Charged dust solutions for the warp drive spacetime
Osvaldo L. Santos-Pereira, Everton M. C. Abreu, Marcelo B. Ribeiro
aa r X i v : . [ g r- q c ] F e b Charged dust solutions for the warp drive spacetime
Osvaldo L. Santos-Pereira, ∗ Everton M. C. Abreu,
2, 3, 4, † and Marcelo B. Ribeiro
1, 4, 5, ‡ Physics Institute, Universidade Federal do Rio de Janeiro, Brazil Department of Physics, Universidade Federal Rural do Rio de Janeiro, Serop´edica, Brazil Department of Physics, Universidade Federal de Juiz de Fora, Brazil Applied Physics Graduate Program, Physics Institute,Universidade Federal do Rio de Janeiro, Brazil Valongo Observatory, Universidade Federal do Rio de Janeiro, Brazil (Dated: February 11, 2021)
Abstract
The Alcubierre warp drive metric is a spacetime construction where a massive particle locatedinside a spacetime distortion, called warp bubble, travels at velocities arbitrarily higher than thevelocity of light. This theoretically constructed spacetime geometry is a consequence of generalrelativity where global superluminal velocities, also known as warp speeds, are possible, whereaslocal speeds are limited to subluminal ones as required by special relativity. In this work weanalyze the solutions of the Einstein equations having charged dust energy-momentum tensor assource for warp velocities. The Einstein equations with the cosmological constant are writtenand all solutions having energy-momentum tensor components for electromagnetic fields generatedby charged dust are presented, as well as the respective energy conditions. The results show aninterplay between the energy conditions and the electromagnetic field such that in some cases theformer can be satisfied by both positive and negative matter density. In other cases the dominantand null energy conditions are violated. A result connecting the electric energy density with thecosmological constant is also presented, as well as the effects of the electromagnetic field on thebubble dynamics.
PACS numbers: 04.20.Gz; 04.62.+v; 04.90.+eKeywords: warp drive, charged dust, electromagnetic tensor, curved spacetime ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION It has been known for some time that general relativity allows for particles to travel glob-ally with superluminal velocities because special relativity limits the particles velocities tosubluminal ones only locally. In order words, special relativity basically states that the lightspeed limit must be obeyed inside a local light cone. The warp drive metric proposed byAlcubierre [1] satisfies both conditions, since it advances a geometrical construction charac-terized by a spacetime distortion, called warp bubble , such that a particle inside this bubbletravels with superluminal velocity in global terms, whereas locally its speed remains sub-luminal. more specifically, locally the warp bubble guarantees that the particle’s velocityis kept below the light speed, whereas outside the local light cone, created by the bubbledistortion, the whole bubble structure travels with superluminal velocities, or warp speeds.
The warp bubble is created in such a way that the spacetime in front of it is contracted whilebehind the bubble the spacetime is expanded. In its original proposition and first studies itwas foresaw that the warp metric would violate the energy conditions, as well as supposedlyrequiring huge quantities of negative energy density.After Ref. [1] several papers tried to comprehend the main aspects of the warp drivemetric. For example, Ref. [2] discussed some quantum inequalities that should be validas a result of the
Alcubierre warp drive metric , concluding that large amounts of negativeenergy would be needed to convey particles with small masses across small distances at warpspeeds. Therefore, they figured that prohibitive huge quantities of negative energy densitywould be necessary to generate a warp bubble. Also dealing with quantum inequalities, Ref.[3] computed the limits required for the energy values and the bubble parameters necessaryfor the existence of the warp drive. The conclusion reached by this author was that theenergy needed for a warp bubble is ten orders of magnitude greater than the total mass ofall the visible universe, also negative.Looking at the same problem of the physics of superluminal propulsion systems for in-terstellar travel, but from a different viewpoint, Krasnikov [4] analyzed the scenario of amassive particle making a round trip between two distant points in space at speeds fasterthan a photon. He questioned that this is not viable when reasonable conditions for globalhyperbolic spacetimes are made. He analyzed in details some peculiar spacetime topologies,supposing that, for some of them, they need tachyons for superluminal trips to occur. He alsoassumed the need for a possible particular spacetime prearranged with some devices alongthe travel path, which would be set up and activated as needed in order to the superluminaltravel to occur without tachyons. Such spacetime constructions was called as the
Krasnikovtube by Ref. [5].The metric proposed by Krasnikov was further generalized by Everett and Roman [5] byconceiving a tube along the path of the particle connecting Earth to a distant star. Insidethis tube the spacetime is flat, however the lightcones are opened out in such a way as toallow the superluminal travel in one direction. One of the issues analyzed in Ref. [5] is thatsince the Krasnikov tube does not involve closed timelike curves we are able to construct atwo way non-overlapping tube system such that it would work as a time machine. It was alsodemonstrated in Ref. [5] that a great quantity of negative energy density is necessary for theKrasnikov tube to function. These authors also used the generalized Krasnikov tube metric2o compute an energy-momentum tensor (EMT) which would be positive in some particularregions.Further studies concerning the metric proposed by Everett and Roman [5] were carried outby Lobo and Crawford [6, 7]. They investigated in detail the metric and the respective EMTobtained from it and if it were possible for a superluminal travel to exist without violatingthe weak energy condition. Quantum inequalities used in Ref. [5] were also explored.Van de Broeck [8] demonstrated that a minor modification of the Alcubierre geometrywould reduce the total energy necessary for the creation of the warp bubble. By introducinga modification of the original warp drive metric the total negative mass-energy necessaryto describe the spacetime distortion capable of warp speeds would be of the order of somesolar masses. Natario [9] questioned if both the expansion and contraction of the space ofthe bubble is a matter of choice. He suggested a new version of the warp drive metric withzero expansion. Lobo and Visser [10] discussed that for the Alcubierre warp drive, and itsversion proposed in Ref. [9], the center of the bubble must be massless. They presenteda linearized theory for both concepts and found out that even for low speeds the negativeenergy stored in the warp fields must be just a relevant part of the mass of the particle atthe center of the warp bubble. White [11, 12] depicted how a warp field interferometer couldbe implemented at the Advanced Propulsion Physics Laboratory. Lee and Cleaver [13, 14]looked at how external radiation might affect the Alcubierre warp bubble, possibly making itenergetically unsustainable, and how a proposed warp field interferometer could not detectspacetime distortions. Mattingly et al. [15] discussed curvature invariants in the Natariowarp drive.One aspect that is often overlooked regarding the Alcubierre warp drive metric is that theoriginal proposal did not come from solving the Einstein field equations (Alcubierre 2018,private communication), but as a geometrical construction theoretically capable of generatingwarp speeds. Indeed, the original proposal of the warp drive metric was not accompaniedby a dynamical equation, which is the case when a metric is reached from solutions of theEinstein equations.In a previous paper [16] we started from this realization and then endeavored to actuallysolve the Einstein equations using the simplest possible mass-energy distribution, incoherentmatter or dust, as a starter in order to verify if this distribution were actually capable ofcreating a superluminal warp field, that is, a warp bubble. Although the results went backto vacuum, the solutions have indeed generated a dynamical equation for the warp metricregulating function β (see below), which was found to obey in a particular case the Burgersequation for inviscid fluid with shockwaves in the form of plane waves [see also Ref. 17].In this work we intend to pursue a similar path, that is, to discuss solutions of the Einsteinequations having the Alcubierre warp drive geometry and considering a non null cosmologicalconstant. To do that we considered a charged dust capable of generating an electromagneticfield, wrote the EMT for both components, solved the equations and wrote the respectiveenergy conditions. The results showed an interplay between the energy conditions and theelectromagnetic field such that in some cases the former can be satisfied by both positiveand negative matter density. In other cases the dominant and null energy conditions areviolated. A result connecting the electric energy density with the cosmological constant isalso presented, as well as the effects of the electromagnetic field on the bubble dynamics.3he plan of the paper is as follows. Section 2 presents both a brief review of the warpdrive metric and electromagnetism in curved spacetimes necessary for a self contained paper.In Section 3 we calculated the electromagnetic EMT and the respective Einstein tensorcomponents. In Section 4 we analyzed the energy conditions and Section 5 is dedicated tothe investigation of conditions concerning the electric and magnetic fields under the warpdrive metric. Section 6 presents our conclusions and final remarks.
2. BASIC CONCEPTS2.1. Warp drive metric
From special relativity it is well known that nothing moves faster than light, but in thescope of general relativity, the spacetime is not flat anymore, but dynamic. This allows thepossibility of exotic solutions such as wormholes and warp drive, for example. This last oneis a way of going from a point A to a point B in space in times arbitrarily smaller than thelight would take to travel between those points.In [1] it was described a possible way that a mass particle could travel from one point toanother in spacetime measured by an external observer in a time interval smaller than thelight would travel the same distance. In few words, consider a particle that leaves an inertialreference frame A , which remains at rest, towards another inertial reference frame, also atrest, named B , at a distance D from A . The particle is inside a bubble that can modify thespacetime in a way that the space behind the bubble is expanded whereas it is contractedin front of if. This dynamics allows the particle inside the bubble to travel the distance D in a time less than D/c , where c is the speed of light, as measured by external observersdistant from the bubble, although it is still moving inside a light cone, which means that theparticle does not travel faster than light locally.The warp drive metric [1], using Cartesian coordinates x µ = ( t, x, y, z ), is given by ds = − (1 − β ) dt − β ( r s , t ) dx dt + dx + dy + dz , (2.1)where the term β ( r s , t ) is the shift vector, a boost in the x -direction where the particledescribes its trajectory inside the bubble. The shift vector is given by β ( r s , t ) = v s ( t ) f ( r s ) , (2.2)where v s ( t ) = dx s /dt is the warp bubble speed, x s is the coordinate for the center of thebubble, f = f ( r s ) is the regulating function [1], which regulates the warp bubble shape. Theparameter r s is the radius of the bubble given by r s = p [ x − x s ( t )] + y + z . (2.3)In this work we shall assume µ = ǫ = 1 and G = c = 1.4 .2. Electromagnetism in curved spacetime In this section we will calculate the EMT for this warp drive metric. But first we willintroduce the electromagnetic energy-momentum tensor (EEMT) T αβ using, of course, theelectromagnetic field strength tensor F αβ .It is well known that Maxwell electromagnetism is consistent with special relativity, theLorentz force law, that the Maxwell equations are valid for any inertial reference system,and that they can be put in what is known as the covariant formulation, which means todescribe electromagnetism in special relativity language in a manifestly invariant form underLorentz transformations. This formalism is constructed in a flat spacetime with Minkowskimetric in Cartesian coordinates given by ds = − dt + dx + dy + dz . (2.4)Due to the manifest covariance of the Maxwell equations in spacetime notation, if partialderivatives are replaced by covariant derivatives the extra terms cancels out and the equationsremain the same, making it possible to substitute the Minkowski metric η µν for a curvedspacetime metric g µν in a general curvilinear coordinate system.The 4-gradient is given by the following expression ∂ µ = ∂∂x µ = (cid:18) ∂∂t , −∇ (cid:19) , (2.5)and we have that g αβ , and the 4-gradient in covariant form is ∂ α = g αβ ∂ β .The electric E and magnetic B fields are described through the electromagnetic 4-potential, which is a covariant 4-vector with the electric scalar potential φ as its first com-ponent and the magnetic vector potential A as the other components, A α = ( φ, A ) . (2.6)The electric and magnetic fields above can be written in tensor notation, such as B a = ǫ abc (cid:16) ∂ b A c − ∂ c A b (cid:17) , (2.7) E a = ∂ A a − ∂ a A . (2.8)In matrix form, the field strength F αβ can be written as, F αβ = − E − E − E E − B B E B − B E − B B , (2.9)where the electric field E and the magnetic field B are in vector form E = ( E , E , E ) , (2.10) B = ( B , B , B ) , (2.11)5he covariant form of the electromagnetic tensor in general curved spacetime is given by thefollowing expression, F αβ = g αµ g βν F µν . (2.12)The EMT for curved space time can be written as, where we use here that the signatureis ( − + ++), T αβ = 14 π (cid:18) g αβ F γν F γν − g γν F αγ F βν (cid:19) . (2.13)If an extra term ( µ u α u β ) is added to EEMT it would describe a charged matter withdensity µ and proper velocity u α [18]. The EMT can be given by T αβ = T (dust) + T (elec) = µ π u α u β + 14 π (cid:18) g αβ F νσ F νσ − F αν F βν (cid:19) , (2.14)which is the EEMT for the dust embedded in an electromagnetic field in a curved spacetime.It is very clear in the last equation what each part of the tensor means.
3. THE ENERGY-MOMENTUM AND EINSTEIN TENSOR COMPONENTS
From now on we will discuss the conditions for the energy density T from the EMT to bepositive and how the radiant matter density µ together with the electromagnetic componentsinfluence the warp drive and how higher orders of the warp drive shift O ( β ) influence thespacetime. We will discuss interesting expressions found from the computation of Einsteinequations that provide us some insight on the plausibility of the warp drive concept. Considering the 4-velocity of the Eulerian observers u α = ( − , , ,
0) [1], the dust matterradiation part µu α u β from Eq. (2.14) has only one non zero component, i.e., T (dust)00 = µ .The non zero and non redundant components of the EEMT are given by4 πT = µ + 12 ( B + E ) + ( B E − B E ) β + β (cid:0) B − B − E − E (cid:1) + ( B E − B E ) β + β (cid:0) E + E (cid:1) , (3.1)4 πT = B E − B E + β (cid:0) B − B + E (cid:1) + ( B E − B E ) β − β (cid:0) E + E (cid:1) (3.2)4 πT = B E β + B B β + B E − B E , (3.3)4 πT = − B E β + B B β − B E + B E , (3.4)6 πT = β (cid:0) E + E (cid:1) + 12 (cid:0) B − B + E − E (cid:1) − ( B E − B E ) β, (3.5)4 πT = − B E β − B B − E E , (3.6)4 πT = B E β − B B − E E , (3.7)4 πT = β (cid:0) E − E (cid:1) − ( B E + B E ) β + 12 (cid:0) B − B + E − E (cid:1) , (3.8)4 πT = E E β + ( B E − B E ) β − ( B B + E E ) , (3.9)4 πT = − (cid:0) E − E (cid:1) β + ( B E + B E ) β + 12 (cid:0) B − B + E − E (cid:1) . (3.10)If we take β → T αν = πµ + ( E + B ) − S − S − S − S − σ − σ − σ − S − σ − σ − σ − S − σ − σ − σ . (3.11)where S , S and S are the vector components of the Poynting vector, S = E × B /µ , and σ ij are the nine components of the Maxwell stress tensor defined by σ ij = ǫ E i E j + 1 µ B i B j − (cid:18) ǫ E + 1 µ B (cid:19) δ ij , (3.12)which will be rewritten as σ ij = E i E j + B i B j − (cid:16) E + B (cid:17) δ ij , (3.13)where i, j = 1 , ,
3. We can notice that considering a warp drive spacetime background,the EEMT components have extra β -correction terms in comparison with Minkowski back-ground. In this scenario we can consider that the warp drive can generate electromagneticfields or at least it reinforces an already existing one, as it can be seen from Eqs. (3.1)to (3.10). In particular, the component T , which is the energy density, has fourth order β -corrections. 7 .2. Energy density T Analyzing the EEMT T term, which is given by Eq. (3.1), it shows that this term is a β -fourth order degree polynomial that has the form T ( β ) = ω (0) + ω (1) β + ω (2) β + ω (3) β + ω (4) β , (3.14)where the coefficients ω ( k ) are given by the expressions ω (0) = µ + 12 ( B + E ) , (3.15) ω (1) = B E − B E , (3.16) ω (2) = 12 (cid:0) B − B − E − E (cid:1) , (3.17) ω (3) = B E − B E , (3.18) ω (4) = 12 (cid:0) E + E (cid:1) . (3.19)As we said above, for β = 0, the warp drive metric becomes the Minkowski metric and theonly non zero component of the T term would be ω as expected from Eq. (3.11). Assuminga “weak” warp drive, that considers only shift vector O ( β ) terms, the condition for β , where T ≈ ω (0) + ω (1) β > β < µB E − B E + 12 B + E B E − B E , (3.20)which means that for this condition the shift vector has an upper bound limit which can bevery large if B E ≈ B E . Now considering an usual warp drive, with higher orders of theshift vector O ( β ), the conditions for T > ω (2) and ω (3) from Eqs.(3.17) and (3.18) are positive, and consequently we have the following conditions B + B > E + E + 2 E + B . (3.21) B E − B E > , (3.22)where we have neglected the term ω (4) in Eq. (3.19), since it is always positive and it ismultiplied by β . Both Eqs. (3.21) and (3.22) show connections between the components ofthe electromagnetic field in a way that the energy density T is positive for the warp drivespacetime in an electromagnetic background with a cosmological constant.8 .3. Einstein tensor components Let us now calculate the Einstein tensor components added by the cosmological constant.Considering G µν = R µν − g µν R − Λ g µν , (3.23)where g µν is the warp drive metric tensor given in Eq. (2.1) and Λ is the cosmologicalconstant. So, we have that G = Λ(1 − β ) −
14 (1 + 3 β ) "(cid:18) ∂β∂y (cid:19) + (cid:18) ∂β∂z (cid:19) − β (cid:18) ∂ β∂y + ∂ β∂z (cid:19) , (3.24) G = Λ β + 34 β "(cid:18) ∂β∂y (cid:19) + (cid:18) ∂β∂z (cid:19) + 12 (cid:18) ∂ β∂y + ∂ β∂z (cid:19) , (3.25) G = − ∂ β∂x∂y − β (cid:18) ∂β∂y ∂β∂x + β ∂ β∂x∂y + ∂ β∂t∂y (cid:19) , (3.26) G = − ∂ β∂x∂z − β (cid:18) ∂β∂z ∂β∂x + β ∂ β∂x∂z + ∂ β∂t∂z (cid:19) , (3.27) G = Λ − "(cid:18) ∂β∂y (cid:19) + (cid:18) ∂β∂z (cid:19) , (3.28) G = 12 (cid:18) ∂β∂y ∂β∂x + β ∂ β∂x∂y + ∂ β∂t∂y (cid:19) , (3.29) G = 12 (cid:18) ∂β∂z ∂β∂x + β ∂ β∂x∂z + ∂ β∂t∂z (cid:19) , (3.30) G = 12 ∂β∂z ∂β∂y , (3.31) G = − Λ − ∂∂x (cid:20) ∂β∂t + 12 ∂∂x ( β ) (cid:21) − "(cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) , (3.32) G = − Λ − ∂∂x (cid:20) ∂β∂t + 12 ∂∂x ( β ) (cid:21) + 14 "(cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) . (3.33)After a long algebraic manipulation of these expressions and considering the Einsteinequations G µν = 8 πT µν , we obtain the following set of partial differential equations43 Λ = 8 π (cid:20) T + 2 βT + (cid:18) β − (cid:19) T (cid:21) , (3.34)12 ∂ β∂y + 12 ∂ β∂z = 8 π ( T + βT ) , (3.35)9 (cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) − Λ = 8 πT , (3.36)12 (cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) = 8 π ( T − T ) , (3.37) − ∂∂x (cid:18) ∂β∂t + 12 ∂∂x ( β ) (cid:19) −
2Λ = 8 π ( T + T ) , (3.38) − ∂ β∂x∂y = 8 π ( T + βT ) , (3.39) − ∂ β∂x∂z = 8 π ( T + βT ) , (3.40)12 ∂β∂y ∂β∂z = 8 πT , (3.41) − (cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) + Λ = 8 π (cid:0) T + 2 βT + β T (cid:1) . (3.42)Note that the set of partial differential equations from Eq. (3.34) to Eq. (3.42) are quitecumbersome to solve if we consider the electromagnetic components that appear in theEEMT components T µν , even if we consider them as constants components.
4. ENERGY CONDITIONS
In this section we will calculate the energy conditions for the EEMT and we will seethat the inequalities written above will be satisfied. We will find connections between thecomponents of the electromagnetic field, the radiant matter density µ and the shift vector β . For the weak energy condition, the EMT at each point of the spacetime must satisfy thecondition T ασ u α u σ ≥ , (4.1)where, for any timelike vector u ( u α u α < k ( k α k α = 0), for an observerwith unit tangent vector v at a certain point of the spacetime, the local energy densitymeasured by any observer is non-negative [19]. For the EEMT, the expression T ασ u α u σ isgiven by T ασ u α u σ = 12 (cid:16) E + E (cid:17) β − (cid:16) B E − B E (cid:17) β + 12 (cid:16) B + E (cid:17) + µ . (4.2)10otice that Eq. (4.2) is a quadratic function of β . So, to be solved the discriminant ofthis quadratic equation must be positive, namely (cid:16) B E − B E (cid:17) − (cid:16) E + E (cid:17) (cid:20) (cid:16) B + E (cid:17) + µ (cid:21) > , (4.3)and we have for the matter density µ that,0 < µ < ( B E − B E ) E + E ) − (cid:16) B + E (cid:17) , (4.4)which shows a condition between the electromagnetic components, (cid:16) B E − B E (cid:17) > (cid:16) B + E (cid:17)(cid:16) E − E (cid:17) and that the matter density must have a positive inferior minimum value, since the r.h.s. ofthe inequality in Eq. (4.4) must be always positive. This result tells us that the weak energycondition must be satisfied if we consider both positive and negative matter density. Solvingexactly the inequality in Eq. (4.2) we have that β − < β < β + (4.5)where β ± = B E − B E E − E ± vuut" B E − B E E − E − B + E + 2 µE − E (4.6)and considering a “weak” warp drive where β ≈ β > B E − B E " (cid:16) B + E (cid:17) + µ , (4.7)which tells us that in this case, the shift vector has a limiting minimum value condition.This is an interesting result since it implies that even for lower order of β the warp bubblespeed is not limited by the weak energy condition. Notice that if B E → B E , then theright hand side of this last equation assumes unlimited values, which could be observed infact in low strength electromagnetic fields and low density of radiant matter. For every timelike vector u a the following inequalities must be satisfied T αβ u α u β ≥ , and F α F α ≤ , (4.8)where F α = T αβ u β is a non-spacelike vector. We can realize that these conditions meanthat for any observer, the local energy density appears to be non-negative and the local11nergy flow vector is non-spacelike. In any orthonormal basis the energy dominates theother components of the EMT, T ≥ | T ab | , for each a, b. (4.9)Evaluating the first condition T αβ u α u β ≥ F α F α , it will be given by a fourth degreepolynomial on β given by the following expression, F α F α = ω (4) β + ω (3) β + ω (2) β + ω (1) β + ω (0) , (4.10)where ω (0) , . . . , ω (4) are implicit functions of the spacetime coordinates ( t, x, y, z ) and also ex-plicit functions in terms of the electromagnetic field components. Notice that the subindexesof ω ( k ) ’s are not tensor indexes and the terms ω ( k ) β k are not tensor contractions. The coef-ficients are given by ω (4) = 14 (cid:0) E + E (cid:1) , (4.11) ω (3) = ( E + E )( E B − E B ) , (4.12) ω (2) = 12 ( E + E )( E − B − µ ) − ( B E − B E ) , (4.13) ω (1) = ( E B − E B )( B − E + 2 µ ) , (4.14) ω (0) = −
14 ( B + E ) − ( E + B ) µ − µ −
12 ( B E + B E + B E ) + 12 ( B E − B E ) + 12 ( B E − B E ) + 12 ( B E − B E ) . (4.15)Since F α F α is a fourth order polynomial on β with complicated expressions for its componentsit is a rather challenging work to calculate all the roots for this polynomial and to find thegeneral requirements for the dominant energy condition to be true. However, a simple andpragmatic way to impose the validity for this energy condition is to require that all thecoefficients ω ( k ) to be a positive value. The strong energy condition is given by (cid:18) T αβ − T g αβ (cid:19) u α u β ≥ , (4.16)12or any timelike vector u . The expression from the strong energy condition in Eq. (4.16) forthe EEMT and the warp drive metric is given by the following equation (cid:0) E + E (cid:1) β − B E − B E ) β + B + E + µ ≥ , (4.17)which is a quadratic β -inequality. For the strong energy condition to be satisfied it isnecessary that the discriminant of the above quadratic equation be real, that is( B E − B E ) − (cid:0) E + E (cid:1) h B + E + µ i < , (4.18)which is very similar to the equation we found for the weak and first dominant energyconditions with the difference of a multiplication by a factor of 2 µ ≤ ( B E − B E ) ( E + E ) − (cid:0) B + E (cid:1) , (4.19)which has the same meaning of the matter density to satisfy the strong energy condition asin Eq. (4.4) for the weak energy condition. The matter density has an inferior bound and itmay be negative. But the strong energy condition could still be satisfied as long as µ is lessthan or equal to the right hand side of the inequality in Eq. (4.19). The null energy conditions are satisfied for the null vector k since the following inequalitymust be satisfied T ασ k α k σ ≥ , for any null vector k α . (4.20)Assuming the following vector k α = ( a, b, ,
0) where a and b can be determined by solvingthe equation k α k α = 0. Hence, we can find two possible results connecting a and b , a = bβ + 1 or a = bβ − . (4.21)Considering this last result, the calculation of T ασ k α k σ is given by T ασ k α k σ = T k k + 2 T k k + T k k . (4.22)Substituting Eq. (4.21) and that k = a and k = b into Eq. (4.22), the result of the nullenergy condition for the EEMT is a complicated function of β given by T ασ k α k σ = ω (4) β + ω (3) β + ω (2) β + ω (1) β + ω (0) , (4.23)where the coefficients ω ( k ) are also functions of β given by the expressions ω (4) = a E + E ) , (4.24) ω (3) = a ( B E − B E ) − ab ( E + E ) , (4.25)13 (2) = 12 (cid:2) a ( B − B − E ) + 4 ab ( B E − B E ) + ( b − a )( E + E ) (cid:3) , (4.26) ω (1) = ab ( B − B − B + E ) + ( a − b )( B E − B E ) , (4.27) ω (0) = a B + E ) + b B − B + E − E ) + 2 ab ( B E − B E ) + a µ , (4.28)which are hard-working algebraic expressions concerning the null energy condition. The nextstep would be to solve them in a general way and to find the specific connections for theenergy condition to be valid.
5. ANOTHER EXPRESSION OF THE ENERGY MOMENTUM TENSOR
In this section we will introduce a specific observer that simplifies the EMT components.We will use this approach as a laboratory relative to the warp drive in an electromagneticfield background before analyzing and solving the generalized Einstein equations.
Let us consider that the electric and magnetic fields are orthogonal to each other, likein an electromagnetic wave. In addition, they do not obey the wave equation, but they dosatisfy the relation below, E i B i = 0 . (5.1)Now, let us assume that from the point of view of this observer the electric field has only onecomponent, E , which is the only one non zero component and it points to the direction ofthe particle trajectory inside the warp bubble. Hence, from Eq. (5.1), it is straightforwardthat B = 0, but we can still have the other magnetic components. Having said that, thenon zero components of the EEMT can be computed such that,4 πT = µ + 12 ( B + E ) + β (cid:0) B − E (cid:1) , (5.2)4 πT = β (cid:0) E − B (cid:1) , (5.3)4 πT = B E , (5.4)4 πT = − B E , (5.5)14 πT = 12 (cid:0) B − E (cid:1) , (5.6)4 πT = 12 (cid:0) B − B + E (cid:1) , (5.7)4 πT = − B B , (5.8)4 πT = 12 (cid:0) B − B + E (cid:1) . (5.9)Notice that B = B + B and now that E = E . Now we will use our specific choice of the electromagnetic field, namely, E = ( E , , B = (0 , B , B ) in the energy conditions. We have demonstrated in the last section thatboth the weak and strong energy conditions could be satisfied in a general manner, but thedominant and null conditions require a hard-working calculation to show that they are valid. Using the results obtained in the last section together with the specific choice of theelectromagnetic field, we can find that the weak energy condition from Eq. (4.2) can besatisfied if T ασ u α u σ = 12 (cid:0) B + E (cid:1) + µ ≥ , (5.10)which means that even with no radiant matter density, i.e., µ = 0, the weak energy inequalityis still non-negative. Besides, even with a negative matter density, it could still be positivefor this energy condition to be satisfied if the electromagnetic field strength ( B + E ) / µ as in Eq. (5.10). From Eq. (4.17) the inequality simplifies to (cid:18) T αβ − T g αβ (cid:19) u α u β = B + E + µ ≥ , (5.11)and it is clear that, since B + E is obviously always positive, the strong energy condition isvalid for the simplified choice of electromagnetic field since the matter density µ is positive. Inthe case of a negative matter density, the modulo of electromagnetic energy density B + E must be larger than the negative matter density µ .15 .2.3. Dominant energy condition for the simplified choice of EEMT For the dominant energy inequality, the first condition T αβ u α u β ≥
0, still gives us thesame result as the one from the weak energy inequality. But now with this simplified choicefor the electromagnetic field, the second requirement for the dominant energy condition isgiven by F α F α = − µ − ( E + B ) µ −
12 ( E − B ) − E B , (5.12)which is a quadratic function of the matter density µ . The sign of this expression will dependon the value of the electromagnetic components, of course. But, for real solutions, we musthave that E B ≥ ( E − B ) / Recovering the null energy condition from Eq. (4.23), we have that T ασ k α k σ = a B − E ) β + ab ( E − B ) β + 12 ( a + b ) B + 12 ( a − b ) E + a µ , (5.13)Disregarding the fact that a and b are functions of β , we will use the assumption thatEq. (5.13) is a quadratic function of β and we will impose that the discriminant of thisequation must be positive or zero for real solutions. Hence, the null energy condition mightbe satisfied for any value of β if a ( E − B )( µ + B + E ) ≥ . (5.14)which means that, considering a positive µ , the main condition is E ≥ B .Table I summarizes all the necessary requirements for the energy conditions to be validsimultaneously for the specific simplified choice of the electromagnetic field. TABLE I: Summary for the energy conditionsEnergy condition ResultsWeak (cid:0) B + E (cid:1) + µ ≥ B + E + µ ≥ E B ≥ ( E − B ) / E ≥ B , for µ > . For the simplified choice of the electromagnetic field, the set of partial differential Eqs.(3.34) to (3.42), that resulted from algebraic simplifications of the Einstein equations are43 Λ = 8 π (cid:18) µ + 13 B + 23 E (cid:19) , (5.15)162 ∂ β∂y + 12 ∂ β∂z = 0 , (5.16) − (cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) − Λ = 4 π (cid:0) B − E (cid:1) , (5.17)12 (cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) = 8 π (cid:0) B − B (cid:1) , (5.18) − ∂∂x (cid:18) ∂β∂t + 12 ∂∂x ( β ) (cid:19) −
2Λ = 8 πE , (5.19) − ∂ β∂x∂y = 8 πB E , (5.20) − ∂ β∂x∂z = − πB E , (5.21)12 ∂β∂y ∂β∂z = − πB B , (5.22) − (cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) + Λ = 4 π (cid:0) µ + E + B (cid:1) , (5.23) In this section we will calculate the divergence of the EEMT considering that the compo-nents of the electromagnetic field are functions of the spacetime coordinates x µ . The resultsare as follows, T α ; α = − ∂ ( βB ) ∂x − ∂ ( B + E ) ∂t − ∂ ( B E ) ∂z + ∂ ( B E ) ∂y − ∂ ( µβ ) ∂x − ∂µ∂t , (5.24) T α ; α = 12 ∂ ( B − E ) ∂x , (5.25) T α ; α = − ∂ ( βB E ) ∂x − ∂ ( B B ) ∂z − ∂ ( B E ) ∂t + 12 ∂∂y ( B − B + E ) , (5.26) T α ; α = ∂ ( βB E ) ∂x + ∂ ( B E ) ∂t − ∂ ( B B ) ∂y + 12 ∂∂z ( B − B + E ) . (5.27)Considering that E , B and B are constants, Eqs. (5.24) to (5.27) can be written suchas T α ; α = − B ∂β∂x − ∂ ( µβ ) ∂x − ∂µ∂t (5.28)17 α ; α = 0 (5.29) T α ; α = − B E ∂β∂x (5.30) T α ; α = B E ∂β∂x (5.31)Imposing that the EEMT must be conserved, i.e., the divergence must be zero, which showsthat Eq. (5.28) is a continuity equation, but Eq. (5.30) implies that either B = 0, or E = 0or ∂β∂x = 0, and Eq. (5.31) implies that either B = 0, or E = 0 or ∂β∂x = 0. Next we willanalyze how each one of these cases can affect the set of Eqs. (5.15) to (5.23). ∂β∂x = 0 and E = 0 These conditions are satisfied simultaneously and it can be seen from Eq. (5.19). More-over, Eqs. (5.20) and (5.21) are identically zero and the set of Einstein differential equationsfrom Eq. (5.15) to (5.23) can be written such asΛ = 6 πµ + 2 πB , (5.32) ∂ β∂y + ∂ β∂z = 0 , (5.33) (cid:18) ∂β∂y (cid:19) + (cid:18) ∂β∂z (cid:19) = −
43 Λ − π B , (5.34) (cid:18) ∂β∂y (cid:19) − (cid:18) ∂β∂z (cid:19) = 16 π (cid:0) B − B (cid:1) , (5.35) ∂β∂y ∂β∂z = − πB B , (5.36) (cid:18) ∂β∂y (cid:19) + (cid:18) ∂β∂z (cid:19) = − − π (cid:0) µ + B (cid:1) . (5.37)The above set of equations imply that the cosmological constant is null and the followingrelation between the electromagnetic field and the matter density µ = − B . (5.38)So, the matter density will always be negative for this case, but the energy conditions willstill be satisfied. The shift vector will not depend on the x spacetime coordinate, and thereis no Burgers equation, i.e., no shock wave, but β is function of ( t, y, z ) and it also satisfiesthe Laplace equation according to Eq. (5.33). Both Eqs. (5.34) and (5.37) are specific cases18f the well known Eikonal equation, and they imply that the solution for the shift vector isnot unique and may have a complex component. For this case the EMT in matrix form isgiven by, T αν = − µ (1 + β ) − βB − βB − B ( B − B ) − B B − B B − ( B − B ) . (5.39)It is clear that since the matter density is negative, i.e., µ <
0, then the energy densityfor the energy momentum tensor is positive, namely, T > B = 0 and B = 0 For this case it is straightforward to see that B = B = 0 implies that ∂β∂y = 0 and ∂β∂z = 0from Eqs. (5.20) to (5.22). The set of equations for this case is the followingΛ = 4 πE , (5.40) µ = 0 , (5.41) − ∂∂x (cid:18) ∂β∂t + 12 ∂∂x ( β ) (cid:19) = 2Λ + 8 πE . (5.42)This result violates both null and dominant energy conditions, but it is an interestingtheoretical result since both matter density µ and magnetic field B are null, but the cos-mological constant is positive and proportional to the electric field energy as seen in Eq.(5.40). T µν = − E (1 − β ) − βE − βE − E E
00 0 0 E . (5.43)which is the final and symmetric form of the EEMT. Notice the presence of only the electricfield and the shift function since B = 0.
6. CONCLUSIONS AND FINAL REMARKS
In this work we have investigated the solutions of the Einstein equations for the Alcubierrewarp drive metric with the choice of dust and electromagnetic field energy-momentum tensor(EMT) as possible sources of global superluminal particle speeds, that is, warp velocities.The Einstein equations were analyzed and all results concerning the components of the elec-tromagnetic field were discussed. The energy conditions were presented as functions of theelectromagnetic field components, and we have established conditions on these componentssuch that they obey, or not, the energy conditions the warp drive metric. The connections19ound between electromagnetic field and the superluminal effects of the warp drive resultingfrom the Einstein field equations, namely, being solutions of them, are new. We have alsodiscussed the potential dynamic consequences of these results on the warp bubble dynamics.We found that the energy conditions can be satisfied for positive and negative matterdensity, requirements which were summarized in a table. We also showed that the nulldivergence of the electromagnetic tensor results in some interesting cases, such as, the matterdensity becoming negative and proportional to the magnetic field energy density, the shiftvector β being a function of only ( t, y, z ), and the absence of the Burgers equation as foundin Ref. [16]. Nevertheless, a specific case of Eikonal equation has to be solved, leading to awave equation, in order to find solutions for β . In another case we found a violation of boththe dominant and null energy conditions, the matter density and the magnetic field are nulland the cosmological constant is proportional to the electric energy density. Acknowledgments
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