Posts Tagged ‘hyperbolic spacetime’

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.

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