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rtomes
2007-Oct-07, 01:06 AM
In the now closed ATM thread http://www.bautforum.com/against-mainstream/63081-harmonics-theory.html I made some statements about the possible existence of standing waves in general relativity that were challenged by Nereid. This was near the end of the 30 days and the matter was not sufficiently addressed by me. However it did alert me to the fact that things that I had assumed would be accepted facts about some aspects of GR in the large scale structure of the universe were in fact not so. Since then I have made several attempts to get answers on these issues from physicists without a great deal of success. The questions were sent to sci.physics.research and did not appear (probably a moderator decision). I asked a physicist who said I could only have one question. Then I tried http://www.physicsforums.com/ where the thread http://www.physicsforums.com/showthread.php?p=1457174 has had a couple of replies but they are not really helping a heck of a lot. So I thought that I would try here.

I would appreciate if some learned folk could answer these questions to increase my understanding.

1. Does GR support standing wave solutions to the equations?

2. Do physicists study these standing wave solutions?

3. Is it correct to describe GR equations as wave equations? (e.g. as Maxwell's equations and Schroedinger equations are)

4. Do Black holes have vibrational modes that depend mainly on the event horizon radius R or 2*pi*R?

5. Would such vibrational modes have periods calculated as R/c and 2*pi*R/c or are relativistic adjustments needed?

6. Are periods related to these vibrational modes observed in galactic core black holes? I know that we cannot observe the gravitational waves, I am referring to associated emissions, such as light, that we might observe.

7. Are GR standing waves ever considered as possibly relevant to large scale structure in the Universe?

Any other comments along similar lines are most welcome.

Thanks
Ray Tomes

Michael Noonan
2007-Oct-07, 01:31 AM
Hello Ray,

I wonder if harmonics could also encompass the use of discord in the way Beethoven did? You see ever since the question came up about locust I have been intrigued by the difference between our countries level of abundance.

Australia is very sparse and vegetation is not always easy to access so a vector that enjoyed a harmonic could in practice become extinct due to being too successful.

Whereas New Zealand is a beautiful place and abundant in vegetation so the locust there do not need to swarm to do an emergency evac to greener pastures. Here in Australia I believe the cycle is every seven years which align with the least harmonic number in the sequence.

I am sorry I don't have the answers you need but I hope this thought is in some way of use to you.

Cheers Michael N

Celestial Mechanic
2007-Oct-07, 04:46 AM
Hello Ray,

I wonder if harmonics could also encompass the use of discord in the way Beethoven did? You see ever since the question came up about locust [sic] [Snip!]

Australia is very sparse [Snip!]

Whereas New Zealand is a beautiful place [Snip!]

I am sorry I don't have the answers you need but I hope this thought is in some way of use to you.
This is mostly off-topic and not very coherent.

Tim Thompson
2007-Oct-07, 05:08 AM
1. Does GR support standing wave solutions to the equations?
2. Do physicists study these standing wave solutions?
3. Is it correct to describe GR equations as wave equations? (e.g. as Maxwell's equations and Schroedinger equations are)

Unconditional yes to 1 & 2. Yes to 3 with the caveat that the equations of general relativity include waves (gravitational waves), but also include quite a lot that is not waves. For examples of physicists studying standing waves in GR see Bondi, 2004 (http://adsabs.harvard.edu/abs/2004RSPSA.460..463B); Andrade, et al., 2004 (http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRVDAQ000070000006064001000001&idtype=cvips&gifs=yes); ; Beetle, Bromley & Price, 2006 (http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRVDAQ000074000002024013000001&idtype=cvips&gifs=yes); Beetle, et al., 2007 (http://adsabs.harvard.edu/abs/2007arXiv0708.1141B).



4. Do Black holes have vibrational modes that depend mainly on the event horizon radius R or 2*pi*R?
5. Would such vibrational modes have periods calculated as R/c and 2*pi*R/c or are relativistic adjustments needed?

No to both questions, so far as I can tell. Black holes do have ringing modes, but they are not so easily described (see, i.e., A numerical study of the quasinormal mode excitation of Kerr black holes (http://arxiv.org/abs/gr-qc/0608091); Dorband, et al., Physical Review D 74(8): paper no. 084028, 2006).



6. Are periods related to these vibrational modes observed in galactic core black holes? I know that we cannot observe the gravitational waves, I am referring to associated emissions, such as light, that we might observe.

I am unaware of any such observations, and I doubt that it would be possible. The gravitational wave amplitudes for the ringing modes of a black hole are swamped by the other physics going on, so such tiny vibrations would be lost in the noise.


7. Are GR standing waves ever considered as possibly relevant to large scale structure in the Universe?

As far as I can tell, no. The standing wave studies I gave above all deal with standing waves resulting from the interference between outgoing & incoming waves for inspiraling binaries. I have not seen any application to large scale structure, and would not expect standing wave to be relevant there anyway. Large scale structure is related to constructive interference of sound waves ("baryon acoustic oscillations") in the photon-baryon fluid of the early universe. The standing waves in the inspiraling binaries are really quasi standing waves, as they cannot be stable. One would normally expect true, stable standing waves in an oscillating cavity (the classic example of the organ pipe comes to mind). In the absence of such, one would not expect true standing waves.

That about exhausts my current knowledge of the subject.

Celestial Mechanic
2007-Oct-07, 05:22 AM
[Snip!] 1. Does GR support standing wave solutions to the equations?

2. Do physicists study these standing wave solutions?
Off the top of my head I would say "yes". A quick Google showed a few entries for something called the "periodic standing wave approximation".

3. Is it correct to describe GR equations as wave equations? (e.g. as Maxwell's equations and Schroedinger equations are.)
The wave equations in electromagnetism are homogeneous solutions to the field equations (the ones with sources in them), obtained by setting the sources to zero. Since the EM equations are linear, the solutions of the homogeneous equations can be added to the particular solutions (for the sources) and still be solutions of the Maxwell equations. I side with you that there are such things as non-linear wave equations, and the wave equations of GR are non-linear. I fear that many of our problems with regard to the detection of gravity waves comes from the fact that we are forgetting about non-linearity. (Most approaches to gravity waves use linearized equations, see Gravitation by Misner, Thorne, and Wheeler, for example.)

Setting the source equal to zero converts the Einstein field equation to Gij=0, and there is a huge literature on these solutions, called Einstein spaces. I don't know much about these solutions, but I would be surprised if there weren't any periodic solutions.

4. Do Black holes have vibrational modes that depend mainly on the event horizon radius R or 2*pi*R?

5. Would such vibrational modes have periods calculated as R/c and 2*pi*R/c or are relativistic adjustments needed?
I don't think relativistic adjustments would be needed, at least in the case of non-rotating black holes. Black holes are already as relativistic as you can get! :D

Don't forget that black holes can also have angular momentum. From dimensional arguments we see that sqrt(c5/G/J) has dimensions of frequency, so any frequencies of a black hole are going to be functions of this and c/R.

6. Are periods related to these vibrational modes observed in galactic core black holes? I know that we cannot observe the gravitational waves, I am referring to associated emissions, such as light, that we might observe.
What we observe from a black hole is the x-ray radiation of the accretion disk. It has orbital dynamics determined by the metric and I don't see any way for vibrational modes of the black hole to be strong enough to affect the orbits of particles in the accretion disk in an observable way. I could be wrong on this.

7. Are GR standing waves ever considered as possibly relevant to large scale structure in the Universe?
Not sure.

publius
2007-Oct-07, 05:33 AM
This is most complex subject. The mathematics of the EFE are not trivial, in fact it is some of the most complex mathematics there is. To truly understand it requires a lot of work, and a lot of mathematical prowess. It is well beyond me, but I will make a pitiful attempt to answer this question. To answer it, we need to know what the EFE is. That is, what is that quantity we are attempting to solve for.

That quantity is the metric, and what that is, I can only say, "something that describes the geometry of space-time". It is nothing like any Newtonian notion of a gravitional field. Far from it. For a comparison to Maxwell, you can think of the metric as playing the role of a potential. But it is a rank-2 tensor potential, and the notion of a "field" to be found from first derivatives of that potential is not well defined. This analogy is very rough, just very general notion. The EM potential is a vector (rank-1) quantity. Gravity cannot be described by a vector field. It must be (at least) a rank-2 tensor.

The EFE gives us a very complicated, and very non-linear tensor differential equation relating the metric to the sources, the stress-energy tensor. Mass-energy is not the only contributor to the field. Momentum (pressure) as well currents of both are as well.

Can you call it a wave equation? Well, you can, but you've got to appreciate there is more to, much, much more than any simple scalar or vector wave equations.

To solve for the metric, you've got to choose a coordinate system, and there is much complexity and much freedom involved in that. The spirit of GR shows us that how we describe things has to components, invariant physics, and coordinate choice. The latter is arbitrary, but it determines greatly how we explain things. What things look like. Yet it is arbitrary. When a metric is found, one worries about the invariant aspects of the space-time it describes more than any coordinate specifics.

And so, the nature of any "waves" one may fathom, or just choose to call things such, depends on those coordinate choices.

Metric solutions exist that can be described as "standing waves" in a given coordinate system, as well as travelling waves. But these are waves in a tensor "thing", the geometry of space-time and go beyond any mere vector thing. They have nothing to do with anything like your "harmonics" theory idea in ATM.

Is there any relevance to Cosmology there. Well, here is something:

http://background.uchicago.edu/~whu/araa/node23.html

What is being discussed there is well beyond my ability to comprehend, other than the effects of something that be described as a standing metric wave perturbation on the CMB. The source of that perturbation is apparently "something relating to inflation". This is something to be explained by the high priests, not by a piker like me.

-Richard

Ken G
2007-Oct-07, 02:33 PM
I don't have much to add to what has been said already, except to say that I don't even know how someone would meaningfully apply the concept of a "standing wave" solution in GR in a way that isn't trivial. What confuses things is that there are actually two very different ways of getting a standing wave-- one which is to have two equal-strength waves propagating in opposite directions such that the spatial dependence separates completely from the time dependence, like can happen in a cavity like a flute, and the other is to choose a reference frame that suppresses the time dependence entirely. The former is the "nontrivial" way to do it, but the latter way seems more trivial, yet appears to be what people are talking about in the infalling binaries in GR (see Tim Thompson's links).

In other words, the usual meaning for a "wave" solution is something that depends on space and time such that it is not a separate function of space and time, but rather a single function of a combination of space and time, where the combination of space and time looks like something that propagates. But if you simply call that combination what you mean by "space", then there's no variation left for time-- it's a standing wave. So any nondispersive solution to a wave equation could be viewed as a standing wave in GR, the only issue is whether or not it is admissible to call your combined coordinate "space". You can't for a normal light wave, because space would need to be propagating at c, and we know that leads to all kinds of nasty singularities. But in other cases, you don't get singularities, apparently-- like for the inspiralling binaries of Tim Thompson's links, where what you mean by space is in the co-rotating frame with the binary. What's not clear to me is if those situations suppress all time dependence, or if some separable sinusoidal dependence remains-- more like oppositely propagating waves. Bondi talks about inward and outward waves, but co-orbiting reference frames would normally suppress all time dependence, so I don't know. In the CMB that publius was talking about, space is probably the comoving frame (it usually is in cosmology), and I have no idea how that interacts with inflation to give standing waves, but it seems like this might actually be the other type of standing wave-- where there's still sinusoidal time dependence, but it separates completely from the spatial dependence.

rtomes
2007-Oct-07, 07:06 PM
Many thanks for these answers which, along with some of the material in wikipedia, make it much clearer to me that this is an area that is very difficult to do calculations because of the complexity of the maths so that in general people have to use approximations. I just wonder to what extent GR simulations (e.g. http://www.google.co.nz/search?q=general+relativity+simulations ) are a realistic means to study what I am interested in which is large scale 3-dimensional standing waves? (Well, I guess someone might say 4-dimensional). If there is software that might be suitable for this type of study then I guess that would be a lot easier than trying to manipulate the maths which is certainly beyond my capabilities.

My primary interest is large scale waves (e.g Mpc upwards), and it does seem that much of the research is concerned with conditions around black holes and even then with ripples rather than the whole structure modes. I am guessing that the dissipation that happens due to non-linearity will be much faster for these smaller waves but haven't managed to find anything on the rate of dissipation that I can understand.

Assuming that larger waves would dissipate more slowly then the questions relate to how that dissipation occurs? It seems to me that from consideration of symmetry, if there were a large repetitive 3D structure in the universe then at least some of this dissipation ought to be by way of production of harmonics of the waves. Is that right? If not, where would the energy go? How would it manifest?

Nereid
2007-Oct-08, 01:20 AM
I would like to emphasise a point already made by several: the topic 'standing waves in GR' does not lend itself to simple, intuitive understanding in which word picture analogies with standing waves in organ pipes (for example) are used to draw conclusions about some ~Mpc-scale phenomena in cosmology/astrophysics.

Or, saying much the same thing from the other direction, working out what sort of ~Mpc-scale GW (gravitational wave) phenomena may arise in an LCDM (or other!) universe would almost certainly require rolling up your sleeves and getting stuck into the relevant math.

From the observational side, there is a LIGO team looking for 'stochastic gravitational-wave backgrounds'; one of their papers is "Searching for a Stochastic Background of Gravitational Waves with LIGO (http://arxiv.org/abs/astro-ph/0608606)". The introduction section points to some earlier papers that may be of direct relevance to the question in the OP:
Many possible sources of stochastic GW background have been proposed and several experiments have searched for it (see (Maggiore 2000; Allen 1996) for reviews). Some of the proposed theoretical models are cosmological in nature, such as the amplification of quantum vacuum fluctuations during inflation (Grishchuk 1975), (Grishchuk 1997), (Starobinsky 1979), pre-big-bang models (Gasperini & Veneziano 1993), (Gasperini & Veneziano 2003), (Buonanno et al. 1997), phase transitions (Kosowsky et al. 1992), (Apreda et al. 2002), and cosmic strings (Caldwell & Allen 1992), (Damour & Vilenkin 2000), (Damour & Vilenkin 2005). Others are astrophysical in nature, such as rotating neutron stars (Regimbau & de Freitas Pacheco 2001), supernovae (Coward et al. 2002) or low-mass X-ray binaries (Cooray 2004).

A number of experiments have been used to constrain the spectrum of GW background at different frequencies. Currently, the most stringent constraints arise from large-angle correlations in the cosmic microwave background (CMB) (Allen & Koranda 1994; Turner 1997), from the arrival times of millisecond pulsar signals (Jenet et al. 2006), from Doppler tracking of the Cassini spacecraft (Armstrong et al. 2003), and from resonant bar GW detectors, such as Explorer and Nautilus (Astone et al. 1999). An indirect bound can be placed on the total energy carried by gravitational waves at the time of the Big-Bang Nucleosynthesis (BBN) using the BBN model and observations (Kolb & Turner 1990; Maggiore 2000; Allen 1996). Similarly, (Smith et al. 2006a) used the CMB and matter spectra to constrain the total energy density of gravitational waves at the time of photon decoupling.One advantage, if it can be called that, of these observations is that they constrain the GW energy (density), over a wide range of frequencies and spatial scales.

rtomes
2008-Jan-14, 11:40 PM
I was just rereading this and having some further thoughts which naturally leads to further questions.

The sort of waves that I am interested in are very large structure rather than some sort of contained wave. If there is a "container" then it is the universe as a whole, or at least that is the starting point. Perhaps the nearest thing in the literature are discussions such as a bouncing or oscillating universe. This would appear to me to be a starting point for space-time wave structures. Therefore many of the answers refer to much lower frequencies than I am interested in.

I am then interested in the various low order harmonics of standing waves that fill the universe. It does seem that these are not seriously studied. Perhaps that is because the maths is too difficult as has certainly been attested to here. Perhaps it is also because people are not aware that this is a very rich potential source of information about the entire structure of the universe at all scales. And I do mean all scales, from galactic super-clusters through to sub-atomic particles, the scales at which structure appears can be predicted if it can be shown that non-linear waves exist and therefore lose energy to their harmonics over time. I will say no more about that here, but direct those interested to read the now closed ATM thread http://www.bautforum.com/against-mainstream/63081-harmonics-theory.html

My remaining question then is, if the entire universe (should it be finite) is taken as the "container" and we consider an oscillating or bouncing universe, what standing wave structures are likely to exist?

Ken G
2008-Jan-15, 02:04 AM
Well, a "bouncing" universe would itself be considered a standing wave structure of infinite wavelength, in the sense of having a periodic time behavior that separates completely from the space behavior. What the significance of that might be for our universe is not terribly clear-- there's no evidence we have a bouncing universe.

rtomes
2008-Jan-16, 03:03 AM
While a bouncing universe is certainly not proven, it would be fair to say that it is not disproven also wouldn't it? It might not be favoured by the majority, but papers are still published in the Journals from time to time.

Would it be fair to say that if the universe were bouncing then there would be standing waves in GR (maybe time, space and space-time modes) and that at present these modes are not studied, probably because the maths is too complicated?

I am looking for a fair statement that cosmologists and physicists would not disagree with I can make as an introduction to my Harmonics Theory work that shows that there is no evidence that it is incompatible with GR although it certainly is not consistent with Big Bang Cosmology as it indicates an age for the universe that exceeds 10^23 years.

Of course there are other alternatives to bouncing if red shift is ex-plained by something like Narlikar's variable particle mass theory. Bouncing is itself a near Big Bang concept.

Ken G
2008-Jan-16, 03:18 AM
While a bouncing universe is certainly not proven, it would be fair to say that it is not disproven also wouldn't it?It would be fair to say there is zero evidence in favor of it, and plenty opposed. But we never get to know if that's what "really happened" or not-- we get to choose our best model for understanding our history, and right now, that is not a bouncing universe.

It might not be favoured by the majority, but papers are still published in the Journals from time to time.It's not a majority/minority issue, it's an evidence issue. Or in this case, a lack of evidence issue. But there's no harm in pursuing "what if" scenarios, who knows what tomorrow's data will bring-- ergo the papers.


Would it be fair to say that if the universe were bouncing then there would be standing waves in GR (maybe time, space and space-time modes) and that at present these modes are not studied, probably because the maths is too complicated?I don't know how complicated the maths would be.


I am looking for a fair statement that cosmologists and physicists would not disagree with I can make as an introduction to my Harmonics Theory work that shows that there is no evidence that it is incompatible with GR although it certainly is not consistent with Big Bang Cosmology as it indicates an age for the universe that exceeds 10^23 years.I think if you can build a theory entirely from possible solutions to GR, no one can take great issue with it. Even if the data doesn't support it, at least it's one way to find out what kind of data one would need to say one way or the other. Models aren't bad when they are wrong, they are bad when they are false.


Of course there are other alternatives to bouncing if red shift is ex-plained by something like Narlikar's variable particle mass theory. Bouncing is itself a near Big Bang concept.Yes, bouncing is just a matter of observed boundary conditions, the physics is all pretty much the same (except for the thermodynamic problems of how to "reset" a universe, but no one knows the thermodynamics of the Planck era anyway).