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MOND-related formula

  1. Apr 24, 2007 #1

    Jonathan Scott

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    Gold Member

    In GR calculations below the cosmological scale, it is
    conventional to assume that a sufficient distance from
    the central object, space becomes flat. However, when
    that object is a galaxy or similar, it seems to me that it
    might be better to assume that the boundary is more
    like the 3-D equivalent of a cone, constituting a fraction
    (m/M) of the total solid angle needed to close the
    universe, where m is the local mass and M is the mass
    of the universe.

    If one assumes that the area of an enclosing sphere
    has been decreased by a factor (1-m/M), then linear
    dimensions have been decreased by approximately
    the square root of that, (1-m/2M). That forms the
    cosine of the angle by which the "conical" space
    diverges from being flat, so the sine and hence the
    angle relative to flat space is approximately sqrt(m/M).
    The curvature of the "cone" is 1/r times this, and it
    seems plausible that for slow-moving objects this
    could give rise to an additional acceleration relative
    to flat space of c^2/r sqrt(m/M). Note that this
    formula was only reached by somewhat imprecise
    analogies, so the above is not actually a rigorous
    derivation, and even if the analogies are valid it
    could still hide factors of 2 or similar. However,
    the resulting formula seems quite promising.

    In MOND, when the acceleration due to normal
    gravity becomes low enough, a different term
    in the acceleration comes into effect which is of
    the form sqrt(G m a_0)/r. This matches the
    above formula if the MOND acceleration parameter
    a_0 is equal to c^4/GM. The experimental value of
    a_0 is around 1.2*10^-10 ms^-2 so this matches the
    formula if the mass of the universe is approximately
    10^54 kilograms. This is certainly around the right
    order of magnitude, which seems very interesting,
    given that this formula was derived from an idea
    relating to the shape of space and the closure of the
    universe, unlike MOND itself which is (as far as I
    know) purely empirical at present.

    In this case, the extra acceleration would merely be
    added to the Newtonian acceleration, which in the
    MOND formalism is formally equivalent to using an
    interpolation function of the following form, assuming
    my calculations were correct:

    mu(x) = (sqrt(1+1/4x)-sqrt(1/4x))^2

    (I found it quite surprising that the above
    expression is equal to x when x is small, as I
    would not have guessed that at first glance).

    Does anyone know whether this interpolation
    function (based on adding the accelerations together)
    is considered viable with current galaxy data?

    This formula c^2/r sqrt(m/M) has another curious
    feature, which is however far from cosmological.
    I found this when I was investigating under what
    conditions the MOND and Newtonian accelerations
    are comparable. Specifically, consider the
    acceleration at the surface of a particle of mass
    m and Compton radius r = hbar/mc, and consider
    when it is equal to the "conical space" acceleration:

    Gm/r^2 = c^2/r sqrt(m/M)

    Moving some factors of c and r around we get

    Gm/rc^2 = sqrt(m/M)

    If we substitute the Compton radius expression
    for r, we get:

    Gm^2/(hbar c) = sqrt(m/M)

    Squaring and rearranging this, we get

    m^3 = ((hbar c/G)^2)/M


    m = cube root of (((hbar c/G)^2)/M)

    If we use M = 10^54 kg from matching the MOND
    result, this gives

    m = approx 34 MeV/c^2

    That is, the mass for which these two acceleration
    expressions are equal is around 65 times the mass
    of the electron, around the right order of magnitude
    for all common particles.

    No, I don't know whether either of the above results
    (MOND or particle) is physically meaningful, but I
    just thought they both seemed rather interesting.
  2. jcsd
  3. Jun 26, 2007 #2

    Jonathan Scott

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    Gold Member

    I've subsequently found that if MOND effects simply involved adding in the MOND acceleration, using the previously mentioned interpolation function, then anomalous results attributable to MOND would probably have already been detected in solar system experiments, as the effect would have been stronger than the known "Pioneer anomaly", and it might well also have been detected in Cavendish-type laboratory experiments to measure G (as mentioned in another thread). However, I would still be interested to know of any specific evidence which definitely rules out such local MOND effects.

    It appears that the proponents of MOND maintain that the MOND effect only "switches on" when the overall potential gradient due to all fields is less than the critical acceleration. However, it is unclear how stars could then be affected by MOND when almost all of the component particles within the star are within a gravitational field which far exceeds the MOND threshold.
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