Two diverse models of embedding class one
aa r X i v : . [ g r- q c ] M a y Two diverse models of embedding class one
Peter K. F. Kuhfittig* ∗ Department of Mathematics, Milwaukee School of Engineering,Milwaukee, Wisconsin 53202-3109, USA
Abstract
Embedding theorems have continued to be a topic of interest in the general theoryof relativity since these help connect the classical theory to higher-dimensional man-ifolds. This paper deals with spacetimes of embedding class one, i.e., spacetimesthat can be embedded in a five-dimensional flat spacetime. These ideas are appliedto two diverse models, a complete solution for a charged wormhole admitting a one-parameter group of conformal motions and a new model to explain the flat rotationcurves in spiral galaxies without the need for dark matter.
Keywords:
Wormholes, Dark matter, Conformal symmetry, Embedding class one
Highlights: • The embedding of curved spacetime in higher-dimensional flat spacetime is discussed. • The focus is on two diverse models of embedding class one. • The first model is a complete wormhole solution including the junction conditions. • The second is a new model to explain the flat galactic rotation curves without theneed for dark matter.
Embedding theorems have a long history in the general theory of relativity, as exemplifiedby the induced-matter theory discussed in Ref. [1]. Of particular interest to us arespacetimes of embedding class one. Here we recall that an n -dimensional Riemannianspace is said to be of class m if n + m is the lowest dimension of the flat space in whichthe given space can be embedded. It is well known that the exterior Schwarzschild solutionis a Riemannian space of class two. Following Refs. [2], we assume a spherically symmetricmetric of class two that will be reduced to class one by a suitable transformation discussedin Sec. 2. Other useful references are [3, 4, 5].These ideas will be applied to two very different models, a complete solution for acharged wormhole admitting a one-parameter group of conformal motions and a newmodel to explain the flat rotation curves in spiral galaxies without the need for darkmatter. ∗ E-mail: kuhfi[email protected] The embedding
The discussion in Ref. [2] begins with the static and spherically symmetric line element(in units in which G = c = 1) ds = e ν ( r ) dt − e λ ( r ) dr − r (cid:0) dθ + sin θ dφ (cid:1) , (1)where λ and ν are functions of the radial coordinate r . It is shown that this metric ofclass two can be reduced to class one and thereby embedded in the five-dimensional flatspacetime ds = − (cid:0) dz (cid:1) − (cid:0) dz (cid:1) − (cid:0) dz (cid:1) − (cid:0) dz (cid:1) + (cid:0) dz (cid:1) . (2)This reduction is accomplished by the following transformation: z = r sin θ cos φ , z = r sin θ sin φ , z = r cos θ , z = √ K e ν cosh t √ K , and z = √ K e ν sinh t √ K . The differentialsof these components are dz = dr sin θ cos φ + r cos θ cos φ dθ − r sin θ sin φ dφ, (3) dz = dr sin θ sin φ + r cos θ sin φ dθ + r sin θ cos φ dφ, (4) dz = dr cos θ − r sin θ dθ, (5) dz = √ K e ν ν ′ t √ K dr + e ν sinh t √ K dt, (6)and dz = √ K e ν ν ′ t √ K dr + e ν cosh t √ K dt, (7)where the prime denotes differentiation with respect to the radial coordinate r . To facil-itate the substitution into Eq. (2), we first obtain the expressions for − ( dz ) − ( dz ) − ( dz ) and − ( dz ) + ( dz ) : − (cid:0) dz (cid:1) − (cid:0) dz (cid:1) − (cid:0) dz (cid:1) = − dr − r (cid:0) dθ + sin θ dφ (cid:1) , (8) − (cid:0) dz (cid:1) + (cid:0) dz (cid:1) = e ν dt − K e ν ν ′ dr . (9)Substituting Eqs. (8) and (9) in Eq. (2), we obtain the metric ds = e ν dt − (cid:18) K e ν ν ′ (cid:19) dr − r (cid:0) dθ + sin θ dφ (cid:1) . (10)So metric (10) is equivalent to metric (1) if e λ = 1 + K e ν ν ′ , K > . (11)This condition is equivalent to the condition derived by Karmarkar [6] in terms of theRiemann curvature tensor components: R = R R + R R R , R = 0 . (See Ref. [7] for details.) 2 Conformal Killing vectors
As noted above, we assume a static spherically symmetric spacetime admitting a one-parameter group of conformal motions, which are motions along which the metric tensorremains invariant up to a scale factor. Equivalently, there exist conformal Killing vectorssuch that L ξ g µν = g ην ξ η ; µ + g µη ξ η ; ν = ψ ( r ) g µν , (12)where the left-hand side is the Lie derivative of the metric tensor and ψ ( r ) is the conformalfactor [8, 9]. The metric tensor g µν is conformally mapped into itself along the vector ξ , which generates the conformal symmetry. This type of symmetry has been used todescribe relativistic stellar-type objects, as discussed in Refs. [10, 11]. Additional newgeometric and kinematical insights are described in Refs. [12, 13, 14, 15, 16]. Two earlierstudies assumed non-static conformal symmetry [9, 16]. Another significant observationis that the Kerr black hole is conformally symmetric [17].To study the effect of conformal symmetry, it is convenient to use line element (1)with the opposite signature [18, 19]: ds = − e ν ( r ) dt + e λ ( r ) dr + r ( dθ + sin θ dφ ) . (13)Using this form, the Einstein field equations become e − λ (cid:18) λ ′ r − r (cid:19) + 1 r = 8 πρ, (14) e − λ (cid:18) r + ν ′ r (cid:19) − r = 8 πp r , (15)and 12 e − λ (cid:20)
12 ( ν ′ ) + ν ′′ − λ ′ ν ′ + 1 r ( ν ′ − λ ′ ) (cid:21) = 8 πp t . (16)Following Herrera and Ponce de Le´on [10], we can simplify the analysis by requiringthat ξ α U α = 0, where U α is the four-velocity of the perfect fluid distribution, so that fluidflow lines are mapped conformally onto fluid flow lines. From the assumption of sphericalsymmetry, it now follows that ξ = ξ = ξ = 0 [10]. Eq. (12) then yields the followingresults: ξ ν ′ = ψ, (17) ξ = ψr , (18)and ξ λ ′ + 2 ξ , = ψ. (19)From Eqs. (17) and (18), we then obtain ν ′ = 2 /r and thus e ν = Cr , (20)3here C is an integration constant. Now from Eq. (18) we get ξ , = 12 ( ψ ′ r + ψ ) . Substituting in Eq. (19) and using ν ′ = 2 /r , simplification yields λ ′ = − ψ ′ ψ . Finally, solving for λ , we have e λ = (cid:18) Bψ (cid:19) , (21)where B is another integration constant. When substituting into Eqs. (14)-(16), itbecomes apparent that B is merely a scale factor, so that we may assume that B = 1.We then get e − λ = ψ (22)and the Einstein field equations can be rewritten as follows:1 r (cid:0) − ψ (cid:1) − ( ψ ) ′ r = 8 πρ, (23)1 r (cid:0) ψ − (cid:1) = 8 πp r , (24)and ψ r + ( ψ ) ′ r = 8 πp t . (25) Wormholes are handles or tunnels in spacetime connecting widely separated regions ofour Universe or entirely different universes. While there were a number of forerunners,actual physical structures suitable for interstellar travel was first proposed by Morris andThorne [20]. Such wormholes can be described by the static and spherically symmetricline element ds = − e r ) dt + dr − b ( r ) /r + r ( dθ + sin θ dφ ) , (26)using units in which c = G = 1. Here Φ = Φ( r ) is called the redshift function , whichmust be everywhere finite to avoid an event horizon. The function b = b ( r ) is calledthe shape function since it determines the spatial shape of the wormhole when viewed,for example, in an embedding diagram [20]. The spherical surface r = r is the throat of the wormhole. The shape function must satisfy the following conditions: b ( r ) = r , b ( r ) < r for r > r and b ′ ( r ) ≤
1, called the flare-out condition . For a Morris-Thornewormhole, this condition can only be satisfied by violating the null energy condition,thereby becoming the primary condition for the existence of a traversable wormhole.4he discussion in Ref. [20] was based on the following strategy: specify the geometricconditions required for a traversable wormhole and then either manufacture or searchthe Universe for matter or fields that will produce the corresponding energy-momentumtensor. One of our goals in this paper is to reverse this strategy by showing that theconditions described are sufficient for producing a complete solution, i.e., for obtainingboth Φ = Φ( r ) and b = b ( r ), as well as the necessary junction conditions. The motivation for a wormhole with a constant charge Q , first proposed by Kim and Lee[21], was provided by the Reissner-Nordstr¨om black hole ds = − (cid:18) − Mr + Q r (cid:19) dt + (cid:18) − Mr + Q r (cid:19) − dr + r ( dθ + sin θ dφ ) , suggesting that e λ ( r ) = (cid:18) − b ( r ) r + Q r (cid:19) − . (27)Charged wormholes are also discussed in Refs. [22, 23].Since we are assuming conformal symmetry, we have e ν = Cr from Eq. (20). More-over, from Eq. (22), ψ = 1 − b ( r ) r + Q r . (28)We also assume that b = b ( r ) satisfies the usual conditions for a shape function: letting r = r be the radius of the throat, we require that b ( r ) = r and b ( r ) <
1, while b ( r ) < r for r > r . Returning to Sec. 2, we know from Eq. (11) that in view of Eq. (27) e λ ( r ) = 11 − b ( r ) r + Q r = 1 + 14 Ke ν ( r ) [ ν ′ ( r )] . (29)Moreover, from Eq. (20), e ν = Cr , we have1 = (cid:18) − b ( r ) r + Q r (cid:19) (cid:18) K ( Cr ) 4 r (cid:19) since ( ν ′ ) = 4 /r . Hence 11 + KC = 1 − b ( r ) r + Q r . (Observe that since b ( r ) = r , Q cannot be zero.) Solving for b ( r ), we get b ( r ) = r (cid:18) Q r −
11 + KC (cid:19) . (30)5he condition b ( r ) = r now leads to 1 + KC = r /Q and C = 1 K (cid:18) r Q − (cid:19) . (31)The result is b ( r ) = r (cid:18) Q r − Q r (cid:19) . (32)This result, in turn, leads to b ′ ( r ) = 1 − Q r < , (33)provided that r > Q .The flare-out condition is thereby satisfied, but unlike a Morris-Thorne wormhole,satisfying this condition does not automatically result in a violation of the null energycondition (NEC). To see why, recall that the NEC states that for the energy-momentumtensor T αβ , T αβ k α k β ≥ k α . Consider the radial outgoing null vector(1 , , , b = b ( r ) were the shape function of a regular Morris-Thorne wormhole,Eq. (26), we would have [20]8 πρ + 8 πp r = b ′ ( r ) − b ( r ) /rr + 2Φ ′ r (cid:18) − b ( r ) r (cid:19) . So at r = r , 8 πρ ( r ) + 8 πp r ( r ) < b ′ ( r ) <
1. The problem is that our metrichas been altered due to the embedding, Eq. (11), i.e., e λ ( r ) = 1 + Ke ν ( r ) ( ν ′ ) . Asa result, we are no longer dealing with the same null vector: since b ( r ) has changed,so has 8 πρ ( r ) + 8 πp r ( r ). On the other hand, we also made use of Eq. (27), e λ ( r ) =(1 − b ( r ) /r + Q /r ) − , which gives us the effective shape function used in Ref. [23]: b eff ( r ) = b ( r ) − Q r . (34)This form suggests that Eq. (30) be modified as follows: b ( r ) = r (cid:18) Q r − KC (cid:19) + Q r . (35)This modification needs to be justified by showing that b ( r ) in Eq. (35) satisfies all therequired conditions. To that end, let us denote the throat by r = r , so that b ( r ) = r .This condition yields 1 + KC = r / Q and C = 1 K (cid:18) r Q − (cid:19) . (36)Substituting in Eq. (35), we get b ( r ) = r (cid:18) Q r − Q r (cid:19) + Q r . (37)6inally, the flare-out condition is also met: b ′ ( r ) = 1 − Q r < . (Since we want b ′ ( r ) to be positive, we also require that r > Q .)As noted earlier, we still need to check the violation of the null energy condition: FromEqs. (1), (22), and (27), e − λ ( r ) = 1 − b ( r ) r + Q r = ψ ( r ) . Substituting b ( r ), we get ψ ( r ) = 1 − (cid:18) Q r − Q r (cid:19) − Q r + Q r = − Q r + 3 Q r . (38)Thus ψ ( r ) = Q r . (39)Also, ( ψ ( r )) ′ = 4 Q r . (40)Returning now to the Einstein field equations (23) and (24),8 π ( ρ + p r ) | r = r = 1 r (cid:20) ψ ( r ) − ( ψ ( r )) ′ r (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = r = 1 r (cid:18) Q r (cid:19) − Q r = − Q r < . (41)So the null energy condition is indeed violated.Returning now to Eq. (36), it is clear that the free parameter K can be used todetermine the constant C and hence e ν , which, in turn yields the redshift function.To complete the solution, we still need to consider the following: we can see from Eq.(20), e ν = Cr , that our wormhole spacetime is not asymptotically flat. So the wormholematerial must be cut off at some r = a and joined to an exterior Schwarzschild spacetime, ds = − (cid:18) − Mr (cid:19) dt + dr − M/r + r ( dθ + sin θ dφ ) . (42)Thus e ν ( a ) = Ca = 1 − Ma , (43)where M is the mass of the wormhole as seen by a distant observer and C is obtainedfrom Eq. (36). It follows that the cut-off r = a is implicitly determined by the equation1 K (cid:18) r Q − (cid:19) a = 1 − Ma . (44)So given M , Q , and r , the free parameter K determines the radius of the junctionsurface. Eq. (44) will have a real solution if K is sufficiently large. (Plausible valuesmight be 10 m and 10 m arising in the discussion of compact stellar objects in Ref.[5].) 7 Flat galactic rotation curves
One goal in many modified gravitational theories is to explain the peculiar behavior ofgalactic rotation curves without postulating the existence of dark matter, whether this benoncommutative geometry [24, 25] or f ( R ) modified gravity [26]. The basic problem isthat test particles move with constant tangential velocity v φ in a circular path sufficientlyfar from the galactic core. Taking the observed rotation curves as input, it is well knownthat e ν = B r l , (45)where l = 2 v φ and B ia an integration constant [27]. In addition, it is shown in Ref. [28]that in the presumed dark-matter dominated region, v φ ≈
300 km / s = 10 − for a typicalgalaxy. So l = 0 . m is themass of a star, v φ the constant tangential velocity, and m the mass of everything else.Multiplying m by the centripetal acceleration yields m v φ r = m m Gr , (46)where G is Newton’s gravitational constant. The result is (since G = 1) m = rv φ . (47)Eq. (47) essentially characterizes the dark-matter hypothesis, but, as noted above,other explanations are possible. As a starting point, suppose we consider [30] e λ ( r ) = 11 − m ( r ) /r . Then m ( r ) = r (1 − e − λ ). Since e − λ → r → ∞ , we can assume that m ( r ) iaapproximately constant over a large range of r . In other words, m ( r ) = Cr for someconstant C . Unfortunately, by Eq. (47), C has to be approximately equal to v φ . Thisobstacle can be overcome by the embedding theory in Sec. 2. Using Eq. (11), m ( r ) = 12 r (cid:18) −
11 + Ke ν ( ν ′ ) (cid:19) , (48)the free parameter K gives us the extra degree of freedom to produce the correct values,provided, of course, that e ν and ( ν ′ ) are indeed approximately constant. To that end, letus return to Refs. [27]-[29], which deal with typical galaxies, including our own. Supposewe assume for now that B = 1 in Eq. (45). So with our own galaxy in mind, let usconsider the range from 8 kps to 50 kps, associated with flat galactic rotation curves.Then r l ranges from (26 000 × . × ) . ≈ . . ×
26 000 × . × ) . ≈ . . These calculations show that the value of B has little effect, so that e ν does remainapproximately constant.The values for ( ν ′ ) are much less robust. However, of great help in this situation isthat B drops out entirely: from e ν = B r l , we have( ν ′ ) = l r . So we can obtain an adequate approximation for the above range, while in the range 16kps to 30 kps, the resulting values for ( ν ′ ) are essentially fixed: 4 . × − m − and1 . × − m − , respectively. For these values, K ≈ m .With this choice of K , Eq. (48) reduces to m ( r ) = v φ r , thereby producing anotheralternative to the dark-matter hypothesis. This outcome may be viewed as the analogueof the induced-matter theory in Ref. [1], i.e., one could maintain that the five-dimensionalflat spacetime impinges on our Universe to produce the effect that we normally interpretas dark matter. Remark:
The existence of two models from the same embedding theory invites thefollowing speculation: according to Brownstein and Moffet [31], a significant amount ofdark matter is missing in the Bullet Cluster 1E0657-558. This is also the cluster that hassupposedly shown that dark matter actually exists. The main argument in Ref. [31] isthat this phenomenon can be explained by means of a modified gravitational theory, towhich we could add the present embedding theory. On the other hand, if the dark-matterhypothesis is to be retained and if some of the dark matter is indeed missing, then theexistence of a conformally symmetric charged wormhole may be the preferred explanation:the Bullet Cluster consists of two colliding galaxies moving at very high velocities, so thatthe dark matter could be literally driven into the wormhole.
It is well known that a curved spacetime can be embedded in a higher-dimensional flatspacetime. A spacetime is said to be of class m if n + m is the lowest dimension of theflat space in which the given space can be embedded. Following Ref. [2], we assume aspherically symmetric metric of class two that can be reduced to class one by a suitabletransformation.These ideas have been applied to two completely different models, a new solution fora charged wormhole admitting a one-parameter group of conformal motions and a newmodel to explain the flat rotation curves in spiral galaxies without the need for darkmatter. The existence of the latter model can be attributed to the free parameter K inthe embedding theory. In the former case, the free parameter K plays an equally criticalrole in obtaining a complete wormhole solution: K helps determine the redshift and shapefunctions, as well as the radius of the junction interface that joins the interior solution toan exterior Schwarzschild spacetime. 9 eferenceseferences