4. Anisotropic Solution of Einstein Maxwell Action

We now attempt to construct anisotropic solutions of 5 dimensional Einstein Maxwell action along with matter density. Our action in this section is

(26)

Here, notation g denotes the determinant of the metric which we choose to have one negative signature along time direction. Ricci scalar is denoted by R. The electromagnetic potential and field strength are denoted by and, respectively. We also incorporate source for electromagnetic field denoted as as well as a matter density denoted by with a four velocity profile. The matter term in action is conspired to give the correct energy momentum tensor for a pressureless dust i.e.. The Maxwell equation is

(27)

The metric fluctuations of the action leads to following Einstein equations.

We further attempt to find solutions of these set of equations using a method employed earlier to find generalizations of Majumdar-Papapetrou metrics [22] [23] . We note that the Majumdar-Papapetrou metrics are 4 dimensional extremal solutions of Einstein-Maxwell equations of the type

(28)

along with an electromagnetic flux of the kind. The function V is required by Einstein equations to be a harmonic function of the 3 dimensional flat subspace. When searching for solutions in dimensions, we generalize the metric ansatz to be of type

(29)

The metric depends on spatial coordinates only. It leads to the following Ricci tensor,

(30)

The indices denote spatial coordinates only. Here, notation denotes Laplacian defined over the internal space with metric. The Ricci tensor component is found to be vanishing. When we write the corresponding energy momentum tensor and try to satisfy Einstein equations, we notice that there is no analog of any term like in the expression of. Such a term, if kept, will require us to solve complicated non-linear equations. One chooses a relation between parameters m and n, so as to make such term vanish i.e.. Returning to our interest of 5 dimensional metrics, we find that we should take and. Thus, our metric ansatz becomes

(31)

We next evaluate the components of the Einstein tensor and they are found to be

(32)

We choose our internal subspace to be the anisotropic Ricci flat space that we obtained in last section i.e. Equation (25). Thus, Ricci tensor and Ricci scalar vanishes. The Einstein tensor in our case thus reduces to

(33)

Next, we make an ansatz for the electromagnetic potential. We assume it to be along the time direction

(34)

We also make an ansatz for the four velocity of the matter density. We assume matter to be at rest i.e.

(35)

Such a choice also ensures that. The energy momentum tensor component from electromagnetic field is given in terms of field strength tensor as

(36)

The matter energy density contribution to energy momentum tensor is

(37)

The non trivial components of Einstein equations are

We notice that the term explicitly proportional to in Equation (38) is same as (tt) component Einstein equation as in (38). Canceling it, we get

(38)

It can be solved easily if we take the function A proportional to function V. The equation fixes the relation to be. Then the rest of Einstein equations simplifies to

(39)

In terms of, the above equation can be written as

(40)

We next make an ansatz for the source of the electromagnetic field. We take only the time component of to be non trivial. Along with the above choice for electromagnetic potential, the Maxwell equation takes a form

(41)

This equation will be consistent with the Equation (40), if we choose

(42)

Thus we are left with a single equation viz. Equation (40), which is a non homogenous Laplacian equation. We now proceed to solve it for some suitable choices of matter density in the next section.

5. Anisotropic Solutions with Chosen Source Profiles

5.1. Polynomial Solutions

In the previous section, the Einstein equations were reduced to a single non-homogenous harmonic equation which is sourced by the density profile of the matter field. The equation now left to solve is

(43)

We choose our spatial subspace to be same as the anisotropic 4 dimensional subspace which was obtained in the section (3). Therefore,

(44)

The determinant of the metric is. We will henceforth denote simply by x. For simplicity, we assume the function to be a function of u and x only. Then Equation (43) becomes,

(45)

We next define polar coordinates and. The equation appears in polar form as

(46)

where we have taken to be independent of, i.e. we restrict ourselves to the lowest harmonic. Now the equation can be made amenable to analytical results by suitably choosing the profiles for the matter density. We next choose

(47)

where, n is a positive integer greater than 2 and c is a constant. Such a form of energy density is physically reasonable as it vanishes smoothly to zero when one proceeds towards infinity. Then Equation (46) becomes,

(48)

One can easily solve it to obtain

(49)

We can restrict ourselves to polynomial form by choosing. This leads to

(50)

We further proceed by making a particular choice of. Thus, we get

(51)

Then, the metric now appears as

(52)

But, this solution shows two essential singularity where Kretschmann tensor diverges. They are and. Thus there are two naked singularities in this solution. We next consider a fictitious matter whose density profile is negative by replacing the constant c with. This sends the second singularity at to a negative value of r, thus out of the considered spacetime. Choosing constant, the metric now appears as

(53)

We find that the (tt) component of the metric for small values of r is

(54)

Thus, this solution has a horizon near, which also hides the essential singularity residing at. We expect this solution to be of extremal type as the same is true for all such previous generalizations of Majumdar-Papapetrou metrics.

5.2. Sine Gordon Solution

One can get here an equation of Sine Gordon type by a different choice of matter density. First, we choose a different radial coordinate

Then the Equation (46) becomes

(55)

Next, we choose matter density profile to be of form

(56)

The above equation then reduces to a Sine Gordon equation,

(57)

It admits a solution

(58)

The resultant metric looks like

(59)

The range for coordinate r is restricted from to. This ensures that is finite everywhere except at one end, where one encounters a singularity.

6. Asymptotically Anti de Sitter Space

We further explore the properties of the Ricci flat 4 dimensional metric obtained in section (3). We consider a massive (but not back-reacting) field living in this space. Recent advances in gauge/gravity correspondence allows us to approximate its two point correlations for strong self coupling of the field. The correspondence states that the generating functional of the correlations of a 4 dimensional strongly coupled field theory defined on the boundary of a 5 dimensional asymptotically anti de Sitter space is same as the partition function of the latter theory of gravity. The boundary values of fields in the gravitational theory are coupled to sources of appropriate fields in field theory [32] -[35] . The method to calculate correlator in our 4 dimensional space will be to embed it in an anti de Sitter (AdS) space with one higher dimension such that it behaves as its boundary. The two point correlator of the massive field can be related to geodesics traveling in higher dimensional AdS space. We expect that these correlators will capture the effect of singularity in the same way as cosmological singularities are demonstrated to show such behavior in similar setups [24] -[30] . The embedding for our case is

(60)

We note that the above metric is an Einstein metric and its curvature invariants are

Thus, the metric shows a singularity at. Another interesting feature of this embedding is the limit. In this limit, the metric becomes

(61)

with the following curvature properties.

(62)

This is Euclidean anti de Sitter space and it can be verified that the metric in Equation (61) is the same as AdS in Poincare coordinates after a change of variables. [Take]. The metric also possesses a scaling symmetry,

(63)

Since the metric asymptotically becomes Euclidean anti de Sitter, it can be conjectured along the lines of AdS/CFT duality that it can be explored to learn about an Euclidean Yang Mills theory living in an anisotropic background in its strong coupling limit. Simplest quantity that can be calculated to probe the field theory is the two point correlation function for operators with high conformal dimensions. This correlation can be approximately given in terms of the length of the geodesics with endpoints on the boundary [24] .

We next calculate the geodesics connecting two endpoints on the boundary at different u coordinates. In this section, will denote the affine parameter along the geodesic. The geodesic equations can be obtained by extremizing a Lagrangian

where denotes the expression

Here, denotes the rate of change of y along the affine parameter. The above Lagrangian is independent of y and z, so it results in two constants of motion as follows

(64)

If we write this expression explicitly in terms of constant of motion X, we get

Similarly, one can obtain another constant of motion Y as

Simplifying the above two equations leads to

(65)

If we denote the expression

then the remaining geodesic equations can be written as

(66)

With a little manipulation, we can find a conserved quantity along the geodesic as

(67)

This constraint considerably simplifies the dynamical equation for variable t, which can now be solved to obtain

(68)

where D is another integration constant. We note that t becomes zero for both. Thus, as the affine parameter varies from to, the geodesic drops from boundary into bulk and again comes back to boundary. We also notice that the relation also sets the maximum depth the geodesic from the boundary can reach.

The maximum value of the scale invariant quantity along the geodesic is We next calculate the

length of the geodesic. To deal with divergences, we put a cutoff for the affine parameter. We choose it to vary between to, where is chosen to be very small. This cut off is related to the ultra violet cutoff along the radial direction. According to Equation (63), if the cutoff along t is given in terms of a scale invariant quantity as for a very small real number, then it is related to according to Equation (68) as

. It can be inverted for very small to give. The geodesic reaches boundary at both ends as approaches 0. We then calculate the length of the geodesic hanging in the bulk as

(69)

Using the identity Equation (67), we simplify it to

(70)

The latter part can be identified as the divergent contribution due to asymptotic AdS geometry and can be dropped during regularization. We next calculate the length of the geodesic on the boundary by using the boundary metric.

Thus, can be approximated to be, i.e. twice the radial distance of the turning point from the boundary. We calculate

(71)

Thus, the two point correlators for higher dimensional operators living on the boundary metric is expected to show a behavior [24]

(72)

Here the mass of the field (m) is also approximated to be same as conformal dimension () for large values. The term above will be most dominating term of the correlator since higher mass will suppress the contribution of quantum fluctuations. The above result shows that the behaviour of the two point correlator is same as the case of a flat space for smaller values of the separation of the points i.e.. We can neglect the effect of the second factor in such cases. The result is expected for a conformal field theory on a conformally flat spacetime. As the separation increases i.e. approaches 1, the second factor starts contributing significantly and the correlation function vanishes as a power law. We thus feel the effect of background anisotropic space for large separations.

7. Conclusion

We have constructed an anisotropic 4 dimensional asymptotically flat Riemannian metric which is also Ricci flat. We later incorporated it in a 5 dimensional space time along with matter density and electromagnetic flux using a method similar to Majumdar-Papapetrou way of constructing extremal solutions. With certain choices of matter density profiles, we were able to construct explicit solutions. The singularity problem seems to be addressed for a case of a fictitious choice of matter density for which we arrived at a solution with a horizon hiding the singularity. Using our method, interesting anisotropic solutions in higher dimensions can be constructed by further investigating different types of Lagrangians for their properties. Given the importance of anisotropic Bianchi VII space-times for their relations with condensed matter systems, we proceeded to study their properties. We embedded the metric in 5 dimensions analogous to the flat space embedding in anti de Sitter space manifested in the Poincare metric. Apart from the 3 directions along the 3 non commutative Killing vectors, we have 2 other spatial directions; one, the u-direction along which the metric coefficients vary. The other is the radial direction towards the bulk of the AdS space. The subspace for a fixed u and its neighbourhood can be considered locally AdS where the bulk metric encodes the properties of the field theory lying on its boundary, which is itself anisotropic. We examined the effect of anisotropy on the properties of the quantum field theory on such background by approximating the two point correlator of two operators with high conformal dimensions. For small separation between the operator positions, we saw a dependence similar to a conformal theory living on the flat space. The effect of curvature of the background is not reflected in this case. However for large separations, the two point correlator vanishes as a power law of the separation between the operator positions. We expect that the large distance behavior is unique to our case and encodes the effect of singularity into it.

Acknowledgements

The authors acknowledge the financial support of DST grant number SR/FTP/PS-149/2011.

Cite this paper

Mahato, M. and Singh, A. (2016) Novel Bianchi VII Space Times and Their Properties.*Journal of Modern Physics*, **7**, 445-457. doi: 10.4236/jmp.2016.75046.

Mahato, M. and Singh, A. (2016) Novel Bianchi VII Space Times and Their Properties.

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[1] Iizuka, N., Kachru, S., Kundu, N., Narayan, P., Sircar, N. and Trivedi, S.P. (2012) Journal of High Energy Physics, 1207, 193.

http://dx.doi.org/10.1007/JHEP07(2012)193

[2] Iizuka, N., Kachru, S., Kundu, N., Narayan, P., Sircar, N., Trivedi, S.P. and Wang, H. (2013) Journal of High Energy Physics, 1303, 126.

http://dx.doi.org/10.1007/JHEP03(2013)126

[3] Kachru, S., Kundu, N., Saha, A., Samanta, R. and Trivedi, S.P. (2014) Journal of High Energy Physics, 1403, 074.

[4] Kachru, S., Liu, X. and Mulligan, M. (2008) Physical Review D, 78, Article ID: 106005.

http://dx.doi.org/10.1103/PhysRevD.78.106005

[5] Son, D.T. (2008) Physical Review D, 78, Article ID: 046003.

http://dx.doi.org/10.1103/PhysRevD.78.046003

[6] Balasubramanian, K. and McGreevy, J. (2008) Physical Review Letters, 101, Article ID: 061601.

http://dx.doi.org/10.1103/PhysRevLett.101.061601

[7] Danielsson, U.H. and Thorlacius, L. (2009) Journal of High Energy Physics, 0903, 070.

[8] Hartnoll, S.A., Polchinski, J., Silverstein, E. and Tong, D. (2010) Journal of High Energy Physics, 1004, 120.

http://dx.doi.org/10.1007/JHEP04(2010)120

[9] Balasubramanian, K. and Narayan, K. (2010) Journal of High Energy Physics, 1008, 014.

[10] Donos, A. and Gauntlett, J.P. (2010) Journal of High Energy Physics, 1012, 002.

[11] Singh, H. (2010) Journal of High Energy Physics, 1012, 061.

[12] Gregory, R., Parameswaran, S.L., Tasinato, G. and Zavala, I. (2010) Journal of High Energy Physics, 1012, 047.

[13] Cassani, D. and Faedo, A.F. (2011) Journal of High Energy Physics, 1105, 013.

[14] Lu, H., Pang, Y., Pope, C.N. and Vazquez-Poritz, J.F. (2012) Physical Review D, 86, Article ID: 044011.

http://dx.doi.org/10.1103/PhysRevD.86.044011

[15] Shu, F.W., Lin, K., Wang, A. and Wu, Q. (2014) Journal of High Energy Physics, 1404, 056.

[16] Nakamura, S., Ooguri, H. and Park, C.S. (2010) Physical Review D, 81, Article ID: 044018.

http://dx.doi.org/10.1103/PhysRevD.81.044018

[17] Ooguri, H. and Park, C.S. (2010) Physical Review D, 82, Article ID: 126001.

http://dx.doi.org/10.1103/PhysRevD.82.126001

[18] Donos, A. and Gauntlett, J.P. (2012) Physical Review Letters, 108, Article ID: 211601.

http://dx.doi.org/10.1103/PhysRevLett.108.211601

[19] Donos, A. and Gauntlett, J.P. (2012) Physical Review D, 86, Article ID: 064010.

http://dx.doi.org/10.1103/PhysRevD.86.064010

[20] Myers, R.C. (1987) Physical Review D, 35, Article ID: 455.

http://dx.doi.org/10.1103/PhysRevD.35.455

[21] Gibbons, G.W. and Warnick, C.M. (2009) Physical Review D, 79, Article ID: 064031.

http://dx.doi.org/10.1103/PhysRevD.79.064031

[22] Varela, V. (2003) General Relativity and Gravitation, 35, Article ID: 1815.

http://dx.doi.org/10.1023/A:1026014114308

[23] Frolov, V.P. and Zelnikov, A. (2012) Physical Review D, 85, Article ID: 064032.

http://dx.doi.org/10.1103/PhysRevD.85.064032

[24] Hertog, T. and Horowitz, G.T. (2004) Journal of High Energy Physics, 0407, 073

http://dx.doi.org/10.1088/1126-6708/2004/07/073

[25] Hertog, T. and Horowitz, G.T. (2005) Journal of High Energy Physics, 0504, 005

[26] Awad, A., Das, S.R., Nampuri, S., Narayan, K. and Trivedi, S.P. (2009) Physical Review D, 79, Article ID: 046004.

http://dx.doi.org/10.1103/PhysRevD.79.046004

[27] Fischetti, S., Kastor, D. and Traschen, J. (2014) Classical and Quantum Gravity, 31, Article ID: 235007

http://dx.doi.org/10.1088/0264-9381/31/23/235007

[28] Engelhardt, N., Hertog, T. and Horowitz, G.T. (2014) Physical Review Letters, 113, Article ID: 121602

http://dx.doi.org/10.1103/PhysRevLett.113.121602

[29] Engelhardt, N., Hertog, T. and Horowitz, G.T. (2015) Journal of High Energy Physics, 1507, 044.

http://dx.doi.org/10.1007/JHEP07(2015)044

[30] Banerjee, S., Bhowmick, S., Chatterjee, S. and Mukherji, S. (2015) Journal of High Energy Physics, 1506, 043.

http://dx.doi.org/10.1007/JHEP06(2015)043

[31] Landau, L.D. and Lifshitz, E.M. (1975) The Classical Theory of Fields. Vol. 2, 4th Edition, Butterworth-Heinemann, Oxford.

[32] Maldacena, J.M. (1999) International Journal of Theoretical Physics, 38, 1113. [(1998) Advances in Theoretical and Mathematical Physics, 2, 231].

http://dx.doi.org/10.1023/A:1026654312961

[33] Witten, E. (1998) Advances in Theoretical and Mathematical Physics, 2, 253.

[34] Gubser, S.S., Klebanov, I.R. and Polyakov, A.M. (1998) Physics Letters B, 428, 105.

http://dx.doi.org/10.1016/S0370-2693(98)00377-3

[35] Freedman, D.Z., Mathur, S.D., Matusis, A. and Rastelli, L. (1999) Nuclear Physics B, 546, 96.

http://dx.doi.org/10.1016/S0550-3213(99)00053-X