Black Holes Wormholes & the Development ofJBIS, a Dynamic Vol. 58, Warp pp.xxx-xxx, Drive Metric 2005
Black Holes Wormholes & The Development of a Dynamic Warp Drive Metric
KELVIN F. LONG 20 Fulmer Road, Beckton, London, E16 3TF, UK.
Using The General Theory of Relativity the properties of spacetime metrics are introduced in the context of Black Holes and Wormholes. Applying the same mathematical tools the development of a dynamic warp drive metric is then described as a possible solution to the interstellar distance barrier. The large negative energy density calculated from the stress-energy tensor required for a reasonable warp bubble radius is also dis- cussed. By demonstrating some of the historical and successful applications of metrics to astrophysics, this paper is intended to provide an introduction to the theory behind the warp drive and stimulate increased academic consideration of the subject.
Keywords: Black holes, wormholes, warp drive, general relativity, metric
1. Background
Since the publication of The General Theory of Relativ- ity nearly a century ago our understanding of gravity ρ 0 0 0 and knowledge of astrophysics has undergone a revo- 0 px 0 0 Tµν = [1.2] lution. In particular, the existence of black holes, an 0 0 p 0 y object once considered not worthy of serious consid- 0 0 0 pz eration for the end point of stellar collapse has now become a core area of astrophysical research. Gen- The terms on the left hand side of Eq.(1.1) de- eral Relativity defines the geometrical warpage of scribe the geometry of spacetime in the presence of spacetime within the vicinity of a matter distribution gravity. For empty space Gµν = 0. The Ricci curvature employing coordinate independent tensor equations - tensor Rµν is found from the Riemann curvature ten- a mathematical language capable of handling multiple α sor Rδµν using the process of contraction. The sca- dimensions with ease. General Relativity supplies us lar curvature R is obtained by further contraction of with the differential Einstein field equation for deter- the Ricci tensor mining the metric gµν contained within the Einstein ten- µν sor Gµν from a given stress-energy tensor Tµν - which is R = g Rµν (1.3) the source of gravity. Thus, the spacetime geometry can be obtained once the energy or matter distribution In spacetime straight lines are curves on geodes- is specified, or put another way the Einstein field equa- ics. The curvature caused by the presence of gravity tion specifies how the presence of matter curves manifests itself in the deviation of one geodesic from spacetime. Einstein used the description ‘gravity is a another and two particles or bodies that may initially manifestation of spacetime curvature’. John Wheeler sums be moving through spacetime on parallel and neigh- up general relativity uniquely as “space tells matter how bouring geodesics will be forced apart or closer to- to move, and matter tells space how to curve” [1]. The gether by spacetime curvature. A geodesic can be Einstein field equation was published in 1915 and has defined as a curve that parallel transports its tan- the following form gent vector along itself.
1 8πG The birth of relativity has seen the removal of the Gµν = Rµν − gµν R = − Tµν (1.1) 2 c4 old Newtonian idea of space and time being sepa- rate entities and a union of the two - ‘spacetime’ has Where the components of the stress-energy ten- been affirmed. The proper time τ and proper dis- sor contain the energy density ρ and principal pres- tance s between events (coordinates) are invariant sures px, py, pz. for all observers and can be written as a line element
1 Kelvin F. Long
× 30 or metric. This is written depending on whether the solar masses (1Ms ~ 1.989 10 kg) it is not theoreti- time interval is dominant (timelike) τ2 = t2 - x2 or the cally possible for the star to end its life in a cold distance interval is dominant (spacelike) s2 = x2 - t2. degeneracy-supported state, that is where the star We note that for light x = t = 0 (nulllike). The metric is supported by the pressure of an ideal gas of de- can also be expressed in differential tensor form generate electrons (i.e White Dwarf with M<1.4Ms) or where the indices µ,ν goes from 0 to 3. an ideal gas of degenerate neutrons (i.e neutron star
with M<3Ms). Instead, a more massive body will un- 33 2 = µ ν dergo complete collapse to form a black hole, where ds ∑∑gµν dx dx (1.4) µν the escape velocity exceeds even the speed of light – hence the name. To describe the events on a plane Where the Einstein summation convention is im- that passes through the centre of a spherically sym- plied by repeated index metric (non-spinning) centre of gravitational attrac- tion one defines the Schwarzschild metric in polar 2 µ ν ds = gµν dx dx (1.5) coordinate form
Re-writing this in matrix form − r r 1 ds2 = −1− g dt 2 + 1− g 0 g g g g dx r r 00 01 02 03 (2.1) 1 2 2 2 2 2 g10 g11 g12 g13 dx dr + r (dθ + sin θdφ ) ds2 = [dx0dx1dx2dx3 ] × g g g g 2 20 21 22 23 dx So that the non-zero components of the metric g30 g31 g32 g33 dx3 are
(1.6) −1 r r = − − g = − g g00 1 , g11 1 In a four dimensional spacetime (t, x, y, z), the 16 r r component metric is reduced to a symmetric ten = 2 = 2 2 θ g22 r , g33 r sin component metric gµν of signature [-1,+1,+1,+1] or The radius from which even light cannot escape is −1 0 0 0 known as the gravitational radius r and for a mass m g 0 +1 0 0 is given by gµν = (1.7) 0 0 +1 0 = 2Gm 0 0 0 +1 rg (2.2) c 2
Solving for the components of the spacetime met- Where the gravitational constant ric G = 6.673x10-11 m3kg-1s2
2 = − 2 + 2 + 2 + 2 8 -1 ds g00dx0 g11dx1 g22dx2 g33dx3 and the speed of light c = 2.9979x10 ms . Alternatively Eq.(2.2) may be written in geometrodynamic units This specifies the spacetime interval between ≡ ≡ with G c 1 so that rg = 2m. A consequence of events by use of a flat (no curvature present) Eq.(2.1) is that from the perspective of an external Minkowski spacetime. Written in Cartesian coordi- → observer, as the stellar surface r rg, an event hori- nates this is zon is formed such that no events internal to the horizon can be observed from outside, and this in- 2 = − 2 2 + 2 + 2 + 2 (1.8) ds c dt dx dy dz cludes preventing an observer actually seeing the star fall within the event horizon. This gives an ap- In special relativity although the concepts of length pearance of the star being frozen at this radius. Once and time do not remain invariant for different observ- the star has fallen within its gravitational radius t → 0 ers, the quantity ds2 is invariant for all observers. The and r → ∞ thus collapsing to a point of infinite density metric can also be re-written in a more useful polar and forming the black hole singularity. coordinate form as
ds2 = −dt 2 + dr 2 + r 2 (dθ 2 + sin2 θdφ 2 ) (1.9) In 1935 Albert Einstein and Nathan Rosen pub- lished a paper [2] which demonstrated one of the 2. Black Holes & Wormholes first mathematical descriptions of a wormhole by performing a coordinate transformation (letting r = For a star with a mass greater than around three u2+2m) on Eq.(2.1) that removed the region contain-
2 Black Holes Wormholes & the Development of a Dynamic Warp Drive Metric ing the curvature singularity. The solution is a math- demonstrated in this paper that these powerful tools ematical representation of physical space by a space can be applied to another problem which is as much of two asymptotically flat sheets (∞, -∞) connected to do with practical ‘future’ engineering as it is a by a bridge (at u = 0) or Schwarzschild wormhole problem of physics - namely that of interstellar travel. with a ‘throat’ at A(u=0)=16pm2. The Schwarzschild wormhole is described by the metric It is clear that with current solid or liquid chemical rocket fuels (i.e Nitrocellulose or Nitroglycerine ho- u 2 mogeneous mixtures and Hydrogen or Kerosene + ds2 = − dt 2 + 4(u 2 + 2m)du2 + (u 2 + 2m)2 dΩ2 u 2 + 2m Oxygen) [8] travel to the nearest stars will take thou- (2.3) sands of years. Alternative proposals are being con- sidered and in particular, NASA’s Breakthrough Pro- Where dΩ2 = dθ2 + sin2 θdφ2 and dr2=4u2du2. The next pulsion Physics Program [9] is looking at (i) propul- steps in the development of a theoretical wormhole sion that requires no propellant mass (ii) propulsion led to further research notably by scientists such as that attains the maximum transit speeds physically John Wheeler in the 1950’s [3] and later by Morris possible (iii) breakthrough methods of energy pro- and Thorne in the 1980’s [4] who constructed a met- duction to power such devices. However, the current ric to describe a spherically symmetric and static and most realistic theoretical proposals attain speeds wormhole with a proper circumference 2πr with the that are only a fraction of the speed of light mainly coordinate r decreasing from -∞ to a minimum value using Newtons action-reaction principle by mass ejection. Some examples are microwave sail (~0.02c), at ro where the throat is located and then increasing ∞ nuclear fission (~0.15c), matter-antimatter (~0.2c) from ro to + . This solution has the distinctive feature of being horizon-less and is described by the metric and nuclear fusion (~0.3c) [10]. The star Epsilon Eridani is an orange K type star located at a distance 2 D = 10.8ly away (or 3.31parsecs) in the constellation 2 2φ (r) 2 dr 2 2 ds = −e dt + + r dΩ (2.4) 1− b(r) / r Eridanus. One can calculate that a vehicle travelling at a fraction p of light speed the return journey time The terms φ(r) and b(r) describes the shape of the between Epsilon Eridani and Earth is given by wormhole and are known as the ‘redshift function’ and ‘shape function’ respectively. × 15 = 2D(ly) 9.4605 10 m Tret (years) p ly One of the hurdles towards achieving a travers- able wormhole is the requirement of negative energy c year 8 −1 3 to hold open the wormhole throat. This requires nega- 2.9979×10 ms 31536×10 s tive mass, so many scientists dismiss wormholes as an unreal possibility, although a considerable body This results in 1080years, 144years, 108years and of research has been conducted in recent years [5]. 72years for microwave sail, nuclear fission, matter- Decades after the first wormhole solutions were dem- antimatter and nuclear fusion respectively. Although onstrated there is now a large body of scientists these estimates demonstrate promise, the actual seriously considering the physics problems that technology is far from reaching fruition. Eventually, wormholes present. Astronomers have even consid- humankind may master these breakthrough propul- ered the possibility that if natural wormholes existed sion concepts but they still require journey times then the presence of negative energy could produce longer than the average human lifespan. The need a unique signature corresponding to chromaticity for an alternative is therefore evident. effects from microlensing events [6]. At the very least they are considered by physicists to be an interest- The real dream is the development of a propulsion ing insight into what the laws of physics will allow system that is capable of traversing the galaxy at within the limits of our current understanding. speeds that would be practical (~few years travel Wheeler has expressed an optimistic view of worm- time) to an expansionist human civilisation undergo- holes “If relativity is correct, and if it allows for worm- ing significant exponential population growth (~1.3% holes, then somewhere, somehow, wormholes must exist per year as of 2004 and expected to rise to 43billion - or so I want to believe.” [7]. by 2150) [11]. With the realism that the presence of a matter distribution results in the warpage of 3. The Warp Drive spacetime, which then manifests itself as gravity, the suggestion has arose that it may be possible to With the above historical developments in theoreti- warp spacetime in a way that effectively allows a cal physics derived from general relativity it shall be vehicle to move across enormous stellar distances
3 Kelvin F. Long in a short transit time. This essentially implies for
Eq.(1.1) that a metric gµν is first specified correspond- ing to the specific spacetime warpage and the com- ponents of the stress-energy tensor Tµν are then com- puted. This concept is elegantly expressed by Pekkov [12] “Since the shape of a free body’s worldline is deter- mined by the geometry of spacetime a local change of spacetime geometry will affect a body’s worldline i.e a body’s state of motion”. The remainder of this paper will examine some of the novel developments of this astounding proposal since the beginning of warp σ field theory as an academic science around ten years Fig. 1 warp drive ‘top hat function for the case when = 8 and R = v = 1. ago. s
→ ∞ 3.1 Alcubierre Warp Drive Metric Minkowski spacetime is recovered as rs . The term s is a wall thickness free parameter. A vehicle will be
In 1994 Miguel Alcubierre [13] designed a metric located at rs = 0. From the line element Eq.(3.1) the that resembled a spherical warp bubble of radius R covariant components of the metric are given by in spacetime. The idea is that any vehicle would be kept inside the warp bubble sitting at rest with re- f (r ) 2 v 2 −1 − f (r )v 0 0 s s s s spect to the interior, whilst the local spacetime in − + = f (rs )vs 1 0 0 front of the bubble would contract and the spacetime g µν (3.4) 0 0 +1 0 behind the warp bubble would expand by an equal + amount - a so called bipolar distortion. The warp 0 0 0 1 bubble moves forward with a speed determined by the contraction/expansion rate. This allows the warp The contravariant components of the metric are bubble interior (and the vehicle) to be moved through found by first calculating the determi- = spacetime faster than the speed of light at a velocity nant det g µν g µν of Eq.(3.4). A matrix is then formed consisting of the cofactors of each compo- vs(t) with respect to the distant universe (as seen by a distant observer). That is to say, within the warp field nent. Transposing the matrix to obtain the adjoint the speed of light limit has been raised so that locally and then dividing by the determinant to find the in- the vehicle does not violate the central tenet of spe- verse of Eq.(3.4) gives the contravariant components µν cial relativity. However, the speed of the vehicle may g of the metric. Solving the left hand side of Eq.(1.1) be considerably higher than the nominal value of c one can then compute the Einstein components of outside of the warp field. The Alcubierre metric is the Alcubierre warp drive metric. i.e
2 = − 2 + − 2 + 2 + 2 2 2 ds dt [dx vs (t) f (rs (t))dt] dy dz 1 ∂f (r ) ∂f (r ) G = − v 2 s + s 00 s ∂ ∂ (3.5) (3.1) 4 x y
The radius r (t) is a measure of the distance from s 3.2 Wall Thickness the centre of the warp bubble In a further development of the Alcubierre metric = − 2 + 2 + 2 rs (t) (x xs (t)) y z (3.2) Laurence Ford et al [14] included a warp bubble wall thickness term ∆ to relate to the free parameter σ in Where x (t) = the trajectory of the bubble centre, s the Alcubierre original form of the shape function v (t)=bubble velocity. The term f(r ) is a geometrical s s Eq.(3.3). The Alcubierre shape function is modified step type shape function that has a ‘top hat’ appear- by use of a piece-wise continuous function ance as shown in fig. 1 and is given by ∆ < − = σ + − σ − rs R f p.c (rs ) 1 = tanh[ (rs R)] tanh[ (rs R)] 2 f (rs ) (3.3) 2 tanh[σR] ∆ ∆ 1 ∆ R − < r < R + f (r ) = − (r − R − ) 2 s 2 p.c s ∆ s 2 The requirement on the type of shape function to ∆ > + = describe the bubble is that it is equal to unity inside rs R f p.c (rs ) 0 ≤ 2 the warp bubble (f(rs)=1 at rs R) and goes to zero → → outside the warp bubble (f(rs) 0 as rs R) so that flat (3.6)
4 Black Holes Wormholes & the Development of a Dynamic Warp Drive Metric
Then computing the piece wise continuous function tive energy densities is called ‘exotic’ and is not derivative as well as the derivative of the shape func- available within the known universe. Also, negative tion Eq.(3.3), both of which are equal at rs=R so that energy requires negative mass and as implied by Eq.(4.1) classical physics forbids this, although quan- df (r ) df (rs ) = p.c s tum theory does allow for the existence of negative drs drs energy densities. A similar difficulty exists for worm- holes, in that they also require negative energy den- df (r ) p.c s = − 1 sities to hold open the traversable throat. As noted ∆ drs by Ford et al [16] negative energy should not be con- fused with antimatter (which has positive energy) or df (r ) σ sec h(2σR) s = with the energy usually associated with the σ drs 2 tanh( R) cosmological constant which although has a nega- tive pressure, also has positive energy. Then using trigonometric identities and re-arrang- ing one obtains The quantum inequality for a free massless scalar field in four dimensional Minkowski spacetime is given [1+ tanh 2 (σR)]2 ∆ = (3.7) by Ford et al [15] to be 2σ tanh(σR) ∞ µ ν σ σ → ∆ τ < Tµν n n > 3 Note that if R is large then tanh( R) 1 and ~2/s. o ∫ dτ ≥ − π τ 2 +τ 2 π 2τ 4 (4.3) The use of the piece wise continuous function Eq.(3.6) −∞ 0 32 0 ∆ means that inside the warp bubble where rs
µ ν τ Tµν n n ≥ 0 (4.1) If 0 is small compared to the time scale over which the bubble velocity changes then it is appro- The major difficulty with the Alcubierre warp drive priate to consider the warp bubble velocity to be metric as defined in Eq.(3.1) is that it violates several constant vs(t)~vb during the sampling time. In Eq.(4.4) ∆ of these energy conditions. Using Eq.(3.2) for rs(t) vb is a multiple of the speed of light and is a multiple and substituting ρ2=y2+z2 as the radial distance ρ per- of the Planck length. It is common practice within pendicular to the x-axis, then using the Einstein ten- curved space quantum field theory to write all lengths sor G00 component Eq.(3.5) of the metric, in and masses in multiples of the Planck length. The geometrodynamic notation all observers will see an use of h = G = c = 1 units implies that 1Lp corresponds energy density given by Eq.(1.1) of the form to a mass of 2.18 x 10-5 g. Calculating
2 2 2 2 ∆ ≤ 75v L or ∆ ≤ 10 v L (4.5) µν 1 1 v (t)ρ df (r ) b p b p T n n = T 00 = G00 = − s s µ ν π π 2 8 8 4rs (t) drs Where Lp has been explicitly introduced in the (4.2) equation so that ∆ has the units of meters and ig- nored the constants in Eq.(4.4) to get an order of This energy density violates the WEC because it is magnitude estimate as is usually quoted in the litera- negative everywhere. Matter that gives rise to nega- ture. Eq.(4.5) demonstrates that the wall thickness
5 Kelvin F. Long cannot be much more than the Planck length (∆ ~102 with vb=1). The exception to this is in the limit of a very -35 8 -1 large velocity. (i.e. If Lp ~10 m and vb= c ~10 ms then ∆ ~10-25 which is much larger than the Planck length). By taking the integral of the local matter energy density Eq.(4.2) over the proper volume and using the piece wise continuous shape function Eq.(3.6) an expression for the amount of negative energy required to form the warp bubble is derived
∞∆2 + 1 df (r) 1 R / 2 −1 E = − v2 ∫∫r 2 dr = − v2 r 2 dr 12 b dr 12 b ∆ Fig. 2 Negative energy (log) required to form warp bubble of 0 R−∆ / 2 given radius.
2 1 −1 (R + ∆ / 2)3 (R − ∆ / 2)3 = − 2 − 52 46 35 E vb ~10 joules, ~10 joules and ~10 joules respectively. 12 ∆ 3 3 To put these energy requirements into perspective fur- ther, if one had a negative mass equivalent to the mass Using the trigonometric identity for (R±∆/2)3 33 of the Earth (ME ~10 grams) this would only be suffi- cient to form a warp bubble radius of ~10-18m. This is 2 ∆ = − 1 2 R + one thousandth the Compton wavelength of the proton E vb (4.6) 12 ∆ 12 λ -15 ( c,p 1..3214x10 m). Clearly, with our current under- standing unless we are dealing with dimensions smaller Ignoring the right hand term in the bracket than the atomic scale, the warp drive appears to re- quire ‘physically unobtainable energies’. 1 v2R2 E ≈ − b (4.7) 12 ∆ 5. Van Den Broeck (VDB) Metric
Note that for the warp bubble energy requirements Van Den Broeck [17] applied a trivial modification to ∝ ν 2 2 ∆-1 E b , R , . Multiplying the top and bottom of the the Alcubierre geometry by keeping the surface area right hand side of Eq.(4.7) by L and using Eq.(4.5) p of the warp bubble microscopically small while si- multaneously expanding the spatial volume inside 1 E ≈ − v L R2 = −3.19×1066 v L R2 the bubble by a factor 1+α. × 2 2 b p b p 12 10 Lp 2 = − 2 + 2 − 2 + 2 + 2 (4.8) ds dt B (rs )[(dx vs f (rs )dt) dy dz ] (5.1) In conventional mgs units the mass m in grams required for a given radius R of warp bubble is then The function f(rs) has the following properties with ~ ~ easily calculated from the relation R > R + ∆
−5 < = 2.18×10 g rs R f (rs ) 1 m = −3.19×1066 v L R2 b p (4.9) R ≤ r < R + ∆ 0 < f (r ) ≤ 1 1Lp s s (5.2) + ∆ ≤ = R rs f (rs ) 0 Similarly, the required energy in joules is calcu- lated and shown in figure 2 as a log plot. The structure of the VDB warp bubble can be described as “a small Alcubierre bubble (with thick- − 2 ∆ 2.18×10 5 g 1kg 1 ness located at radius R from the centre) surrounds E = −3.19×1066 v L R2 ~ 3 b p a neck leading to a ‘pocket’ (with thickness ∆ located 1Lp 10 g 1c ~ at radius R from the centre) with a large internal vol- (4.10) ume, with a flat region in the middle.” Following on from the convention used by Van Den Broeck the regions Thus, a 100m warp bubble would require a negative are labelled IV, III, II and I respectively where regions mass of 6.95x1065 grams or a negative energy of I and III are flat. 6.95x1062 joules. Similarly, the negative energy require- ments for a warp bubble radius of 1m, 1cm 1mm, The restriction that all lengths should be larger 1micron (10-6m) and 1 Compton electron wavelength than the Planck length L is applied and the values ~ ~ p λ -12 58 54 α 17 ∆ -15 -15 -15 ( c,e = 2.4262x10 m) are ~10 joules, ~10 joules, = 10 , = 10 m, R = 10 m and R = 3 x 10 m
6 Black Holes Wormholes & the Development of a Dynamic Warp Drive Metric
(radius to outer surface) are assumed. Then using (superluminal) and the metric component g00 = 0. the same approach as detailed by Ford et al [14] and Because of this horizon, events inside the warp bub- Eq.(4.6) the negative energy is calculated for an in- ble will not be able to causally influence events on ner pocket radius of 100m the other side of the surface (i.e information cannot be sent from inside the warp bubble to the region at ≈ − × 29 EIV 6.3 10 j the front). This implies that the ship is not able to control the warp bubble itself. Again, Hoiland re- E ≈ −1.4×1030 j II sponds that the ship would be causally connected to This energy requirement for the VDB metric (which part of the warp bubble matter shell controlling the corresponds to ~1033g negative mass) is still ex- speed, which is totally contained inside the horizons. tremely large but does demonstrate a considerable If the ship were to drop out of warp then this region ()≈ would cause a further tachyonic (v > c) region out- reduction EAL 2EVDB compared to the estimate obtained from a purely Alcubierre type metric. side of the event horizon to disconnect.
6. Other Warp Drive Metric Work 6.2 The Krasnikov Tube
6.1 Natario Metric An obvious but interesting observation made by Coule [21] is that in order to construct a warp drive like In January 2002 Jose Natario published a paper [18] spacetime you must first select a mattter source to that claimed the expansion and contraction of support this geometry - in other words, how do you spacetime associated with the Alcubierre warp drive manipulate spacetime. Naturally, to manipulate planet was purely a marginal consequence of the choice of or star size masses is far beyond our current techno- the metric type used with the shift vector taken to lie logical expertise. Although ideas do exist [22] as to in the direction of motion. He argued that it is possi- how this could be achieved, such as by using so ble by choosing a vector field where the shift vector called ‘electro-gravi-magnetics’ theory, which relates is divergence free (∇ ⋅ β = ) and where any compres- 0 to modifying the vacuum polarizability by applying sion in the radial direction is exactly balanced by an electromagnetic fields (although on a much smaller expansion in the perpendicular direction, to construct scale). In order to construct a warp drive one would a spacetime with zero expansion that effectively just need to distribute a matter field ahead of the vehicle slides through spacetime in order to create the geometric structure one re- quires - a so called ‘need one to make one’ paradox. 2 = − 2 + − β − ds dt [d x (x, y, z zo (t))dt] To address this and to circumvent the problem of the (6.1) − β − ship being causually disconnected from the [d x (x, y, z zo (t))dt] Alcubierre spacetime Krasnikov [23] has proposed However, the Alcubierre metric in the form given that any modifications to the spacetime for by Eq.(3.1) was not intended to be a realistic warp superluminal speeds should necessarily occur in the field, but merely to demonstrate the application of causal future of the ship. That is, a ‘tube’ that has flat general relativity to the warp drive proposal. Natario spacetime within but with opened out light cones, also comments “light rays emitted in directions approach- must be laid along the ships path connecting both ing the horizon will become increasingly blue shifted, the departure point and arrival point of the journey reaching infinite blue shift at the horizon”. along the entire trajectory of the bubble. Such a tube of spacetime is described by a metric of the form In response Hoiland [19, 20] argues that if you properly take into account the effects the expanded ds2 = −(dt − dx)(dt + kdx) + dρ 2 + ρ 2dφ 2 (6.2) Lorentz invariant frame (warp bubble) has upon the incoming photons and their lightcones then it is seen Where k(r) is an increasing function of 2 2 that they are not infinitely blue shifted (and therefore ρ = y + z . Although the Krasnikov tube is cer- as hot) as Natario has claimed. Hoiland also argues tainly an intriguing idea, like warp drive and worm- that when applying a proper Cauchy (equal time sur- holes calculations show that the total negative energy face) evolution it is seen that the Newtonian frame in density is also unphysically large (~1061g for a 1 m the static inner region of the warp bubble would long by 1 m wide tube) and the maintenance of the provide a boundary which acts as a shield to any tube will require wall thicknesses of order a few hyper-accelerated incoming photons. Planck lengths. It was described in section 5 how the Van Den Broeck metric significantly reduced the warp Natario also points out the presence of horizons bubble energy requirements by the inclusion of an when the ship is going faster than light speed internal bubble to the Alcubierre spacetime. With the
7 Kelvin F. Long same intention, Gravel et al [24] have introduced an ble ever be created in a ‘laboratory environment’. Po- inner tube within the original Krasnikov tube de- tentially, using physics instrumentation and technical scribed by the metric diagnostics to explore spacetime warpage, this could lead to the generation of a new and distinct field of 2 2 2 2 ds = −(dt − dx)(dt + kdx) + B (dρ + ρdφ ) (6.3) physics. Followed by the formation of teams of scien- tific expertise familiar with the laws of general relativ- Where B is an internal expansion function. For a ity, quantum field theory and particle physics assem- similar tube of length 1m the negative energy re- bled from academic and government institutions. Ex- quirement is reduced requiring ~1030g of negative perimental results could then be scaled up to gauge mass ()E ≈ 2E . KRAS GRAV likely requirements for macroscopic sized warp bub- bles. Admittedly, this is a fantastic possibility, but a 6.3 Low Speed Limitations sufficiently inspiring vision to continue pushing the boundaries of warp drive research. However, one ac- By formulating the Alcubierre and Natario warp drive knowledges a recent comment by Visser [25] to “sound spacetimes Visser et al [25] has applied linearized a strong cautionary note against over-enthusiastic misinter- (accurate to first order in v ) gravity to the weak field s pretation of the technological situation.” warp drive taking the bubble velocity to be non-rela- tivistic (v << c). When a weak gravitational field is 8. Conclusions present then spacetime is closely approximated by flat Minkowski spacetime of special relativity and By using the same mathematical tools to describe the metric has the form the physical properties of black holes and worm-
g µν ≡ ηµν + hµν (6.4) holes, the creation of a dynamic warp drive metric has been advanced in recent years. The large nega-
Where hµn << 1. In the previous sections and analysis tive energy requirement and Planck length sized wall of warp bubble metrics, the vehicle was treated as a thickness constraint seem to be two of the biggest point particle and massless moving along a geodesic in physics obstacles towards the realisation of inter- spacetime. In the linearized theory the vehicle is treated stellar travel. Another challenge is the enormous task as a finite mass within the warp bubble. Using this of practically creating a warped spacetime struc- linearized theory a model was constructed of the warp ture via a matter distribution, currently far beyond drive spacetime in which the bubble interacts with the our technical knowledge. The dream of achieving a finite mass vehicle. It was found that the total amount fully operational spacecraft that employs a propul- of energy condition violating matter must be a certain sion system based upon the warp drive principle is fraction of the positive mass of the vehicle contained truly daunting. Although the theoretical studies dis- within the warp bubble. The authors demonstrated that cussed here demonstrates the potential, there is cur- non-perturbative exact solutions of the warp drive rently little evidence to suggest that such a system is spacetimes violate the classical energy conditions not practically possible. However, the warp drive metric just at superluminal speeds but also at low warp bubble has undergone significant developments in the last speeds. If the volume integral is positive by the WEC ten years since the seminal paper by Alcubierre and then a low speed limitation on the Alculbierre warp there is every reason to be optimistic about improve- drive Eq.(3.1) results ments for the future. Perhaps a deeper understand- ing of the physics constraints and any potential solu- 2 2σ ≤ v R M ship (6.5) tions will soon arise as long as scientist continue to apply their mathematical insights into this most in- Where σ ~1/∆. This limitation also applies to the teresting a possibility. Since we do not yet have a full Natario metric Eq.(6.1). The constraint that the net understanding of the quantum nature of gravity or a total energy stored in the warp field is less than the unified theory of physics, the complete fallibility of total mass-energy of the vehicle places a restriction the warp drive cannot be ruled out at this time. on the velocity of the warp bubble implying that the bubble velocity will be very low. In this paper it has been shown how using general ∆ relativity black holes and wormholes have become 2 M ship v ≤ (6.6) part of ‘accepted’ mainstream scientific research. It R2 is hoped that warp drive research will cease to become 7. Laboratory Scaling Experiments just an interesting ‘gedanken’ experiment within the next century and will be taken seriously by both quantum field As a final speculative note, it is interesting to consider theorists and general relativists as a plausible application what would happen should an atomic sized warp bub- to interstellar travel. In the interests of brevity some
8 Black Holes Wormholes & the Development of a Dynamic Warp Drive Metric warp drive research has been omitted in this paper, Michael Pfenning and Lawrence Ford for responding although the work discussed herein does represent to questions relating to warp field theory. The author the initial and important developments within the field is grateful for the assistance of Graham Hill, Alan as required reading. Dawes and David Johnson in comprehending some of the mathematics. Gemma Perrin and Jonathan 9. Acknowledgements Brooks are thanked for their continued support. This paper is dedicated to Max. Any errors are the au- The author would like to thank Edward Halerewicz, thors responsibility.
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(Received 6 December 2004; 26 January 2005)
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