Looking more closely at Milgrom’s Scholarpedia entry on
MOND, I found something else that I didn’t like. It was the method by which he arrives at an
equation that I used in the previous post, A Minor Departure from MOND, namely g(in
the MOND regime)=√(GMa0)/r.
I was walking the dogs actually, mulling over things, and
realised that I couldn’t for the life of me remember how I arrived at that
equation. I must have seen it, got stuck
in a mental alleyway and just automatically applied it. Very embarrassing.
While it works, and seems to work better from
one perspective with the different value of a0, it won’t wash
if there’s no derivation. And there’s no
derivation. This is the numerology that
I was complaining about a few posts ago.
What Milgrom writes is: “() A0 is
the “scale invariant” gravitational constant that replaces G in
the deep-MOND limit. The fact that
only A0 and M can appear in the deep-MOND
limit dictates, in itself, that in the spherically symmetric, asymptotic limit
we must have g∝(MA0)1/2/r,
since this is the only expression with the dimensions of acceleration that can
be formed from M, A0, and r.” The term A0 had been introduced
earlier in the text: “A0 is the “scale invariant” gravitational
constant that replaces G in the deep-MOND limit. It might have
been more appropriate to introduce this limit and A0 first,
and then introduce a0≡A0/G as
delineating the boundary between the G-controlled standard dynamics
and the A0-controlled deep-MOND limit.”
The problem I have is that, in Towards a physical interpretation of MOND's
a0, I considered critical density of our universe, and that very
specifically uses the Gravitational Constant (G), and I consider the gravitational
acceleration at the surface of a Schwarzschild black hole with the same density
as that critical density, and that equation also very specifically uses G. However, the resultant acceleration would be
right on the border between “the G-controlled standard dynamics and
the A0-controlled deep-MOND limit”, so there’s an issue right there.
There’s also an issue with the fact that forces are vector
quantities, in the case of gravity directly towards the centre of mass (although
due to the summing and negation of sub-forces created by every element of the
mass).
When considering the surface of a black hole, the gravitational
force is towards the centre of the mass of the black hole. Now, in earlier posts, I have indicated that
the density of the universe is the same as the density of a Schwarzschild black
hole with a radius equivalent to the age of the universe times the speed of
light. What I have never said, at any
point, is that the universe is inside a black hole.
My position has been more that the universe *is*
a black hole, which may seem rather esoteric, but the point is that I don’t consider
there to be an outside in which there would be a black hole inside of which our
universe would sit. To the extent that there
is a universe in which our universe is nestled, that “outer” universe is on the
other side of the Big Bang. So it’s not
so much a “where” question, but rather a “when” question.
But even then, it’s not correct to say that the “outer”
universe is in our past, because time in that universe was/is orthogonal to our
time, and in the same way the spatial dimensions of the “outer” universe were/are
orthogonal to our spatial dimensions.
(I know this is difficult to grasp initially, but this video may go some way to explaining a version of the concept.)
This introduces another issue. If we could, in any way, consider our universe
to be a black hole in an “outer” universe, then our universe would be smeared
across the surface of that black hole and any gravitational force due to the
total mass of that black hole would be orthogonal to our spatial dimensions.
So, while it’s tempting to consider a value of a0
that is linked to the mass of a black hole with the dimensions of a FUGE universe, it doesn’t seem
supportable.
I had tried a method, considering the curvature of the “fabric
of spacetime”, but I suspect that it introduces more problems than it solves.
An image like this illustrates curvature of two dimensions, but it represents curvature of three dimensions. We could eliminate another dimension, to get something like this:
In this image, the notional gravitation that a0
would represent would be a vector field throughout with a downwards
trajectory. Without a mass deforming spacetime,
that vector field would be orthogonal to it, but with any deformation, there
would be a component that is not orthogonal.
It made sense at the time, since it does tie the effect of a
gravitational force that should be uniform throughout the universe to a mass that
is deforming spacetime but I don’t have any confidence that it works, since the
upshot would be additional deformation, which could have a
potential runaway effect.
Someone else might have an idea as to how this could work, even if it seems to me to be a dead-end.
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