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In mathematical physics, a closed timelike curve (CTC) is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point.

'Inside the inner horizon (of a charged/rotating black hole) there is a toroidal region around the ring singularity within which the light cone in t−ϕ coordinates opens up to the point that ϕ as well as t are time-like coordinates. The direction of increasing proper time along t is t increasing, and along ϕ is ϕ decreasing, which is retrograde. Within the toroidal region, there exist time-like trajectories that go either forwards or backwards in coordinate time t as they wind retrograde around the toroidal tunnel. Because the ϕ coordinate is periodic, these timelike curves connect not only the past to the future (the usual case), but also the future to the past, which violates causality. In particular, as first pointed out by Carter (1968), there exist closed time-like curves (CTCs), trajectories that connect to themselves, connecting their own future to their own past, and repeating interminably...'

Equations

Black hole metric-

[tex]c^2 d\tau^2 = g_{tt}c^2dt^2\ +\ g_{rr}dr^2\ +\ g_{\theta\theta}d\theta^2\ +\ g_{\phi \phi} d\phi^2\ +\ 2 g_{t \phi}cdt d\phi[/tex]

where for Kerr-Newman metric-

[tex]g_{tt}=-1+(2Mr-Q^2)/\rho^2[/tex]

[tex]g_{rr}=\rho^2/\Delta[/tex]

[tex]g_{\theta\theta}=\rho^2[/tex]

[tex]g_{\phi\phi}=(r^2+a^2+[(2Mr-Q^2)a^2 \sin^2\theta]/\rho^2) \sin^2\theta[/tex]

[tex]g_{t\phi}=(Q^2-2Mra) \sin^2\theta/\rho^2[/tex]

where

[tex]\rho^2=r^2+a^2 \cos^2\theta[/tex]

[tex]\Delta=r^2-2Mr+Q^2+a^2[/tex]

and

[tex]M=Gm/c^2,\ a=J/mc,\ Q=C\sqrt(G k_e)/c^2[/tex]

The ergosphere boundaries which denote space rotating faster than c are at-

[tex]g_{tt}=0[/tex]

which occurs where

[tex]\Delta=a^2 \sin^2 \theta[/tex]

In the uncharged Kerr geometry when [itex]\theta=\pi/2[/itex], within the inner horizon, this occurs at r=0 (i.e. at the ring singularity edge) but in the Kerr-Newman geometry occurs away from the ring singularity with positive radius.

Within a charged/rotating black hole, the boundary of the CTC toroidal region is at-

[tex]g_{\phi \phi}=0[/tex]

which occurs where

[tex]\frac{R^4}{\Delta}=a^2 \sin^2 \theta[/tex]

where [itex]R=\sqrt(r^2+a^2)[/itex]

In the uncharged Kerr geometry the CTC torus is entirely at negative radius, r<0 (i.e. within the ring singularity), but in the Kerr-Newman geometry the CTC torus extends to positive radius.

Turnaround (i.e. infall velocity becomes zero) and counter-rotation occurs at-

[tex]g_{t \phi}=0[/tex]

which occurs where

[tex]r_0=\frac{Q^2}{2M}[/tex]

in both Reissner-Nordstrom (charged, non-rotating) and Kerr-Newman geometry.

In the uncharged Kerr geometry [itex]r_0[/itex] is at the ring singularity, r=0, but in the Kerr-Newman geometry, [itex]r_0[/itex] has a positive radius.

Extended explanation

[itex]g_{tt}=0[/itex] defines the boundary of ergospheres, [itex]g_{t\phi}=0[/itex] defines the turnaround/counter-rotation radius, and [itex]g_{\phi\phi}=0[/itex] defines the boundary of the toroidal region containing closed timelike curves. For a BH to have both charge and spin, [itex]Q^2+a^2\leq M^2[/itex] applies. For a black hole with Q=0.8 and a=0.6, the BH would be extremal and the CTCs would not be hidden by an event horizon.

'...one might be tempted to think that perhaps the cause of the counter-rotational phenomenon (in Kerr geometry) might be found in the fact that the co-rotating area (ergosphere) and the counter-rotating area (Time Machine) are separated by the ring-singularity. So perhaps the counter-rotation can be explained by saying that the ring-singularity acts like a mirror turning directions into their negatives. Such a symmetry about the ring-singularity would still not account for the rotational sense of the ring-singularity itself; in particular it would keep us wondering why the singularity co-rotates with the ergosphere but counter-rotates with the CTCs. The simplest way of seeing that such mirroring about the singular region cannot possibly give a hint for the diametric revolutions of the ergosphere and the Time Machine is by looking at Kerr-Newman spacetimes where the co-rotational and counter-rotational areas are no longer separated by the ring-singularity.' (Source page 7)

Note: This entry looks at CTCs within black holes only, CTCs can appear in other solutions to the Einstein field equation of general relativity. These include:

the Misner space (which is Minkowski space orbifolded by a discrete boost)

the van Stockum dust (which models a cylindrically symmetric configuration of dust)

the Gödel lambdadust (which models a dust with a carefully chosen cosmological constant term)

the Tipler cylinder (a cylindrically symmetric metric with CTCs)

Bonnor Steadman solutions describing laboratory situations such as two spinning balls

J. Richard Gott has proposed a mechanism for creating CTCs using cosmic strings.

(source)

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# What are closed timelike curves

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