Posts Tagged ‘cosmology’

Normal distribution applied to Milky Way galaxy M-Sigma relation and bulge star data

July 18, 2013

Image

This superposition is supposed to show how the M-sigma relation could be applied to a given galaxy (the Milky Way). The vertical blue lines represent the positions of + and – sigma, the standard deviation of the normal distribution. The horizontal green line is positioned at the points where the blue lines intersect the distribution curve. Values are read at the green line from the vertical velocity dispersion axis.

It is seen that the M-sigma velocity dispersion for the Milky Way is about 100-103 km/s which we can use to estimate the M-sigma mass of the MW central supermassive black hole.

Msigma

This graph was made by the author of  “The M-Sigma Relation” in Wikipedia. I am trying to track his identity. No luck yet.

Take a look:

Two ten-billion-solar-mass black holes at the centers of giant elliptical galaxies

McConnell, Nicholas J.; Ma, Chung-Pei; Gebhardt, Karl; Wright, Shelley A.; Murphy, Jeremy D.; Lauer, Tod R.; Graham, James R.; Richstone, Douglas O.

Nature, Volume 480, Issue 7376, pp. 215-218 (2011)

 “Observational work conducted over the past few decades indicates that all massive galaxies have supermassive black holes at their centres. Although the luminosities and brightness fluctuations of quasars in the early Universe suggest that some were powered by black holes with masses greater than 10 billion solar masses, the remnants of these objects have not been found in the nearby Universe. The giant elliptical galaxy Messier 87 hosts the hitherto most massive known black hole, which has a mass of 6.3 billion solar masses. Here we report that NGC 3842, the brightest galaxy in a cluster at a distance from Earth of 98 megaparsecs, has a central black hole with a mass of 9.7 billion solar masses, and that a black hole of comparable or greater mass is present in NGC 4889, the brightest galaxy in the Coma cluster (at a distance of 103 megaparsecs). These two black holes are significantly more massive than predicted by linearly extrapolating the widely used correlations between black-hole mass and the stellar velocity dispersion or bulge luminosity of the host galaxy. Although these correlations remain useful for predicting black-hole masses in less massive elliptical galaxies, our measurements suggest that different evolutionary processes influence the growth of the largest galaxies and their black holes.”

    


  Of course, M-sigma works best when it is confined to galaxies of a given class. Maybe giant ellipticals constitute another such class.

  The M-sigma relation (those widely used correlations) may be written[i],[ii] :

  (1)        

         a)   M  =  Mbh  =  3.1 (σ/200 km s-1)4 x 108  Mʘ  =  M.     

  A current study, based on published black hole masses in nearby galaxies, gives[iii]

         b)   M  =  Mbh  =  1.9 (σ/200 km s-1)5.1 x 108  Mʘ  =  M.

  (2)                  Solar mass [iv]  =   Mʘ   =    1.98855 x 1030 kg                                   

  (3)                    M   =  Mbh  =   M●   =   r* v2/κG  from eqn. (2) in the paper, by the Postulate

                 Mbh  =  r* σ2/κG  =  3.1×108 (σ/200,000 m s-1)4 Mʘ       

                                      vσ  =  “σ” by the Postulate

  or

  (4)                  Mbh  =  r* σ 2/κG  =  3.1×108 (σ/200,000 m s-1)5 Mʘ   =   M  

                 Milky Way mass, Mmw  =  7×1011 M[v]

                                                or     = 1–1.5×1012 M[vi]

                                             with “Dark Matter “ contributing  

   We cannot have it both ways. Either bulge stars obey standard Kepler (SK) or adapted Kepler (AK). Which?  Is it a mixture of SK and AK as in eq. (6) of

   the paper? The Author of the paper dislikes the mixture, as it appears in eq. (6). But, such questions are good. They make for lots more research.

   So, astrophysicists, cosmologists and their grad students should love the Postulate.

[ii]     Ferrarese, F. and Merritt, D. (2000), A Fundamental Relation between Supermassive Black Holes and Their Host Galaxies, The Astrophysical Journal, 539, L9-L12

[iii]    McConnell, N. J. et al. (2011), Two ten-billion-solar-mass black holes at the centres of giant elliptical galaxies, Nature, 480, 215-218

[v]        Milky Way Mass  7×1011 M   Reid, M. J. et al. (2009). “Trigonometric parallaxes of massive star-forming regions. VI. Galactic structure, fundamental parameters, and noncircular motions”. The Astrophysical Journal 700: 137–148, solar mass M  =  1.9891×1030 kg,   Mmw  =  1.4 x 1042 kg   computed by conventional methods

[vi]    Milky Way Mass including “Dark Matter” 1–1.5×1012 M McMillan, P. J. (July 2011). “Mass models of the Milky Way”. Monthly Notices of the Royal Astronomical Society 414 (3): 2446–2457.  solar mass M  =  1.9891×1030 kg,  Mmw  =  2-3 x 1042 kg  by conventional methods

Wikipedia, rotation velocity  =  v,  AVD

 

Estimates for the mass of the Milky Way vary, depending upon the method and data used. At the low end of the estimate range, the mass of the Milky Way is 5.8×1011 solar masses (M), somewhat smaller than the Andromeda Galaxy. Measurements using the Very Long Baseline Array in 2009 found velocities as large as 254 km/s for stars at the outer edge of the Milky Way, higher than the previously accepted value of 220 km/s. As the orbital velocity depends on the total mass inside the orbital radius, this suggests that the Milky Way is more massive, roughly equaling the mass of Andromeda Galaxy at 7×1011 M within 50 kiloparsecs (160,000 ly) of its center. A 2010 measurement of the radial velocity of halo stars finds the mass enclosed within 80 kiloparsecs is 7×1011 MBut, we cannot apply standard Kepler or a correlation diagram based on unadapted Kepler, to stars that obviously do not follow Kepler’s laws, as is exemplified by the flat MW velocity dispersion diagram. But we go ahead anyway as if we haven’t a clue and do not understand. Most of the mass of the Galaxy appears to be matter of unknown form which interacts with other matter through gravitational but not electromagnetic forces; this is dubbed dark matter. A dark matter halo is spread out relatively uniformly to a distance beyond one hundred kiloparsecs from the Galactic Center. Mathematical models of the Milky Way suggests that the total mass of the entire Galaxy lies in the range 1-1.5×1012 M.

 

Galactic rotation,    velocity = v,  AVD

The stars and gas in the Galaxy rotate about its center differentially, meaning that the rotation period varies with location. As is typical for spiral galaxies, the distribution of mass in the Milky Way Galaxy is such that the orbital speed of most stars in the Galaxy does not depend strongly on their distance from the center. Away from the central bulge or outer rim, the typical stellar orbital speed is between 210 and 240 km/s. Hence the orbital period of the typical star is directly proportional only to the length of the path traveled. This is unlike the situation within the Solar System, where two-body gravitational dynamics dominate and different orbits have significantly different velocities associated with them. The rotation curve (shown in the figure) describes this rotation.

If the Galaxy contained only the mass observed in stars, gas, and other baryonic (ordinary) matter, the rotation speed would decrease with distance from the center. However, the observed curve is relatively flat, indicating that there is additional mass that cannot be detected directly with electromagnetic radiation. This inconsistency is attributed to dark matter. Alternatively, a minority of astronomers propose that a modification of the law of gravity may explain the observed rotation curve. The constant rotation speed of most of the Galaxy means that objects further from the Galactic center take longer to orbit the center than objects closer in. But, in fact, they orbit faster than they would if they followed Kepler’s 3rd  law. This is actually the problem. If they orbited according to Kepler’s 3rd, they would orbit so slowly as they neared the galactic rim that the spiral arms would wrap backward multiple times around the galactic center like the mainspring of an old windup clock. So, we can actually see the anomalous velocity dispersion at work when we observe a spiral galaxy.

Black Hole Singularities and the Possibility of Two Dimensional Gravity

March 9, 2013

According to general relativity, in a 3-D universe with time, the gravitational field of all compact objects behaves as if the objects are point masses and the field strength must decline as 1/r2. In a 3-D universe, therefore it is said, it is impossible to support a hyperbolic 1/r gravitational field. But, black holes are different.

Why bother with the whole concept of black holes if they are not different? Collapse of matter into a black hole must not only create a singularity (within the limits imposed by the Heisenberg uncertainty principle) but, the spin rate or orbital frequency of in-falling matter of the black hole must also increase without bound as radius r decreases to values near zero below the event horizon. Attempts to explain away these singularities on the basis of a non-existent quantum gravity scheme are vacuous extrapolations of tentative hypotheses that amount to pure conjecture.

Black hole singularities exist. Einstein through Schwarzchild and others say so. Who claims to be more brilliant than these fellows? I appeal to authority here only because it seems to be the only thing that impresses some. If you want to claim that BH singularities are mere artifacts of an inadequate theory, show me the Math.

Black holes are different. When matter and energy collapse under an infinitely strong gravitational field to a point mass that is as tiny as may be necessary to explain its properties (not necessarily to zero, the true meaning of infinity), the result is a phase change. Spacetime phase changes are S.O.P. in the repertoire of theoretical cosmologists, like Alan Guth. Let us adhere to the hydrodynamics metaphor used by Einstein in his development of GR. Flat spacetime is a massless superfluid. Helium IV is a superfluid but, it is not massless.

To extend the metaphor, it is not hard to imagine that spacetime could undergo a phase change, just as helium IV may. In a black hole this change involves a reduction in dimensionality. This is about the only change available to it. Analysis of the equations of GR shows that gravitational strength, Fg is proportional to 1/r(n-1) where n = number of spatial dimensions. In a 3-D universe, Fg declines as 1/r2. In a 2-D universe, Fg declines as 1/r.

Using basic geometry, GR allows gravitational fields that decline as 1/r(n-1) where n is defined as above. So, if there is to be a hyperbolic SBH gravitational field it must occur in 2-D.  The gravitational field strength in its local limit is defined as flux through a surface element of a n-sphere; that surface area behaves as 1/r(n-1). One can find the derivation of this relation in any good geometry textbook.

So, a black hole must use the gravitational energy of in-falling matter to raise its gravitational potential, the gravitational energy level, to the 2-D “state”. We are starting to talk quantum language now.

The shape of this singular BH gravitational field strength diagram, as it is a 2-D entity embedded in a 3-D space, is a nominally flat disk or platter with a potentially infinite radius. Unlike Kerr, I call this topology of the event horizon a “spin disk” because it arises from the infinite rotational and orbital spin rate of matter that has in-fallen toward the singularity. As a spacetime entity, this new phase ignores the event horizon and propagates outward to beyond the edge of the galaxy. It emanates from the central core of the galaxy wherein resides any central supermassive black hole.

Here is one of the non-intuitive consequences of GR. It is known that matter in-falling toward the event horizon must experience time dilation. From our external perspective, we would perceive time for this matter as having slowed and even stopped at the event horizon. Viewed from any point outside the event horizon, time really does stop there. But from its own perspective time does not stop and such matter does indeed drop through the event horizon where it may take part in whatever processes it might (time reversed or not).

There is simultaneously an inverse square gravitational field set up by this time-frozen matter at the event horizon and an inverse gravitational field set up by this same matter that has already in-fallen to the singularity. There is no violation of conservation laws here because no object can feel these separate effects simultaneously. If an object orbits the galactic center in the plane of the galactic disk, it feels the inverse 1/r field. If it orbits on a trajectory not aligned with the galactic plane, if it orbits chaotically, it feels the inverse square 1/r2 field.

This has consequences for the analysis of the orbital motion of close-in Milky Way bulge stars like S2 for the determination of the MW’s supermassive black hole mass according to Kepler’s laws. Kepler is valid for the 2-D case as well as for the 3-D case. But, it has to be adjusted for the 1/r gravitational field as does Newton’s law. No deep relativistic calculations are needed. One can determine what changes must be made in Newton’s law and Kepler’s laws by inspection.

Above, I explained how a 2-D gravitational field can exist in our 3-D universe. It must be associated with a black hole having an infinite spin rate as well as infinite density and infinite gravitational field strength. Within the bounds of Heisenberg uncertainty, these singularities must exist. There is no point in trying to explain them away using some kind of unfalsifiable overly advanced unintelligible gravitational quantum sophistry.

I show that the hyperbolic 1/r inverse gravitational field can exist as a spin disk surrounding any black hole with said disk extending far beyond the event horizon toward infinite radius. This explains MOND and the anomalous velocity dispersion because hyperbolic 1/r gravity means that orbital velocity, v, around a galactic center containing a black hole, v = (GMbh)1/2. That is, v becomes constant dependent only on Mbh and G. This v is not only constant for a given galaxy, it is constant from galaxy to galaxy. This means that GMbh must itself be a constant, implying a new fundamental physical constant. Remember that Noether’s theorem concerning invariants, fields and symmetry is engaged whenever a fundamental constant is invoked.

But, G may not be the same G that applies in 3 dimensions. So, I call it G*. Besides by an extension of GR, one might get G* from the M‑sigma relation as well as by the anomalous velocity dispersion. But, the mass of the central galactic supermassive black holes must first be refigured on the basis of the hyperbolic field if very many of the orbits of the bulge stars that were used to get Mbh were coincident with the galactic plane. If all or most of these orbits were chaotic and not aligned with the galactic plane, the BH mass determinations may be okay.

Perhaps the main expression of the Postulate should be written v  =  (G*Mbh)1/2 from 2-D Newton’s law. Then G*  =  v2/Mbh for any galaxy with any size supermassive black hole with its associated vavd  =  vσ  =  v. In order for G* to be constant, v2 and Mbh must vary inversely. We see that, from the M-sigma relation, they do.

So, we have a relation between Mbh and v as vavd or vσ that is linear. So, the slope is Mbh/v  =  a constant . Then Mbh/v2  =  a similar constant and v2/Mbh  =  a constant also. The Postulate says G*Mbh  =  v2 and so G*  =  v2/Mbh, as above.

When doing any calculation involving the hyperbolic (1/r) gravitational field, remember that Fg  =  G*Mm/rr1 where r1 is the unit vector of r. This ensures dimensional integrity.

In the M-sigma relation, the exponent of v or σ has been expressed as being equal to either 4 or 5 with constants adjusted as appropriate to compensate for the difference. In the past, this was done arbitrarily, for convenience. There is now theoretical grounds to choose the exponent of v such that exp(v) = 2 when using the M-sigma relation for bulge stars orbiting in the plane of the galactic disk. When analyzing the AVD one finds that vavd  =  virtually a constant within and among galaxies. When the scatter in the data is accounted for, the Postulate predicts that this will be found to be untrue and vavd2  =  G*Mbh, especially when the effect of embedded massive BHs in the body of each galaxy is made up for.

I think that σ is not distant enough from the singular center for the hyperbolic field to overcome chaotic effects. Perhaps v should be based on 2σ or on the velocities of stars orbiting in the plane of the galactic disk just beyond 98% of the average orbital radii of bulge stars. This suggests the wonderful possibility for much more research.

The Postulate explains both the anomalous velocity dispersion and the M‑sigma effect.

It may not be quite this simple because all spiral galaxies have massive black holes embedded within them, besides their central black hole. Because of the flat hyperbolic (1/r) gravitational field’s longer reach, the gravitational, spacetime spin disks of these BHs would tend to align with that of the central black hole’s so that their hyperbolic fields would combine. So G* got from vavd may not be so pure and pristine.

The meaning of the hyperbolic gravitational field of black holes is that MOND (Modified Newtonian Dynamics, suggested by Mordehai Milgrom) is explained without recourse to Dark Matter or to modifications of Newtonian dynamics. Newton and Kepler must be understood in two dimensions, that is all.

All of the observations that are said to support Dark Matter as being, say, a huge halo of WIMPs engulfing galaxies and galactic clusters also support the hyperbolic (1/r) gravitational field postulate, even the Bullet Cluster effect.  Dark Matter 3-D maps obtained by analysis of gravitational lensing also follow logically from the Postulate.

The hyperbolic (1/r) supermassive black hole gravitational field is indeed a postulate. This means that there can be no argument against it. It must be taken at face value and carried to its logical extreme whereupon it will be either reduced to absurdity or else found to be correct.

When extrapolated to the entire universe, the hyperbolic field mimics Dark Energy too. If Alan Guth’s inflaton particle originated in 2-D space and began to roll down its own hyper-gravitational super-potential slope toward a lower energy 3-D state, the higher energy 2-D potential energy would be progressively transformed in a time dependent quantum-like transition to the new 3-D “ground state”. This potential energy would show up as apparently increasing kinematic momentum of all stars and galaxies in the universe. That is, the universe would appear to be expanding at an accelerating rate.

This is an exciting idea because the whole universe is thus to be regarded as a quantum object. It may provide a route to a falsifiable certifiable theory of quantum gravity because 2-D gravity does not lead to a gravitational catastrophe as r tends to zero, due to Heisenberg uncertainty’s prior restraint in this case. And, it is renormalizable, a prerequisite for any quantum theory of gravity. This Postulate may point to a means to prove the existence of the multiverse. If Guth is right, Hugh Everett could be right.

Dark Matter is an unnecessary ad hoc fix

January 11, 2012


The singularity at the center of a black hole must be unique and have testable consequences.

Dark Matter is an unnecessary ad hoc fix to fill in the blanks in the Friedmann model under the FLRW metric. Galactic supermassive black-holes exist as true physical singularities according to the Kretschmann invariant and Schwartzchild’s analysis of his spacetime metric under GR. Therefore, as point masses, they must possess a hyperbolic (1/kr) gravitational field, NOT a field that falls off as 1/r2. Now, k = constant = 1m, S.I., for dimensional integrity. It is not true that GR cannot tolerate hyperbolic spacetime geometries. “The universe is hyperbolic.” said Albert Einstein in his classic paper of 1915. An hyperbolic field will give constant orbital acceleration to orbiting bodies as far from the center of a black-hole as we might like to measure. This means that bodies near the periphery of a galaxy should seem to move at constant velocity because rotational acceleration does not drop to near zero there as with a 1/r2 inverse square law, it becomes consant. This constant velocity distribution effect has actually been measured and has given rise to the notion of Dark Matter.

Gravitation does not fall nearest to zero between galaxies in a cluster either. So they too can bend light and affect redshifts in ways that mimic Dark Matter. The rotation of galaxies in clusters is also influenced by the black-holes that they contain with their 1/kr gravitational potential profiles. The not quite counterbalanced redshift effects in the Sunyaev-Zeldovich phenomenon are influenced by the hyperbolic galactic and galactic cluster gravitational fields that exist as light falls out of such clusters and super-clusters into a large void and as it climbs out of it again after the universe has expanded by another billion light years or more.

Scientists are mapping, not Dark Matter, but the huge extent of the network of hyperbolic galactic and super-galactic gravitational fields that behave like Dark Matter because of the mathematical properties of the hyperbolic gravitational field are similar to that expected for Dark Matter.

Primordial massive and supermassive black-holes with their 1/kr galactic gravitational fields can also mimic the “halos” of dark Matter that are postulated to have existed just after the big bang and before the emission of the cosmic microwave background. There is nothing that Dark Matter explains that cannot be accounted for just as well or better by the hyperbolic black hole gravitational field.

The hyperbolic 1/kr supermassive black-hole galactic gravitational field explains “the Dark Matter Effect” without Dark Matter and it is more parsimonious and is a falsifiable hypothesis, unlike Dark Matter which is revised every time no Dark Matter is found.

The conditions for validity of Birkhoff’s Theorem are not met for real black-holes. Therefore, Birkhoff’s Theorem does not apply. It sometimes may be used as a first approximation, but it cannot be depended upon as a rigid rule for precise calculations. “The physics near at the extreme curvature of a black-hole singularity is not well defined”. This covers Birkhoff’s too.

It does too matter how the internal mass is distributed if it is contained within a single point. Then, in fact, it is NOT distributed at all! This is the point of Kretschmann’s invariant and Schwartzschild’s GR analysis of the consequences of his metric. Ordinarily, the distribution would not matter. But, a singularity must be different. If this is not explicitly acknowledged in some way, then to say there is a singularity with such intense curvature of spacetime in its vicinity that the laws of physics must begin to break down is a meaningless fatuous gesture to humility. It is false humility if it has no ameliorating effect on professional arrogance. Please, do not just restate Birkhoff.

I contend there is a loophole here or a gross misinterpretation. The consensus interpretation of Birkhoff and of Schwartzschild/Kretschmann cannot both be true at the same time. There must be a measureable consequence of the presence of a singularity that is beyond imaginary untestable gedanken experiments. The test is the hyperbolic gravitational field. It results in a nonzero constant rotational velocity distribution effect in spiral galaxies, ellipticals, globulars and galactic clusters. This is easier to believe than Dark Matter.

The very same phenomena that are used to argue for Dark Matter can be used to argue for the hyperbolic field. So, it is testable. But, how do we choose between them? I think that Occam ’s razor is the principle of choice here. WIMPS and neutralinos and the other oddball particles that have been proposed require ad hoc additions to theories or their complete rewrites. The hyperbolic field is far simpler. All that is needed is acknowledgement that the black hole singularity is unique. No rewrite of GR. No undetectable new heavy particles that get given self-serving, revised, lower detection limits every time they are determined to be really undetectable.

There seems to be a tendency of cosmologists to think inside the box. They never really consider anything that is outside the consensus. So too do journal editors rely on  conventional wisdom. They would all have been supremely comfortable with the Pope’s decision to censor Galileo.

“Cosmologists are always wrong, but never in doubt.”   Lev Landau