A Casually Connected Superluminal Warp Drive

A causally connected faster than light Warp Drive spacetime_ F. Loupy R. Heldz D. Waitex E. Halerewicz, Jr.{ M. Stabnok M. Kuntzman__ R. Simsyy January 28, 2002 Originally appeared in arXiv: gr-qc/0202021 Abstract The authors will demonstrate, that while horizons do not exist for warp drive space-times traveling at sub-light velocities, horizons begin to develop when a warp drive space-time reaches light speed velocities. They will show that the control region of a warp driven ship lie within the portion of the warped region that is still causally connected to the ship, even at faster than light velocities. This allows a ship to slow to sub-light velocities. Furthermore, the warped regions, which are causally disconnected from a warp ship, have no correlation to the ship’s velocity.

1 Introduction One 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 the ship 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 its velocity. In this work, we will show, that while part of the warped region becomes causally disconnected from the ship at faster than light speeds, the behavior of that part does not depend on the ship’s speed and can be engineered while the ship is still sub-light. Also, the control region of the ship's velocity remains in the portion of the warped region that is still casually connected to the ship (see figure 3).

2 Two-dimensional warp drive In 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; current URL: http://www.geocities.com/halgravity/atpg.html [email protected];Lusitania Companhia de Seguros SA, Rua de S Domingos a Lapa 35 1200 Lisboa Portugal; research independent of employer. [email protected] [email protected] {[email protected] [email protected] [email protected] [email protected];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: 2 2 2 ds2 = A2 dt2 — (v sS(rs)) dt -vsS(rs)dxdt-dx (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 ESAAHiscock ship frame metric can be written from:

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 the Alcubierre 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 becomes A(rs) = 1 rs < R -(delta/2) A(rs) =Kappa R-(delta/2) R + (delta/2) (10) Kappa is a large number, when not near< R -(delta/2) and R+(delta/2). We note that the function, A, can not be a function of the speed, but requires exotic matter to have this property The Pfenning-Piecewise function f(rs) is defined as: f(rs) = 1 rs < R-(delta/2) f(rs) = ? +(R-rs )/delta R-(delta/2)R + (delta/2) (11) The ESAA ship frame Piecewise function S(rs)=1-f(rs) is: S(rs) = 0 rs < R-(delta/2) S(rs) = = ? -(R-rs )/delta R-(delta/2)R + (delta/2) (12) The ESAA-Hiscock horizon (7) is: H(rs) = 1 rs < R-(delta/2) H(rs) = A2 -v2 ( ?

-(R-rs )/delta)2

R-(delta/2)
H(rs) = A2 - v2 rs >R + (delta/2) (13) We will study now the behavior of the ESAA-Hiscock Horizon function for three cases: 1-ship sub-light (v < 1) 2-ship at light speed (v = 1) 3-ship faster than light (v > 1) To more easily discuss the effects of v and A on H, we will define three regions which corresponding to the three ranges that rs takes.

3.1 Sub-light ship velocities

There are no horizons for the proposed space-time (9), with the functions (10,11,12,13), when v<1. Since A is always great than or equal 1, and S(rs) is never greater than 1, in all of the three regions covered by the H function, H is always greater than 0. Hence the ship is causally connected to all of the warp shells.

3.2 Light speed ship velocities In this case, we need to examine all of the regions. In region 1, as before, the ship will be always connected to the region starting from rs = 0 (ship location) to rs = R + (delta/2). In region 2, a horizon will form where A goes to 1, which is at R+delta/2. The ship becomes causally disconnected from the region beyond R+(delta/2), which is region 3. The part of the speed control still lies in the region between R-delta/2 to R-delta/2, so the ship can change the speed and Decrease its speed to sub-light if needed.

3.3 Faster than light ship velocities Again, in region 1, as before, the ship will be always causally connected to the region from rs = 0 to rs = R + (delta/2). From equation 7, we see that a horizon will appear somewhere between R-delta/2 to R+delta/2 depending on v and the value of A at rs. That part of region 2 and all of region 3 is causally disconnected form the ship. Assuming that the behavior of A was engineered when the ship was at sub-light speeds, the capacity to slow the ship down to sub-light speeds should still exist.

4 Stress-Energy momentum tensor Consider the Stress-Energy momentum tensor for a ship frame ESAA-warp metric T 00 = vs2 /32Pi (dS(rs)/drs) 2

(sigma/rs)2

/A4

( 14)

Substituting dS(rs)/drs with -1/delta or -sigma/2 yields: 2 T 00 = vs /32Pi

2

2 4

(sigma /2 rs) /A

(15 )

5 Remote Observer Horizons The remote frame ESAA warp drive metric is given by: ds2 = -A2 dt2 + [dx -vsf(rs)dt]2 ( 16) As stated previously, A = 1 in regions 1 and 3 and in the region 2 is some large value. The function f(rs) has the ordinary values for the Top Hat function. In the calculation of the horizons, we will first examine the behavior of the A function , and then the horizons as seen in the remote frame.

5.1 The Preprogrammed A function Although we set up to define A by "pre-programmed exotic matter", which does not change when the ship pass from sub-light to faster than light velocity (see figure 3), we have not defined A. With a Pfenning-Piecewise behavior of A, in region 3, A still has to have a large value to keep this region causally connected to the ship, when vs > 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 the function f(rs) starts to decrease from 1 to 0. It is in that region where the exotic matter resides. If we define: 2 A =[(1 + tanh[sigma(rs -R)] ^-N]/2 ( 17 ) 2

N is an arbitrary exponent designed to reduce the stress-energy requirements. This expression can 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-Hiscock horizon to the outermost layers of the warped region, this should make the speed more

controllable by the ship. The major part of the warped region is connected to the ship so the ship can reduce to sub-light velocities. 2However

from a dimensional point of view N = R=_, such that N becomes a measure of shell thickness.

5

5.2 remote frame horizons We now define a Hiscock horizon function for the remote frame, in the analogous fashion as for the ship’s 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 if vs = 1 the horizon appears for the remote frame. This region (1) while connected to the ship frame becomes causally disconnected from the remote frame. if vs > 1, then somewhere in region 2, a horizon appears which is causally disconnected from the remote frame while connected to the ship frame, and vice versa. If we utilize the top hat function (15) for the warped region R _ (_=2) _ rs _ R + (_=2) then one 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 to the remote frame. The remote frame can detect part of the Pfenning warped region. If the ship changes its speed, then the remote frame will observe the changing speed. Thus, a signal sent by the ship can go up to rs = R + (delta/2) and a signal sent by remote observer can go inward tot rs = R -(delta2). Therefore part of the region between R -(delta/2) < rs
6 Conclusions: In this work, the authors have demonstrated how the lapse function, A, defined as a Pfenning-Piecewise function, can resolve the faster than light control problem of the warp drive. A is assumed not to change its behavior when the ship passes from sub-light to faster than light, although we do not provide a source for the nature of A. This will investigated in a future work.

Acknowledgements: The Authors of this work would like to express the most profound and sincere gratitude to Miguel Alcubierre for his time and patience during all the phases of development of this work.

References [1] F.Loup, D.Waite, and E.Halerewicz, Jr. Reduced total energy requirements for a modi_ed Alcubierre warp drive space-time gr-qc/0107097. [2] W.Hiscock. Quantum e_ects in the alcubierre warp drive space-time. Class. Quantum Grav., 14: L183{88, 1997.

gr-qc/9707024 [3] M.Pfenning. Quantum inequality restrictions on negative energy densities in curved spacetimes. gr-qc/9805037 Figure 2: Luminal horizon formation. The red region represents where a horizon will form once a warp drive Space-time [1] reaches luminial velocities. 8 Figure 3: Faster than light warp bubble frame regions. The blue region is the remote frame horizon, the yellow region is the Pfenning region, and the red region is the ship frame horizon.