Entropy Product Formula for Gravitational Instanton

We investigate the entropy product formula for various gravitational instantons. We speculate that due to the mass-independent features of the said instantons they are \emph{universal} as well as they are \emph{quantized}. For isolated Euclidean Schwarzschild black hole, these properties simply \emph{fail}.


Introduction
There has been a strong interest in microscopic interpretation of black hole (BH) entropy [1,2,3,4,5,6,7,8] in terms of D-branes come due to the work by Strominger and Vapa [9]. In d-dimension Euclidean quantum gravity, this entropy is due to the (d − 2)dimensional fixed point sets of the imaginary time translation Killing vector. There are many fixed point sets which could also give rise to BH entropy.
Thus in the present work, we wish to investigate the entropy product formula for various gravitational instantons. Instantons are non-singular and having imaginary time. They arises in quantum field thory in evaluating the functional integral, in which the functional integral is Wick rotated and expressed as an integral over Euclidean field configuarations. They are the solutions of the Euclidean Einstein equations. They have the signature of the form (+ + ++). There are two types of instantons discoverd so far. One is an asymptotically locally flat(ALF), which was first discoverd by Hawking in 1977 [29] and the other is an asymptotically locally Euclidean(ALE), discovered by Gibbons and Hawking in 1978 [30]. The examples of ALE classes are Flat space, Eguchi-Hanson [31] and multi-instanton [30].
The ALF class of solutions are asymptotically flat in the 3D sense, the fourth, imaginary -time, direction being periodic. The surfaces of large radii could be think of as an S 1 bundle over S 2 . The product bundle corresponds to the asymptotically flat(AF) solutions which include the Euclidean Schwarzschild and Euclidean Kerr solutions [32,33]. The twisted bundles correspond to the multi-Taub-NUT solution [29], and the Taub-Bolt solution first discoverd by Page [34].
In our previous work [27,28,18], we investigated the properties of inner and outer horizon thermodynamics of Taub-NUT (Newman-Unti-Tamburino) BH, Kerr-Taub-NUT BH and Kerr-Newman-Taub-NUT BH in four dimensional Lorentzian geometry. The failure of First law of BH thermodynamics and Smarr-Gibbs-Duhem relation for Taub-NUT and Kerr-Taub-NUT BH in the Lorentzian regime gives the motivation behind this work. What happens one can go from Lorentzian geometry to Riemannian geometry? This is the prime aim in this work. By studing the properties of these instantons what would be the effects on the BH entropy product formula due to the non-trivial NUT parameter?
In general relativity, the non-trivial value for the BH entropy is due to the presence of the fixed point set of the periodic imaginary time Killing vector. The fixed point set here we considered actually is the BH horizons (H ± ). Here H + is called event horizon and H − is called the Cauchy horizon. In four dimension, such fixed point sets are of two types, isolated points or zero dimensional which we call NUTs and two surfaces or two dimensional which we call Bolts. Thus one can thought Bolts as being the analogue of electric type mass-monopoles and the NUTs as being gravitational dyons endowed with a real electric type mass-monopole and an imaginary magnetic type mass-monopole. The presence of magnetic type mass introduces a Dirac string like singularity in the spacetime which is so called Misner string was first pointed out by Misner in his paper on the Lorentzian Taub-NUT spacetime [35]. A Misner string is a coordinate singularity which can be considered as a manifestation of a "non-trivial topological twisting" [36] of the manifold (M, g ab ). This twist is parametrized by a topological term, the NUT charge.
In our previous investigation [18], we have taken the metric in a Lorentzian spacetime in 3 + 1 split form as In this work, we are interested to study the metric in a Riemannian spacetime which could be written in 3 + 1 split form: Here all quantities are independent of t or τ . The Wick rotation that transforms from one case to other is by the transformation t → iτ and w i → iω i . F can be thought of as an electric type potential, and ω i or w i as a magnetic-type vector potential. The associated magnetic field is H ij = ∂ i ω j − ∂ j ω i and it is gauge invariant. This could be used to define a magnetic monopole moment called the NUT charge n [30]. If n = 0, the fibration could not be trivial. In the Lorentzian geometry, these fixed point sets are the two-dimensional Boyer bifurcation sets of event horizon [5,37]. In Riemannian geometry, these fixed point sets are of two types: zero dimensional point or NUTs, and two dimensional surfaces or Bolts [30]. A NUT possesses a pair of surface gravities κ 1 and κ 2 . p and q are a pair of coprime integers such that κ 1 β is the period of the imaginary time coordinate. The structure of the paper is as follows. In Sec. 2, we have considered the Euclidean Schwarzschild BH. In Sec. 3, we have investigated the properties of Self dual Taub-NUT instantons. In Sec. 4, we have studied the mass-independent properties of Taub-BoltiInstantons. In Sec. 5, we have described the properties of Eguchi-Hanson instantons. In Sec. 6, we have examined the properties of Taub-NUT-AdS Spacetime. In Sec. 7, we have studied the entropy product formula for Taub-Bolt-AdS spacetime and finally in Sec. 8, we have examined the product rules for Dyonic Taub-NUT-AdS and Taub-Bolt-AdS spacetime.

2
Euclidean Schwarzschild metric: To give a warm up, let us consider the Schwarzschild BH(where G = c = 1) in Euclidean form as The apperent singularity at the event horizon r + = 2M can be removed by identifying τ with a period β = 8πM [33,29]. The radial coordinate has the range 2M ≤ r ≤ ∞.
Then the topology of the manifold is where the O(2) corresponds to translations in the periodically identified imaginary time τ and the O(3) corresponds to rotations of the θ and φ coordinates. The Killing vector ∂ τ has unit magnitude at large radius and has a Bolt on the horizon r + = 2M which is a 2-sphere S 2 of area The surface gravity is given by Thus the BH temperature is Thus for an isolated Euclidean Schwarzschild BH the area product would be which tells us that the product is dependent on mass and thus it is not universal. Also it is not quantized. The Euclidean action derived in [29] From that one could derived the entropy as in [29] Thus the entropy product for isolated Euclidean Schwarzschild BH should be Indeed, it is not universal nor it is quantized.

Self dual Taub-NUT Instantons:
In this section we shall calculate the entropy product and area product of four dimensional Taub-NUT spacetime. It is of ALF. ALF metrics have a NUT charge,or magnetic type mass, n, as well as the ordinary electric type mass, M. The NUT charge is βc 1 8π , where c 1 is the first Chern number of the U(1) bundle over the sphere at infinity, in the orbit space Ξ. If c 1 = 0, then the boundary at infinity is S 1 × S 2 and the spacetime is AF. The BH metrics are saddle points in the path integral for the partition function. Thus, if c 1 = 0, the boundary at infinity is a squashed S 3 , and the metric could not be analytically continued to a Lorentzian metric. The squashed S 3 is the three-dimensional space on which the boundary conformal field theory will be compactified, with β identified with the inverse temperature i.e. T = 1/β.
Hawking [29] first given an examples of gravitational instanton was the self-dual Taub-NUT metric described by It is ALF with a central NUT. The self-dual Taub-NUT instanton has M = n and the anti-self-dual instanton has M = −n. The value r = n is now a zero in F (r). The (θ, φ) two-sphere has a zero area at r = n, so the zero in F (r) is a zero-dimensional fixed point of ∂ φ , a NUT. In order to make the solution regular, we take the region r ≥ n and let the period of τ be 8πn. The metric has a NUT at r = n, with a Misner string running along the z-axis from the NUT out to infinity i.e. n ≤ r ≤ ∞.
We know from the idea of path-integral formulation of quantum gravity which tells us that the Euclidean action derived in [29] where Z is the partition function of an ensemble with the path integral taken over all metrics g and matter field φ that are appropriately identified with the period β of τ . Therefore the entropy could be derived as It is indeed mass independent and thus it is universal. The surface gravity is calculated to be Thus the BH temperature should read off Now we see what happens the above results for another instantons that is Taub-Bolt.
It is a non-self-dual, non-compact solution of the vacuum Euclidean Einstein equations. In order to make the solution regular we have restricted in the region r = r + ≥ 2n, and the Euclidean time has period β = 8πn. Asymptotically, the Taub-Bolt instanton behaves similar manner as the Taub-NUT, so it is ALF. Since we are setting the fixed point is at r = r + = 2n, therefore the area of the S 2 does not vanish there and the fixed point set is 2-dimensional, thus it is a Bolt of area Thus the area product for Taub-Bolt instanton will be Thus the area product does independent of mass and also quantized. Similarly, the action was calculated in [38] Thus the entropy was derived by the universal formula

Eguchi-Hanson Instantons:
A non-compact instanton which is a limiting case of the Taub-NUT solution is the Eguchi-Hanson metric [31], The instanton is regular if we consider the region r ≥ n, and let τ has period 8πn. The metric is ALE type. There is a Bolt of area at r = n is given by which gives rise to a Misner string along the z-axis. Thus the product is universal and could be quantized. The Euclidean action was derived in [38] Thus entropy corresponds to We now turn our attention for the Taub-NUT and Taub-Bolt geometries in four dimensional locally AdS spacetime. The spacetimes have a global non-trivial topology due to the fact that one of the Killing vector has a zero dimensional fixed point set called NUT or a two-dimensional fixed point set called Bolt. Moreover, these four dimensional spacetimes have have Euclidean sections which can not be exactly matched to AdS spacetime at infinity. 6 Taub-NUT-AdS Spacetime: In this section we shall consider the spacetime which are only locally asymptotically AdS and with non-trivial topology. The metric on the Euclidean section of this family of solutions could be written as [39,40] where and ℓ 2 = −3 Λ , with Λ < 0 being the cosmological constant. Here M is a (generalized) mass parameter and r is a radial coordinate. Also, τ , the analytically continued time i.e. Euclidean time, parametrizes a circle S 1 , which is fibered over the two-sphere S 2 , with coordinates θ, φ. The non-trivial fibration is a consequence of a non-vanishing NUT parameter n.
There are some restrictions [41] for existence of a regular NUT parameter. Firstly, in order to ensure that the fixed point set is zero dimensional, it is necessary that the Killing vector ∂ τ has a fixed point which occurs precisely when the area of the two-sphere is zero size. Secondly, in order for the Dirac-Misner string [35] to be unobservable, it is necessary that the period of τ be β = 8πn. To avoid the conical singularity, we must check H ′ (r + = n) = 1 2n . Thirdly, the mass parameter M must be M = n − 4n 3 ℓ 2 . After simplifying the metric coefficients, we obtain H(r) = r − n r + n 1 + (r − n)(r + 3n) ℓ 2 (31) and the range of the radial coordinate becomes n ≤ r ≤ ∞. For our requirement, the Euclidean action for this spacetime was calculated in [42,43] and the entropy will be Thus the entropy product should be It is independence of mass parameter and does depend on NUT parameter and cosmological constant. Thus the entropy product is universal for Taub-NUT-Ads spacetime. The Hawking temperature T + = 1 8πn is same as Taub-NUT BH. The first law of thermodynamics is also satisfied as dM = T + dS.

7
Taub-Bolt-AdS Spacetime: For Taub-Bolt-AdS, the metric has the same form as in (29) but the fixed point set here is two dimensional or Bolt and with additional restrictions are the metric coefficients H(r) vanish at r = r b > n. In order to have a regular Bolt at r = r b , the following conditions must be satisfied: (i) H(r b ) = 0, (ii) H ′ (r b ) = 1 2n and the numerator of H(r) at r = r b being a single one. From the condition (i), we get the mass parameter at r = r b : Then we find [41] H ′ (r b ) = 3 ℓ 2 To satisfy the condition (ii) we must have the quadratic equation for r b : which gives the solution for r b in two branches The discriminat of the above equation must be negative for r b to be real and for r b > n we obtain the following inequality for n: The Euclidean action was computed in [43] I = 4πn Now it can be easily derive the entropy via the universal entropy formula: 8 Dyonic Taub-NUT-AdS and Taub-Bolt-AdS Spacetime: The general form of the metric for dyonic Taub-NUT-Ads spacetime [44,45,46] is given by where, The gauge field reads off A ≡ A µ dx µ = qr r 2 − n 2 + ν r 2 + n 2 r 2 − n 2 (dτ − 2n cos θdφ) , The conditions of smoothness of the Euclidean section implies that the parameter q is related to the parameter ν gives a deformation from the uncharged system. When these parameters go to to zero value, we obtain simply Taub-NUT-Ads spacetime. In order to have a regular position of NUT or Bolt at r = r ± , we set N (r) = 0 and also the gauge field A must be regular at that point. Thus we obtain the mass parameter as and The electric charge and potential at infinity corresponds to Now the Euclidean action for the above spacetime was calculated in [47](in units where G = c = 1) The entropy was calculated as: When we set r ± = r n = n, we get a dyonic NUT spacetime. For this spacetime the above calculations reduced to and Finally, the entropy is given by S ± = 4πn 2 1 − 6 n 2 ℓ 2 + 2ν 2 .
Thus the entropy product formula for dyonic Taub-NUT is The product formula is indeed independent of mass and depends on n, ℓ and ν. For a dyonic Bolt, we set r ± = r b and satisfies the fourth order equation for r b : Let r b = r b± be the solution of the equation. Then we find the action as previously Similarly the entropy is given by S ± = 2π [3r 4 b + (ℓ 2 − 12n 2 ) r 2 b + n 2 (ℓ 2 − 3n 2 )] r 2 b + (r 4 b + 4n 2 r 2 b − n 4 ) ℓ 2 ν 2 (3r 2 b − 3n 2 + ℓ 2 ) r 2 b + (r 2 b − n 2 ) ℓ 2 ν 2 .
After substituating the value of r b± in the entropy product formula, it seems that the product is independent of mass and depends on n, ℓ, ν for dyonic Taub-Bolt instanton.

Conclusion:
We have studied the mass-independent feature for various gravitational instantons. This universal feature gives us strong indication to understanding the microscopic properties of BH entropy. It would be interesting if one considered the entropy product formula for other instantos like Multi-Taub NUT, Non-self dual Taub-NUT, S 4 , CP 2 , S 2 × S 2 and Twisted S 2 × S 2 . We expect these instantons also gives us universal features.