LenseâThirring precession in PlebaÅskiâDemiaÅski Spacetimes

Eur. Phys. J. C (2013) 73:2536DOI 10.1140/epjc/s10052-013-2536-1

Regular Article - Theoretical Physics

Lense–Thirring precession in Plebanski–Demianski Spacetimes

Chandrachur Chakraborty1,a, Partha Pratim Pradhan2,b

1Saha Institute of Nuclear Physics, Kolkata 700064, India2Department of Physics, Vivekananda Satabarshiki Mahavidyalaya, Manikpara, West Midnapur, West Bengal 721513, India

Received: 25 June 2013 / Revised: 27 July 2013© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Abstract An exact expression of Lense–Thirring preces-sion rate is derived for non-extremal and extremal Plebanski–Demianski spacetimes. This formula is used to find the ex-act Lense–Thirring precession rate in various axisymmetricspacetimes i.e., Kerr–Newman, Kerr–de Sitter etc. We alsoshow that if the Kerr parameter vanishes in the Plebanski–Demianski spacetime, the Lense–Thirring precession doesnot vanish due to the existence of NUT charge. To derive theLense–Thirring precession rate in the extremal Plebanski–Demianski spacetime, we first derive the general extremalcondition for Plebanski–Demianski spacetimes. This gen-eral result could be applied to obtain the extremal limit inany stationary and axisymmetric spacetimes.

1 Introduction

The axisymmetric vacuum solution of the Einstein equa-tions are used to describe the various characteristics of dif-ferent spacetimes. The most important and physical space-time is Kerr spacetime [1], describing a rotating black holewhich possesses a finite angular momentum J . The Kerrspacetime with a finite charge Q is expressed as a Kerr–Newman black hole. Actually inclusion of Cosmologicalconstant may arises some complexity in calculations. With-out this particular constant the spacetimes possess two hori-zons, namely event horizon and Cauchy horizon. But thepresence of the cosmological constant leads to an extrahorizon—the cosmological horizon. All of these spacetimescan be taken as the special cases of the most general axisym-metric spacetime of Petrov type D which was first given byPlebanski and Demianski (PD) [2]. This spacetime containsseven parameters—acceleration, mass (M), Kerr parame-ter (a, angular momentum per unit mass), electric charge(Qc), magnetic charge (Qm), NUT parameter (n), and the

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Cosmological constant. At present, this is the most generalknown axially symmetric vacuum solution of the Einsteinfield equation. This solution is important at the present timebecause it is now being used in many ways. People work-ing in semi-classical quantum gravity have used this type ofmetric to investigate the pair production of black holes incosmological backgrounds ([3]). Some people are workingto extend this type of solution to higher dimensions. But thisspacetime is still not well-understood at the classical levelof general relativity. In particular, the physical significanceof the parameters employed in the original forms are onlyproperly identified in the most simplified special cases andthe most general PD metric covers all spacetimes which arewell-known to us till now.

In this paper, we shall investigate an important effectof classical General Relativity in PD spacetime: the drag-ging of inertial frames which was discovered by Lense andThirring [4]. It is well-known that any axisymmetric andstationary spacetime with angular momentum (rotation) isknown to exhibit an effect called Lense–Thirring (LT) pre-cession whereby locally inertial frames are dragged alongthe rotating spacetime, making any test gyroscope in suchspacetimes precess with a certain frequency called the LTprecession frequency [4].

More generally, we can say that frame-dragging effectis the property of all stationary spacetimes which may ormay not be axisymmetric [5]. It has been discussed in detailabout the LT effect in non-axisymmetric stationary space-times in a very recent paper by Chakraborty and Majumdar[6] in where the authors have shown that only Kerr parame-ter is not responsible for the LT precession, NUT parameteris equally important to continue the frame-dragging effect.

Hackmann and Lämmerzahl have shown that the LT pre-cession vanishes (Eq. (45) of [7]) in PD spacetimes (withvanishing acceleration of the gravitating source), if the Kerrparameter a = 0. But it is not the actual scenario. We shallshow in our present work, the LT precession does not vanishdue to the presence of NUT charge n (angular momentum

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of 9 Eur. Phys. J. C (2013) 73:2536

monopole [8]) in the PD spacetimes with zero angular mo-mentum (J = a = 0).

It may be noted that the observations of LT precessiondue to locally inertial frame-dragging have so far focusedon spacetimes where the curvatures are small enough; e.g.,the LT precession in the earth’s gravitational field which wasprobed recently by Gravity Probe B. There has been so farno attempt to measure LT precession effects due to frame-dragging in strong gravity regimes [9]. Our LT precessionformulas for different spacetimes are valid in strong as wellas weak gravitational field as we do not make any approxi-mation to derive the LT precession rate. We also note that theearth’s gravitational is described by Kerr geometry which isphysically reliable. But the most general axisymmetric PDspacetime is not physically reliable till now because the ex-istence of some parameters (e.g., NUT parameter (n), Qe ,Qm) of PD spacetimes are not yet proved.

Lynden-Bell and Nouri-Zonos [10] studied a Newtoniananalogue of monopole spacetimes and discussed the obser-vational possibilities for (gravito)magnetic monopoles (n).In fact, the spectra of supernova, quasars, or active galac-tic nuclei might be good candidates to infer the existenceof gravito-magnetic monopoles [11]. The Kerr–Taub-NUTsolutions representing relativistic thin disks are of great as-trophysical importance since they can be used as modelsof certain galaxies, accretion disks, the superposition of ablack hole and a galaxy or an accretion disk as in the case ofquasars.

In this manuscript, we will derive the exact LT preces-sion rates in non-extremal PD spacetimes without invokingthe weak field approximation. We will also find the exact LTprecession rates for the other axisymmetric stationary space-times i.e. Kerr–Newman, Kerr–de Sitter etc. We will furthershow that the non-vanishing LT precession in ‘zero angu-lar momentum’ PD spacetimes and find the extremal limitfor PD spacetime including others axisymmetric and station-ary spacetimes. Finally, we will find the exact LT precessionrates in the extremal PD spacetime and others axisymmetricextremal spacetimes.

The structure of the paper as follows. In Sect. 2, we re-view the general Lense–Thirring precession formula in sta-tionary and axisymmetric spacetimes and derive the exactLense–Thirring precession rate in PD spacetimes with van-ishing acceleration of the gravitating source and discussthe exact LT precession rates in some other stationary, ax-isymmetric spacetimes as special cases of PD spacetimes. Ifthe Kerr parameter vanishes in PD spacetimes, the frame-dragging effect does not vanish due to the existence of NUTcharge. It is shown in a subsections of Sect. 2, as a specialcase of LT precession in PD spacetimes. In Sect. 3, we derivethe more general extremal condition for PD spacetimes anddiscuss the exact LT precession rates in PD spacetimes andalso other various extremal axisymmetric spacetimes as the

special cases of PD spacetimes. A short discussion closesthe paper in Sect. 4.

2 Derivation of the Lense–Thirring PrecessionFrequency

The exact LT frequency of precession of test gyroscopesin strongly curved stationary spacetimes, analyzed withina Copernican frame, is expressed as a co-vector given interms of the timelike Killing vector fields K of the stationaryspacetime, as (in the notation of Eq. (1.159) of Ref. [5])

Ω = 1

2K2∗ (K ∧ dK) (1)

or

Ωμ = 1

2K2ηνρσ

μ Kν∂ρKσ , (2)

where ημνρσ represent the components of the volume-formin spacetime and K & Ω denote the one-form of K & Ω ,respectively. Ω will vanish if and only if (K ∧ dK) does.This happens only in a static spacetime. Using the coordi-nate basis form of K = ∂0, the co-vector components areeasily seen to be Kμ = gμ0. This co-vector could also bewritten in the following form:

K = g00dx0 + g0idxi (3)

Now

dK = g00,kdxk ∧ dx0 + g0i,kdxk ∧ dxi, (4)

(K ∧ dK) = (g00g0i,j − g0ig00,j )dx0 ∧ dxj ∧ dxi

+ g0kg0i,j dxk ∧ dxj ∧ dxi, (5)

(K ∧ dK) = (g00g0i,j − g0ig00,j ) ∗ (dx0 ∧ dxj ∧ dxi

)

+ g0kg0i,j ∗ (dxk ∧ dxj ∧ dxi

)(6)

It can be shown that

(dx0 ∧ dxj ∧ dxi

) = η0jilglμdxμ = − 1√−gεjilglμdxμ

∗ (dxk ∧ dxj ∧ dxi

) = ηkji0g0μdxμ = − 1√−gεkjig0μdxμ

(7)

From Eq. (6) we get

∗(K ∧ dK)

= εij l√−g

[(g00g0i,j − g0ig00,j )

(gl0dx0 + glkdxk

)

− g0lg0i,j

(g00dx0 + g0kdxk

)](8)

Eur. Phys. J. C (2013) 73:2536 of 9

Simplifying the above equation we find

∗(K ∧ dK)

= εij l√−g

[g0i,j (g00gkl − g0kg0l) − g0igklg00,j

]dxk (9)

Now, using Eq. (2) we find that the spatial components ofthe precession rate (in the chosen frame) is

Ωk = 1

2

εij l

g00√−g

[g0i,j (g00gkl − g0kg0l) − g0igklg00,j

](10)

[using K2 = g00].The vector field corresponding to the LT precession co-

vector in (10) can be expressed as

Ω = Ωμ∂μ

= gμνΩν∂μ

= gμkΩk∂μ [as Ω0 = 0]= g0kΩk∂0 + gnkΩk∂n

= 1

2

εij l√−g

[g0i,j

(∂l − g0l

g00∂0

)− g0i

g00g00,j ∂l

](11)

For the axisymmetric spacetime, the only non-vanishingcomponent is g0i = g0φ , i = φ and j, l = r, θ ; substitutingthese in Eq. (11), the LT precession frequency vector is ob-tained as

ΩLT = 1

2√−g

[(g0φ,r − g0φ

g00g00,r

)∂θ

−(

g0φ,θ − g0φ

g00g00,θ

)∂r

](12)

3 Non-extremal Case

3.1 Plebanski–Demianski Spacetimes

The PD spacetime is the most general axially symmetricvacuum solution of the Einstein equation. The line elementof six parameters (as we consider vanishing acceleration)PD spacetimes can be written as (taking G = c = 1) [7]

ds2 = − �

p2(dt − Adφ)2 + p2

�dr2 + p2

Ξdθ2

+ Ξ

p2sin2 θ(a dt − B dφ)2 (13)

where

p2 = r2 + (n − a cos θ)2,

A = a sin2 θ + 2n cos θ, B = r2 + a2 + n2

� = (r2 + a2 − n2)

(1 − 1

2

(r2 + 3n2)

)

− 2Mr + Q2c + Q2

m − 4n2r2

2

Ξ = 1 + a2 cos2 θ

2− 4an cos θ

2

(14)

1 2 = Λ denotes the Cosmological constant divided by three

(the Cosmological constant Λ′ appears in the metric as Λ′3

which we can take in a more simplified form as Λ′3 = Λ),

represents the Plebanski–Demianski(–de Sitter)1 spacetimesand if 2 is replaced by − 2, it represents the PD–AdSspacetimes. So, our metric gμν is thus following:

gμν =

⎛

⎜⎜⎜⎜⎝

− 1p2 (� − a2Ξ sin2 θ) 0 0 1

p2 (A� − aBΞ sin2 θ)

0 p2

�0 0

0 0 p2

Ξ0

1p2 (A� − aBΞ sin2 θ) 0 0 1

p2 (−A2� + B2Ξ sin2 θ)

⎞

⎟⎟⎟⎟⎠

(15)

The various metric components can be read off from theabove metric (15). Likewise,

√−g = p2 sin θ (16)

1In PD metric, Λ appears as either “positive” (+Λ) or “negative”(−Λ). In the whole paper, the PD metric with +Λ represents thePlebanski–Demianski–de Sitter or PD–dS spacetimes and the PD met-ric with −Λ represents the Plebanski–Demianski–anti-de Sitter or PD–AdS spacetimes.

Substituting the metric components into Eq. (12) we caneasily get the LT precession rate in PD spacetimes. But thereis a problem in that formulation since the precession formulais in coordinate basis. So, we should transform the preces-sion frequency formula from the coordinate basis to the or-thonormal ‘Copernican’ basis: first note that

ΩLT = Ωθ∂θ + Ωr∂r (17)

Ω2LT = grr

(Ωr

)2 + gθθ

(Ωθ

)2 (18)

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Next, in the orthonormal ‘Copernican’ basis at rest inthe rotating spacetime, with our choice of polar coordinates,ΩLT can be written as

ΩLT = √grrΩ

r r + √gθθΩ

θ θ (19)

where θ is the unit vector along the direction θ . Our finalresult of LT precession in non-accelerating PD spacetime isthen

ΩPDLT =

√�

p

[a(Ξ cos θ + (2n − a cos θ) a

2 sin2 θ)

� − a2Ξ sin2 θ

− a cos θ − n

p2

]r +

√Ξ

pa sin θ

×[r − M − r

2 (a2 + 2r2 + 6n2)

� − a2Ξ sin2 θ− r

p2

]θ (20)

From the above expression we can easily derive the LTprecession rates for various axisymmetric stationary space-times as special cases of PD spacetime.

3.2 Special Cases

(a) Schwarzschild and Schwarzschild–de Sitter spacetimes:As the Schwarzschild and Schwarzschild–de Sitter space-times both are static and a = Λ = Qc = Qm = n = 0, theinertial frames are not dragged along it. So, we cannot seeany LT effect in these spacetimes. This is a very well-knownfeature of static spacetimes.

(b) Kerr Spacetimes: LT precession rate for non-extremalKerr spacetimes is already discussed in detail in the paper byChakraborty and Majumdar [6]. Setting Λ = Qc = Qm =n = 0 in Eq. (20) we can recover that result (Eq. (19) of [6])which is applicable for Kerr spacetimes.

(c) Kerr–Newman spacetime: Rotating black hole space-times with electric charge Qc and magnetic charge Qm aredescribed by Kerr–Newman metric, which is quite importantin General Relativity. Setting Λ = n = 0 in Eq. (20), we caneasily get the LT precession in Kerr–Newman spacetime. Itis thus (taking Q2

c + Q2m = Q2),

ΩKNLT = a

ρ3(ρ2 − 2Mr + Q2)

[√�

(2Mr − Q2) cos θ r

+ (M

(2r2 − ρ2) + rQ2) sin θ θ

](21)

In the Kerr–Newman spacetime,

� = r2 − 2Mr + a2 + Q2 and ρ2 = r2 + a2 cos2 θ (22)

From the expression (21) of LT precession in Kerr–Newmanspacetime we can see that at the polar plane, the LT preces-

sion vanishes for the orbit r = Q2

2M, though the spacetime is

rotating (a �= 0). So, if a gyroscope rotates in a polar orbit

of radius r = Q2

2Min this spacetime, the gyroscope does not

experience any frame-dragging effect. So, if any experimentis performed in future by which we cannot see any LT pre-cession in that spacetime, it may be happened that the spec-ified spacetime is a Kerr–Newman black hole and the gyro-

scope is rotating in a polar orbit whose radius is r = Q2

2M.

This is a very interesting feature of the Kerr–Newman ge-ometry. Though the spacetime is rotating with the angularmomentum J , the nearby frames are not dragged along it.Without this particular orbit the LT precession is continuedin everywhere in this spacetimes.

(d) Kerr–de Sitter spacetimes: Kerr–de Sitter spacetimeis more realistic, when we do not neglect the Cosmologicalconstant parameter (though its value is very small, it may bevery useful in some cases, where we need to very precisecalculation). Setting n = Qc = Qm = 0, we get the follow-ing expression for Kerr–de Sitter spacetime:

ΩKdSLT = a

ρ3(ρ2 − 2Mr − 1 2 (a4 + a2r2 − a4 sin2 θ cos2 θ))

×[√

�

(2Mr + 1

2ρ4

)cos θ r

+ √Ξ

[M

(2r2 − ρ2)

+ r

2

{a4 + a2r2 − a4 sin2 θ cos2 θ

− ρ2(a2 + 2r2)}]

sin θ θ

](23)

where

� = (r2 + a2)

(1 − r2

2

)− 2Mr and Ξ = 1 + a2

2cos2 θ

(24)

3.3 Non-vanishing Lense–Thirring Precession in ‘ZeroAngular Momentum’ Plebanski–DemianskiSpacetimes

This subsection can be regarded as a special case of non-accelerating PD spacetime in where we take that PD space-time is not rotating, we mean the Kerr parameter a = 0. Ina very recent paper, Hackmann and Lämmerzahl show thatLT effect vanishes (Eq. (45) of [7]) due to the vanishing Kerrparameter. But we can see easily from Eq. (20) that if a van-ishes in PD spacetime, the LT precession rate will be

ΩPDLT |a=0 = n

√�|a=0

p3r (25)

Eur. Phys. J. C (2013) 73:2536 of 9

where

�|a=0 = (r2 − n2)

(1 − 1

2

(r2 + 3n2)

)

− 2Mr + Q2c + Q2

m − 4n2r2

2and

p2 = r2 + n2

(26)

So, the LT precession does not vanish due to the van-ishing Kerr parameter. The above expression reveals thatNUT charge n is responsible for the LT precession in ‘zeroangular momentum’ PD spacetimes. Here, M representsthe “gravito-electric mass” or ‘mass’ and n represents the“gravito-magnetic mass” or ‘dual’ (or ‘magnetic’) mass [10]of this spacetime. It is obvious that the spacetime is not in-variant under time reversal t → −t , signifying that it musthave a sort of ‘rotational sense’ which is analogous to a mag-netic monopole in electrodynamics. One is thus led to theconclusion that the source of the non-vanishing LT preces-sion in the “rotational sense” arising from a non-vanishingNUT charge. Without the NUT charge, the spacetime isclearly hypersurface orthogonal and frame-dragging effectvanishes. This ‘dual’ mass has been investigated in detail inRef. [12, 13] and it is also referred as an ‘angular momentummonopole [8] in Taub-NUT spacetime. This implies that theinertial frame dragging seen here in such a spacetime can beidentified as a gravito-magnetic effect.

In [7] the authors have investigated that the timelikegeodesic equations in the PD spacetimes. The orbital planeprecession frequency (Ωφ −Ωθ) is computed, following theearlier work of Ref. [11, 14, 15], and a vanishing result en-sues. This result is then interpreted as a signature for a nullLT precession in the ‘zero angular momentum’ PD space-time.

We would like to say that what we have focused in thepresent paper is quite different from the ‘orbital plane pre-cession’ considered in [7]. Using a ‘Copernican’ frame, wecalculate the precession of a gyroscope which is moving inan arbitrary integral curve (not necessarily geodesic). Withinthis frame, a torqueless gyro in a stationary but not staticspacetime held fixed by a support force applied to its cen-ter of mass undergoes LT precession. Since the Copernicanframe does not rotate (by construction) relative to the inertialframes at asymptotic infinity (“fixed stars”), the observedprecession rate in the Copernican frame also gives the pre-cession rate of the gyro relative to the fixed stars. It is thus,more an intrinsic property of the classical spin of the space-time (as a torqueless gyro must necessarily possess), in thesense of a dual mass, rather than an orbital plane precessioneffect for timelike geodesics in a Taub-NUT spacetime.

In our case, we consider the gyroscope equation [5] in anarbitrary integral curve

∇uS = 〈S,a〉u (27)

where a = ∇uu is the acceleration, u is the four velocityand S indicates the spacelike classical spin four vector Sα =(0,S) of the gyroscope. For geodesics a = 0 ⇒ ∇uS = 0.

In contrast, Hackmann and Lämmerzahl [7] considerthe behavior of massive test particles with vanishing spinS = 0, and compute the orbital plane precession rate for suchparticles, obtaining a vanishing result. We are thus led toconclude that since two different situations are being con-sidered, there is no inconsistency between our results andtheirs.

We note that the detailed analyses on LT precession inKerr–Taub-NUT, Taub-NUT [16–18] and massless Taub-NUT spacetimes has been done in [6].

4 Extremal Case

4.1 Extremal Plebanski–Demianski Spacetime

In this section, we would like to describe the LT preces-sion in extremal PD spacetime, whose non-extremal caseis already described in the previous section. To get the ex-tremal limit in PD spacetimes we should first determine theradius of the horizons rh which can be determined by setting�|r=rh = 0. We can make a comparison of coefficients in

� = − 1

2r4 +

(1 − a2

2− 6n2

2

)r2 − 2Mr

+[(

a2 − n2)(

1 − 3n2

2

)+ Q2

c + Q2m

]

= − 1

2

[r4 + (

a2 + 6n2 − 2)r2 + 2M 2r + b]

= − 1

2Π4

i=1(r − rhi) (28)

where

b = (a2 − n2)(3n2 − 2) − 2(Q2

c + Q2m

)(29)

and rhi (i = 1,2,3,4) denotes the zeros of �. From thiscomparison we can conclude that for the PD–AdS (whenΛ is negative) black hole, there are two separated positivehorizons at most, and � is positive outside the outer horizonof the PD black hole. In the same way, we can conclude forthe PD–dS (when Λ is positive) black hole that there arethree separated positive horizons at most, and � is negativeoutside the outer horizon of the black hole. Both in the abovecases, when the two horizons of the PD black hole coincide,the black hole is extremal [19].

If we consider the extremal PD black hole, we have tomake a comparison of coefficients in

� = (r − x)2(a2r2 + a1r + a0

)

= − 1

2

[r4 + (

a2 + 6n2 − 2)r2 + 2Mr 2 + b]

(30)

of 9 Eur. Phys. J. C (2013) 73:2536

with a0, a1, a2 being real [7]. From this comparison we canget the following for the PD–AdS spacetime:

bA

x2− 3x2 = a2 + 6n2 + 2 (31)

x3 − bA

x= −M 2 (32)

where bA represents the value of b at PD–AdS spacetimes:

bA = (a2 − n2)(3n2 + 2) + 2(Q2

c + Q2m

)(33)

Solving Eq. (31) for x, we get

x =√

1

6

[−( 2 + a2 + 6n2

) +√(

2 + a2 + 6n2)2 + 12bA

]

(34)

Similarly, we can obtain for the PD–dS black hole

b

x2− 3x2 = a2 + 6n2 − 2 (35)

x3 − b

x= M 2 (36)

In these equations x is positive and related to the coin-cided horizon of the extremal PD–dS black hole.

Solving Eq. (35) for x, we get

x+ =√

1

6

[ 2 − a2 − 6n2 +

√( 2 − a2 − 6n2

)2 + 12b]

(37)

x− =√

1

6

[ 2 − a2 − 6n2 −

√( 2 − a2 − 6n2

)2 + 12b]

(38)

where x+ and x− indicate the outer horizon and inner hori-zons, respectively.

This can be seen by calculating

d2�

dr2

∣∣∣∣r=x+

= − 2

2

√( 2 − a2 − 6n2

)2 + 12b (39)

and

d2�

dr2

∣∣∣∣r=x−

= 2

2

√( 2 − a2 − 6n2

)2 + 12b (40)

For the PD–dS black hole, on the outer extremal hori-zon, d�

dr= 0 and d2�

dr2 < 0 and on the inner extremal hori-

zon d2�

dr2 > 0. Now, we can solve M and a from the twoEqs. (35), (36):

M = x[x4 + 2x2(3n2 − 2) + (3n2 − 2)(7n2 − 2) + 2(Q2c + Q2

m)] 2( 2 + x2 − 3n2)

(41)

a2e = 3x4 + (6n2 − 2)x2 + n2(3n2 − 2) + 2(Q2

c + Q2m)

(3n2 − 2 − x2)(42)

From the above values of a2e and M , we get

a2e = −M 2 [3x4 + (6n2 − 2)x2 + n2(3n2 − 2) + 2(Q2

c + Q2m)]

x[x4 + 2x2(3n2 − 2) + (3n2 − 2)(7n2 − 2) + 2(Q2c + Q2

m)] (43)

The ranges of x and ae are determined from the followingexpressions:

x2 <

( 2 +

√ 2 − 12Q2

6− n2

)(44)

0 < a2e

<[(

7 2 − 24n2)

−√(

7 2 − 24n2)2 − (

4 − 12 2(Q2

c + Q2m

))](45)

Due to the presence of the Cosmological constant, thereexist four roots of x in Eq. (42). When the Cosmological

constant 1l2

→ 0, Eq. (41) and Eq. (42) reduces to

x = M (46)

and,

a2e = x2 + n2 − Q2

c − Q2m (47)

or,

a2e = M2 + n2 − Q2

c − Q2m (48)

respectively.

Eur. Phys. J. C (2013) 73:2536 of 9

Now, the line element of extremal PD spacetimes can bewritten as

ds2 = −�e

p2e

(dt − Ae dφ)2 + p2e

�e

dr2

+ p2e

Ξe

dθ2 + Ξe

p2e

sin2 θ(ae dt − Be dφ)2 (49)

and the final LT precession rate in the extremal PD space-time is

ΩePDLT =

√�e

pe

[ae(Ξe cos θ + (2n − ae cos θ) ae

2 sin2 θ)

�e − a2eΞe sin2 θ

− ae cos θ − n

p2e

]r

+√

Ξe

pe

ae sin θ

[r − M − r

2 (a2e + 2r2 + 6n2)

�e − a2eΞe sin2 θ

− r

p2e

]θ (50)

where

�e = − 1

2(r − x)2

(r2 + 2rx + b

x2

),

Ξe = 1 + a2e

2cos2 θ − 4aen

2cos θ,

pe = r2 + (n − ae cos θ)2,

Ae = ae sin2 θ + 2n cos θ,

Be = r2 + a2e + n2

(51)

and the value of ae is determined from Eq. (42) and the rangeof x and ae (‘e’ stands for the extremal case) are determinedfrom Eq. (44) and (45), respectively. It could be noted that,for the extremal PD–dS spacetimes, there are upper limitingvalues for angular momentum and extremal horizon of theblack hole. Substituting all the above mentioned values andranges in Eq. (50), we get the exact LT precession rate inextremal PD spacetimes.

4.2 Extremal Kerr Spacetime

Substituting ae = M and Λ = 1 2 = Qc = Qm = n = 0 in

Eq. (50) we can easily get the LT precession rate in extremalKerr spacetime:

ΩeKLT = M2[2r(r − M) cos θ r + (r2 − M2 cos2 θ) sin θ θ ]

(r2 + M2 cos2 θ)32 (r2 − 2Mr + M2 cos2 θ)

(52)

The above result is also coming from Eq. (50), the generalLT precession rate for extremal PD spacetime.

Case I: On the polar region, i.e. θ = 0, the ΩLT becomes

ΩeKLT = 2M2r

(r2 + M2)3/2(r − M)(53)

Case II: On the equator, i.e. θ = π/2, the ΩLT becomes

ΩeKLT = M2

r2(r − 2M)(54)

It could be easily seen from Eq. (53) that LT precession di-verges at r = M , since r = M is the only direct ISCO inextremal Kerr geometry which coincides with the principalnull geodesic generator of the horizon [20] and it is also theradius of the horizon which is a null surface. As the LT pre-cession is not defined at the null surfaces and the spacelikesurfaces, the general LT precession formula is derived onlyconsidering that the observer is at rest in a timelike Killingvector field. We have not also incorporated the LT effect forany null geodesics. So, our formula is valid only for r > M

(outside the horizon).

4.3 Extremal Kerr–Newman Spacetime

Substituting a2 = M2 − (Q2c + Q2

m) = M2 − Q2 and Λ =n = 0 in Eq. (21), we can obtain the following LT precessionrate for extremal KN black hole:

ΩeKNLT =

√M2 − Q2

ρ3(ρ2 − 2Mr + Q2)

[(r −M)

(2Mr −Q2) cos θ r

+ (M

(2r2 − ρ2) + rQ2) sin θ θ

](55)

where

ρ2 = r2 + (M2 − Q2) cos2 θ (56)

From the above expression (Eq. (55)), we can make a similarcomment again, like the extremal Kerr–Newman black holethat the gyroscope which is rotating in a polar orbit of ra-

dius r = Q2

2Mcannot experience any frame-dragging effect.

Apparently, it seems that this argument is also true for thegyroscope which is rotating at r = M orbit. But this is nottrue, because this is the horizon of extremal Kerr–Newmanspacetime. So, r = M is a null surface. The general formula(Eq. (12)) which we have considered in our whole paper, isvalid only in timelike spacetimes (outside the horizon), notin any null or spacelike regions.

4.4 Extremal Kerr–de Sitter Spacetime

The extremal Kerr–de Sitter spacetime is interesting be-cause it involves the Cosmological constant. Setting n =

of 9 Eur. Phys. J. C (2013) 73:2536

Qc = Qm = 0 in Eq. (20), we can find the following ex-pression of LT precession at extremal Kerr–de Sitter space-times:

ΩeKdSLT = ae

ρ3(ρ2 −2Mr − 1 2 (a4

e +a2e r

2 −a4e sin2 θ cos2 θ))

×[√

�e

(2Mr + ρ4

2

)cos θ r

+ √Ξe

[M

(2r2 − ρ2) + r

2

{a4e + a2

e r2

− a4e sin2 θ cos2 θ − ρ2(a2

e + 2r2)}]

sin θ θ

]

(57)

where

ρ2 = r2 + a2e cos2 θ, Ξe = 1 + a2

e

2cos2 θ (58)

and

�e = − 1

2(r − x)2

(r2 + 2xr − a2 2

x2

)

Using Eqs. (42), (41), we see that the values for a and M

are

a2e = ( 2 − 3x2)x2

( 2 + x2)

M = x( 2 − x2)2

2( 2 + x2)

(59)

where the horizons are at

x+ =√

1

6

[ 2 − a2 +

√( 2 − a2

)2 − 12a2 2]

x− =√

1

6

[ 2 − a2 −

√( 2 − a2

)2 − 12a2 2]

(60)

where x+ and x− indicate the outer horizon and inner hori-zon. The ranges of a2

e and x are the following:

0 < a2e < (7 − 4

√3) 2 and x2 < 2/3 (61)

which is already discussed in [19]. Substituting all the abovevalues in Eq. (57) and taking the ranges of ae and x, we ob-tain the exact LT precession rate in extremal Kerr–dS space-times.

4.5 Extremal Kerr–Taub-NUT spacetime

To derive the extremal limit in Kerr–Taub-NUT spacetimewe set

� = r2 − 2Mr + a2 − n2 = 0 (62)

Solving for r , we get two horizons which are located atr± = M ± √

M2 + n2 − a2. So, we find that the extremalcondition (r+ = r−) for Kerr–Taub-NUT spacetimes is a2

e =M2 + n2. If we set Qc = Qm = Λ = 0 and M2 + n2 = a2

e

in Eq. (20), we get the following exact LT precession rate atextremal Kerr–Taub-NUT spacetime:

ΩeKTNLT = (r − M)

p

[√M2 + n2 cos θ

p2 − 2Mr − n2

−√

M2 + n2 cos θ − n

p2

]r

+√

M2 + n2 sin θ

p

[r − M

p2 − 2Mr − n2− r

p2

]θ

(63)

where p2 = r2 + (n ∓ √M2 + n2 cos θ)2.

5 Discussion

In this work we have explicitly derived the LT precessionfrequencies for extremal and non-extremal PD spacetime.The PD family of solutions are the solutions of Einsteinfield equations which contain a number of well known blackhole solutions. Among them are included Schwarzschild,Schwarzschild–de Sitter, Kerr, Kerr–AdS, Kerr–Newman,Kerr–Taub-NUT etc. We observe that the LT precession fre-quency strongly depends upon the different parameters likemass M , spin a, Cosmological constant Λ, NUT charge n,electric charge Qc , and magnetic charge Qm.

An interesting point is that LT precession occurs solelydue to the “dual mass”. This “dual mass” is equivalent toangular momentum monopole (n) of NUT spacetime. Forour completeness we have also deduced the LT precessionfor extremal PD spacetime and also for others axisymmet-ric PD-like spacetimes. To get the LT precession rate in ex-tremal Kerr–Taub-NUT–de-Sitter spacetime, the basic pro-cedure is the same as the PD spacetimes with the addi-tional requirement Q2

c + Q2m = 0 in Eqs. (41) and (42)

and the range of x2 < ( 2

3 − n2) and a2e has a range of

0 < a2e < (7 2 − 24n2) − 4

√3( 2 − 3n2)( 2 − 4n2).

In the case of Taub-NUT spacetime, the horizons arelocated at r± = M ± √

M2 + n2. For extremal Taub-NUTspacetime, it has to satisfy the condition r+ = r− or M2 +n2 = 0. The solution of this equation is either M = in orn = iM (where i = √−1) which is not possible as M andn are always real for Taub-NUT black hole. So, there is notany valid extremal condition at Taub-NUT spacetime (withM or without M) and we could not obtain any real LT pre-cession rate due to the frame-dragging effect in extremalTaub-NUT spacetime. Since, the direct ISCO coincides with

Eur. Phys. J. C (2013) 73:2536 of 9

the principal null geodesic generator [20] in extremal Kerrspacetimes, we are unable to discuss the LT precession atthat particular geodesic. So this formula is not valid for thedomain of r ≤ M for the extremal Kerr spacetimes. Thus ourformula is valid only for r > M . In general, the general for-mula for LT precession in stationary spacetime is valid onlyoutside the horizon, as the observer is in timelike Killingvector field. The formula is not valid on the horizon and in-side the horizon. We will discuss about this problem in thenear future.

We also expect that the physically unobserved spacetimeswill be found in future and the strong gravity LT precessionwill be measured in those physically unknown spacetimes.

Acknowledgements We would like to thank Prof. Parthasarathi Ma-jumdar for his encouragement and guidance, besides the long black-board discussions on this topic. One of us (CC) is grateful to the De-partment of Atomic Energy (DAE, Govt. of India) for the financialassistance.

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http://arxiv.org/abs/arXiv:1304.6936

LenseâThirring precession in PlebaÅskiâDemiaÅski Spacetimes

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