Where did the energy in the CMB go to?

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The discussion revolves around the energy loss of cosmic microwave background (CMB) photons due to the expansion of the universe, which causes redshift and reduces their energy. Participants debate the implications of this energy loss in the context of the first law of thermodynamics and general relativity (GR), noting that GR does not conserve energy in the traditional sense. It is suggested that while the number of CMB photons remains relatively constant, their total energy decreases over time, leading to questions about where this lost energy goes. Some argue that the energy is absorbed into the cosmological field, while others emphasize that GR's framework complicates the conservation of energy concept. Ultimately, the conversation highlights the ongoing challenges in understanding energy dynamics in an expanding universe.
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This may seem like a silly question, but it has me puzzled.

Theory tells us that as space expands then photons (and I guess all form of energy?) in that space are red-shifted, ie their wavelength increases, ie they lose energy. The most obvious example of this is the CMB, which consists of maybe 10^88 photons (in the observable universe) "left-over" from the Big Bang. At the time of decoupling from matter (300,000 years after the Big Bang?) each of these photons was of very high energy (low wavelength), but as the universe (space) expanded each of these photons was red-shifted (lost energy), to end up as the microwave background that we see today.

If the first law of thermodynamics is to be obeyed (mass/energy must be conserved), what happened to the energy that these photons lost?
 
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Not lost, just stretched out by expansion. Does a rubber band lose mass when you stretch it? Of course not, it just has less mass [energy] per inch.
 
Chronos said:
Not lost, just stretched out by expansion. Does a rubber band lose mass when you stretch it? Of course not, it just has less mass [energy] per inch.

But if the number of CMB photons remains pretty much constant over time, and if typical photon wavelengths are increasing with time, then the product of number of photons and average energy per photon has to be decreasing, doesn't it?

At any rate, I find this a neat question!
 
Chronos said:
Not lost, just stretched out by expansion. Does a rubber band lose mass when you stretch it? Of course not, it just has less mass [energy] per inch.
sorry Chronos, your analogy doesn't apply in this case.

the total energy in a photon is proportional to its frequency E=hv. (the reason why your analogy does not apply is because the total mass of the rubber band is independent of how much the rubber band is stretched - this is not the case when we think of the relationship between energy and photon wavelengths)

as space expanded, the wavlength of each CMB photon was increased (frequency reduced).

hence the energy of each CMB photon was reduced.

the number of CMB photons has not changed much since decoupling with matter, hence the total energy locked up in the CMB has reduced over time.

this energy must have gone somewhere... but where?
 
The energy is used to power expansion.
 
Chronos said:
The energy is used to power expansion.

what does that mean?

does that imply the expansion "needs energy" from the CMB in order to "power it" - in which case what happens if we have a universe with no CMB, does it stop expanding? i don't think so.

if anything, the CMB contributes to the total mass-energy of the universe and should slow down the expansion rather than "power it".
 
Chronos said:
The energy is used to power expansion.

Even when the expansion is the result of space itself being created? Maybe so, I don't know.
 
Chronos said:
The energy is used to power expansion.

Although I'm not sure of this, I think that it does the opposite (slow the universe down). In a matter-dominated universe, the kinetic energy of the matter is gradually converted into potential energy, slowing the expansion. A radiation-dominated universe will decelerate, just like a matter-dominated one, but the light must move at the speed of light, so you can't take its kinetic energy. Instead, you reduce its frequency.
 
SpaceTiger said:
Although I'm not sure of this, I think that it does the opposite (slow the universe down). In a matter-dominated universe, the kinetic energy of the matter is gradually converted into potential energy, slowing the expansion. A radiation-dominated universe will decelerate, just like a matter-dominated one, but the light must move at the speed of light, so you can't take its kinetic energy. Instead, you reduce its frequency.
Ahhh. OK, yes. The energy of the photons is converted into potential energy... because the expansion is pulling all the photons further apart hence (since they all attract each other gravitationally) the total gravitational potential energy must be increasing as the universe expands, and the energy of each CMB photon is reduced accordingly, so that everything remains balanced. Is that it?
 
  • #10
moving finger said:
... the total gravitational potential energy must be increasing as the universe expands, and the energy of each CMB photon is reduced accordingly, so that everything remains balanced. Is that it?

I think so. I've never directly worked with general relativity, so I'm not sure if my semi-classical understanding of the problem applies here.
 
  • #11
SpaceTiger said:
I think so. I've never directly worked with general relativity, so I'm not sure if my semi-classical understanding of the problem applies here.
It doesn't!

It is important not to inconsistently mix up Newtonian and GR gravitational concepts as has happened in the above posts. GR does not in general conserve energy, in particular it does not do so in the cosmological solution of a non-static evolving universe.

The total energy of the CMB does decrease with time as the photon number (apart from the few absorbed by matter) is conserved.

Classically gravitational potential energy is that energy used in lifting a body against a gravitational force. In GR that force is replaced by curvature. In particular work has to be done to accelerate a body out of its freely falling inertial frame, and weight is the reaction to an inertial force. Therefore classical gravitational potential energy is radically re-interpreted in GR and should not be used to explain either cosmological or gravitational red shift.

The question of “Where does the energy of a CMB photon go?” is similar to the question of “Where does the energy of a gravitationally red shifted photon go?” The GR answer is, "into the field", whatever that means.

GR just loses the CMB energy into the cosmological field with no mechanism to explain how that happens or where it might be re-located.

Remember GR does not in general conserve energy, it conserves energy-momentum instead and that is different. It is the curvature of space-time that violates the conservation of energy, which is why the energy of a system can only be properly defined in GR in the absence of curvature, where the space-time is asymptotically flat.

Garth
 
  • #12
Interesting. You have pointed out a problem in both models.
 
  • #13
Sorry, but the redshifting of CMB photons has nothing to do with gravity. You get the same redshifting in the Milne universe where the effects of gravity are ignored.

Of course if you ignore the idea of space stretching and say that the photons are redshifted because their source was moving rapidly away from us then there isn't a problem.
 
  • #14
chronon you have to be consistent, in this case with the principles of GR.
chronon said:
Sorry, but the redshifting of CMB photons has nothing to do with gravity. You get the same redshifting in the Milne universe where the effects of gravity are ignored.
According to the principles of GR what is 'gravity' except the effect of 'curvature', and what is cosmological expansion except the effect of 'curvature'? The Milne universe may not have matter but it still has 'curvature', that is if you want to remain consistent to GR. Empty space has 'gravity', i.e. 'curvature'!
Of course if you ignore the idea of space stretching and say that the photons are redshifted because their source was moving rapidly away from us then there isn't a problem.
According to the principles of GR are the sources moving through space-time producing a Doppler effect, or is it space-time itself that is 'stretching'?

Garth
 
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  • #15
Garth said:
chronon you have to be consistent, in this case with the principles of GR.According to the principles of GR what is 'gravity' except the effect of 'curvature', and what is cosmological expansion except the effect of 'curvature'? The Milne universe may not have matter but it still has 'curvature', that is if you want to remain consistent to GR. Empty space has 'gravity', i.e. 'curvature'!
Gravity is the curvature of space-time. The space-time of the Milne universe is flat. You might consider the space to be curved, but that depends on the choice of the coordinate system.
Garth said:
According to the principles of GR are the sources moving through space-time producing a Doppler effect, or is it space-time itself that is 'stretching'? Garth
Sources don't move through space-time. Space-time doesn't stretch. These are time-based verbs.
 
  • #16
I agree with Garth. No one, to my knowledge, has ever explained where the lost CMB energy went. there is no energy-conservation law in GR, but GR is currently the prevailing largescale theory of spacetime and other theories just apply locally or approximately. So as far as anyone knows ENERGY CONSERVATION IS NOT TRUE except in some restricted local or approximate sense.

The CMB is an excellent example to show that energy conservation is not true, since no one can find the lost energy.

The concept of energy is a growing, evolving idea. Feynman's "dennis the menace" story illustrates this (I don't have a link). So one can conjecture that someday the concept of energy will be extended to something we don't know about, that we can measure, and people will measure it and say ahah THAT is where the missing CMB energy went, so they will have "found" the missing energy and conservation will be vindicated.

but that is only a conjecture. there would need to be an improved theory of spacetime, to replace GR. Because in GR the CMB energy when the photons get stretched out just goes away. It is a wellknown fact that GR does not support the energy conservation law.

So moving finger question is not at all silly. It seems very sensible to be puzzled. I've quoted several posts, including Garth's to keep track of the main points in the thread.
moving finger said:
This may seem like a silly question, but it has me puzzled.

Theory tells us that as space expands then photons (and I guess all form of energy?) in that space are red-shifted, ie their wavelength increases, ie they lose energy. The most obvious example of this is the CMB, which consists of maybe 10^88 photons (in the observable universe) "left-over" from the Big Bang. At the time of decoupling from matter (300,000 years after the Big Bang?) each of these photons was of very high energy (low wavelength), but as the universe (space) expanded each of these photons was red-shifted (lost energy), to end up as the microwave background that we see today.

If the first law of thermodynamics is to be obeyed (mass/energy must be conserved), what happened to the energy that these photons lost?

SpaceTiger said:
... I've never directly worked with general relativity, so I'm not sure if my semi-classical understanding of the problem applies here.

Garth said:
It doesn't!

It is important not to inconsistently mix up Newtonian and GR gravitational concepts as has happened in the above posts. GR does not in general conserve energy, in particular it does not do so in the cosmological solution of a non-static evolving universe.

The total energy of the CMB does decrease with time as the photon number (apart from the few absorbed by matter) is conserved.

Classically gravitational potential energy is that energy used in lifting a body against a gravitational force. In GR that force is replaced by curvature. In particular work has to be done to accelerate a body out of its freely falling inertial frame, and weight is the reaction to an inertial force. Therefore classical gravitational potential energy is radically re-interpreted in GR and should not be used to explain either cosmological or gravitational red shift.

The question of “Where does the energy of a CMB photon go?” is similar to the question of “Where does the energy of a gravitationally red shifted photon go?” The GR answer is, "into the field", whatever that means.

GR just loses the CMB energy into the cosmological field with no mechanism to explain how that happens or where it might be re-located.

Remember GR does not in general conserve energy, it conserves energy-momentum instead and that is different. It is the curvature of space-time that violates the conservation of energy, which is why the energy of a system can only be properly defined in GR in the absence of curvature, where the space-time is asymptotically flat.

Garth
 
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  • #17
It is fun to calculate how much the lost energy is. I did this a couple of years ago and found that (unless I made a mistake) the lost CMB energy is within a factor of ten or so of the estimated "dark energy" density.

that is, if you take a cubic kilometer it has estimated about 0.6 joules of "dark energy" in it, according to astronomers.

but if you look at all the CMB photons in that cubic kilometer and add up their energy it is something roughly on the order of 1/10000 of that, like
on order of 0.00006 joules.

now each one of those photons has lost 999/1000 of his energy by being stretched out by the expansion of space

so therefore, if you gave each photon, in that cubic kilometer, BACK the energy that he has lost by redshifting, then within a factor of ten or so there would be about the same CMB energy in the cubic kilometer as there is supposed to be dark energy.

but there is no theory that allows lost CMB energy to reappear as dark energy. indeed I am skeptical that dark energy exists. it is not proved.
also one can be skeptical of the estimate of how much dark energy there is, assuming it exists----the 0.6 joule per cubic km.

I will recalculate the energy per cubic kilometer of the CMB and see how it compares, however. this is just for fun and not serious.
 
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  • #18
calculation in natural units:
WMAP estimate of dark energy density is 0.85E-120
current CMB temperature 0.96E-31
current CMB energy density is (pi2/15) (0.96E-31)4

order of magnitude about E-124

but redshift of CMB is 1100, so each photon started life 1100 times more energetic, so if you gave them back their lost energy the density would be E-121

so about a tenth of "dark energy" density

... comparable anyway, same ballpark
 
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  • #19
marcus said:
so about a tenth of "dark energy" density

... comparable anyway, same ballpark

Actually, I think this is a natural consequence of the standard model (sort of). In order for the universe to remain flat, the total energy density must be critical. In the standard model, the currently dominant form of energy is dark energy, while in the past it was matter and radiation, respectively. During its era, of course, radiation had an energy density comparable to critical, but since then the comoving energy density has dropped due to the redshift. If it weren't for that redshift, the radiation would still have an energy density comparable to critical density and, therefore, the dark energy density.

Now, it's no coincidence that the matter-radiation transition and decoupling occur at about the same time (within a factor of a few), because recombination is related to the fact that matter is becoming the dominant constituent. Thus, the comoving energy density lost since decoupling is roughly equivalent to that lost since matter-radiation equality which, by definition, is equivalent to that lost by redshift of photons. In light of this, your calculation is perhaps a bit less surprising.

That said, however, we have no idea what the dark energy is, so it's possible that you're right (redshift -> dark energy). I doubt it, though.
 
  • #20
chronon said:
Gravity is the curvature of space-time. The space-time of the Milne universe is flat.
The Milne universe is hyperbolic, k = -1, to my knowledge that means it isn't flat or even conformally flat! In the cosmological solution to Einsteins field equation the space-time of an empty universe has curvature.
You might consider the space to be curved, but that depends on the choice of the coordinate system.
Sources don't move through space-time. Space-time doesn't stretch. These are time-based verbs.
True, I was reflecting your own choice of words back to you, again, be consistent!

However from our frame of reference and our temporal "time-based" existence we can ask whether these sources are moving away from us and cosmological red shift is doppler in nature or whether space-time "stretches", i.e. the world lines of distant observers co-moving with the cosmological frame of reference in which the universe (CMB) is globally isotropic and homogeneous diverge from each other.

Garth
 
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  • #21
marcus must remember "the marcus conjecture"
https://www.physicsforums.com/archive/topic/t-2220_The_Marcus_Conjecture.html
that is, the energy of the CMB photons being transformed into dark energy

But we can also consider the inverse mechanism: vacuum energy (a candidate to dark energy) decaying into CMB photons
http://arxiv.org/abs/astro-ph/0409451

So, perhaps this is like the water cycle here in Earth, water from the sea evaporating, then falling like rain, and being channeled again to the sea by streams and rivers.

Only a mad thought. Possibly the reality of what happens is very different
 
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  • #22
Again I'll try to emphasise the difference between space and spacetime. General relativity tells us about the intrinsic curvature of spacetime. It allows considerable freedom of how we coordinatise spacetime. What we think of as space depends upon our choice of coordinate system.
Garth said:
The Milne universe is hyperbolic, k = -1, to my knowledge that means it isn't flat or even conformally flat! In the cosmological solution to Einsteins field equation the space-time of an empty universe has curvature.
The space in the Milne universe is hyperbolic in the usual coordinate system. The spacetime is flat. The case which is usually referred to as 'flat', k=0, has curved spacetime, flat space (in the usual coordinate system)
Garth said:
True, I was reflecting your own choice of words back to you, again, be consistent!
When I talk about movement or stretching, this is with respect to space, not spacetime
 
  • #23
marcus must remember "the marcus conjecture"
Its as good as any i have seen, if one wants to go beyond "established",
models, yea bring out your dead, least they rot and infect ye.
 
  • #24
An observer on an oil rig counts 10 coast bound waves per minute passing by the rig. An observer passing by the rig on a coast bound vessel counts 2 waves per minute passing by his vessel. Is there a missing energy problem here?
 
  • #25
Yes I'd have to go with the doppler effect on that one. Maybe we just see them as red but if you looked at them from the opposite side they would be violet or something.
 
  • #26
floped perfect said:
Yes I'd have to go with the doppler effect on that one...

be careful though, it is possible you are mistaken.
something taught in first few weeks of a cosmology course is
the generally accepted idea that the cosmological redshift is not a
doppler effect.

that is, the redshift is not due to the motion of the source at the time of emission, and it is not due to to our motion at the time we receive the signal. (there are small doppler effects due to motion that are superimposed on the main redshift)

the usual story is that the redshift is chiefly caused by the expansion of space during the time that the light is in transit.

there are some cosmology lecture notes available online that probably cover this---by Ned Wright or by Eric Linder. I have to go, can't supply links at the moment.

anyway caution is advisable. attributing redshift to doppler puts you in
a kind of marginal or minority place AFAIK, you could be right but it would involve a major overhaul of the current agreed on model of the universe to make you right
 
  • #27
  • #28
Are galaxies really moving away from us or is space just expanding?
This depends on how you measure things, or your choice of coordinates.
This seams reasonable, but there is one aspect I do not understand. If we consider that space is expanding, then to compute the (aparent) recession speed from redshift one should make use of a formula derived from the Friedmann equation and the Hubble law (redshift -> comoving distance from the Friedmann equation -> recession speed from Hubble's law). But If we consider that the galaxies are really moving away from us, then the recession speed can be directly derived making use of the Doppler effect. This second method leads to a very different result (no superluminal recession speeds are possible with the Doppler effect formula). Inserting then the obtained speed into Hubble's law, one would obtain a very different comoving distance than the one obtained making use of the first method. Since distances might be independently measured with other methods, doesn't this mean that both descriptions are not equivalent?
 
  • #29
But that is just wrong. Space is expanding and there is no way to get around that. You can apply a doppler interpretation to redshift, but guess what? You get a totally unrealistic age for the universe. Something has to give. I chose the concordant version.
 
  • #30
hellfire said:
Are galaxies really moving away from us or is space just expanding?
This depends on how you measure things, or your choice of coordinates.

This seams reasonable, but there is one aspect I do not understand. If we consider that space is expanding, then to compute the (aparent) recession speed from redshift one should make use of a formula derived from the Friedmann equation and the Hubble law (redshift -> comoving distance from the Friedmann equation -> recession speed from Hubble's law). But If we consider that the galaxies are really moving away from us, then the recession speed can be directly derived making use of the Doppler effect. This second method leads to a very different result (no superluminal recession speeds are possible with the Doppler effect formula). Inserting then the obtained speed into Hubble's law, one would obtain a very different comoving distance than the one obtained making use of the first method. Since distances might be independently measured with other methods, doesn't this mean that both descriptions are not equivalent?

I agree it sounds reasonable that you can change coordinates but I wouldn't know how to translate that into a redshift.

Just for illustration sake, I put the CMB redshift z = 1100 into Morgan's calculator and got

1. then (when light was emitted from the Last Scattering Surface)
LSS distance from us was 40 million LY
LSS recession speed was 57c

2. now (when we receive the microwaves)
LSS distance from us is 45.5 billion LY
LSS recession speed is currently 3.3c

So I can see how you might construct coordinates so that the atoms that emitted a particular CMB photon are actually moving away at 57 times speed of light when they emit
and are now moving away (when we get the light) at 3.3 times the speed of light. And all this time space does not expand. that is reasonable I guess.
It is just substituting the atoms motion thru space for the usual expansion picture.

But I do NOT see how one could translate that motion into a DOPPLER effect. The relativistic doppler formula that i know applies only to speeds less than c. But the recession speeds we are dealing with are all the time greater than c. So what doppler formula would one use?

Maybe I am missing something and someone will explain. But it seems much easier to understand the redshift as a non-doppler effect of the expansion of space. And AFAIK this is the usual way people teach it and talk about it.

It seems reasonable that one could introduce new coordinates so all the expansion is represented as motion, but then (like hellfire was asking maybe) how do the Friedmann equations work? What do they even look like? We are told that expansion started off very fast and was slowing down for the first 10 billion years or so and is now speeding up. This can be explained in the Friedmann equation model as due to changing matter and dark energy density, or the cosmological constant. But I do not see how, if one throws the Friedmann model out and the Friedmann-Robertson-Walker coordinates one gets anything except some rather artificial construct in which the changing expansion rate is either impossible to represent or else has to be put in by hand. Must confess that I haven't thought about that kind of coordinate change so several people here are probably in better position to clarify this.
 
  • #31
Chronos said:
But that is just wrong. Space is expanding and there is no way to get around that. You can apply a doppler interpretation to redshift, but guess what? You get a totally unrealistic age for the universe. Something has to give. I chose the concordant version.

I'm suspect Chronos is right about this. (notwithstanding I do not fully understand how one would re-coordinatize so that one would represent the recession of galaxies as motion and interpret the redshift as doppler.)
 
  • #32
General relativity determines the shape of spacetime, but doesn't specify how you coordinatise it. The usual choice is to take a time coordinate corresponding to the proper time of comoving objects. My feeling is that this is just going back to the idea of absolute time, and it is better to use a coordinate system in which the speed of light is taken to be constant. Then nothing travels faster than light. I've written some ideas on how this could be done at: http://www.chronon.org/Articles/milne_cosmology.html

It is important to distinguish between the actual physics of the situation and the effects of the choice of coordinates. The trouble with the proper time system is that it leads to effects which people interpret as physical, such as stretching space. (I don't think that the slowing down from 57c to 3.3c is physical either). The constant speed of light system has some strange properties as well, but these are the more familiar puzzles of special relativity.

The best way to distinguish between the physical effects of gravity in GR, and the artifacts of the coordinate system is to consider the Milne universe, where there is no gravity. Take a look at Matt McIrvin's page on the subject: http://world.std.com/~mmcirvin/milne.html
 
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  • #33
marcus said:
But I do NOT see how one could translate that motion into a DOPPLER effect. The relativistic doppler formula that i know applies only to speeds less than c. But the recession speeds we are dealing with are all the time greater than c. So what doppler formula would one use?
I agree with you marcus, and I want to add the following for clarification. In my understanding, the only fact that one obtains different recession speeds with both methods is not the problem, since relating redshift to recession speed may be a matter of definition (how to infer about recession speeds without measuring redshifts?). But, as long as (1.) it exists a relation between speed and distance (Hubble's law) and (2.) there exist independent methods to determine distances, then both interpretations of the Hubble flow cannot be equivalent.
 
  • #34
chronon said:
General relativity determines the shape of spacetime, but doesn't specify how you coordinatise it. The usual choice is to take a time coordinate corresponding to the proper time of comoving objects. My feeling is that this is just going back to the idea of absolute time, and it is better to use a coordinate system in which the speed of light is taken to be constant. Then nothing travels faster than light. I've written some ideas on how this could be done at: http://www.chronon.org/Articles/milne_cosmology.html
Thank you for the link to your page. I've got to spend some time tracking down Milne.

chronon's page said:
Well I hope I've shown you that the usual way of looking at the universe isn't the only one, and one can use SR coordinates without disagreeing with general relativity. Next time you hear of something strange going on in cosmology remember to think 'Is this just because of the choice of coordinate system?'.
 
  • #35
Pardon me for being impolite. Chronon is espousing a bunch of crap.
 
  • #36
Chronos said:
Pardon me for being impolite. Chronon is espousing a bunch of crap.
I am a late-comer to the cosmology table (middle-aged amateur astronomer). The best way for me to learn is to compare models (new and old) and see where they differ in assumptions, methodology, etc. You can learn as much or more from a failed experiment as you can from a successful one, if you are willing to take the time to study the process and the observations and determine where the model's predictive power failed. It is very easy to adopt "status quo" thinking and avoid considering non-mainstream ideas, but that habit leads to stagnation and the perpetuation of error.

Feynman on epistemology said:
...Another of the qualities of science is that it teaches the value of rational thought as well as the importance of freedom of thought; the positive results that come from doubting that the lessons are all true. You must here distinguish--especially in teaching--the science from the forms or procedures that are sometimes used in developing science. It is easy to say, "We write, experiment, and observe, and do this or that." You can copy that form exactly. But great religions are dissipated by following form without remembering the direct content of the teaching of the great leaders. In the same way, it is possible to follow form and call it science, but that is pseudo-science. In this way, we all suffer from the kind of tyranny we have today in the many institutions that have come under the influence of pseudoscientific advisers.

We have many studies in teaching, for example, in which people make observations, make lists, do statistics, and so on, but these do not thereby become established science, established knowledge. They are merely an imitative form of science analogous to the South Sea Islanders' airfields--radio towers, etc., made out of wood. The islanders expect a great airplane to arrive. They even build wooden airplanes of the same shape as they see in the foreigners' airfields around them, but strangely enough, their wood planes do not fly. The result of this pseudoscientific imitation is to produce experts, which many of you are. [But] you teachers, who are really teaching children at the bottom of the heap, can maybe doubt the experts. As a matter of fact, I can also define science another way: Science is the belief in the ignorance of experts.

When someone says, "Science teaches such and such," he is using the word incorrectly. Science doesn't teach anything; experience teaches it. If they say to you, "Science has shown such and such," you might ask, "How does science show it? How did the scientists find out? How? What? Where?"

It should not be "science has shown" but "this experiment, this effect, has shown." And you have as much right as anyone else, upon hearing about the experiments--but be patient and listen to all the evidence--to judge whether a sensible conclusion has been arrived at.

In a field which is so complicated [as education] that true science is not yet able to get anywhere, we have to rely on a kind of old-fashioned wisdom, a kind of definite straightforwardness. I am trying to inspire the teacher at the bottom to have some hope and some self-confidence in common sense and natural intelligence. The experts who are leading you may be wrong.

I have probably ruined the system, and the students that are coming into Caltech no longer will be any good. I think we live in an unscientific age in which almost all the buffeting of communications and television--words, books, and so on--are unscientific. As a result, there is a considerable amount of intellectual tyranny in the name of science.

Finally, with regard to this time-binding, a man cannot live beyond the grave. Each generation that discovers something from its experience must pass that on, but it must pass that on with a delicate balance of respect and disrespect, so that the [human] race--now that it is aware of the disease to which it is liable--does not inflict its errors too rigidly on its youth, but it does pass on the accumulated wisdom, plus the wisdom that it may not be wisdom.

It is necessary to teach both to accept and to reject the past with a kind of balance that takes considerable skill. Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers of the preceding generation.

So carry on. Thank you.

I'm pretty keen on epistemology, so it's enlightening to find out whether a cosmologist has tried to make a "clean-slate" start, and if not, what prior assumptions he has embraced. Milne apparently started with Special Relativity and the things then-known by observational astronomers, and constructed a cosmological model from those priors that did not require curved space-time. His model may not have been entirely correct, but it appears to have been quite valuable to the progression of cosmology, including his "cosmological principal", which is a crucial feature of the standard model. Again, Chronon, thanks for the link.
 
  • #37
turbo-1 said:
Thank you for the link to your page. I've got to spend some time tracking down Milne.
When I was writing the web page I read the book:
Relativity, gravitation and world-structure by Edward Arthur Milne

You might be interested in the notes I made about this book. (My intention is eventually to add 'further reading' notes for each page to my website, but I haven't got round to it yet)
Interesting how he derives results from 1st principles. Doesn't like curved space. Rejects finite, unbounded universe - real/image distinction - somewhat muddled. (Obviously never played Asteroids.)
Particle horizons= continuous creation (sees Particle horizons as now) .
At one stage seems to adopt the 'cancelling forces' view of gravity, as Newton did. However, later he seems to criticize Newton's adoption of this point of view.
A random spread of velocities means that some particles will gravitate towards their place of rest. He sees this as a possible explanation of Dust clouds, and of cosmic rays. However, he does have particles reaching the velocity of light, so something is surely wrong.
So Milne was trying to find an alternative to General relativity, whereas today the 'Milne Universe' is thought of as a particular model within the framework of general relativity.
 
  • #38
chronon said:
Again I'll try to emphasise the difference between space and spacetime. General relativity tells us about the intrinsic curvature of spacetime. It allows considerable freedom of how we coordinatise spacetime. What we think of as space depends upon our choice of coordinate system.The space in the Milne universe is hyperbolic in the usual coordinate system. The spacetime is flat. The case which is usually referred to as 'flat', k=0, has curved spacetime, flat space (in the usual coordinate system)When I talk about movement or stretching, this is with respect to space, not spacetime

I have taken some time to answer this post as I have been very busy and I needed time to do a little calculation and look up your links.

First thank you for correcting my slack choice of words, space stretches in the expanding universe, I am well aware of the difference between the curvature and evolution of space and the curvature of space-time. It was late and I was hasty in my post. :blushing: However, what space-time is doing in the standard modes of the R-W metric Friedmann models is a more complicated question.

You have to distinguish between the Milne model, the empty limit of the Friedmann models of homogeneous and isotropic GR, and Milne Cosmology or 'Kinematic Relativity' which is a non-GR alternative cosmology and of which I have been aware for some time and to which I referred in my paper.

It is in Kinematic Relativity that space-time is flat. In the empty Friedmann model k = -1 and R = t, the 'Milne model', the universe spatially expands linearly and 3D space is hyperbolic embedded in a 4D manifold. The space-time of this model can be said to be not 'flat', in the sense that not all the components of its Riemann tensor Rabcd are zero, even though it can be embedded into flat Minkowski space-time by a transformation of coordinates.

The point I was making was that in the standard GR cosmological solution 'empty space' had a non-trivial structure.
 
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  • #39
Garth said:
It is in Kinematic Relativity that space-time is flat. In the empty Friedmann model k = -1 and R = t, the 'Milne model', the universe spatially expands linearly and 3D space is hyperbolic embedded in a 4D manifold. The space-time of this model can be said to be not 'flat', in the sense that not all the components of its Riemann tensor Rabcd are zero, even though it can be embedded into flat Minkowski space-time by a transformation of coordinates.
I'm not convinced about that - I think the Riemann tensor is zero. However, I'm rather out of practice in doing GR calculations, so I may have to think about it for a while. (I tried Googling: Milne "Riemann Tensor" , but all the hopeful looking quotes were to subscription only websites of journals)
 
  • #40
chronon said:
I'm not convinced about that - I think the Riemann tensor is zero. However, I'm rather out of practice in doing GR calculations, so I may have to think about it for a while. (I tried Googling: Milne "Riemann Tensor" , but all the hopeful looking quotes were to subscription only websites of journals)
I believe all components of the Riemann tensor are zero in the Milne universe. The Weyl tensor is zero, as it is zero in any Robertson-Walker model, and the components of the Ricci tensor are also zero, since \ddot a = 0 and \dot a = \sqrt{-k}. This makes sense, since all components of Tuv are zero and space is homogeneous and isotropic. Thus space-time is flat (although it expands) and space is curved k = -1.
 
  • #41
chronon said:
I'm not convinced about that - I think the Riemann tensor is zero. However, I'm rather out of practice in doing GR calculations, so I may have to think about it for a while. (I tried Googling: Milne "Riemann Tensor" , but all the hopeful looking quotes were to subscription only websites of journals)
Here's a way to find (at least sometimes!) copies of these papers without paying $20-30 per article. At the journal page, highlight the title of the article and Google search on that title. This process is GREATLY simplified if you use Firefox as your browser - just highlight the title, right-click on it, and in the pop-up menu choose "Search Web for..." Firefox will open your default search engine in a new tab (preserving the page you are on currently in another tab).

Even better, you can go to the Mozilla site and add Google Scholar to your tool bar and even make it your default search engine. Google Scholar will pull up the pay-as-you-go journals, but it will usually also pull up free archives, and it will identify citations and do other nice things.
 
  • #42
For the metric
<br /> a(t)^2 (d \chi^2 + sinh(\chi)^2(d\theta^2 + sin(\theta)^2 d\phi^2)<br />

which should be the FRW metric of interest (with k=-1, i.e. negatively curved spacelike hypersurfaces)

with a(t) = t

I get all components of R = 0, which implies that the Ricci and the Einstein are also zero (which means it's the zero density soulution as desired). Which also means that while the space-like hypersurfaces which perserve isotropy aren't flat, the space-time as a whole is flat, as other posters have remarked (and Google turns up a number of web pages which say the same thing).

However, while this is a flat spacetime, I don't think it satisfies the necessary conditions for conserving energy ("asymptotic flatness"). In spite of the similarity in names, flat does not necessarily imply asymptotically flat.
 
  • #43
hellfire said:
I believe all components of the Riemann tensor are zero in the Milne universe. The Weyl tensor is zero, as it is zero in any Robertson-Walker model, and the components of the Ricci tensor are also zero, since \ddot a = 0 and \dot a = \sqrt{-k}. This makes sense, since all components of Tuv are zero and space is homogeneous and isotropic. Thus space-time is flat (although it expands) and space is curved k = -1.
and chronon and pervect

The Ricci tensor is the trace or contracted form of the Riemann and that is zero throughout
Rabad = Rbd = 0,
its components can cancel each other out
R0b0d + R1b1d + R2b2d + R3b3d = 0, so one can have
Rbd = 0 even if an individual
Rabad is not zero. Of course the trace can vanish even with an individual non zero Rabcd if it is 'off-diagonal'.

As I said above, the solution to the empty GR cosmological case is not the same as SR flat Minkowski space-time, it has a structure imposed by the cosmological boundary conditions imposed on the Einstein field equation.

And hellfire as we have recently established it is 3D space that expands not 4D space- time!


Garth
 
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  • #44
Thank you for your clarification Garth. You may be right. I did not perform the calculations and I was relying only on my (probably bad) intuition. To calculate all components of the Riemann tensor is actually a lot of work; may be you could tell me which one(s) you expect to be non-zero, that I can convince myself...
 
  • #45
In order to be sure I have just calculated all 256 components of the Riemann tensor for the Milne universe metric! (Allowing for asymmetries of course). Whereas what I have said about all the components of the Riemnann tensor not neccessarily being zero for the all the components of the Ricci tensor to be zero is in general correct, in the particular case of the Milne metric the components of the Riemann tensor are zero. The Milne space-time is flat and I apologise for misleading anyone. :blushing:
Garth
 
  • #46
So to recap, the solution to the empty GR cosmological case is not the same as SR, its space hypersurface has a structure imposed by the cosmological boundary conditions imposed on the Einstein field equation. Although it is rather academic to talk about such a non-Euclidean 3D geometry in an empty universe as there would be nobody around to measure it.

The reason I thought some of the components of the Riemann had to be non-zero was the (incorrect) intuition that that had to be so in order for its space-time to impose the hyperbolic geometry on its space foliations.

Garth
 
  • #47
Garth said:
Whereas what I have said about all the components of the Riemnann tensor not neccessarily being zero for the all the components of the Ricci tensor to be zero is in general correct, in the particular case of the Milne metric the components of the Riemann tensor are zero.
Anf if I was correct in my reasoning, this is due to the fact that space is homogeneous and isotropic. Since we know that the Ricci tensor is zero, then we have e.g. R00 = R0000 + R1010 + R2020 + R3030, the first being zero due to the antisymmetry of the Riemann tensor and the others are zero because they must be equal. Correct?
 
  • #48
Garth said:
In order to be sure I have just calculated all 256 components of the Riemann tensor for the Milne universe metric!

What program did you use to get your result? I used GRTensorII, and the metric I posted, to get the same result - that all the coefficients of the Riemann were zero. I'm assuming you did use a program, it would be very messy not to mention tedious to carry out that calculation by hand.
 
  • #49
I still naively assert the CMB energy was never lost, merely diluted by expansion.
 
  • #50
pervect said:
What program did you use to get your result? I used GRTensorII, and the metric I posted, to get the same result - that all the coefficients of the Riemann were zero. I'm assuming you did use a program, it would be very messy not to mention tedious to carry out that calculation by hand.
The pencil and paper programme! Its not too messy if you do it methodically, I had a few exciting moments with non-zero results, which demanded double checking only to find that in fact they were zero, and that gave me a deep feeling and confidence in the Riemann tensor. Its like walking or cycling rather than riding in a car, even though its slower you get a feel for the countryside.
For my sins I am now working my way through the Riemann for a general Friedmann model to answer hellfire's question, I'll let you know how I get on!

Garth
 

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