Thursday, March 18, 2010

Modified Newtonian Dynamics


Albert Einstein easily derived and wrote a relativistic differential equation that was guaranteed to reproduce Newton's Law of Gravity when it was integrated. He could have just as easily written a differential equation that reproduced MOND, but he didn't. He just didn't. He had no reason to do so since the various MOND observations had not been made. So, let no-one say that MOND contravenes relativity.

So, relativistically speaking, if we consider the definition of a black-hole, it must be accompanied by a very characteristic and very different gravitational field. Because it is a singularity, a single point-mass with infinite density (at the limit as the center is approached), it must possess a gravitational field that is defined and determined by its single point, its singular property. Its gravitational field must therefore approach an asymptote at radius = 0 and, and by symmetry, it must approach another asymptote at radius = infinity. This is the definition of a hyperbolic field, NOT one that follows Newton's inverse square law, which is parabolic.

If black-holes posses hyperbolic gravitational fields, then there is no mystery in MOND. No big deal. Look at this image of a whiteboard derivation of the hyperbolic black-hole (HBH) gravitational acceleration and velocity profile near a galaxy containing a supermassive black-hole at its center. On the whiteboard, I also have written a synopsis of the MOND development.

NOTE: In the HBH segment, r must be multiplied by k. Here, k = 1m (S.I.) for dimensional integrity. In other words, F = GMm/kr, k = 1m (S.I.) all the extensions of F have the same caveat.




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Of course, something must be done to compensate for the fact that (GM)^1/2 does not have the dimensions of a velocity, G x M not being divided by r. This could be resolved with the adjustable parameter ratio b/d having a dimension the same as 1/r. OR, better yet, multiply r by the absolute value of the unit vector of r, retaining the unit. This would be a mathematical acknowledgement that we are dealing with a singularity.

For MOND, Newtonian gravitational acceleration is denoted aN. The variant acceleration due to MOND expresses aN in terms of an acceleration that is modified by the function μ(a/a0), which is equal to 1 when the radius from the center is small enough for the overall gravitational acceleration to be large relative to the MOND constant, a0. When the putative acceleration is small relative to a0, μ is equal to a0 such that the equations on the lower left are satisfied. This happens when radius r is large enough for a2/a0 = aN = GM/r, and the total force of gravitational attraction enters the MOND regime. These values for r are equal to and beyond the point where the velocity distribution of stars encircling the galaxy in its outer regions becomes constant.

Remember, M = the mass of an entire galaxy with its black hole. M2 = the supermassive black-hole mass while M1 = mass of the stars in the disk inside the radius r, to a given star. This radius might enclose more than 95% of all the stars in the disk and so may as well be considered to enclose all of them. But, a graphical model would have to take a sliding percent enclosure into account in order to plot a theoretical velocity distribution. So, it will be a little while before I actually do this. Maybe I’ll just assume a Gaussian distribution for the number of stars with a given average velocity in the disc and a different Gaussian for the stars in the spheroidal bulge. The bulge must be chaotic and so it would follow different statistics. This is why the net velocity fades to zero in the bulge.

According to Newton’s inverse square law, velocity should fall off rapidly toward zero in this outer zone as the periphery is approached. But, observations show that it does not. Instead, it falls off much more slowly and becomes constant according to the vMOND equation on the lower right.

But, the hyperbolic field contribution to the overall galactic gravitational field shown as the blue curve, y2, in the graph above, gives precisely the same result. Thus, vMOND = vHBH. If it is admitted that black-holes are different, that they have special properties, among them that they are gravitational singularities, then this is not too surprising. In fact, it should have been thought of before, and it probably has. But, it is now being overlooked. I hope this effort on my part may prevent this oversight from being propagated indefinitely. Sorry, there will never be a Nobel for MOND.

Note that the usual depiction of the velocity profile of a spiral galaxy shows velocities rising to a maximum as one moves toward the center whereupon they fall off virtually to zero as one gets very close to r = 0. My simpler velocity distribution profile is for just one or for a very few stars. The standard picture of a maximum and fall-off in velocity occurs because stars get crowded approaching the center and their orbital paths become chaotic. As one moves closer in, just as many stars move clockwise as move counterclockwise (and on more nearly perpendicular trajectories relative to the galactic plane) and the net velocity declines. The stellar distribution becomes more spheroidal too, resulting in the classic galactic “bulge”. When I formulate my model, I will have to take all this into account as well. This is going to be fun.

It is interesting to imagine what a galaxy would look like if purely Newtonian F = ma = GMm/r2 ≠ GMm/r for large r. Then, v ≠ (GM)^1/2 = constant. Instead, rotational velocity would rapidly fall off toward zero as r increases without bound. The stars would lag much further and further behind and the spiral arms would wrap around the center much more tightly, like the mainspring of an old wind-up clock. So, one can actually see the MOND effect by just looking.

This means that there may well be no such thing as dark matter or twin matter or any such Baroque complications festooning the simple picture of the universe that we, as scientists, should be looking for. It is in the nature of human beings to overdo. Dark matter and MOND are in danger of becoming vastly overdone.

So, now let me engage in a little bit of my own overdoing.

If this version of MOND is correct, there will never be a measurement of a0 obtained in the laboratory in a supersensitive Cavendish experiment. That is, not unless we can produce some sufficiently long-lived black-holes in the lab. Nor will evidence be found inside or just outside the solar system. HBH MOND requires participation of a black-hole. This prediction will never go over very well with a lot of important people, so it will not be readily accepted.

In elliptical and globular galaxies wherein the MOND effect may be observed, HBH MOND will require that one or more supermassive black-holes shall be found, naked black-holes at that. The intragalactic black-hole presence in galactic clusters and superclusters may not be enough to account for MOND in these objects either, so some naked black holes may well be found to be embedded within them too. Nobody is looking for them now, so it is not surprising that they have not yet been found.

The hyperbolic field concept can be extended to the entire universe, too. If it can be verified, it may go a long way toward an accounting for the mistaken interpretations of acceleration and dark energy in the universe. This would require that the primordial black-hole, the mother of all black-holes or MOAB, must have persisted in some form, probably greatly diminished, for a long time after it started to decay into the universe that we see now. In other words, the highly excited inflaton particle may have taken some time to deconvolve, decompose and collapse, so it may even still persist today, along with its hyperbolic field in which we may still be embedded.

But, if the inflaton paticle decayed and its hyperbolic gravitational field collapsed with it, a transitional Newtonian field rose up to replace it almost instantly. These would be other cases where entities expanded or contracted at many times the speed of light. In other words, the inflation process had reflections or reverberations that we still see today. The old hyperbolic field is not finished collapsing. The new Newtonian field is not finished rising. This would have subtle relativistic effects in their own right.

Let’s just hitch a ride on a really fast space-ship heading on a straight line for deep space in any direction with a destination rather more than 13.72 billion light years away and we just may be able to find out. But, it will take us considerably more than 13.72 billion years. More than 27.44 billions years. Still, surely by then, the MOAB hyperbolic field will have completely dissipated with little or no residual effects.

The good news is, if strange, weird science is needed to justify grant funding, this is it.

1 comment:

Gary Kent said...

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