What Are Closed Timelike Curves in Black Holes?

In summary, closed timelike curves (CTCs) are theoretical constructs that describe the possibility of time travel within the framework of Einstein's theory of general relativity. CTCs work by bending the fabric of spacetime, allowing an object to travel back in time and interact with its past self. While mathematically possible, their existence has not been proven or observed in the real world and raises questions about causality and free will. If possible, CTCs could have significant implications for our understanding of reality and the possibility of altering the course of history. They are often related to wormholes, which are hypothetical tunnels through spacetime that connect two distant points and could serve as a means of traversing CTCs.
  • #1
19,547
10,291
Definition/Summary

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)

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
 
Astronomy news on Phys.org
  • #2
Closed timelike curves (CTCs) are worldlines in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point. In the case of a charged/rotating black hole, the boundary of the CTC toroidal region is defined by g_{\phi \phi}=0, where \frac{R^4}{\Delta}=a^2 \sin^2 \theta, R=\sqrt(r^2+a^2), and \Delta=r^2-2Mr+Q^2+a^2. The turnaround/counter-rotation radius is defined by g_{t \phi}=0, which occurs where r_0=\frac{Q^2}{2M}. If the black hole has Q=0.8 and a=0.6, the BH would be extremal and the CTCs would not be hidden by an event horizon. CTCs can also appear in other solutions to the Einstein field equation of general relativity, such as Misner space, van Stockum dust, Gödel lambdadust, Tipler cylinder, Bonnor Steadman solutions, or cosmic strings.
 

FAQ: What Are Closed Timelike Curves in Black Holes?

What are closed timelike curves?

Closed timelike curves (CTCs) are theoretical constructs that describe the possibility of time travel within the framework of Einstein's theory of general relativity. These are paths through spacetime that loop back on themselves, allowing an object to return to its own past.

How do closed timelike curves work?

CTCs work by bending the fabric of spacetime in such a way that an object can travel back in time and interact with its past self. This is achieved by accelerating an object to speeds close to the speed of light and moving it along a curved path, causing time to bend and create a loop.

Are closed timelike curves scientifically possible?

While closed timelike curves are mathematically possible within the framework of general relativity, their existence has not been proven or observed in the real world. Additionally, they are theoretically problematic as they raise questions about causality and the concept of free will.

What are some potential implications of closed timelike curves?

If closed timelike curves are possible, it could open up the possibility of traveling back in time and altering events in the past. This could have significant implications for our understanding of causality, the nature of reality, and the possibility of altering the course of history.

How are closed timelike curves related to wormholes?

Closed timelike curves are often theorized to exist within the context of wormholes, which are hypothetical tunnels through spacetime that connect two distant points. In some theories, wormholes could serve as a means of traversing closed timelike curves and traveling through time.

Back
Top