Can Higher Order Curvature Theories of Gravity Solve the Dark Matter Puzzle?

[turbo]

I think you have it backwards. To quote John Baez: "... quantum field theory and general relativity have really different attitudes towards the energy density of the vacuum. The reason is that quantum field theory only cares about energy differences. If you can only measure energy differences, you can't determine the energy density of the vacuum."

It's a matter of semantics, and a bit tricky, but look further down the page. He claims that measurement can be made within GR by studying the curvature of the universe and calculation of the ZPE energy can be made in QFT while ignoring gravitation. The "measurement" in this case is not a measurement at all, but an extrapolation of absolute ZPE energy from GR's concept of curved space-time and the observed flatness of our Universe. This is not a "measurement" in any real sense. We can only measure energy differences from an established standard, and since the ZPE is the ground state of our Universe, it's absolute energy density will not be directly measurable by us. Baez acknowledges this in a rather indirect fashion, saying that to have any faith in the accuracy of the "measurement":

To believe any of these measurements are right, one must have some faith in general relativity, because that's the theory which we use to relate spacetime curvature to energy density. For the measurements that attempt to determine an actual value for the energy density of spacetime one must have more faith in general relativity, and also other assumptions about cosmology.

My polarized ZPE model eliminates the need for the "curved space-time" explanation for gravity, upon which this "measurement" relies, and which has been shown to be deficient at galactic and cluster scales.

 

[turbo]

Here is a paper that models quantum vacuum fields as non-gravitating in the absence of perturbing masses. While the author is working in the GR framework and is somewhat bound by its precepts, his perception of the function of the vacuum fields is very close to my model of gravitation via the polarized ZPE fields. In addition, he makes the suggestion that the polarized vacuum fields in the presence of pertubating mass/energy may fulfill the role of non-baryonic dark matter.

http://citebase.eprints.org/cgi-bin/citations?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F0304061

 

[ohwilleke]

this is from Willeke post #18


this is an interesting idea. i wish you or somebody would write up something along these lines, with some sample values of things calculated.

Just to put a little finer point on it (and disregarding "general vicinity" MOND gravity effects which may not be trivial as described before and would slow time down even more), the gravitational redshift from a spherical mass M with radius R is 1+z=1/SQRT(1-2GM/c^2R) *1*. I haven't seen how this is derived but would assume that it is an integral of gravitational redshift with respect to dx (distance from the star) over x=R to x=infinity. Now, if MOND applies, you could really fairly naiively break this integral into two parts, one would be the integral used to derive the term above integrated from x=R to x=the radius at which GM/R^2=a sub zero (call it R sub a), and the other integrating from x=R sub a to R observer. Now, the first component is going to be very, very close to 1+z as given by the standard formula for graviational redshift, since under Newtonian gravity the R sub a to R sub infinity contribution to the total should be very small because it falls off proportional to 1/R^2. Basically you are integrating over a 1/R^2 curve. But, in MOND gravity, you are integrating over a 1/R curve.

Now, gravitational potential energy in Newtonian gravity is proportional to GM/r. But, gravitational potential energy in MOND is proportional to GMln r. If gravity were entirely a 1/r force, one would thus expect something like 1+z=1/(SQRT(1-2GM/c^2ln R) *2*, only it isn't even entirely clear to me that the difference between the integral to r observer and r infinity is insignificant. Of course, you'd really have to subtract out of *2* a factor equal to *2* from x=radius of star to x= R sub a, and add in the value of *1* for that range.

If the difference between r observer and r infinitity is insignficant at intergalactic distances then this is a modest matter. It means that the traditional formula *1* systematically understates z by a function of the radius of the star regardless of distance.

But, if the distance between r observer and r infinity is significant then in addition to formula *1* and correction term *2* which is a function of star radius, then you will have a third correction representing the integral over MOND gravity of the time dilation from r observer to r infinity which will be a function not only of R and M, but also of r observer. This would have to be subtracted out of *1* plus modified *2*. (But the third correction term subtracted out should be less than *2* since otherwise aggregate MOND graviational effects are less than aggregate Newtonian gravitational effects which doesn't make since given that MOND graviational effects are the same close to the source mass and greater far from the source mass). It would imply that gravitational redshift rather than being more or less independent of distance at intergalactic scales is dependent upon distance, and while I can't do the math on the fly, I'd imagine that this wouldn't be terrrifically hard to do with naiive mond that simply says before a sub zero use 1/r^2 and after it use 1/r, which in a symmetric, single mass case is probably an exact result even though it wouldn't be in an asymmetric or multiple mass case.

The bottom line is that the modified formula should give you more gravity and hence more redshift and hence imply that distance objects are closer than they appear, even without considering distant field effects of other objects in the universe. And, closer objects mean a smaller Hubble's constant which in turn implies less dark energy. Moreover, if there is distance dependence, it would mean that our observations which are most easily confirmed (e.g. by Cephid paralaxes nearby) systematically differ from our observations using Cephid yardsticks at great distances because the Chepids at great distances look farther away than they really are.

 

[turbo]

ohwilleke, I admire your math skills and appreciate your willingness to say "what if" in regard to concepts that seem to have become codified in the standard model. Have you explored the mechanics of gravitation that MOND is modeling with such accuracy? I have a mechanism for gravitation that is workable on both GR and quantuum scales, and is entirely falsifiable (something sadly lacking in much of the standard model!).

I have known for years (due to correspondence with a former associate of Hubble) that Edwin Hubble was not comfortable with equating redshift with "recessional velocity". He discovered the relationship between redshift and apparent distance, but was not the "father of the Big Bang" as he often is portrayed in the popular press. We must bear in mind that Einstein's CC was an attempt to make GR fit his preferred steady state cosmology, and that Hubble was working in a similar theoretical setting, with a presumed steady-state universe. Once redshift was popularly interpreted as cosmological recession, though, it was inevitable that mathemeticians would extrapolate back to a presumed singularity, assign an age to the universe, etc, etc, and the Big Bang was invented.

Because of all the fudge factors that have been loaded onto Big Bang cosmology to keep it afloat, I decided to start back at Newton and GR (with appropriate epistemological skepticism) and try to model a cosmology that incorporates only things we know exist, or that are experimentally supported to a reasonable standard of confidence. My background as an optician gave me the physics to model "gravitational" lensing properly (EM waves encountering density variations in the propagating media, NOT particle-like photons following null-geodesics in curved space-time) and once that was in place, it took only about 6 months to develop a model that logically models gravity and inertia (not properly addressed in GR) and dispenses with DM, DE, gravitons, Higgs bosons, and the overly-simplistic ad-hoc space-time curvature model of GR.

The only mechanism required to resolve these problems is a differential in the gravitational infall rate of matter vs. antimatter. This mechanism provides the means by which the virtual pairs of the quantum vacuum fields are aligned, and all the above effects are derived naturally from that field polarization.

I know..."does it cure cancer and solve world hunger, too?", but this one mechanism is falsifiable (by a portion of the Athena project) and if it is proven to be true, it will provide a dynamical basis for quantum gravity - the biggest hurdle to grand unification.

 

[ohwilleke]

turbo-1, given your interest in zero point electromagnetic fields, you might find this MOND analysis interesting.

What is gravitational potential in "naiive MOND" (i.e. conversion from Newtonian to MOND gravity exactly at a sub zero with no smoothing function)?

Well, Newtonian gravitational accelleration is -GM/r^2. So, Newtonian gravational potential energy is GM/r. Now, Newtonian gravitational accelleration equals a sub zero at r equals SQRT(GM/a_0). So, gravitational potential at that point, gravitational potential energy is SQRT(G*M*a_0).

Now, what is MOND potential energy? Well, in general MOND gravitational accelleration is -SQRT(G*M*a_0)/r. Well, let's integrate that from r= SQRT(G*M*a_0) to r observer. The result is SQRT(G*M*a_0)*[ln SQRT(GM/a_0)]-SQRT(G*M*A_0)*ln r observer + C (an integration constant). Now, at the critical MOND radius, this, of course, is 0+C (integrating from a point to itself should always equal 0 + C). So, what is C? It is Newtonian gravitational potential energy at the critical radius, i.e. SQRT(G*M*a_0).

So gravitational potential energy in naiive MOND, is precisely:

SQRT(G*M*a_0)*(1+[ln(SQRT(GM/a_0)]-ln r)

An interesting fact about this formula is that it allows graviational potential energy at a sufficiently great distance to be negative! When does this happen?

ln (r_critical)=1+ln(SQRT(GM/a_0))

This in turn, implies that:

e^ln(r_crit)=e^(1+ln(SQRT(GM/a_0))

This means that:

r_crit=e*e^ln(SQRT(GM/a_0))

This implies that:

r_crit=e*SQRT(GM/a_o).

Now e=2.718. . .
a_0=1.2*10-10 m/s^2
G=6.672*10-11 N*m^2/kg^2

So r_crit, at which gravitational potential energy is approximately equal to zero is approximately 0.67*SQRT(M). Now, I've been sloppy about dimensions here, but let us assume for the sake of argument that I have it right, since I used MKS units all along for my constants.

Among the interesting results of this analysis:

The potential energy of the Sun hits zero at about 10^15 meters, which is 1/10th of a light year.

Now, all sophisticated relativistic expressions of MOND preserve the center of gravity principal. And, while naiive MOND should not generally match sophisticated MOND, the two should be nearly identical in the isolated case where you have a single, symmetrical point mass at a given distance. Now, since the universe, in a big bang theory is finite and has a center of mass, it should be possible to figure naiive MOND potential from its center of mass.

Now the canonical value for the mass of the universe (which includes dark matter and dark energy and all sorts of other stuff I don't think exists) is 3*10^51 kg. So r_critical for the universe as a whole is roughly 6.3*10^25 meters. And, the canonical radius of the universe is 10^26 meters. Now, if you ditch the 27% of the universe that is allegedly dark matter, you find that r_critical for the universe equals the r_universe. Which implies that, while there may be locally negative graviational potential energies (not clear one way or that other on that with our simple modeling), that the universe has a whole does not have globally negative gravitational potential energy, which is a nice Machian kind of result, and ZPE folks tend to embrace Machian approaches.

Now, maybe the universe wide figures are all wet. If you are a ZPE guy, another of your purposes in life is to consider the possibility that there is some negative energy field that is out there to cancel out your positive ZPE field, right? Well, MOND gravity, gives you a formula to create a field that has, at least potentially, locally negative potential energy, and given the way it is derived, it would be fair to say that this negative potential energy is measured against a sort of ground state (in the universal example particularly). How could negative gravitational potential energy be physical? One way to do it would be to assume that the zero it started at wasn't zero after all, and that zero was actually the ZPE baseline, which is positive. Adjust the size of the universe a little and the net negative graviational field for the universe can balance out the ZPE for the universe.

What do you think?

 

[turbo]

If you are a ZPE guy, another of your purposes in life is to consider the possibility that there is some negative energy field that is out there to cancel out your positive ZPE field, right? Well, MOND gravity, gives you a formula to create a field that has, at least potentially, locally negative potential energy, and given the way it is derived, it would be fair to say that this negative potential energy is measured against a sort of ground state (in the universal example particularly). How could negative gravitational potential energy be physical? One way to do it would be to assume that the zero it started at wasn't zero after all, and that zero was actually the ZPE baseline, which is positive. Adjust the size of the universe a little and the net negative graviational field for the universe can balance out the ZPE for the universe.

What do you think?

Thank you for putting this so succinctly. I have have been involved in a running battle (actually, a nice elucidating discussion ) as you know, regarding the validity of expressing vacuum energy as an absolute number relative to QFT's theoretical empty field, instead of expressing it relative to the ground state energy of our universe (which is essential to my model). Indeed "negative gravitational potential" relative to the vacuum energy baseline (not negative in absolute terms, like negative in relation to the theoretical empty quantum field) is eminently acheivable in our universe. Simply restrict the formation of some portion of the ZPE spectrum. This is done in Casimir devices by producing an area bounded by a gap that is smaller than the wavelengths of the portion of the ZPE spectrum that you wish to suppress.

Regarding suppression of ZPE field energy, it is possible (but not yet integrated into my model, because I haven't thought it all through) that the ZPE spectrum in a given space fine-tunes itself to the local fine structure of space-time, promoting longer-wavelength, less energetic pairs in relaxed fields (massless) and promoting the production of shorter-wavelength, more energetic pairs in densified fields (perturbed by mass). Up to this point, I have concentrated on the mechanics of vacuum field polarization and have not delved too deeply into field spectra. The prospect of the ZPE fields featuring intrinsic cutoffs (or at least manifesting in preferred spectral distributions) is intriguing, though, and may hold promise.

As you suspect, my model shows an unperturbed (massless) ZPE field to be relaxed and gravitationally neutral, but when the field is in the presence of matter, it assumes a preferred orientation. In the presence of mass, the ZPE field is polarized and densified - it exists at a higher energy state than unperturbed ZPE and it gravitatates to mass and is also self attractive. I believe that MOND is modeling the gravitational and inertial effects of the polarized ZPE field on the galactic scale. That is intuitive at this point, since I do not have the math skills to prove it. The densification and gravitation of the ZPE field in clusters provides the optical density gradient necessary to explain cluster lensing without space-time curvature and also supplies the gravitational binding force necessary to hold the clusters together without DM. My model has met with some scepticism (did I say some?), but when you can introduce one concept (mass polarizes ZPE fields) and thereby eliminate a half-dozen cosmological "fudge factors" and resolve the countless discordant observations that engendered them, it's far too attractive to ignore. Of course, it doesn't help that my math skills are poor, which makes it tough to communicate with people for whom math is their mother tongue. If the Athena project demonstrates a differential in the gravitational infall rates of matter vs. antimatter, I hope to be able to persuade someone to take a closer look at polarized ZPE, though.

 

[ohwilleke]

How does your ZPE approach differ, if at all, from Wesson's?

http://www.calphysics.org/inertia.html

 

[ohwilleke]

Well, Newtonian gravitational accelleration is -GM/r^2. So, Newtonian gravational potential energy is GM/r. Now, Newtonian gravitational accelleration equals a sub zero at r equals SQRT(GM/a_0). So, gravitational potential at that point, gravitational potential energy is SQRT(G*M*a_0).

Now, what is MOND potential energy? Well, in general MOND gravitational accelleration is -SQRT(G*M*a_0)/r. Well, let's integrate that from r= SQRT(G*M*a_0) to r observer. The result is SQRT(G*M*a_0)*[ln SQRT(GM/a_0)]-SQRT(G*M*A_0)*ln r observer + C (an integration constant). Now, at the critical MOND radius, this, of course, is 0+C (integrating from a point to itself should always equal 0 + C). So, what is C? It is Newtonian gravitational potential energy at the critical radius, i.e. SQRT(G*M*a_0).

So gravitational potential energy in naiive MOND, is precisely:

SQRT(G*M*a_0)*(1+[ln(SQRT(GM/a_0)]-ln r)

I'll pre-emptively note that my analysis was not quite right. The conclusion is right (unless I made another mistake) but I am really just adding two definite integrals (from to to the r at which a_o is reachd, and from a_0 to r_observer) which together span from r=0 to r_observer, and definite integrals don't get integration constants.

Also, it is interesting to note that the ratio of r at a_o and r_critical (i.e. where gravitational potential energy is zero), appears to be independent of mass. The ratio is precisely equal to e.

 

[turbo]

How does your ZPE approach differ, if at all, from Wesson's?

http://www.calphysics.org/inertia.html

Well, I actually found the CIPA site some months after I began modeling gravitational lensing as a classical optical effect (EM waves interacting with density gradients in the media along their propogation path). I had already determined to my own satisfaction that to achieve this, ZPE fields must gravitate and I had worked out a means by which the fields would be polarized in the presence of matter. When I found the CIPA site (including Wesson's overview of the work in the ZPE field, BTW) I was very surprised to find that key researchers in the field (Puthoff, Haisch, Rueda...) had already decided that the ZPE fields do not gravitate, or at least that their gravitation must be suppressed to avoid extreme space-time curvature.

My approach differs in that I treat the ZPE EM fields as if they are real fields, capable of being polarized and capable of a range of energy densities. In my model, unperturbed (massless) ZPE fields are randomly oriented and relaxed and do not gravitate, but in the presence of mass, they become polarized and densified and they do gravitate. I was frankly puzzled by the fact that despite their modeling of the Machian-like nature of inertia relative to the ZPE field, these researchers were willing to ignore the possible role of the ZPE fields in gravitation. A simple treatment of the ZPE EM fields as real fields (capable of being polarized and capable of energy differentials) would have given them the basic tools needed to model ZPE gravitation. Of course, many of the ZPE folks of that era had been funded (by NASA and a few private groups) to do research into exploiting the potential energy of the ZPE fields for power generation, space propulsion, etc, so perhaps we shouldn't expect that their research would be brought to bear on cosmological problems in GR. My efforts have been entirely directed to resolving cosmological problems, which is why I needed to model gravitation.