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The metric around a straight, static and infinitely long cosmic string is flat except for the topological deficit in the angular coordinate, as described for example here. As a result, there is no gravitational force on surrounding matter. The deficit angle is also expected to be quite small, of order $10^{-6}$ radians.

My question is: what would happen if such a string was on a collision course with earth ? assuming that it interacts only gravitationally, will it simply pass through? or will there be some observable effect ?

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Interesting question. I found this paper:

Motohashi & Suyama, "Detecting cosmic string passage through the Earth by consequent global earthquake". Astrophys. Space Sci. 348, 565-569 (2013).

which discusses the effects of a cosmic string passing through or near earth. The conclusion is that it will likely generate global earthquake whose magnitude depends on string tension -- from potentially catastrophic, to barely noticeable.

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  • $\begingroup$ Thanks! this actually leads me to another question: would there be similar observable effects if the string passes through the sun (e.g. changes in the sun brightness?) and could such effects be detected on other stars ? or is the probability of a cosmic string actually hitting a star too small ? $\endgroup$
    – J. Delaney
    Commented Feb 24, 2022 at 20:47
  • $\begingroup$ @J.Delaney They mention that this is applicable to stars too, so maybe some oscillations in the brightness. The probability of a string passing through stars is very small of course, because of how rare the galaxies and stars are, but more so because we already know that cosmic strings are extremely rare, even if exist $\endgroup$
    – Kosm
    Commented Feb 24, 2022 at 20:54
  • $\begingroup$ @J.Delaney it's more likely that we'll see them though lensing, or gravitational waves -- either stochastic background or bursts $\endgroup$
    – Kosm
    Commented Feb 24, 2022 at 20:57

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