Rotating thin shells in (2+1)-dimensional asymptotically AdS spacetimes: Mechanical properties, Machian effects, and energy conditions
aa r X i v : . [ g r- q c ] J un ROTATING THIN SHELLS IN (2+1)-DIMENSIONALASYMPTOTICALLY ADS SPACETIMES: MECHANICALPROPERTIES, MACHIAN EFFECTS, AND ENERGY CONDITIONS
JOS´E P. S. LEMOS
Centro Multidisciplinar de Astrof´ısica - CENTRA, Departamento de F´ısica, Instituto SuperiorT´ecnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais 1Lisboa, 1049-001, [email protected]
FRANCISCO J. LOPES
Centro Multidisciplinar de Astrof´ısica - CENTRA, Departamento de F´ısica, Instituto SuperiorT´ecnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais 1Lisboa, 1049-001, [email protected]
MASATO MINAMITSUJI
Centro Multidisciplinar de Astrof´ısica - CENTRA, Departamento de F´ısica, Instituto SuperiorT´ecnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais 1Lisboa, 1049-001, [email protected]
A rotating thin shell in a (2+1)-dimensional asymptotically AdS spacetime is studied.The spacetime exterior to the shell is the rotating BTZ spacetime and the interior isthe empty spacetime with a cosmological constant. Through the Einstein equation in(2+1)-dimensions and the corresponding junction conditions we calculate the dynamicalrelevant quantities, namely, the rest energy-density, the pressure, and the angular mo-mentum flux density. We also analyze the matter in a frame where its energy-momentumtensor has a perfect fluid form. In addition, we show that Machian effects, such as thedragging of inertial frames, also occur in rotating (2+1)-dimensional spacetimes. Theweak and the dominant energy condition for these shells are discussed.
Keywords : thin shell; general relativity in 2+1 dimensions; anti-de Sitter spacetime; blackhole
1. Introduction
In (2+1)-dimensional general relativity with a cosmological constant, spacetimeswith no matter have no propagating degrees of freedom, so that vacuum space-times have local constant curvature. Nonetheless, in the presence of a negativecosmological constant, there is a nontrivial black hole solution, the Ba˜nados, Teit-elboim, Zanelli (BTZ) black hole.
1, 2
In its most general form the BTZ black holeis a stationary solution representing a rotating black hole, with mass and angularmomentum. For zero rotation the BTZ black hole is static. The BTZ black hole, Jos´e P. S. Lemos, Francisco J. Lopes, Masato Minamitsuji having a negative cosmological constant, is an asympotically anti-de Sitter (AdS)solution.Besides being a pure vacuum solution of a spacetime with a negative cosmologicalconstant, the BTZ solution is also a solution exterior to a (2+1)-dimensional matterconfiguration. A simple matter configuration is a thin shell. Collapsing nonrotatingthin shells with an exterior static BTZ solution have been studied in
3, 4 . Staticthin shells with an exterior static BTZ solution were worked out in . Gravitationalcollapse and configurations of rotating thin shells with an exterior stationary BTZsolution was analyzed in .Here, we want to study rotating thin shells for which the exterior spacetimeis the stationary BTZ solution and the interior spacetime is the empty spacetimewith a cosmological constant. We translate to 2+1 dimensions the framework ofPoisson to treat rotating shells in (3+1)-dimensional general relativity , and thus,through the Einstein equation in (2+1)-dimensions and the corresponding junctionconditions, we calculate the dynamical relevant quantities.This paper is organized as follows. In Sec. 2 we perform an analysis of the(2+1)-dimensional spacetime with a negative cosmological constant generated by arotating thin matter shell. The exterior is the BTZ solution, the interior is the emptyspacetime, and the energy-density, the pressure, and the angular momentum fluxdensity of the matter in the shell are calculated. In Sec. 3 we analyze the thin shell’smechanical properties in a frame where one can detect the energy-momentum tensoras a perfect fluid. In Sec. 4 we study rotating properties of the spacetime and findthat (2+1)-dimensional spacetimes can display Machian effects, such as the draggingof inertial frames. In Sec. 5 we discuss the weak and dominant energy conditionsfor the thin shell matter. We conclude in Sec. 6. In an Appendix we rederive theenergy-momentum tensor in a perfect fluid form using a different approach.
2. The thin shell spacetime: BTZ outside, AdS inside2.1.
Einstein equation
In 2+1 dimensions, the Einstein equation is G αβ + Λ g αβ = 8 πG T αβ , (1)where G αβ is the Einstein tensor, Λ is the cosmological constant, g αβ is the space-time metric, G is the gravitacional constant in 2+1 dimensions, and T αβ is theenergy-momentum tensor of the matter fields. We will choose units with the veloc-ity of light equal to one, so that G has units of inverse of mass. Greek indices runfrom 0 , ,
2, with 0 being the time index. Since the background spacetime is AdSwhich has a negative cosmological constant, we define the AdS length l through theequation − Λ = 1 /l . (2)We consider a timelike shell with radius R in a (2+1)-dimensional spacetime. In otating thin shells in (2+1)-dimensional spacetimes two spatial dimensions a shell is a ring. This ring divides spacetime into two regions,an outer region and an inner region. The outside region
The outside region is described by an exterior metric given by the BTZ line ele-ment
1, 2 ds o = g o αβ dx αo dx βo = − (cid:18) r l − Gm + 16 G J r (cid:19) dt o + dr (cid:18) r l − Gm + 16 G J r (cid:19) + r (cid:18) dφ − GJr dt o (cid:19) , r > R , (3)written in a circularly symmetric outer coordinate system x αo = ( t o , r, φ ), and where m is the Arnowitt-Deser-Misner (ADM) mass and J is the spacetime angular mo-mentum. The inside region
The interior region is the vacuum state empty spacetime, a variant of pure AdSspacetime, with metric given by
1, 2 ds i = g i αβ dx α dx β = − ρ l dt i + l ρ dρ + ρ dψ , ρ < R , (4)written in a circularly symmetric inner coordinate system x αi = ( t i , ρ, ψ ). The shell-ring region: the matching and junction conditions
Here we analyze the matching region, i.e., the ring. From the outside, the metric atthe shell Σ, r = R , is from Eq. (3) ds = − (cid:18) R l − Gm + 16 G J R (cid:19) dt o + R (cid:18) dφ − GJR dt o (cid:19) , r = R . (5)Now, we want to remove the off-diagonal term in the induced metric (5), as viewedfrom the outer region. For that we go to a corotating frame by defining a new polarcoordinate ψ = φ − Ω t o . (6)This makes the induced metric diagonal if the angular velocity Ω is choosen to beΩ = 4 GJR . (7)Then, in coordinates ( t o , ψ ), the induced metric at Σ is written as ds = − (cid:18) R l − Gm + 16 G J R (cid:19) dt o + R dψ , r = R . (8)
Jos´e P. S. Lemos, Francisco J. Lopes, Masato Minamitsuji
From the inside, the induced metric is obtained by setting ρ = R in Eq. (4), i.e., ds = − R l dt i + R dψ , ρ = R . (9)On the other hand, at the shell, the metric can be written in terms of its ownproper coordinates x a Σ = ( τ, ψ ), with τ being the proper time of the shell. So, theline element ds = h ab dx a Σ dx b Σ , with h ab being the intrinsic metric at the shell, canbe written as ds = − dτ + R dψ . (10)We have still to apply the first and second junction conditions, see, e.g., . Thefirst junction condition states that the induced metric on the shell must be the sameon both sides of the shell and at the shell. From Eqs. (8)-(10) this gives dτ = (cid:18) R l − Gm + 16 G J R (cid:19) dt o = R l dt i . (11)The second junction condition states that S ab = − πG ([ K ab ] − [ K ] h ab ) , (12)where S ab is the energy-momentum tensor of the matter in the shell, K ab = n α ; β e αa e βb is the second fundamental form of the shell, with n α being the normal vector tothe shell, e αa are the tangent vectors to the shell, the semicolon denotes covariantderivative, and a quantity in square brackets denotes the jump from the outside tothe inside. The tangent vectors e αa are such that h ab given through Eq. (10) can bewritten as the metric induced either from the outside or the inside metric g αβ at R ,i.e., h ab = g αβ e αa e βb , where g αβ stands to either g o αβ or g i αβ . The three independentcomponents of the surface energy-momentum tensor can then be calculated usingEq. (12) to give S ττ = − πGl − lR r R l − Gm + 16 G J R ! , (13) S ψψ = 18 πGl R − G l J lR r R l − Gm + 16 G J R − , (14) S τψ = J πR . (15) Properties of the matter of the ring shell
These calculations were performed in the shell’s rest frame coordinate ( τ, ψ ). Inthis frame we define the proper frame quantities, namely, the proper rest energy otating thin shells in (2+1)-dimensional spacetimes density λ , the proper pressure p , and the proper angular momentum flux density j as λ = − S ττ , p = S ψψ and j = S τψ . So, λ = 18 πGl − lR r R l − Gm + 16 G J R ! , (16) p = 18 πGl − R − G l J lR r R l − Gm + 16 G J R , (17) j = J πR . (18)If one considers slowly rotation the terms in J in Eqs. (16) and (17) should bediscarded, keeping the linear term in Eq. (18).An interesting quantity is the total rest mass M of the shell, given by M = 2 πRλ . (19)Then using Eqs. (16) and (19) one gets an expression for the ADM mass-energy m in terms of the other quantities, m = Rl M − GM + 2 G J R . (20)This expression connects shell quantities, namely, M and R with spacetime quan-tities, namely, m , J and l . Properties of the spacetime
Inside the shell the spacetime is empty spacetime. The matching region is the ring-shell region. Outside the shell the spacetime is a rotating BTZ spacetime. The BTZspacetime has two intrinsic radii, given by the zeros of the metric component g rr in Eq. (3). They are the gravitational radius r + and the Cauchy radius r − , whichhave the expressions r + = 2 l s Gm + r G m − G J l , (21) r − = 2 l s Gm − r G m − G J l . (22)The inequality r − ≤ r + always holds, with the equality r − = r + being equivalentto J = ml . Inverting Eqs. (21) and (22) gives m and J in terms of r + and r − , m = r + r − Gl , (23) J = r + r − Gl . (24)
Jos´e P. S. Lemos, Francisco J. Lopes, Masato Minamitsuji
Up to now we have put no constraints on the angular momentum J of theoutside spacetime. There are three cases we should discuss. First, J < ml , which isequivalent to r + > r − , in which case the shell is underspinning. Second, J = ml ,which is equivalent to r + = r − , and the outside spacetime is extremal BTZ and theshell is called extremal. Third, J > ml , in this case both r + and r − are imaginary,and the spacetime and the shell are overspinning. For the first two cases r + ≥ r − , weimpose the mechanical constraint that the shell must be outside its own gravitationalradius, i.e., there are no trapped surfaces or horizons in the spacetime. This meansthat the condition R ≥ r + (25)must hold. For the other case, J > ml , the spacetime is overspinning and the shellcan be taken to R = 0 in which case is a naked singularity. Thus, for this case onehas to impose simply R ≥
0. We will not discuss further this latter case as it is toosimple and the analysis done so far is enough. So in the rest of the paper we assume r + ≥ r − . (26) The metric and matter quantities in terms of r + and r − We have found all the relevant expressions for the ring-shell spacetime and havedisplayed the two intrinsic important radii of the spacetime, r + and r − . It is thusconvenient to write the metric and the matter quantities in terms of r + and r − ,instead of m and J . We thus use Eqs. (21)-(24). From Eq. (3), the exterior BTZmetric can be written as ds o = − l r (cid:2)(cid:0) r − r (cid:1) (cid:0) r − r − (cid:1)(cid:3) dt o + l r dr (cid:0) r − r (cid:1) (cid:0) r − r − (cid:1) + r (cid:16) dφ − r + r − lr dt o (cid:17) , r > R . (27)The angular velocity Ω that makes the exterior metric diagonal at Σ, i.e., Eq. (6),can be written as Ω = r + r − lR . (28)In coordinates ( t o , ψ ), the induced metric at Σ is then ds = − l R (cid:2)(cid:0) R − r (cid:1) (cid:0) R − r − (cid:1)(cid:3) dt o + R dψ , r = R . (29)In the rest frame of the shell, the components of the energy-momentum tensor S ab ,Eqs. (16)-(18) are given by λ = 18 πGl h − R q ( R − r )( R − r − ) i , (30) p = 18 πGl h R − r r − R q ( R − r )( R − r − ) − i , (31) j = r + r − πGlR . (32) otating thin shells in (2+1)-dimensional spacetimes
3. The fluid seen as a perfect fluid3.1.
The perfect fluid energy density and pressure, and the angularvelocity of the reference frame that detects a perfect fluid
We now want to pass to a reference frame where the energy-momentum tensor ofthe matter has the form of a perfect fluid energy-momentum tensor. We will seethat this is possible. Thus, we want that the energy-momentum tensor takes theform S ab = ¯ λ u a u b + ¯ p (cid:0) u a u b + h ab (cid:1) , (33)where h ab is the metric on the shell given in Eq. (10), u a is some velocity field on theshell, and perfect fluid quantities are written as barred quantities, ¯ λ is the perfectfluid energy density and ¯ p its pressure.Due to the circular symmetry and the fact that there are only two componentsfor the velocity u a , one can write it as u a = γ ( τ a + ¯ ωψ a ), for some angular velocity ¯ ω ,and τ a = ∂x a Σ ∂τ , ψ a = ∂x a Σ ∂ψ . Then the normalization condition gives γ = 1 / √ − ¯ ω R and so u a = 1 √ − ¯ ω R ( τ a + ¯ ωψ a ) . (34)Since u a u a = − S ab u b = − ¯ λ u a , (35)meaning that the velocity field u a is an eigenvector of S ab with eigenvalue − ¯ λ . Thisyields two equations which enable to calculate ¯ λ and ¯ ω . Indeed, Eq. (35) yields¯ λ = − S ττ + R ¯ ω S ψψ R ¯ ω and ¯ ω ω R = − S ψτ − S ττ + S ψψ . Having ¯ λ and u a one finds the pressure¯ p by projecting S ab into the direction orthogonal to u a , i.e.,¯ p = ( h ab + u a u b ) S ab . (36)Then using Eqs. (35)-(36) together with previous equations one gets¯ λ = 18 πGl − s R − r R − r − ! , (37)¯ p = 18 πGl s R − r − R − r − ! , (38)¯ ω = r − r + R s R − r R − r − . (39)Note that ¯ ω is the angular velocity of the reference frame which detects a perfectfluid relative to the shell’s proper frame, i.e., dψdτ = ¯ ω . The derivation presented sofar follows . For an alternative derivation see Appendix. Jos´e P. S. Lemos, Francisco J. Lopes, Masato Minamitsuji
4. Effects due to rotation: Machian effects and dragging of theinertial frames4.1.
The several rotations and angular velocities
We can now explore some effects due to rotation. The BTZ metric given by Eq. (3),appropriate for the vacuum region outside of the shell, tends to the AdS metric atinfinity. Pure AdS metric is a nonrotating metric. Thus, infinity is the standard ofa nonrotating frame, the AdS frame, or the fixed stars frame in Machian language.The empty metric given by Eq. (4), appropriate for the vacuum region inside of theshell, is in a coordinate system that is rotating with respect to infinity. Indeed, fromEq. (3), one deduces that the interior region, including the interior neighborhood ofthe ring, has the property that lines with constant ψ move with respect to t o , theglobal AdS time at infinity, with angular velocity dφ/dt o = Ω, where Ω is given inEq. (28) (see also Eq. (7)). Now, since the inside is the empty space metric, lineswith constant ψ represent inertial frame lines for the interior spacetime that rotatewith Ω with respect to AdS infinity.Moreover, the angular velocity ¯ ω , found previously as the angular velocity of theframe that detects perfect fluid quantities, is measured in the frame that rotates withΩ relative to infinity, the frame that has proper time τ . Then, the correspondingangular velocity ω measured in the BTZ global time t o is ω = dψdt o = dτdt o dψdτ = √ ( R − r )( R − r − ) lR ¯ ω = r − lr + − r + r − lR , i.e., ω = 1 l r − r + − l r − r + r R . (40)This ω is the angular velocity of the reference frame which detects a perfect fluidrelative to the shell’s proper frame redshifted to the global time t o . It is alwayspositive since R ≥ r + .In addition, we can calculate the angular velocity of the reference frame at theshell that detects a perfect fluid relative to infinity t o . Call this angular velocity ω ∞ . Then, the angular velocity ω ∞ of the reference frame at the shell that detectsa perfect fluid relative to infinity t o , is clearly the sum of the angular velocity ω of the reference frame that detects a perfect fluid relative to the proper rest framewith time τ redshifted to the global time t o , plus the angular velocity of the shellrelative to the global time t o , i.e., ω ∞ = dφdt = dψdt + Ω = ω + Ω . (41)The angular velocity ω ∞ of the reference frame at the shell that detects a perfectfluid relative to a nonrotaing frame at infinity with time t o , is then ω ∞ = 1 l r − r + , (42)where we have substituted Eqs. (28) and (40) in Eq. (41). Eq. (42) does not dependon R , thus ω ∞ is independent of the shell, it is an intrinsic property of the spacetime. otating thin shells in (2+1)-dimensional spacetimes This does not occur in the corresponding 3+1 dimensional spacetimes .Remarkably, the angular velocity ω ∞ given in Eq. (42) coincides with the hori-zon’s angular velocity of the BTZ black hole. Indeed, the pure BTZ spacetime,given by Eq. (3) for 0 ≤ r < ∞ , possesses a null Killing vector normal to thehorizon given by n α = t αo + ω + φ α , where ω + is the horizon angular velocity
1, 2 .The condition n α n α = 0 gives g + 2 ω + g φ + ω g φφ = 0, i.e., using Eq. (3), G J r − ω + GJ + ω r = 0. The solution is ω + = GJr , i.e., ω + = 1 l r − r + . (43)Thus, the angular velocity of the special reference frame at the shell ω ∞ has thesame expression as the angular velocity of the BTZ pure vacuum horizon ω + as seenby observers at infinity. Machian effects and the dragging of the inertial frames
We can continue to explore the effects due to the rotation of the ring. As we havediscussed, any inner line or interior observer with ψ constant is moving with respectto t o at infinity with angular velocity dφ/dt o = Ω. Now, observers with constant ψ are inertial observers for the empty interior metric and these observers rotatewith respect to AdS infinity with Ω. Let us call the angular velocity of the interiorobservers as Ω in with Ω in = Ω. This Ω in is caused by the presence of the rotationof the shell, and it is called the dragging of inertial frames. From Eq. (28) we have,Ω in = 1 l r − r + r R . (44)We can compare this angular velocity of interior observers with the angular velocityof the shell ω ∞ given in Eq. (42). The ratio between the two angular velocitiesΩ in /ω ∞ is then Ω in ω ∞ = r R . (45)For R large the inside observers rotate with a small fraction of the shell’s referenceframe in question. For R = r + , i.e., when the shell approaches its own gravitationalradius, the inside observers corotate with the shell Ω in ω ∞ = 1, indicating a very strongdragging effect and a prominent example of Mach’s principle.The rotating effect on the inside region relative to infinity caused by the presenceof a rotating shell as an example of the dragging of inertial frames effect is well knownin 3+1 general relativity, see e.g. . We see that in 2+1 dimensions this phenomenonalso occurs. Jos´e P. S. Lemos, Francisco J. Lopes, Masato Minamitsuji
5. Energy Conditions5.1.
The weak energy condition
In Eq. (25) we have imposed that the shell’s radius is always larger than the gravi-tational radius, R ≥ r + . We discuss here the weak and the dominant energy condi-tions. The weak energy condition is automatically satisfied since we have λ and p non-negative. The dominant energy condition
The dominant energy condition can be discussed in the the frame that detects thematter as a perfect fluid, where the quantities ¯ λ and ¯ p are the relevant ones. In thisframe the dominant energy condition states that¯ p ≤ ¯ λ . (46)From Eqs. (37) and (38) this means R − r − ≤ R − r . (47)So, first, Eq. (47) can be obeyed when R → ∞ . In this case one has fromEqs. (37) and (38) that ¯ λ = πGl r − r − R and ¯ p = πGl r − r − R . Now, from Eq. (20)one has that for large R that m = Rl M . So in this limit ¯ λ = M πR , ¯ p = M πR , and¯ p = ¯ λ . For M constant the matter disappears in the R → ∞ limit. However, it canbe the case that M grows proportional to R , so that ¯ λ is constant. In this case thelimit R → ∞ is also well defined and the shell obeys the dominant energy conditionsince ¯ p = ¯ λ .Second, Eq. (47) is satisfied when r + = r − (or J = ml ), i.e., the outer BTZspacetime is extremal and the observers that see a perfect fluid are rotating at theextremality limit, i.e., at the speed of light, R ¯ ω = 1, see Eq. (39). In this case, forthese observers the matter in the shell becomes massless as expected and so ¯ λ = 0as one can check from Eq. (37). Now, the pressure for these observer also vanish,see Eq. (38). This can be interpreted considering that the AdS attraction due to thenegative cosmological constant is now purely balanced by some kind of centrifugalforce due to this maximum rotating speed, making the pressure going to zero. Inthe limit one has ¯ p = ¯ λ and so the dominant energy conditon is satisfied for any R . For all other settings the dominant energy condition is violated. This is due tothe fact that the spacetime is AdS, with a negative cosmological constant, which asseen as a perfect fluid does not obey the dominan energy condition.
6. Conclusions
We have studied the dynamics of a rotating thin matter ring shell in (2+1)-dimensional spacetime with a negative cosmological constant. The outside metric isthe BTZ metric, asymptotically AdS, and the inside metric is the empty spacetimemetric with a negative cosmological constant. We obtained the shell’s rest energy otating thin shells in (2+1)-dimensional spacetimes density, the pressure and angular momentu flux density by using the junction con-ditions. Due to rotation, the shell’s energy-momentum tensor was not in the formof a perfect fluid. We then attempted to write the energy-momentum tensor in aperfect fluid form and obtained the thin shell’s energy density, pressure and angularvelocity for the frame where one sees the thin shell as a perfect fluid with no angu-lar momentum flux. We have found that Machian effects occur in 2+1 dimensionsand that frame dragging is present in these spacetimes. We analyzed the weak anddominant energy conditions of the system. The dominant energy condition impliesthat there are only two valid configurations: the R → ∞ case with its subcases,the trivial M = 0 shell and the nontrivial M = 2 πλR shell, and the extremal case r + → r − (or J = ml ). Appendix: The energy-momentum tensor in a perfect fluid form,another derivation
Here we derive properties of the matter in a frame that sees it as a perfect fluid.The intrinsic metric on the shell is given by ds = − dτ + R dψ . (48)In the rest frame of the shell, the energy-momentum tensor S ab is given by λ = 18 πGl (cid:16) − R q ( R − r )( R − r − ) (cid:17) , (49) p = 18 πGl (cid:16) R − r r − R q ( R − r )( R − r − ) − (cid:17) , (50) j = r + r − πGlR . (51)Now the metric on the shell, Eq. (48), is invariant under the boost¯ τ = γ (cid:16) τ − ¯ ωR ψ (cid:17) , ¯ ψ = γ (cid:16) ψ − ¯ ωτ (cid:17) , (52)with γ ≡ √ − ¯ ω R . (53)Indeed, using Eq. (52) in (48) one gets, ds = − d ¯ τ + R d ¯ ψ . (54)Defining S ¯ τ ¯ τ = − ¯ λ , S ¯ ψ ¯ ψ = ¯ p , and S ¯ τ ¯ ψ = ¯ j , and transforming S ab appropriately,one obtains ¯ λ = γ (cid:16) λ − ωj + ¯ ω R p (cid:17) , (55)¯ p = γ (cid:16) p − ωj + ¯ ω R λ (cid:17) , (56)¯ j = γ n(cid:16) ω R (cid:17) j − ¯ ωR (cid:0) λ + p (cid:1)o . (57) Jos´e P. S. Lemos, Francisco J. Lopes, Masato Minamitsuji
Imposing ¯ j = 0 fixes ¯ λ = 18 πGl (cid:16) − s R − r R − r − (cid:17) , (58)¯ p = 18 πGl (cid:16)s R − r − R − r − (cid:17) , (59)¯ ω = r − Rr + s R − r R − r − , (60)which agreed with the results in Sec. 3.1. Thus, λ = γ (cid:16) ¯ λ + ¯ ω R ¯ p (cid:17) , (61) p = γ (cid:16) ¯ p + ω R ¯ λ (cid:17) , (62) j = ¯ ωR γ (cid:0) ¯ λ + ¯ p (cid:1) , . (63)The set of Eqs. (61)-(62) can be written in a perfect fluid form S ab = ¯ λ u a u b + ¯ p ( u a u b + h ab ) , (64)with u τ = γ, u ψ = γ ¯ ω, u τ = − γ, u ψ = γ ¯ ωR , (65) h ττ = − , h ψψ = R , h τψ = 0 . (66) Acknowledgments
We thank Gon¸calo Quinta and Jorge Rocha for conversations. We thank FCT-Portugal for financial support through Project No. PEst-OE/FIS/UI0099/2014.FJL thanks FCT for financial support through project Incentivo/FIS/UI0099/2014.MM thanks FCT for grant number SFRH/BPD/88299/2012.
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