Re: Formation of Closed Timelike Curves with Morris-Thorne wormholes

From: Adam Getchell (agetchell_at_physics.ucdavis.edu)
Date: 02/18/05 

Date: Fri, 18 Feb 2005 17:42:06 +0000 (UTC)

George Jones wrote:

> Suppose we have a spacetime that has a wormhole, and that spacetime is
> (nearly) flat outside of the wormhole. Suppose further that the throat
> of the wormhole is infinitesimally short.

Hmmm ... I don't think you can make this assumption.

Using the Morris-Thorne wormhole metric:

ds^2=-exp(2*phi(l))*dt^2+dl^2+r(l)^2*(d(theta)^2+sin^2(theta)d(phi)^2))

where l is the proper radial distance (-infinity to +infinity), then a
traveler of length L through the wormhole throat experiences a tidal
acceleration g equal to:

g/L = Abs((1-b/r)(-phi''-(phi')^2)+ (1/(2r^2))*(b'r-b)phi')

where primes denote derivatives wrt l.

(Visser, eq. 13.9)

Since the beings would like to traverse the wormhole without getting
shredded, this effectively sets a minimum "size" (shape and
lapse/redshift function) for the wormhole.

> Because the throat of the wormhole is short (and thin), Every event on
> the worldline of mouth 1 is identified with a corresponding event on the
> worldline of mouth 2. A and A' are one such pair of identified points.

I don't think you can identify events from mouth 1 to events on mouth 2:
the metric is definitely not flat.

In particular, traversing the wormhole from distance -l1 to +l2 takes:

Proper time for traveler: Integral(from -l1 to +l2)*dl*(v*gamma)^-1

Time wrt to static observer: Integral(from -l1 to +l2)*dl*(v*exp(phi))^-1

For the solution with exotic matter limited to the throat in the
original 1988 Morris-Thorne paper, the transit time is about 200 days.

> Which pairs of events are identified in this scenario? At the instant
> that the mouths start to move apart, place a (zeroed) clock in mouth 1.
> The clock is also in mouth 2, again because of the infinitesimal throat
> length. Therefore, the clock measures proper time for both "twins", and

I think in this case the clocks have to be placed at the entrance/exit
to the wormhole, rather than at the throat, so the time should be given
by the second equation wrt to a static observer, i.e. involving
exp(phi). Then (I think) the moving mouth will have the additional
Lorentz factor described.

Although I'm not yet sure that relativistic movement of the wormhole
won't collapse the metric. As a very crude approximation (I don't know
how accurate it is):

For perturbation theory, g ~ g0+ delta(g), where g = wormhole metric,
g0 is the unperturbed Morris-Thorne metric, and delta(g) is the
perturbation.

Roughly speaking, delta(g) < g0, else perturbation theory doesn't
apply. To first order, we can approximate delta(g) by the Lorentz
contraction of the length in the z direction. This gives us the
condition that:

gamma(dz) * z < 2 z

where dz=v, and gamma(dz) is 1/Sqrt(1-v^2/c^2). This gives us dz =
..866 c, so in order for our perturbation to die off, maximum velocity
in the z direction is .86 c.

Another way of saying this is that we want a convergent geometric series:

Sum(n=0 to infinity) a*r^n converges for r < 1

Now, as a SWAG, I'm going to aim for quadratic damping. This means we
want the gamma factor to be no more than 1.5, so that the perturbation
terms go as 1/2^n (remembering the factor of 1 goes to the g0 term).
This arises from integrating the geometric series term by term: Integral
of 1/x^2 --> 1/x, but the series 1/x diverges.

So limiting gamma(dz) < 1.5 yields dz < .74c.

This is why I limited the wormhole speed to ~.7c in my example.

> The following spacetime diagram shows the worldlines of the 2 mouths and
> 3 pairs of identified events. If Han goes into the wormhole at one event
> in the pair, he comes out of the wormhole at the other event of the
> pair. As explained above, there has to be a timelike relationship
> between events in a pair in order to have a CTC.
>
>
> t
>
> |
> |\
> | \
> | \
> | \
> | \C'
> | \
> | \
> | \
> C| \
> | \B'
> | /
> B| /
> | /
> | /
> | /A'
> | /
> A| /
> | /
> | /
> |/
> O----------------------------- x
>
>
> The proper time of any event on the worldline of mouth 1 is just the t
> coordinate of that event. The proper time tau of any event on the
> worldline of mouth 2 is related to the t coordinate of that event by
>
> tau = t*sqrt(1 - v^2). (1)

Where t is the integral I gave above, and I think that properly we
should consider the length contraction of the shape function b for the
moving mouth.

> Suppose a time machine forms at C', i.e., C and C' lightlike related.
> Then,
>
> 1 = (x_C' - x_C)/(t_C' - t_C)
>
> = x_C'/(t_C' - t_C'*sqrt(1 - v^2))
>
> = v*x_C'/[ (2*x_B' - x_C')*(1 - sqrt(1 - v^2))]. (5)
>
> After playing around with (5), I get (maybe incorrectly)
>
> x_C' = x_B'*[1 - sqrt((1 - v)/(1 + v))]. (6)
>
> In your example, v = 7/10 and x_B' = lightyears.
>
> I get x_C' = 58 lightyears. A time machine forms when mouth 2 get within
> 58 lightyears of mouth 1 on the return trip.
>
> I haven't checked the above carefully, so there may be mistakes.

I'll have to work this out with the caveats I've mentioned, thanks!
Unless I'm completely barking up the wrong tree.

> Regards,
> George

Adam

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