**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/r^{2} central force leads very rapidly to a loss of original structure since angular velocities behave like ω∝ 1/r^{2}. 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.

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

- By introducing cylindrical Minkowski coordinates (m
^{0}, m_{1}= ρ cos(φ),m_{2}= ρ sin(φ),m^{3}) and using (m^{0}=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 M^{4}in the cylindrical coordinates is m_{kl}&rightleftarrow; (1,-1,-1,-ρ^{2}). The induced metric of X^{2}in these coordinates has only diagonal components and can be written as(g

_{tt}=1-ρ_{t}^{2}, g_{φφ}=-ρ^{2}-ρ_{φ}^{2}) . - 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^{αβ}m^{k}_{β}m_{kl}j_{A}^{l}(-g_{2}^{1/2})=0 .The conservation laws of momentum and energy hold true and the conservation of angular momentum L

_{3}in direction orthogonal to the galactic plane givesg

^{φφ}ρ^{2}(-g_{2})^{1/2}=1/ρ_{0}.where ρ

_{0}is integration constant. This givesx

_{φ}= +/- x(x^{2}-1)^{1/2}, x= ρ/ρ_{0}.From this it is clear that the solution is well-defined only for ρ≥ ρ

_{0}, which suggests that the branches of the spiral must turn back at ρ=ρ_{0}(x=1). At the limit x→ 1, x_{φ}approaches zero. One might guess that one has a spiral, which rotates around x=1 since dφ/dx diverges but this does not seem to be the case. - The differential equation can be solved explicitly: one has
∫ dx/(x(x
^{2}-1)^{1/2})= +/- φ +φ_{0}.The elementary integral using the substitution x= cosh(u) gives

φ

_{+/-}= φ_{0}+/- arctan(y) ,y=(x

^{2}-1)^{1/2}.The argument of arctan is real only for x≥ 1. Could one define the solution for x<1, where the argument is imaginary? arctan(iy) for real argument y as arctan(iy)= (i/2)ln((1+x)/(1-x)). This would mean that φ is not real.

- The solution has formally infinitely many branches φ
_{+/-,n}differing by an integer multiple of π. However, for a fixed value of +/-, the branches differing by a Δ φ= n2π coincide so that one obtains only 2 branches meeting at the x=1 circle at angles φ_{0}and φ_{0}+π.x→ 1 corresponds to φ

_{+/-,n}→ φ_{0}+/- nπ and x→ ∞ corresponds to φ_{+/-}→ φ_{0}+/- π/2+/- nπ. The variation of φ for a given branch is π/2. - What could be the physical interpretation? The two branches for a fixed sign factor +/- meet x=1 circle at tangles φ
_{0}and φ_{0}+π. Could galactic bar connect these points? Could the diverging value of dφ/dx at x=1 mean that φ increases by φ at this point?It is now known that also in the case of the Milky Way there are only two branches. If this is the case then the two branches plus galactic bar could correspond to a single long cosmic string in the galactic plane which has collided with a transversal cosmic string. On the other hand, there is evidence that there are several structural components involved with the Milky Way.

There is however no spiral structure involved, which suggests that this simple model cannot describe spiral waves.

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.

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