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Superluminosity: The Alcubierre Warp Drive Wilbert Ras,1 Michael Chan,1 and Arie Reinier de Reuver1 1

Department of Physics and Astronomy, University of British Columbia 6224 Agricultural Road, Vancouver, British Columbia, Canada, V6T 1Z1 (Dated: February 16, 2018) In this paper the concept of the Alcubierre warp drive will be investigated. The Alcubierre warp drive is a solution to the Einstein equations which allows for superluminal travel as observed by a stationary observer. The Alcubierre metric is derived and explored from which the volume expansion function is computed and interpreted. This showed that the spacetime around a traveler is contracted in front of the spaceship while expanded behind the spaceship. This would allow this warp drive bubble to transverse at arbitrary speeds. The energy density needed to sustain the warp drive is derived and shown to be negative and unattainable large. While impossible now, there are optimistic thoughts about the Alcubierre warp drive for the future.

I.

INTRODUCTION

Ever since Einstein postulated in 1916 [1] that nothing could travel faster than the speed of light. This really restricts the human reach in the universe. Take for example the potentially habitable planets Trappist (1e, 1f, 1g) at a distance of 40 light years from the earth. [2] Even with time dilation for traveling astronauts the researchers on earth would still be at least 80 years older before any research could be conducted. Therefore, physicists have tried to find a way around this constraint of the speed of light. This ’superluminal’ space travel would allow the exploration of the universe way beyond what is currently possible; a concept famously called ’warp drive’. By definition a warp drive is a solution of the Einstein Equations which allows superluminal travel for a spaceship, while obeying the laws of general relativity as stated by Einstein.[3] Several physicist have derived such warp drives from the Einstein Field Equations [1] such as Miguel Alcubierre [4] and Jos´e Nat´ario [5]. The possibility of such mechanisms have been extensively discussed throughout the years. In this paper the Alcubierre metric will be investigated to create an understanding of its mechanisms, limitations and feasibility. II.

rs =

p

(x − xs )2 + y2 + z 2

(2)

The shape function fs causes the curvature of spacetime inside the warp drive. Alcubierre chose fs to be a top hat function as seen in formula (3). [4] [5]

fs =

tanh(σ(rs + R)) − tanh(σ(rs − R)) 2tanh(σR)

(3)

This is called the Alcubierre Shaping Function (ASF) in which R relates to the radius of the warp drive bubble and the parameter σ relates to the thickness of the warp drive bubble. The function behaves like a step function which satisfies f (rs ) = 1 for rs < R and vanishes for rs > R.[6][7] These regions can be respectively defined as the interior and exterior region of the warp drive ’bubble’. This ASF top hat function is displayed in figure 1. MATLAB [8] was used to construct all plots and execute all computations.

THE ALCUBIERRE WARP DRIVE

In a warp drive the following equation for a line element must hold: [5] ds2 = −dt2 +

3 X (dxi − X i dt)2

(1)

t=1

Whereby X i = (X, Y, Z) are functions. Alcubierre obeyed this equality and chose X to be equal to vs f (rs ) and Y and Z to equal zero. The case is considered where the spaceship only traverses in the x−direction, thus Y and Z are set to zero. vs is the normal derivative of the position the spaceship xs to time t. The parameter rs is the radial coordinate centered around the ship. This is defined by:

FIG. 1. The ASF top hat function.

Substituting (3), (2), in (1) the Alcubierre metric can be constructed:

2

ds2 = −dt2 + (dx − vs f (rs )dt)2 + dy2 + dz 2

(4)

This spacetime metric will be investigated further. First one can note that spacetime in the exterior of the warp bubble is flat and given by the familiar Minkowski spacetime, because f (rs ) = 0. The Cauchy surface, dt = 0, is flat and thereby the 3-geometry too. If the Cauchy surface is flat all the curvature originates from the extrinsic curvature Kij given by: 1 Kij = (δi X j + δj X i ) 2

expanded. This however does not show if the spaceship moves over a time-like curve regardless of its speed. Therefore, in figure 4 the spacetime diagram of a spaceship is plotted. The light cones result from the Alcubierre metric by equating ds2 = 0.[6] dx = ±1 + vs (t)f (rs ) dt

(8)

In figure 3 the spaceship moves over a straight line with slope vs = 0.5, thus with speed equal to half the speed of light.

(5)

The contraction and expansion of the volume element θ is known as the trace of the extrinsic curve tensor and is expressed as: θ = Kii = δi X i

(6)

As previously stated Y = Z = 0 and X = vs f (rs ), thus using the chainrule and choosing xs = 0 the following equation is obtained. xs df (rs ) rs drs (7) xs σ tanh2 (σ(rs − R)) − tanh2 (σ(rs + R)) = vs tanh(σR) rs 2

θ = vs

Choosing the value of the parameters σ = 8 and R = 1 the expansion of the volume elements could be plotted, this can be seen in figure 2.

FIG. 3. The spacetime diagram of a spaceship with the future light cones resulting from the Alcubierre metric with vs = 0.5

As expected at every point along the spaceships world line lies in between the light cones. If we now take vs = 1.5 the following figure is obtained.

FIG. p 2. The volume expansion θ plotted against x and ρ = y 2 + z 2 for a single point on the spaceships world line. This results from the Alcubierre metric where σ = 8 and R = 1.

Figure 2, it can be seen that the volume expansion θ in front of the spaceship is negative, thus spacetime is contracted, while θ behind the spaceship is positive, thus

FIG. 4. The spacetime diagram of a spaceship with the future light cones resulting from the Alcubierre metric with vs = 1.5.

In the graph it is seen that the curve of the spaceship still

3 lies in the light cones for all points in spacetime, while its speed is greater than the speed of light. This is because the spacetime around the spaceship is warped by the ASF and the lightcones are tipped over. Thus, the observers inside the warp bubble can attain an arbitrary speed vs , which could be higher the speed of light, with respect to observers outside the bubble. The postulates of Special Relativity are not violated as travelers in the interior do not exceed the speed of light themselves but travel within their local light cone. It is the fact that spacetime gets warped, contracted and expanded that observers in the interior are able to cover larger distances in shorter time than light, implying vs > c, traveling superluminal. To prove the spaceship moves over a timelike curve we substitute x = xs (t) into the metric (4). This results in dτ = dt

(9)

This proves that regardless of vs the spaceship moves over a time-like trajectory. This also indicates that observers inside the warp bubble do not suffer from relativistic effects since proper time equals coordinate time.[4] To understand the position of the spaceship with respect to the warp drive more clearly one could have a look at figure 5.

within the disturbance of spacetime which travels at an arbitrary velocity. III.

NEGATIVE ENERGY DENSITY

The energy density necessary to support a warp drive follows from the Einstein equations. However, as proven by Nat´ario the energy density to sustain a warp drive is negative, meaning that it can’t be supported by classical matter. [5][6] The energy density ρ necessary for a warp drive is given by [11]: ρ = Tαβ nα nβ =

1 (3) ( R + (K ii )2 − Kij K ij ) 16π

(10)

Herein T 00 equals ρ and (3)R denotes the scalar curvature of the Cauchy surface. However, as mentioned earlier the Cauchy surface is a flat space with no curvature and results in (3)R = 0. Recognizing K ii as the stretch function θ equation (10) results in: T 00 =

1 (θ 2 − Kij K ij ) 16π

(11)

The energy density can actually be computed as only X is nonzero and θ = δx X. With the definition of kij equation (11) can be expressed in equation (5) and can be reduced to: T 00 =

1 [(δx X)2 − (δx X)2 16π 1 1 − 2 ( δy X)2 − 2 ( δz X)2 ] 2 2

(12)

The factor 2 in front of ( 12 δy X) in (12) is due to the fact that the contribution of Kxy K xy is equal to Kyx K yx . Idem for the fourth term. This can be further derived with X = vs f (rs ) to FIG. 5. A simplified top-down view of a spaceship traversing within the Alcubierre warp drive bubble. [9]

It should be noted that the spaceship travels between the ripples (contraction and expansion) in spacetime, and not in them. If this were the case humans could not survive a journey with such a device because of extreme gravitational effects. Looking at the simplified model of the warp drive, displayed in figure 5, one can compare the warp drive to an ordinary spring, considering the spaceship to be in the middle of a section of the spring.[10] When the spring in front of the spaceship is contracted and the section of spring behind the spaceship is expanded. This disturbance in the spring will travel through the whole spring causing the spaceship to travel along with this disturbance between the contracted and expanded sections of the spring.The spaceship never exceeds the speed of light in its own frame, it is stationary

T 00 =

−1 2 df (rs ) 2 (y2 + z 2 ) ) v ( rs2 32π s drs

(13)

This proves that the energy density needed to sustain for the Alcubierre warpdrive is negative. IV.

PROBLEMS WITH THE WARP DRIVE

Although the Alcubierre warp drive is a solution to the Einstein equations, it still brings some difficulties. As previously mentioned negative energy is necessary for the warp drive to work which violates energy conditions. However this could potentially be solved, as stated by Alcubierre, by the existence of Casimir effects as Casimir effects lead to stress tensors which violate the energy conditions as well.

4 Next to the fact that negative energy is needed, the amount of negative energy necessary is problematic as well. For the first design of the Alcubierre drive an energy equivalent of −1064 would be needed for a small interstellar journey [12], while the mass of the observable universe is about 1053 (and positive). This extraordinary amount of negative energy has been reduced (to a few milligrams) in later calculations by Chris van den Broeck [13] [14], Serguei Krasnikov [15], and Harold White [16] by for example changing the shape of the warp drive bubble from a sphere to a torus. Krasnikov also hypothesized if negative energy could not be realized the solution may lie in arranging masses along a predestined pathway that could be set in motion to create the warp drive field. [17] This way the superluminal velocities could still be achieved but only after a pathway is constructed which the vessel needs to follow, like a railroad and a train. Nat´ario stated that when inside the warp drive bubble it would be impossible to send signals to the front of the bubble. [5] This would be problematic as it gives the astronauts no option of stopping or steering the bubble, with therein the spaceship. A colleague of the Introduction to General Relativity course, Julia Thomas, pointed out that if Hawking radiation does in fact exist the faster than light travel of the space ripple would produce great amounts of it, destroying everything around it, including the spaceship inside the warp drive bubble. It has been hypothesized that after a superluminal trip the particles could potentially be dangerously blueshifted and released in enormous energetic gamma- and highenergy-particle rays upon deceleration, destroying everything in front of the warp drive bubble (including the destination). [18] Pfenning Everett stated that for a superluminal warp bubble capable of traveling at tenfold the speed of light has a maximal thickness of 10−32 m (which is close to the Planck length). [19] In the original calculation the amount of negative mass necessary for the warp drive to work was −1064 kg, and to fit that into a 10−32 m wall would be highly impractical. Alcubierre himself acknowledged that his warp drive

could potentially be used to travel back in time, because of the possibility to create closed timelike curves. However this is prohibited by Steven Hawkins chronology protection conjecture. [20] Therefore if the Alcubierre warp drive is used to travel back in time something will potentially go wrong, like an outburst of accumulated energy or the creation of a black hole.

[1] A. Einstein. Die Grundlage der allgemeinen Relativit¨ a tstheorie. Annalen der Physik, 354:769–822, 1916. [2] University of Puerto Rico at Arecibo. Habitable exoplanets catalog. 2015. [3] Fernando Loup. The analysis of harold white applied to the natario warp drive spacetime. from 10 times the mass of the universe to arbitrary low levels of negative energy density using a continuous natario shape function with power factors. warp drives with two warped regions. 2013. [4] Miguel Alcubierre. The warp drive: hyper-fast travel within general relativity. Classical and Quantum Gravity,

11(5):L73, 1994. [5] Jos´ e Nat´ a rio. Warp drive with zero expansion. Classical and Quantum Gravity, 19(6):1157, 2002. [6] James B Hartle. Gravity: An introduction to einsteins general relativity, 2003. [7] Gabriele U Varieschi and Zily Burstein. Conformal gravity and the alcubierre warp drive metric. ISRN Astronomy and Astrophysics, 2013, 2013. [8] MATLAB. version 7.10.0 (R2010a). The MathWorks Inc., Natick, Massachusetts, 2010. [9] Kuochengliao. The alcubierre warp-drive model, 2017. [10] A. DeBenedictis. Warp drive. 1994.

V.

CONCLUSION

The creation of a Alcubierre warp drive is theoretically possible as it obeys the Einstein equations. This means that spacecrafts inside the Alcubierre drive could traverse closed timelike world lines which cause it to travel at superluminal speeds according to observers on earth. In the frame of the spaceship Minkowski spacetime is present, making sure that locally the speed of light is not exceeded, thereby obeying fundamental physical laws. The warp drive works on the concept of contracting spacetime in front of the spaceship and expanding it behind said spaceship. The spaceship itself exists in flat space in between the two spacetime ripples. This disturbance in spacetime could travel at arbitrary speeds, taking the spaceship with it. The volume expansion of spacetime was computed and depicted. In this figure the contraction and expansion of spacetime becomes clear. The price that needs to be paid for the Alcubierre warp drive is negative energy. It was computed that the energy density needed for an Alcubierrre warp drive is a negative one. The amount of negative energy could range from −1064 kg to a few negative milligrams. This could potentially be solved by the existence of other energy condition defying quantum effects, such as the Casimir effect. Other problems are present as well like the degree of maneuverability whilst inside the bubble, extreme blueshifts, Hawking radiation, and the technology needed to produce said warp drive. With the Alcubierre warp drive it might be possible in the future to travel at superluminal speeds but there still is a long way to go to infinity and beyond.

5 [11] Robert M Wald. General relativity. University of Chicago press, 2010. [12] M Pfenning and L Ford. The unphysical nature of warp drive,” gr-qc, 9702026. 1997. [13] Chris Van Den Broeck. Awarp drive’with more reasonable total energy requirements. Classical and Quantum Gravity, 16(12):3973, 1999. [14] Chris Van Den Broeck. Alcubierres warp drive: Problems and prospects. In AIP Conference Proceedings, volume 504, pages 1105–1110. AIP, 2000. [15] Serguei Krasnikov. Quantum inequalities do not forbid spacetime shortcuts. Physical Review D, 67(10):104013,

2003. [16] Stephen Brown. Nasa scientist claims to be on the verge of faster than light warp drive. 2013. [17] SV Krasnikov. Hyperfast travel in general relativity. Physical review D, 57(8):4760, 1998. [18] Brendan McMonigal, Geraint F Lewis, and Philip OByrne. Alcubierre warp drive: On the matter of matter. Physical Review D, 85(6):064024, 2012. [19] Michael John Pfenning and LH Ford. Quantum inequality restrictions on negative energy densities in curved spacetimes. arXiv preprint gr-qc/9805037, 1998. [20] Allen E Everett. Warp drive and causality. Physical Review D, 53(12):7365, 1996....