A simple TGD based model for a spiral galaxy

The origin of the spiral structure of spiral galaxies is one of the poorly understood problems of astrophysics. Independent motions of stars around galaxy in 1/r² central force leads very rapidly to a loss of original structure since angular velocities behave like ω∝ 1/r². 1/ρ central forces caused by cosmic string orthogonal to the galactic plane gives ω ∝ 1/ρ. This suggests that there exists some pre-existing spiral structure which is much denser than the surrounding matter. The formation of stars would occur intensely in these regions and the decay of the dark energy of the cosmic string to ordinary matter would also generate stars rotating around the galaxy as effectively free objects. The spiral structure rotates slowly and in a good approximation keeps in shape so that the structure behaves somewhat like a rigid body.

This view differs from the density wave theory (see this) assumes that this structure is dynamically generated and due to self-gravitation. The density wave would be analogous to a traffic jam. The cars entering the traffic jam slow down and the jam is preserved. It can move but with a much slower velocity than the cars. Density wave theory allows us to understand why star formation occurs intensely in the spiral structure with a high density.

TGD suggests that the structure corresponds to a cosmic string, which has thickened to a monopole flux tube and produced ordinary matter.

1. One possibility is that the galaxy has formed in a topologically unavoidable collising of cosmic string (extremely thin 4-surfaces with 2-D M⁴ projection). The cosmic string orthogonal to the galactic plane would contain the dark en

ergy liberated in its thickening and giving rise to part of galactic dark matter and the galactic blackhole would be associated with it. It would create a 1/ρ gravitational expansion explaining the flat velocity spectrum of distant stars. The cosmic string in the galactic plane would in the same way give rise to the galactic matter at the spiral arms and outside the central region. The galactic bar could correspond to a portion of this string.

A simple model for the string world sheet assignable to the string in the galactic plane is as a minimal surface. In the first approximation, one can neglect the gravitational interaction with the second string and see whether it is possible to obtain a static string with a spiral structure with several branches and having a finite size. Th string carries monopole flux and should be closed and one can consider a shape which is flattened square like flux tube, which has changed its shape in the 1/ρ gravitational field of the long string (ω ∝ 1/ρ) and formed a folded structure. The differential rotation tends to lengthen the string and increase its energy. Hence one expects that string tension slows down differential rotation to almost rigid body rotation. The simplest model is as a minimal stationary string world sheet.

1. By introducing cylindrical Minkowski coordinates (m⁰, m₁= ρ cos(φ),m₂= ρ sin(φ),m³ ) and using (m⁰=t,φ) as coordinates also for the string world sheet, one can write that ansatz in the form ρ=ρ(t,φ). The metric of M⁴ in the cylindrical coordinates is m_kl&rightleftarrow; (1,-1,-1,-ρ²). The induced metric of X² in these coordinates has only diagonal components and can be written as

(g_tt=1-ρ_t², g_φφ=-ρ²-ρ_φ²) .

For a static ansatz one has ρ= ρ(φ) so that the field equation reduces to an ordinary differential equation for ρ. Rotational invariance allows us to solve the equation as a conservation law for the angular momentum component parallel to the normal of the galactic plane. For as general infinitesimal isometry with Lie algebra generator j_A^k the conservation of corresponding charge reads as

∂_α(g^αβm^k_βm_klj_A^l(-g₂^1/2)=0 .

The conservation laws of momentum and energy hold true and the conservation of angular momentum L₃ in direction orthogonal to the galactic plane gives

g^φφρ²(-g₂)^1/2=1/ρ₀ .

where ρ₀ is integration constant. This gives

x_φ= +/- x(1-x²)^1/2 , x= ρ/ρ_{0 .}

From this it is clear that the solution is well-defined only for ρ≤ which suggests that the branches of the spiral must turn back at ρ=ρ₀ (x=1).

The differential equation can be solved explicitly: one has

∫ dx/(x(1-x²)^1/2)= +/- φ +φ₀ .

The integration constant φ₀ can be put to zero and the elementary integral using the substitution x= sin(u) gives

φ= +/- ln[|cos(u)u-u²sin(u) +cot(u)|]

= +/- ln[|arcsin(x) (1-x²)^1/2-arcsin(x)²x +(1-x²)^1/2/x|] . Consider now the general properties of the solution.

1. The value range of x is [0,1] so that the value ranges of arcsin(x) are [0,π] and its shifts by an integer multiple of 2π. This gives rise to a many-valuedness: arcsin_n(x)= arcsin₀(x)+ n2π.

There is an additional double-valuedness due to the fact that one has sin(x+Δ)= sin(x) for Δ=0 and x+Δ = π/2+ arccos(x). At x=π/2 these two roots coincide as comes also clear by looking at the graph of arcsin(x) using the graph of sin(x) in the interval [0,π]. These two branches meet at x=1 which means that single branch turn back.

In an ideal situation this predicts a spiral with an infinite number of branches labelled by integer n and each branch corresponds to a branch turning back at x=1. An infinitely folded closed cosmic string carrying monopole flux would be in question. The number of branches is for physical reasons finite since otherwise the turning points at the circle x=1 would fill the circle densely.

At the limit x→ 0 (center of the galaxy) the term cot(x) becomes infinite and dominates for finite values of n so that exp(φ) approaches infinity meaning that the the spiral for given n has infinitely many branches near origin and meeting at it. The density of matter becomes very high near the origin. This kind of structure could result in a differential rotation of a string around the cosmic string orthogonal to the galactic plane with angular velocity ω ∝ 1/ρ.

One can look at the situation also at x=1. Here the equation for a given branch turning back at φ=φ_n reads as

π/2 – n(n+1/2)4π²= exp(φ_n) .

The turning points for branches fill the circles x=1. Of course, only a finite number of branches is physically possible and should correspond to the number of observed branches. For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: http://matpitka.blogspot.com/2024/03/a-simple-tgd-based-model-for-spiral.html