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A causally connected faster than light Warp Drive space-time_F. Loupy R. Heldz D. Waitex E. Halerewicz, Jr.{ M. Stabnok M. Kuntzman__

R. SimsyyJanuary 28, 2002 Originally appeared in arXiv: gr-qc/0202021Abstract

The authors will demonstrate, that while horizons do not exist for warp drive space-times travelingat sub-light velocities, horizons begin to develop when a warp drive space-time reaches light speedvelocities. They will show that the control region of a warp driven ship lie within the portion of the warpedregion that is still causally connected to the ship, even at faster than light velocities. This allows a ship toslow to sub-light velocities. Furthermore, the warped regions, which are causally disconnected from a warpship, have no correlation to the ships velocity.

1 IntroductionOne of the many objections against warp drives is the appearance of horizons, when a

ship travels at faster than light velocities (see figure 2). The problem is to control the speed of theship at speeds greater than light. If the bubble becomes causally disconnected from the ship,

then observers in the ships frame cannot control the bubble, and the ship cannot reduce itsvelocity. In this work, we will show, that while part of the warped region becomes causallydisconnected from the ship at faster than light speeds, the behavior of that part does not dependon the ships speed and can be engineered while the ship is still sub-light. Also, the control regionof the ship's velocity remains in the portion of the warped region that is still casually connected tothe ship (see figure 3).

2 Two-dimensional warp driveIn order to examine the warp field control problem, we start with the two dimensional

ESAA metric [1] written in the Alcubierre formalism:

ds2= A2 dt2+ [dx -vsf(rs)2dt] (1)_This work was made possible by through the Advanced Theoretical Propulsion Group (ATPG) Collaboration; currentURL:

http://www.geocities.com/halgravity/[email protected];Lusitania Companhia de Seguros SA, Rua de S Domingos a Lapa 35 1200 Lisboa Portugal;research independentof [email protected]@aol.com{[email protected]@[email protected]@yahoo.com;University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 USA

where?

dx = dx+ vsdt and vs =dxs /cdt (2)Substituting in S into equation 2:

1 - f(rs) = S(rs): (3)the line element is:

ds2= A2 dt2 (vsS(rs))2dt2 -vsS(rs)dxdt-dx2 (4)

This is the two-dimensional ESAA space-time required to discuss the horizon problem.

2.1 Two-dimensional ESAA Hiscock metric:In order to continue, we will proceed in a similar manner as did Hiscock [2]. The ESAA-

Hiscock ship frame metric can be written from:


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ds2 = -H(rs)dT2 +A2(rs)dx2 H(rs) (5)The prime on x from now on is implicit.Define dT as:dT = dt -vsS(rs)H(rs)dx (6)Inserting equation 8 into 7 and defining H as the horizon function:

H(rs) = A2-(vsS(rs))2 ( 7)The corresponding line element becomes

ds2= -H(rs)dt2 + 2vsS(rs)dxdt + dx2(8)or

ds2= -[A2 -(vsS(rs))2]dt2+ 2vsS(rs)dxdt + dx2 ( 9)

3 Pfenning-piecewise and Horizon functions:We need to define our functions f,S< and H. The Pfenning integration limits are R -

(delta/2) and R + (delta/2) [3], where we can set delta = 2/sigma and sigma = 14 from theAlcubierre top hat function:

f(rs) = tanh[sigma(rs + R)] -tanh[sigma(rs -R)}/2 tanh(sigma*R)

The values for the Pfenning-Piecewise lapse function becomesA(rs) = 1 rs < R -(delta/2)A(rs) =Kappa R-(delta/2)


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There are no horizons for the proposed space-time (9), with the functions (10,11,12,13),when v 1. We know that the continuous

form of the top hat f(rs) is 1 in the ship and 0 far from it, there exists a open interval, when thefunction f(rs) starts to decrease from 1 to 0. It is in that region where the exotic matter resides.If we define:

2A =[(1 + tanh[sigma(rs -R)] ^-N]/2 ( 17 )

2N is an arbitrary exponentdesigned to reduce the stress-energy requirements. This expressioncan make A be 1 in the ship and far from it while being large in the warped region (region 2).

Below there are numerical simulations (see table 1). By pushing the ESAA-Hiscockhorizon to the outermost layers of the warped region, this should make the speed more


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controllable by the ship. The major part of the warped region is connected to the ship so the shipcan reduce to sub-light velocities.2However from a dimensional point of view N = R=_, such that N becomes a measure of shell thickness.

55.2 remote frame horizonsWe now define a Hiscock horizon function for the remote frame, in the analogous fashion as for

the ships frame.

I(rs) = A2 -(vsf(rs))2 (18)

If vs < 1, the three regions are causally connected to both the ship and remote frame. However ifvs = 1 the horizon appears for the remote frame. This region (1) while connected to the ship framebecomes causally disconnected from the remote frame. if vs > 1, then somewhere in region 2, ahorizon appears which is causally disconnected from the remote frame while connected to theship frame, and vice versa.

If we utilize the top hat function (15) for the warped region R _ (_=2) _ r s_ R + (_=2) thenone has

I(rs) = A2 - (v (1/2+(R-rs)/delta)2 ( 19 )

Providing a large A2> vsf(rs)2, then I(rs) > 0 and this region will be causally connected tothe remote frame. The remote frame can detect part of the Pfenning warped region. If the shipchanges its speed, then the remote frame will observe the changing speed.

Thus, a signal sent by the ship can go up to r s = R + (delta/2) and a signal sent by remoteobserver can go inward tot rs = R -(delta2). Therefore part of the region between R -(delta/2) < rs


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gr-qc/9707024[3] M.Pfenning. Quantum inequality restrictions on negative energy densities in curved space-times. gr-qc/9805037

Figure 2: Luminal horizon formation. The red region represents where a horizon will form once awarp drive

Space-time [1] reaches luminial velocities.8Figure 3: Faster than light warp bubble frame regions. The blue region is the remote framehorizon, the yellow region is the Pfenning region, and the red region is the ship frame horizon.