley suggests that r is proportional to t, so v doesn’t vary: “(so it is Hubble’s “constant”, not the speed of the galaxy that is changing)” comment #3.
This is wrong since, as you look to bigger distances (r) you are seeing smaller times (t) after the big bang.
So r definitely is not proportional to age of universe t. In fact, if the age of the universe is that of the observer’s frame of reference, t is fixed at 13,700 million years. Hubble’s point was that v/r = constant = H, regardless of how far away (back in time) you look. This is why I feel that Dr Oakley’s comments (comments #2 and #3) above are in error.
If an Oxford PhD/D.Phil in quantum field theory can make such an error in looking at the very basics of cosmology, and then come back with personal comments ignoring the science, you can see the problem in communicating the fact that there is an acceleration inherent in the Hubble law!
Nigel,
I can see why people don’t like to get into arguments with you, and unless something changes radically – like you getting a B.A. in physics – I expect that I am going to kick this habit as well. Hubble’s law is a fit. The velocity of galaxies being proportional to their distance roughly fits the observational data. Forget relativity, deceleration and the other stuff: the simplest explanation for this is that the galaxies were all at the same place a time 1/H ago and then flew apart at different (constant, but different) velocities in different directions. Velocity is obviously not the same for all galaxies but t = 1/H is. In other words
v_i = H r_i
where v_i is the speed of galaxy i and r_i is its distance.
which brings us neatly back to the original question…
I thought I answered that in #7… but I suppose I should give some credit to #1, too. If the FRW scale factor (the radius of the balloon) is a(t), the Hubble “constant” is
H = (da/dt)/a
so it is indeed not a constant at all over cosmological timescales. In #7, I used a constant rate of growth for maximum simplicity. In that special case, a = a_0 t, so
H = 1/t
and so
v = H D = D/t
i.e.
dD/D = dt/t
so D is proportional to time; there is no acceleration.
But you can try a more general a = a_0 t^k to get
H = k t^(k-2)
and so
dD/D = k t^(k-2) dt
which integrates to
D ~ exp((k/(k-1)) t^(k-1))
So for instance with k = 1/2 (radiation-dominated universe)
D ~ exp(-sqrt(t))
which decelerates (this remains true for any k < 1).
So the answer to the original puzzle, “Does Hubble’s Law predict that distant galaxies are not just moving away from us, but accelerating?” is:
No, unless the scale factor itself is accelerating.
P.S. Last eq. in #27 should be
D ~ exp(-1/sqrt(t))
Sorry, feel free to fix it.
“I can see why people don’t like to get into arguments with you, and unless something changes radically – like you getting a B.A. in physics – I expect that I am going to kick this habit as well. Hubble’s law is a fit. The velocity of galaxies being proportional to their distance roughly fits the observational data.” – Chris Oakley
You haven’t mentioned the scientific facts at all, you’ve “argued” with me by making personal comments which miss the scientific facts. You’ve ignored entirely everything that I wrote in comment 24.
Look, saying “Hubble’s law is a fit” which we all know doesn’t tell us anything new, because that’s been known since Hubble did it in 1929. I’ve done cosmology and quantum mechanics courses, and I’m studying quantum field theory as time permits.
Hubble’s law: v/r = H. Rearange: v = rH. Differentiate it and you get an acceleration, from the calculus
dv/dt
= d[Hr]/dt
= H(dr/dt) + r(dH/dt)
= H(dr/dt)
= Hv
= H(Hr)
= rH^2
~ 6*10^(-10) ms^(-2) at extreme redshifts approaching the horizon radius.
This was predicted in 1996, and confirmed in 1998. The calculation above was published before confirmation. I’m very well aware (not just from hostility I receive) that it is not in the textbooks, because if it was well known, I wouldn’t need to point it out to people. I’m not pointing this out because I think it’s well known, but because it isn’t well known. It’s counter intuitive which is why it’s not mainstream thinking. If it was totally obvious, someone else would have predicted the acceleration of the universe this way before me. Yet it’s been borne out by factual evidence. It also leads to a simple prediction of the strength of gravitation, which again turns out to be accurate!
SomeRandomGuy,
Thirteen years ago I did a cosmology course that included finding solutions such as the Friedman-Walker-Robertson metric and everything else. General relativity doesn’t predict the small positive cosmological constant to fit te observed expansion of the universe. This approach does.
See my calculation in comment 29 above and note it was done in 1996 two years before Perlmutter’s observations of distant supernovae confirmed it. It’s a different calculation from the Friedmann-Robertson-Walker metric. I work from empirical laws towards falsifiable predictions. This isn’t mainstream fundamental physics, which involves starting with a mathematical speculation like general relativity (or even string theory) that can model just about any universe, and fitting the model to the results of observations ad hoc. That’s one reason why I dropped physics and don’t want a degree in physics. Another reason is the general hostility, prejudice, etc., you receive when trying to have a scientific discussion. It’s not particularly healthy. I’m not claiming to be a paid-up member of the orthodoxy, I’m just pointing out facts that made falsifiable predictions which were confirmed.
Nigel #30, you claim knowledge of FRW, but in #29 you implicitly set dH/dt = 0. As I pointed out in #27, in FRW
H = (da/dt)/a
so for instance with a = a_0 t (linearly growing scale factor)
H = 1/t
As I also pointed out in #30, for a more general
a = a_0 t^k
you get
H = k t^(k-2)
In short, H is NEVER constant in FRW, unless you set k = 2.
Maybe your cosmology course got far enough for you to know that
k = 2/(3*(1+w))
where
w = p/rho
(pressure over energy density) is the equation of state parameter, and as you can see, k = 2 would require w = -2/3, i.e. negative pressure (ah, the joys of inflation).
Be that as it may, I’m sure you can see that k = 2 implies an accelerating scale factor (as does any k > 1) so your “proof” of acceleration is just a roundabout way of saying that the scale factor is accelerating.
This is of course completely consistent with everything which you have already been told.
You are quite right to say that GR does not predict a small cosmological constant. GR says nothing at all about the cosmological constant, other than that it is allowed (at the classical level). It also says nothing about w (the equation of state parameter); that’s a property of matter.
But you are quite wrong to call GR “a mathematical speculation”. GR is a mathematically consistent phenomenological “law” which has been amply confirmed by experiment in the range accessible to us (solar system tests, equivalence principle) but may not be working on very large scales. We don’t know for sure, but if it is to be substituted with something else, that “something else” must at the very least be able to reproduce the successes of GR in the verified regime.
With all due respect, postulating dH/dt = 0 falls a little bit short of that.
SomeRandomGuy,
I did a course in general relativity. Let me explain this to you and also to Chris Oakley very clearly.
Hubble noticed that the ratio v/r = constant = H.
In other words, the velocity of recession increases linearly with distance in spacetime. Hence, when observing the universe by looking out in space (and back in time) in observable spacetime, dH/dt = 0, but if we don’t do that but instead wait around for H to change and then look again in the telescope, we’ll find that H is varying!
In the context of looking out to bigger distances and earlier times after BB, H is a constant, but when we wait around, we’ll find that H is varying as a function of the age of the universe for the observer (not for the observed).
When I did cosmology, the horizon radius of the universe was supposed (from the FRW metric) to be increasing in proportion not to t but to t^(2/3). This slower than linear increase was predicted from the gravitational deceleration on the expansion which was predicted by GR without a CC, i.e. a curved universe of critical density.
This implied that the age of the universe was t = (3/2)/H or H = (2/3)/t.
After it was discovered by Perlmutter in 1998 that the mainstream model was wrong and the universe wasn’t decelerating (because the gravitational deceleration was being cancelled by a repulsive-force type acceleration), the flat geometry of the universe meant that the horizon radius wasn’t expanding as t^(2/3) but merely as t, so the age of the universe in flat geometry is simply
t = 1/H.
This has nothing to do with the Hubble constant varying as we look back in time. It doesn’t! There is no contradiction between dH/dt = 0 for spacetime where t represents earlier epochs in the universe (because as Hubble observed, H is constant when we look back in time because recession velocities vary in proportion to distances or to times past), and H = 1/t.
dH/dt = 0 applies to looking to greater distances in spacetime where the variation of v with r (or time past) means that v/r = constant = H, so dH/dt = 0. When you differentiate such a constant you get zero.
But H = 1/t does not apply to looking to greater distances, because t here is the observer’s time, not the time after the big bang for the object being observed.
The only variation of H you get from H = 1/t is when the age of the universe in our frame of reference varies.
E.g., if you wait for a time of 13,700 million years and then re-measured Hubble’s “constant”, H would have halved.
When you look to greater distances, however, H doesn’t appear to vary! That’s because H when observed in spacetime is a ratio of two things which are both varying in sync: v and r. Because recession velocities increase as you look to greater distances (earlier times), you can’t observe any variation in H with distance. This is so simple, it’s depressing that I have to really keep spelling it out. Again, dH/dt = 0 involves spacetime t for the observed galaxy, while H = 1/t involves observer time t.
“But you are quite wrong to call GR “a mathematical speculation”.”
I’ve gone into the details of the speculations in general relativity here: http://nige.wordpress.com/2008/08/16/authority-problems/
1. The successful predictions of “general relativity” result directly from the inclusion of mass-energy conservation into the gravitational model.
2. General relativity speculatively assumes that the source of gravity is a continuous distribution, not a quantized one consisting of fundamental particles. So the stress-energy tensor has to be supplied by an unrealistically smoothed distribution of matter like a mathematically “perfect fluid”, instead of discrete particles, to act as the source of smooth curvature.
3. General relativity speculatively and implicitly assumes that acceleration is due to spacetime being smoothly curved, instead of there being a quantum field with a series of discrete interactions with gravitons.
4. General relativity’s Ricci curvature tensor is rank-2, so it’s been argued by Pauli and Fietz in the 1930s that gravity is due to spin-2 gravitons, not spin-1 particles like electromagnetism. Spin-1 particle exchange between similar sign gravitational charges (e.g. two masses) would cause repulsion, whereas since attraction occurs, so you need spin-2 to make gravity attractive between two masses. The flaw here is that – while the surrounding universe is electrically neutral for electromagnetism charges (equal positive and negative charges) – it definitely can’t be ignored this way for gravity. Basically you have an immense amount of mass surrounding your apple and Earth, which should be exchanging gravitons with them both. This can lead to spin-1 gravitons producing gravity by pushing together masses that are small compared the mass of the surrounding universe. (Feynman points out in “The Feynman Lectures on Gravitation”, page 30, that gravitons are not necessarily spin-2). This is what I find to be the case, resulting in falsifiable predictions that are checked: http://nige.wordpress.com/2008/01/30/book/
5. General relativity can result in a wide range of metrics depending on what assumptions you make in order to derive those metrics: it’s an endlessly adjustable speculative cosmological model that can model flat and curved universes, and in fact with appropriate ad hoc amounts of dark energy and dark matter it can model anything from endless expansion to collapse.
6. The speculative aspects of general relativity was explained even better by Einstein himself, who stated:
‘I consider it quite possible that physics cannot be based on the [smooth geometric] field principle, i.e., on continuous structures. In that case, nothing remains of my entire castle in the air …’ – Albert Einstein in a letter to friend Michel Besso, 1954.
Nigel, what you are doing now is either
1) totally confused; or
2) dishonest.
Case (2) is that rather than accept that you are wrong, you are now inventing a new definition of H to be used “when observing the universe by looking out in space (and back in time)”, as if that wasn’t the ONLY way H has ever been used.
H is DEFINED as (da/dt)/a, with t DEFINED as the OBSERVER’s time. Or, if you prefer, as the time variable of the FRW metric, the universal time measured by clocks carried by comoving observers (i.e. observers which just go with the flow of universal expansion and don’t move on their own, like the dots painted on the balloon).
The more charitable case (1) is that you are sincerely confused about these basic definitions. If so, maybe Ned Wright’s tutorial on distances in cosmology (and on how people get confused about them) will help:
http://www.astro.ucla.edu/~wright/cosmo_02.htm
No, you are both 1) and 2), because you are confused about the “definition” of the Hubble constant.
(a) Age of universe t = 1/H implies H = 1/t, so the rate of change of H is:
dH/dt = d(1/t)/dt = – 1/t^2
(b) But in spacetime T:
dH/dT = 0
So it’s fundamentally dishonest of you to muddle up the two times!
No amount of confusion and insults against me will make dH/dt = dH/dT. They are not the same. There is not only one possible definition to the Hubble parameter: if you’re dealing with Friedmann’s obsolete law then you have varying H, but if you’re dealing with real physics in real spacetime, you have constant H.
This is because Hubble’s observation was that there is an unvarying ratio v/r = H, it follows H is defined as constant where r = cT, T being time past (if we use capital T for time past to distinguish it from time after big bang). Hubble’s law states:
v/r = v/(cT) = H
= constant regardless of value of T, because v increases in direct proportion to T, keeping observed H in spacetime constant! You still can’t grasp this, and you try to confuse it with Friedmann’s irrelevant (non-spacetime) absolute time since big bang, where H does vary.
Differentiate and you will find that dH/dT = 0 because H is constant as observed in spacetime!
So you’re confusing time after big bang from your obsolete cosmology notes, with spacetime in the Hubble law. The time after big bang is not what we can observe. Please read again what I quoted (maybe on the original blog thread here) from Minkowski wrote in 1908: we have to base physics on spacetime, where distance is proportional to time past because of the speed of light.
Really, for you and others to ignore this and make personal comments based on ignorance about my background or claims that I am being dishonest, is not helpful to the physics! Please understand that the time used in the Friedmann et al metric is not directly observed! Physics calculations in this case needs to be based on observables! Spacetime (i.e. where distance r is related to time past T by r = cT) is observed, and in spacetime the Hubble parameter H is constant because it is the ratio of (velocity)/(radial distance from us, or time past). You seem to be completely confused about this.
Physics needs to build upon facts, not speculative theories. The non-zero dH/dt you get for the Hubble constant varying with absolute time after big bang doesn’t come into what I’m calculating at all, because in spacetime the further the distance, the earlier after the big bang you are seeing. All effects such as light and gravitons will travel to us at velocity c, so in calculating effects we need to treat the physical universe as we observe it, i.e. with time past varying with observable distance. This gives us an effective acceleration for predicting physical facts. The dishonesty and confusion come from the mainstream, I fear.
What your comments – which contain no relevant physics and are just personal attacks based on ignorance – do is to detract attention from the important quantitative success in which the acceleration of the universe was accurately predicted in 1996, years before observation! The kind of confusion you have is not helpful to physics. It’s dishonest to make such comments if you are so confused about the basics. That said, there is a lot of confusion around!
Nigel, let’s try something really simple: look at the graph at the top of this page and tell us which definitions of Hubble “constant”, time and distance it uses.
SomeRandomGuy,
Instead of apologising for your insults, and admitting that you are totally confused and got it wrong, you again just ignore my message and ask a question.
The graph shows that observable distance (double the distance and you’re seeing twice as far back in time) and recession velocity are linearly correlated, i.e. that H = (velocity)/(distance) or H = (velocity)/[(time past)*c] = constant.
Hence, H doesn’t vary because the velocity increases in proportion to distance or to time past. This variation of velocity with time is an effective acceleration.
In my original 1996 8-pages paper predicting the acceleration of the universe, I pointed out Minkowski’s statement that when looking to greater distances we’re seeing the past, and explained that if Hubble in 1929 had tried that he would have predicted the acceleration of the universe.
“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”
– Hermann Minkowski, 1908.
Hubble found (velocity)/(distance) = constant (H) with units of 1/(time). If he had noted that in spacetime (distance) = c*(time past), he would have had the option of finding that (velocity)/(time past) = constant with units of acceleration!
If you want to make the expansion rate of the flat universe absolutely clear in terms of time, it is
v = Hr
= (1/t)*(cT)
= cT/t
where T is time past (which VARIES with observed distance), and t is time after big bang in our frame of reference (so it does NOT vary with distance).
The confusion of youself and Chris is centred on confusing the two times. It’s pretty obvious that you don’t have any real interest in physics, just quoting irrelevant obsolete speculations from your textbook, trying to confuse things, and then handing out insults instead of apologies when your errors are explained to you. But this behaviour is pretty typical in mainstream physics, which is religion.
I remember a fruitless discussion with fellow Electronics World features writer Mike Renardson in 1996. His dismissed it for a different reason to you, because the predicted acceleration of the universe, on the order of 10^{-10} ms^(-2), was extremely small. This prediction and the related prediction of the gravity parameter G was rejected by journals which believed in a large or a zero cosmological constant on the basis of string theory. When it was discovered to be a correct prediction in 1998, people still tried to ignore it and claimed that the small cosmological acceleration of the universe is a “mystery”!
I remember correspondence in which I answered all kinds of concerns Renardson had with the physics in 1996, and then he sent a reply stating “do you really expect me to start believing an unorthodox theory?” I think that this says it all: mainstream physics is a belief-based system, and people need more than factual discussions to convince them of anything new. They need authority as in the stamp of officialdom. Science is supposed to be a matter of facts, but in reality it’s a matter of politics: fame, money, popularity and groupthink. As Chris Oakley wrote above, innovations in physics need sponsorship. Facts don’t speak for themselves. Or people refuse to listen to them unless they come from authority figures, the orthodoxy of religion.
“The variation of velocity with time is effective acceleration”
Right – so if I see a car travelling at 30 kph 30 meters away from me, and one travelling 60 kph 60 metres away from me then it must mean that the nearer one is accelerating and will be travelling at 60 kph when it is 60 meters away – ? I suppose that it is possible … but a much simpler “explanation” is that neither of them is accelerating, but started at the same place (where I am standing) and set off at different speeds. AFAIK we cannot measure galactic acceleration directly so I don’t know how I would prove you wrong. But it certainly is not the most natural explanation.
Bingo! Well done Chris, just the sort of simple analogy I was looking for. I’ll throw up a post on the solution as soon as I get a chance
Nigel #36:
I have no “insults” to apologise for, and I did not ignore your “message”. I asked an elementary question to help you clarify it. Since you failed to answer it, as usual, let’s go through your latest comments in more detail. That’s what you want, right?
Nigel #32:
Wrong. Given a = a_0 t^k, H = k t^(k-2), so k = 2/3 implies H ~ 1/t^(4/3). Now don’t get started shouting that the FRW solution is “obsolete”; in the quote above you are claiming to describe what FRW says.
Nigel #32:
Last time I checked, t^(2/3) was less than t…
Nigel #32:
Wrong. As you have been repeatedly told, on FRW this is only true in the special case a = a_0 t, i.e. linearly growing scale factor. This is NOT the growth rate used by the concordance model.
Nigel #32:
Wrong. As explained in #31, H is constant in time only when k = 2.
Nigel #32:
Here you reveal your confusion about the definitions used when drawing the Hubble diagram. See Ned Wright’s tutorial, linked in #33, where this is explained right after the introductory paragraph about the cosmological principle.
Nigel #32:
Please. The decay of pulsar orbits by gravitational wave emission is explained by “the inclusion of mass-energy conservation into the gravitational model”? Which “gravitational model” is that, exactly?
I will mercifully not say anything more about the rest of your tirade against GR, other than that it’s irrelevant to the issue being discussed, and that it shows that at least in comment #32 you were taking the position that one should not base cosmology on GR.
Nigel #34:
Same error again.
Nigel #34:
What is this mysterious “spacetime T” which you now introduce without definition?
Nigel #34:
Since you had not even defined your mysterious “spacetime T” at this point, I can’t very well have been muddling it up with anything…
Nigel #34:
So now you’re saying that the FRW metric is “obsolete”. This is consistent with #32, which calls GR “speculative”, but it goes even further, since FRW follows from the cosmological principle in any metric theory of gravity. I can only conclude that you are dismissing the entire class of metric theories of gravity.
May I ask how you reconcile this position with the extremely stringent experimental tests of the equivalence principle?
Nigel #34:
In other words, your T is proportional to physical distance as measured in a static, Minkowski spacetime.
Since it appears from the above that you reject all metric theories of gravity, I conclude that you are saying that the universe is correctly described by Minkowski spacetime, and gravity is… what? Newtonian? Mediated by exchange of your “spin 1 gravitons”? And those are also causing repulsion, hence accelerated expansion?
You do realize, of course, that this means the observed expansion would be geometrically like any explosion in 3D, with a central point where it all started? And that since we observe the same rate of expansion in all directions, we would have to be located at that point, i.e. at the center of the universe, for your picture to work?
As an aside, why are your “spin 1 gravitons” causing repulsion only over cosmological distance, and attraction on all scales at least up to galactic? How does that work, exactly, given that exchange of spin 1 bosons is repulsive for equal charges, and you are using mass as charge? Is the mass of the moon of opposite sign to that of Earth? What about the table-top experiments used to test gravity on submillimeter scales, do they too just happen to be made of miraculously balanced amounts of this mass and antimass of yours?
Nigel #34:
See previous point.
“Right – so if I see a car travelling at 30 kph 30 meters away from me, and one travelling 60 kph 60 metres away from me then it must mean that the nearer one is accelerating and will be travelling at 60 kph when it is 60 meters away – ? I suppose that it is possible … but a much simpler “explanation” is that neither of them is accelerating, but started at the same place (where I am standing) and set off at different speeds. AFAIK we cannot measure galactic acceleration directly so I don’t know how I would prove you wrong. But it certainly is not the most natural explanation.” – Chris comment #37
You’re neglecting the time delay in light coming from more distant objects, which is the whole point for the case of cosmology. For cars, the distances are small enough that the delay time in information coming to you is trivial.
For galaxies billions of light years away, you’re seeing them as they were in the past. In this case, as Minkowski states, you have to accept that seeing an object at distance r is the same thing as looking back in time r/c seconds ago.
Two things are varying as you look to greater distances: distance and time. The car analogy ignores the variation in time, which is trivial. But in cosmology this variation in time past is not trivial, and if we differentiate the Hubble velocity correctly we get acceleration
a = dv/dt = d(Hr)/dt = rH^2
which is an accurate prediction made in 1996. It was confirmed. There is no speculation involved in spacetime, the plot of recession velocity versus distance r (or time past T = r/c), or the rules of differentiation. There is also no speculation involved in dH/dt = 0 for the case of looking to greater distances, because the Hubble constant is a constant when we look to bigger distances: v/r is constant because top and bottom are proportional. The Hubble constant is only not a constant when you get away from spacetime and just consider a variation in absolute time. So there is no speculation in predicting the acceleration.
It’s simple physics and mathematics all the way!
SomeRandomGuy,
I explained your errors. Everything you write is wrong/abusive. You have no idea of cosmology. I suggest you read my responses to you.
“a = a_0 t^k, H = k t^(k-2), so k = 2/3 implies H ~ 1/t^(4/3).” – SomeRandomGuy
If the universe’s horizon radius increases as
R ~ t^(2/3), [Equation 1]
then
v = dR/dt = (2/3)t^(-1/3) [Equation 2]
Now, since H = v/R, using Equations 1 and 2 we get:
H = v/R = [(2/3)t^(-1/3)]/[t^(2/3)]
= (2/3)/t
This is my result, H = (2/3)/t. Your result H ~ 1/t^(4/3) is just plain wrong, and I’ve no interest in trying to help you find out why since you are rude and ignorant of physics.
“You do realize, of course, that this means the observed expansion would be geometrically like any explosion in 3D, with a central point where it all started? And that since we observe the same rate of expansion in all directions, we would have to be located at that point, i.e. at the center of the universe, for your picture to work?” – SomeRandomGuy
What I realise is that science is not about prejudice, it’s about facts and making predictions that are subsequently confirmed by observations. If you have a theory based entirely on observed facts that has made checkable unique predictions that have been confirmed, that theory may be correct.
Regarding our place in the universe, I refer you to the largest anisotropy (the cosine variation in the sky) in the microwave background radiation.
In the May 1978 issue of Scientific American (vol. 238, p. 64-74), R. A. Muller of the University of California, Berkeley, published an article about this, titled “The cosmic background radiation and the new aether drift”, stating:
“U-2 observations have revealed anisotropy in the 3 K blackbody radiation which bathes the universe. The radiation is a few millidegrees hotter in the direction of Leo, and cooler in the direction of Aquarius. The spread around the mean describes a cosine curve. Such observations have far reaching implications for both the history of the early universe and in predictions of its future development. Based on the measurements of anisotropy, the entire Milky Way is calculated to move through the intergalactic medium at approximately 600 km/s.”
Most of this 600 km/s velocity is due to our galaxy, the Milky Way, being locally attracted to the larger galaxy Andromeda, so it may be an upper limit on the average speed of the Milky Way mass motion. Now suppose the universe was more like Dr Chris Oakley’s explosion than the curved boundless geometry of mainstream general relativity: since the universe is flat and in any case curvature appears to be a classical approximation to a lot of graviton interactions, if quantum gravity is correct.
Distance is the product of velocity and time, and if we multiply 600 km/s by the age of the universe, we find that the matter in the Milky Way would have moved only 0.3% of the horizon radius of the universe in 13,700 million years.
If the average speed was less than 600 km/s, it would be even closer to the centre of the universe. So it doesn’t pay you to be biased either way. There are lots of problems with multiplying the 600 km/s speed deduced from the major anisotropy in the cosmic microwave background by the age of the universe to obtain our distance from the “centre” or “origin” of the universe. But they can probably all be overcome. Muller’s argument that this is a “new aether drift” in 1978 didn’t catch on, because of relativity. I don’t want to argue about speculations.
I’m mentioning this just as a counter argument to you. I made a fact based prediction that was subsequently confirmed. You then make a comment saying that if it is right it implies we’re at the middle of the universe. Well, I don’t care where we are, only that the prediction works. Copernicus is referred to as a defender of science for arguing that we’re not at the centre of the universe. I think science is quite different to any sort of prejudice: science is not about defending speculations that we are or are not here or there. It’s about establishing facts!
Further support for the result of my calculation in comment #42 above can be found in Marc Lachièze-Rey’s textbook, “Theoretical and Observational Cosmology”, 1999, p 384, available online at:
http://books.google.co.uk/books?id=Lds_QbApCbgC&pg=PA384&lpg=PA384&dq=t2/3+cosmology&source=web&ots=2MHo_RO-ug&sig=2Kl44rBzVTmCCPikkwmVnwhqEw4&hl=en&sa=X&oi=book_result&resnum=7&ct=result
For horizon radius expansion proportional to t^(2/3), you get H = (2/3)/t.
The same also occurs in Lars Bergström and Ariel Goobar, “Cosmology and Particle Astrophysics”, Springer, 2004, p. 202:
http://books.google.co.uk/books?id=XQBJ2he1Cz4C&pg=PA202&lpg=PA202&dq=t2/3+cosmology&source=web&ots=2BV266Qkyc&sig=NExhOnXefqhTAsh0E_zJFV4Dm_I&hl=en&sa=X&oi=book_result&resnum=9&ct=result
“For a flat, matter dominated FLRW [Friedmann-Leimatre-Robertson-Walker] model, a(t) ~ t^(2/3), (da/dt)/a [this (da/dt)/a = H, since a is radius here] = 2/(3t) …”
I’m very busy now and I hope that no more insulting rubbish from the totally ignorant pseudo physicist SomeRandomGuy will appear. I won’t have time to respond to it any more. He doesn’t read anything I write anyway, so what’s the point in trying to explain physics to someone like that?
Before disappearing back to SQL database ASP programming for good, just one more comment about Chris Oakley’s point in comment 37, that we may be seeing stars with different velocities at different distances because stars started from one point and moved with different speeds. This is something I responded to earlier in comment 4 (unfortunately I didn’t turn off italics at one point, so the while section is in italics as a result). Eddington discussed that idea in his book “The Expanding Universe”, referring to papers in Nature which discredited it. The distribution of speeds you need turns out to be contrary to the Maxwell-Boltzmann distribution for a gas like the hydrogen cloud that the universe was soon after the big bang. But let’s assume that is one possibility. That predicts no cosmological acceleration! My argument, a = dv/dt = d(Hr)/dt, does predict cosmological acceleration of the right size, which I think is evidence that it might be right. I then went on to predict the strength of gravity and other things, again using simple facts.
“As an aside, why are your “spin 1 gravitons” causing repulsion only over cosmological distance, and attraction on all scales at least up to galactic? How does that work, exactly, given that exchange of spin 1 bosons is repulsive for equal charges, and you are using mass as charge? Is the mass of the moon of opposite sign to that of Earth?” – SomeRandomGuy
See http://nige.wordpress.com/2008/01/30/book/ for the answer. All masses repel one another by exchanging gravitons. The more mass, the more repulsive charge. If you have two small masses, two planets or nearby galaxies, they will repel each other slightly, but they’re being pushed together harder by gravitons exchanged with the surrounding universe, involving bigger masses and a convergence of gravitons.
The outward radial acceleration from us of mass m is a = dv/dt = rH^2. The second law of motion gives outward force for that mass of F = mrH^2. The third law of motion suggests that there is an equal reaction force, F = mrH^2, directed radially towards us. This quantifies spin-1 graviton predictions for low energy, where only the simplest Feynman diagram contributes significantly to the result, so the path integral becomes very simple and can be evaluated geometrically as Feynman did for QED (see http://nige.wordpress.com/path-integrals/ for a discussion of how this works).
Now the graviton force F = mrH^2 contains mass m and distance r, so it is trivial for relatively small masses and relatively small distances, but is significant for marge masses and/or larger distances.
Two nearby masses get pushed together because they exchange gravitons more forcefully with the surrounding universe than with each other. So the fundamental particles in the Moon are pushed towards the Earth repulsion of immense distant masses more than by graviton impacts from gravitons exchanged between the Earth and the Moon. This is very slightly like LeSage’s pictorial gravity from the Newtonian era (it was said to have been first proposed by Newton’s friend Fatio but Newton didn’t like it because at that time it couldn’t be made to work and make checkable predictions), which was generally discredited because:
1. It couldn’t usefully predict anything checkable like G
2. It wasn’t a gauge theory of virtual radiation (graviton) exchange, so the real radiation it postulated as the exchange radiation would cause drag on moving bodies, heat up bodies until they glowed red how, and would also diffuse into geometric “shadows” to make gravity fall off much faster than the inverse-square law.
http://en.wikipedia.org/wiki/Le_Sage's_theory_of_gravitation
(I also have a discussion of the errors somewhere on my blog.)
The fact-based calculations you’re talking of differs from the LeSage model in that it’s a gauge theory of quantum gravity, which does make predictions of cosmological acceleration, G, and other things, and which does not have the defects of LeSage’s theory.
Nigel #44, you are confusing the “horizon radius” (a.k.a. Hubble or particle horizon) with the FRW scale factor (radius of the 3-sphere).
For heaven’s sake, learn at least the most basic definitions before wasting everybody’s time, yours included.
Nigel #42: Guess what, you’re right about that. I inadvertantly dropped a k in H = (da/dt)/a.
So, correction: given a = a_0 t, the Hubble “constant” is k/t.
So in previous comments, where I say that the Hubble constant is NEVER constant in FRW unless k = 2, read k = 0 (i.e. static universe) instead.
The main points are of course unaffected: you are saying that metric theories of gravity in general and GR in particular are wrong, despite all evidence to the contrary, and should not be used to do cosmology, that spacetime is static and we are located at the center of the universe, and that gravity is mediated by “spin 1 gravitons”, which would make it repulsive between masses of equal sign.
SomeRandomGuy,
I don’t have much time for discussions. If you knew cosmology, you’d have known the fact that t^(2/3) expansion leads to H = (2/3)/t. You don’t know anything about it. Your calculations are irrelevant and wrong.
“The main points are of course unaffected: you are saying that metric theories of gravity in general and GR in particular are wrong, despite all evidence to the contrary, and should not be used to do cosmology, that spacetime is static and we are located at the center of the universe, and that gravity is mediated by “spin 1 gravitons”, which would make it repulsive between masses of equal sign.”
No, I’m not saying that. You’re saying that. What I am saying is a sequence of facts:
Fact 1: the universe is accelerating, confirmed by Perlmutter and others since 1998.
Fact 2: the acceleration was predicted by a = dv/dt = d(rH)/dt = rH^2 back in 1996.
Fact 3: the spin-2 graviton idea relies on a path integral including only two masses: so it forces the exchange of gravitons to have the right spin to cause attraction when exchanged. It makes no falsifiable predictions (string theory of gravitons has a landscape of 10^500 vacua, which can’t be checked).
Fact 4: If you correct the path integral for gravitons so that you include graviton exchange between all mass-energy(gravitational charge) in the universe, instead of just two masses as Pauli and Fietz did in the 1930s when arguing that gravitons have spin-2, you find that gravitons have spin-1 and the basic graviton interaction is the Feynman diagram of virtual radiation being exchanged by analogy to radiation scattering off charge.
Fact 5: This predicts the strength of gravity, and other things.
Your statement that I’m saying that all metric theories of gravity are wrong is in error. I’m saying are facts. (Quite a few metrics of general relativity are useful under certain conditions, where they approximate the underlying quantum gravity dynamics very well.) I’ve no interest in arguing with time-wasting bigots about whether approximate metrics are wrong or right. Life is short and what matters are facts, not uncheckable controversies.
> If you knew cosmology, you’d have known the fact that t^(2/3) expansion leads to H = (2/3)/t.
By this standard, if you knew cosmology, you would have spotted the trivial error immediately.
You didn’t; I had to explain it to you.
If you knew anything about physics at all, you would know that nobody commits formulae for all derived quantities to memory. You learn the fundamenta and derive the rest as needed. Here, I made a trivial error (dropped a k) while doing the derivation in the comment box (admittedly not the safest way). Irritating but not central to the discussion in any way.
> You don’t know anything about it. Your calculations are irrelevant and wrong.
Nigel, you are funny. You swing from declaring FRW and textbooks “obsolete” to quoting them as support for your views in the blink of an eye.
> “The main points are of course unaffected: you are saying that metric theories of gravity in general and GR in particular are
> wrong, despite all evidence to the contrary, and should not be used to do cosmology, that spacetime is static and we are
> located at the center of the universe, and that gravity is mediated by “spin 1 gravitons”, which would make it repulsive
> between masses of equal sign.”
>
> No, I’m not saying that.
Yes you are, although it is perfectly clear by now that you don’t even understand what you are saying.
> You’re saying that. What I am saying is a sequence of facts:
>
> Fact 1: the universe is accelerating, confirmed by Perlmutter and others since 1998.
Actually, far from everybody is prepared to accept this as “fact”. But let’s assume that it is, as understood by cosmologists, Perlmutter included: it means that the scale factor a(t) of the FRW metric is growing at a rate ~t^k, with k > 1. It does NOT mean that FRW is “obsolete”, as you claim.
> Fact 2: the acceleration was predicted by a = dv/dt = d(rH)/dt = rH^2 back in 1996.
The “acceleration” which you “predicted” is based on a trivial misunderstanding which has been explained to you over and over again. You are assuming that GR is wrong, spacetime flat, we located at the center of the universe, and gravity mediated by “spin 1 gravitons”, which would make it repulsive.
> Fact 3: the spin-2 graviton idea relies on a path integral including only two masses:
There is no need to invoke path integrals here. GR is a classical theory, not a quantum field theory. In this context, “spin 2” just reflects the number of spacetime indices carried by the metric.
> so it forces the exchange of gravitons to have the right spin to cause attraction when exchanged.
> It makes no falsifiable predictions (string theory of gravitons has a landscape of 10^500 vacua,
> which can’t be checked).
This is pure gibberish. Again: GR is a classical theory, one of a whole class (metric theories) whose fundamental object (the metric) carries two spacetime indices. This has nothing to do with quanta, i.e. gravitons. String theory is an attempt to construct a quantum theory of gravity which reduces to GR at low energy. Even if you could prove it completely wrong, the validity of GR would be unaffected.
> Fact 4: If you correct the path integral for gravitons so that you include graviton exchange between all
> mass-energy(gravitational charge) in the universe, instead of just two masses as Pauli and Fietz did in the 1930s when
> arguing that gravitons have spin-2, you find that gravitons have spin-1 and the basic graviton interaction is the Feynman
> diagram of virtual radiation being exchanged by analogy to radiation scattering off charge.
More gibberish. There is no consistent quantum field theory of gravity; in the 1930s there wasn’t even a consistent quantum field theory of electrodynamics. Even today, the closest thing would be LQG, which is controversial at best, and most definitely not spin 1.
> Fact 5: This predicts the strength of gravity, and other things.
More gibberish.
> Your statement that I’m saying that all metric theories of gravity are wrong is in error.
You claim that FRW is “obsolete”. FRW follows from the cosmological principle in any metric theory of gravity. Not to mention the fact that the metric carries two spacetime indices, and you keep invoking a mysterious spin 1 gravity.
> I’m saying are facts.
Try “gross misunderstandings”.
> (Quite a few metrics of general relativity are useful under certain conditions, where they approximate
> the underlying quantum gravity
Why, how magnanimous of you. So now GR is no longer wrong?
P.S. Just stumbled across a more orderly discussion of these topics which might be helpful: arXiv:0803.2701.
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“By this standard, if you knew cosmology, you would have spotted the trivial error immediately.”
No, I stated the correct result, and you produced a gibberish “calculation” claiming to insult me with it. Evrything you are writing is just such a lot of gibberish I don’t have the time to respond to it any more. I’m interested in science. Cheers!
On the speculative nature of conjectures concerning spin-2 (attractive or ‘suck’) gravitons, Richard P. Feynman points out in The Feynman Lectures on Gravitation, page 30, that gravitons do not have to be spin-2, which has not been observed.
“Again: GR is a classical theory, one of a whole class (metric theories) whose fundamental object (the metric) carries two spacetime indices. This has nothing to do with quanta, i.e. gravitons. String theory is an attempt to construct a quantum theory of gravity which reduces to GR at low energy. Even if you could prove it completely wrong, the validity of GR would be unaffected.”
I never said that GR is a string theory or a graviton theory. I’ve stated it’s a classical theory which doesn’t include quantum gravity.
“More gibberish. There is no consistent quantum field theory of gravity; in the 1930s there wasn’t even a consistent quantum field theory of electrodynamics. Even today, the closest thing would be LQG, which is controversial at best, and most definitely not spin 1.”
I’ve refereed you to a draft paper, which needs rewriting admittedly when I have time, and I’ve given the reasons there. This is like a Panto when you deny that something I’ve given exists.
“Nigel, you are funny. You swing from declaring FRW and textbooks “obsolete” to quoting them as support for your views in the blink of an eye.”
Wrong, you wanted to discuss the textbooks and then you produced a false calculation, and I used the textbooks to demonstrate that you can’t even do textbook physics without making TRIVIAL errors. You don’t check your own false results but present them as if they discredit me. You don’t apologise for this, you are just a completely unreasonable person.
How on earth can you expect to check physics and new ideas when you can’t even derive a basic formula without making an error? If you had presented it in a humble way saying that you don’t check your claims, then that is one thing. But you didn’t.
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Looking at the expansion of space between galaxies and assuming that space is everywhere expanding at the same rate one might deduce that the galaxies accelarate from each other because there is more and more space expanding between them the further they get from each other.