How to take into account the gravitational interactions in the cosmic string model of spiral galaxy?

What has been neglected in the model is the presence of gravitational force.

1. The thickened string in the galactic plane acts with the cosmic string transversal to the galactic plane and with the matter formed by the decay of the cosmic string in the galactic plane to ordinary matter and reducing its string tension. The first guess is that this gives rise to a gravitational force, which in the first approximation is sum of the force F₁=d(G M(ρ)/ρ) and the force caused F₂= GT/ρ by the cosmic string. The non-relativistic Newton's equation for a point particle in this force field is v²= ρ(F₁+F₂)= d(G M(ρ))+ GM(ρ)/ρ + TG . At short distances, the force caused by matter dominates and at long distances the force due to the long cosmic string dominates.

   One can however argue that if the mass M(r) is generated by the decay of the cosmic string in the galactic plane, one should approximate the galaxy as a single thickened string, at least in the primordial state as a cosmic string.

2. If the planar cosmic string would consist of independent particles, it would decay very rapidly. String tension prevents this. One might however hope that in the first approximation string tension forces initial conditions preserving the identity of the string but that the points otherwise move independently. Note that by Equivalence Principle the decomposition to smaller masses does not depend on the size of the small mass.
3. The intuitive guess is that ince the velocity of rotation increases towards the galactic nucleus, the gravitational force causes a differential rotation of the planar cosmic string. Since the velocity and therefore also angular velocity ω(ρ) increases towards the center of the galaxy, spiral structure is generated. At long distances the velocity of rotation is the same as for distant stars.

The system could be modelled by addition of the gravitational force to the equations of string world sheets as a minimum as an additional radial force. The model would generalize the ordinary Newtonian description of point-like particles in a gravitational field. The above model suggests that convenient coordinates for the string world sheet are (t,φ) and the dynamical variable is the radial coordinate ρ along the rotating string as a function of (t,φ). Numerical modelling along these lines would give partial differential equations. Also now conservation of energy and angular momentum can be used. Could one imagine any elegant solution to the problem.

1. Physical intuition suggests that one should start from the solution without the gravitational force already considered since it looks realistic in some aspects. One should transform the static string to a string in a differential rotation determined by the gravitational forces and forcing only coherent initial conditions for the points of the string so that they all rotate with the velocity. One might even hope that Kepler's law can be used besides conservation laws.
2. Equivalence Principle suggests how one might achieve this at least approximately. Gravitational force is in the Einsteinian description a coordinate force describable in terms of Christoffel symbols. In TGD this force is force in H which can be approximated with M⁴. Could one find a coordinate system of M⁴ in which this coordinate force vanishes? Could a differentially rotating system be the system in which this is the case. This would generalize Einstein's freely falling elevator argument.

Could one obtain this coordinate system by an analog of Lorentz boost in (t,φ) plane by a velocity β(ρ)=v(ρ)/c and leaving ρ and z invariant. This transformation looks like Lorentz transformation for a given ρ but is a transformation to an accelerating system (acceleration is radial).

1. One could consider an infinitesimal variant of thef effective Lorentz boost and exponentiate it to get a flow restricting to the motion of the string defining the string world sheet. The infinitesimal boost would be

   dt= γ(dT- β ρ dΦ) , dφ= (1/ρ)γ(ρ dΦ-β dT) , γ=(1/(1-β²)^1/2 .

   These equations define a flow in (T,φ) plane as an exponentiation of an infinitesimal Lorentz boost for a given value of radial coordinate ρ and one can solve (t,φ) as function of (T,Φ). The intuitive idea is that for ρ given by the static model but with (t,φ) replaced with (T,Φ) this flow reduces to the equations of the static string world sheet. This flow need not be integrable in the entire M⁴. The points (T,ρ) for which Φ differs by a multiple of 2π could correspond to different turns of the spiral rotating around the origin.

   This flow should be integrable in order that the flow lines have interpretation as coordinate lines. It should be possible to write the infinitesimal generator of the Lorentz boosts in (t,φ) plane for a given ρ as a product of scalar function and gradient: j= Ψ dΦ giving dΦ=j/Ψ so that Φ serves as a coordinate. Is it enough to satisfy this condition at the string world sheet at which the condition ρ=ρ(Φ,T) mildens it?

2. It is easy to find how this pseudo Lorentz boost affects the expression of M⁴ metric ds²=dt²-dz²-dρ²-ρ²dφ² by writing the differentials dt, dφ and dρ explicitly:

   ds²=dt²-dz²-dρ²-ρ²dφ² =(γ(dT- β ρ dΦ)²- (γ(ρ dΦ-β dT)² -dρ² -dz² .

   Here ρ(Φ,T) corresponds to the orbits of the point of the string and must satisfy the field equations. Here dρ² expressed in terms of dΦ and dT gives additional contribution to the induced metric.

3. If only the gravitational force of the long cosmic string is taken into account one has β= constant and the analogy with Lorentz books is even stronger.

The wishful conjecture is that these equation satisfy integrability conditions on string world sheet and that the gravitational force disappears from field equations using coordinates (T,Φ) when the velocity parameter corresponds to the expected solution in the external field in coordinate (t,φ).

See article About the recent TGD based view concerning cosmology and astrophysics or the chapter with the same title.

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.