# Gravitational attraction/repulsion of cosmic strings and domain walls

It is well known that straight static cosmic strings don't produce any gravitational effects on test-particles, and that static flat domain walls are repulsive. This can be seen from the linearized theory of Einstein's equation, in the case of static weak fields: $$\tag{1} \square\, \bar{h}_{\mu \nu} = - \, \nabla^2 \, \bar{h}_{\mu \nu} = -\, 8 \pi G \, T_{\mu \nu},$$ where $$\tag{2} \bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \, \eta_{\mu \nu} \, h,$$ and $$\phi = h_{00}/2$$ is the newtonian potential. Playing a bit with these, and the energy-momentum of a static anisotropic fluid, give the following: $$\tag{3} \nabla^2 \, \phi = 4 \pi G (\rho + 2 \sigma + p).$$ For a cosmic string, tangential pressure is negative: $$p = -\, \rho$$, and orthogonal pressure is $$\sigma = 0$$, so the RHS of (3) is 0 (the string doesn't have a gravitational effect). For a cosmic wall we have $$\sigma = -\, \rho$$ and $$p = 0$$ so (3) is negative (the wall is repulsive).

But then I wonder how true is the conclusion. What if the string or wall is moving, twisting and deforming (non-static)?

Is it always true (I don't think so) that any thin cosmic string doesn't attract/repulse? And is it true that any thin cosmic domain wall is repulsive? What is the general case?

EDIT: We need to distinguish two motion states for the strings and walls: static and dynamical (i.e. with arbitrary motions). If the strings/walls are curvy and twisted, but still static, the above equations appear to say that the conclusion is still true (i.e. strings don't attract or repel, while static curvy walls are still repulsive). For dynamical strings/walls, the conclusion appears to be false.

Small-scale dynamics of cosmic strings and membranes, “noise” or “wiggles” can contribute significantly to the energy and gravitational effects of these objects. To describe this we can introduce averaging over appropriately chosen small scales and obtain an effective stress–energy tensor in terms of large scale string/wall behavior.

For cosmic strings the effective equation of state turns out to be independent on the details of excitations and has the form: $$U_\text{eff} T_\text{eff}=T_0^2,$$ where $$U_\text{eff}$$ is effective linear energy density, $$T_\text{eff}$$ is effective tension and $$T_0$$ is the (bare) linear energy density and tension of a unexcited (static) string.

If the effective energy density considerably exceeds the bare value, then the string would produce gravitational attraction mostly determined by this energy density.

The details can be found in the following papers:

• Carter, B. (1990). Integrable equation of state for noisy cosmic string. Physical Review D, 41(12), 3869, doi:10.1103/PhysRevD.41.3869.

• Vilenkin, A. (1990). Effect of small-scale structure on the dynamics of cosmic strings. Physical Review D, 41(10), 3038, doi:10.1103/PhysRevD.41.3038.

Similar wiggle-independent effective equation of state for domain walls (see e.g. here for the discussion): $$\epsilon_\text{eff} \tau^2_\text{eff}=\tau_0^3,$$ is applicable only when the energy density does not significantly exceeds the bare value: $$\epsilon_\text{eff}\approx \tau_0$$. If this condition does not hold, then the effective averaged stress–energy tensor and, consequently, gravitational effect of such wiggly walls depends on the details of excitation spectra. But, again, if energy density exceeds bare values considerably, the net effect would be the gravitational attraction.

One should also keep in mind that generally, wiggly strings and domain walls would lose the energy of their excitations to various types of radiation (gravitational, scalar etc.). And details of such losses depend not only on the spectra of such excitations but also on microscopic structure of these objects, so the thin string/thin wall limit has inherent limitations.

• Does your description applies to static strings and walls, with curves in them, or do they need to move (i.e. be "dynamical") to produce an attractive gravitational field? The static and weak field Einstein equation appears to say that the strings doesn't have any gravitational field (so don't attract) if they don't move (but they could be curved and twisted in any ways, if I didn't made any mistake). Also, I still get repulsion for static walls, even if they are curved in any way.
– Cham
Jan 1, 2021 at 21:36
• I find it odd that a curvy/twisted static string doesn't generate any gravitational field outside the string. If the string have a static knot in the middle of a straight part, would it produce an attraction toward that knot? Or maybe static curved configurations are unstable and should move because there's a tension in the string?
– Cham
Jan 1, 2021 at 21:46
• @Cham: String has to minimize Nambu–Goto action, so a static string would be like a beam of light: it might bend if the background metric is curved but it is as straight as possible. If string has twists and curves at initial moment in (almost) flat space, then those twists would evolve in time (propagating with the speed of light). Jan 2, 2021 at 6:11
• Yes, I now think it’s all clear.
– Cham
Jan 2, 2021 at 6:17
• Do you agree that a straight cosmic string uniformly moving in space would be attractive, while it isn’t producing any gravity if it’s not moving (relative to some inertial observer). ?
– Cham
Jan 2, 2021 at 6:22