Eur. Phys. J. C (2013) 73:2587DOI 10.1140/epjc/s10052-013-2587-3

Regular Article - Theoretical Physics

About the coordinate time for photons in Lifshitz space-times

J.R. Villanueva1,2,a, Yerko Vásquez3,4,b

1Departamento de Física y Astronomía, Facultad de Ciencias, Universidad de Valparaíso, Gran Bretaña 1111, Valparaíso, Chile2Centro de Astrofísica de Valparaíso, Gran Bretaña 1111, Playa Ancha, Valparaíso, Chile3Departamento de Ciencias Físicas, Facultad de Ingeniería, Ciencias y Administración, Universidad de La Frontera,Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chile

4Departamento de Físicas, Facultad de Ciencias, Universidad de La Serena, Avenida Cisternas 1200, La Serena, Chile

Received: 31 July 2013 / Published online: 16 October 2013© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Abstract In this paper we studied the behavior of radialphotons from the point of view of the coordinate time in(asymptotically) Lifshitz space-times, and we found a gen-eralization to the result reported in previous works by Cruzet al. (Eur. Phys. J. C 73:7, 2013), Olivares et al. (As-trophys. Space Sci. 347:83–89, 2013), and Olivares et al.(arXiv:1306.5285). We demonstrate that all asymptoticallyLifshitz space-times characterized by a lapse function f (r)

which tends to one when r → ∞, present the same be-havior, in the sense that an external observer will see thatphotons arrive at spatial infinity in a finite coordinate time.Also, we show that radial photons in the proper system can-not determine the presence of the black hole in the regionr+ < r < ∞, because the proper time as a result is indepen-dent of the lapse function f (r).

1 Introduction

Lifshitz space-times have proven to be important from aholographic point of view. They represent gravity duals ofnon-relativistic systems and they are of particular interestin the studies of critical exponent theory and phase transi-tions [1, 2]. In these space-times the spatial and the tempo-ral coordinates scale in different ways, reflecting the sym-metries of boundary field theory.

Lifshitz vacua space-times in d dimensions are repre-sented by the line element

ds2 = − r2z

�2zdt2 + �2

r2dr2 + r2 dx2, (1)

where z is the dynamical exponent, � is the only lengthscale in the geometry and x stands for a spatial (d − 2)-dimensional vector.

a e-mail: jose.villanuevalob@uv.clb e-mail: yerko.vasquez@ufrontera.cl

This metric is invariant under the following change ofcoordinates:

t → λzt, x → λx, r → r/λ. (2)

In this way, here z is understood as a measure of theanisotropy between spatial and temporal coordinates. Forz = 1 we recover the AdS metric in Poincaré coordinates.

Black hole solutions whose asymptotic behavior is givenby metric (1) have been studied recently. Lifshitz black holeswith a flat transverse section in four dimensions with z = 2have been reported in [3, 4], a topological Lifshitz blackhole with z = 2 was found in [5], a black hole with spheri-cal horizon with z = 4 was found in [6], a three-dimensionalLifshitz black hole with z = 3 was reported in [7], blackholes of Lovelock theory can be found in [8].

Recently, in [9–11] the authors showed that radial pho-tons in those Lifshitz backgrounds can reach the asymp-totic region in a finite coordinate time, as seen by an ex-ternal observer. Motivated by these results we analyzed Lif-shitz space-times with an arbitrary dynamical exponent z,and showed that this behavior still holds.

The paper is organized as follows. In Sect. 2 we ob-tain the fundamental equations for massless particles in the(asymptotically) Lifshitz space-time, then we analyze the ra-dial motion in terms of the coordinate time. Also, we analyzedifferent classes of lapse function that appear in the litera-ture. In Sect. 3 we analyze the radial motion of photons interms of the proper time. Finally, in Sect. 4 we discuss ourresults and make conclusions.

2 Generic Lifshitz black hole space-time

The metric of a generic Lifshitz space-times can be writtenas

ds2 = − r2z

�2zf (r) dt2 + �2

r2

dr2

f (r)+ r2

�2d �x2 (3)

Page 2 of 5 Eur. Phys. J. C (2013) 73:2587

where �x is a (d − 2)-dimensional vector, and the functionf (r) depends only on the radial coordinate. Furthermore,this function takes the value f (r) = 1 when r → ∞, whichmeans that the metrics are asymptotically Lifshitz. So, theLagrangian associated with the metric (3) for radial photonsis given by [9, 12]

2L = − r2z

�2zf (r) t2 + �2

r2

r2

f (r)= 0, (4)

which is independent of the space-time dimension, and thedot represents a derivative with respect to an affine parame-ter along the trajectory.

Thus, we can immediately obtain the following quadra-ture:

dr

dt= ± rz+1

�z+1f (r), (5)

which will be solved by explicitly giving the radial functionf (r). Therefore, assuming that photons are placed in r = R0

when t = 0, we can write the above equation as

t = ±�z+1∫ r

R0

dr ′

r ′z+1f (r ′). (6)

In order to obtain a full description, we will study differentspace-times characterized by the radial function f (r) sepa-rately.

2.1 Lifshitz vacua space-time

This case represents a space-time without black holes, i.e.,the case with f (r) = 1 [3]. Thus, Eq. (6) is written as

t (r) = ±�z+1∫ r

R0

dr ′

r ′z+1, (7)

so, a straightforward integration yields

T (X) = ±[(

1

X0

)z

−(

1

X

)z], (8)

where T = t/� and X = r/� are dimensionless variables,and X0 = R0/�. Thus, an external observer sees that pho-tons moving from X = X0 to X = 0 need an infinite coordi-nate time to do so, whereas photons moving from X0 to thespatial infinite needs a finite time, T1, given by

T1 = limX→∞T (X) =

(1

X0

)z

. (9)

In Fig. 1 we plot the behavior of the dimensionless functiongiven by Eq. (8).

We know from the above analysis that for Lifshitz vacuaspace-times, the integral

t1 = �z+1∫ ∞

R0

dr

rz+1(10)

Fig. 1 Plot for the dimensionless coordinate time, T ≡ t/�, of pho-tons generated by the lapse function f (X) = 1 given by Eq. (8), whereX ≡ r/� and X0 ≡ R0/�. Here we presents several curves for differentvalues of the dynamical exponent z. This shows that, as seen by a sys-tem external to photons, they will fall asymptotically to origin X = 0.On the other hand, this external observer will see that photons arrive atthe spatial infinite in a finite coordinate time, T1, given by Eq. (9)

converges. Now, we would like to know if the following in-tegral converges:

t1 = �z+1∫ ∞

R0

dr

rz+1f (r), (11)

for a generic f (r), with f (r) → 1 when r → ∞.In order to determine this, we apply the limit compari-

son test for improper integrals. First, defining f1(r) = 1rz+1

and f2(r) = 1rz+1f (r)

, both positive definite in the interval[R0,∞).

Now, taking the limit we obtain

a = limr→∞

f1(r)

f2(r)= lim

r→∞f (r) = 1.

This limit is a = 1 for asymptotically Lifshitz black holes;therefore, in accordance with the criteria for integral con-vergence, the integral (11) is convergent. This means thatit takes a finite coordinate time to reach the asymptotic re-gion of radial photons. In the next sections we will con-sider generic functions f (r) and evaluate explicitly the inte-grals (11) for arbitrary z > 0.

2.2 Asymptotically Lifshitz black hole I

Here we will consider the family of asymptotically Lifshitzblack holes whose radial function have the form

f (r) = 1 − rn+rn

. (12)

Here r+ is the so-called event horizon of the Lifshitz blackhole, and n is a real number which may depend (or not)on the dynamical exponent z. Functions of this kind canbe found, for example, in [3, 5, 7, 13]. Therefore, we canrewrite Eq. (6) as

t (r) = ±�z+1∫ r

R0

dr ′

r ′z−(n−1) (r ′n − rn+), (13)

Eur. Phys. J. C (2013) 73:2587 Page 3 of 5

in which case, a straightforward integration leads us to thesolution

T (X) = ±1

z

[Ω(X0;X+, z, n) − Ω(X;X+, z, n)

], (14)

where the function Ω is given explicitly by

Ω(y;X+, z, n) = 1

yz 2F1

(1,

z

n,n + z

n,Xn+yn

), (15)

and X+ ≡ r+/�. In Fig. 2 we plot the behavior of the dimen-sionless function given by Eq. (14).

It can be ascertained here that an external observer willsee that photons take an infinite coordinate time to arriveat the event horizon, which is a common fact with Einstein’sspace-times (S, SdS, SAdS, etc.). Moreover, again we obtainthe situation described in the previous example, i.e., there isa finite coordinate time in which photons arrive at the spatialinfinite given by

T1 = limX→∞T (X) = 2F1(1, z

n, n+z

n,

Xn+Xn

0)

zXz0

. (16)

Fig. 2 Plot for the dimensionless coordinate time, T ≡ t/�, of photonsgenerated by the lapse function f (X) = 1−Xn+/Xn given by Eq. (14),where X ≡ r/�, X+ ≡ r+/� and X0 ≡ R0/�. Here we present severalcurves for different values of the dynamical exponent z and the param-eter n. This shows that, as seen by a system external to photons, theywill fall asymptotically to event horizon X = X+. On the other hand,this external observer will see that photons arrive at the spatial infinitein a finite coordinate time, T1, given by Eq. (16)

Recently, this behavior has been reported in [9] for z = 3and n = 2, in [10] for z = 2 and n = 2, and in [11] for z = 2and n = 4.

2.3 Asymptotically Lifshitz black hole II

In addition to the case above, let us consider the followingfunction:

f (r) =(

1 − rn+rn

)(1 + g

rn

), (17)

where r+ is the event horizon and g is any real constant. Thiskind of lapse function can be founded, for example, in [6].Therefore Eq. (6) can be written as

t (r) = ±�z+1∫ r

R0

dr ′

r ′z−(2n−1) (r ′n − rn+) (r ′n + g), (18)

and its dimensionless solution is given by

T (X) = ±Ψ (X0,X+, z, n, g) − Ψ (X,X+, z, n, g)

z (g + Xn+), (19)

where the function Ψ is given explicitly by

Ψ (y,X+, z, n, g)

= Xn+yz 2F1

(1,

z

n,n + z

n,Xn+yn

)

+ g

yz 2F1

(1,

z

n,n + z

n,− g

yn

), (20)

and g ≡ g/�n. In Fig. 3 we plot the behavior of the dimen-sionless function given by Eq. (19). Again, we found thesame qualitative behavior as preceding cases. An externalobserver will see that photons take an infinite coordinatetime to arrive at event horizon, while they arrive at the spa-tial infinite in a finite coordinate time given by

T1 = limX→∞T (X) = Ψ (X0,X+, z, n, g)

z (g + Xn+). (21)

Fig. 3 Plot for the dimensionless coordinate time, T ≡ t/�, of pho-tons generated by the lapse function f (X) = (1−Xn+/Xn)(1+ g/Xn)

given by Eq. (19), where X ≡ r/�, X+ ≡ r+/�, X0 ≡ R0/� andg ≡ g/�n. Left panel: curves with z = 3 and n = 2 for different val-ues of the constant g; Middle panel: curves with n = 2 and g = 3 fordifferent values of the dynamical exponent, z; Right panel: curves with

g = 2 and z = 3 for different values of the parameter n. This showsthat, as seen by a system external to photons, they will fall asymp-totically to event horizon X = X+. On the other hand, this externalobserver will see that photons arrive at the spatial infinite in a finitecoordinate time, T1, given by Eq. (19)

Page 4 of 5 Eur. Phys. J. C (2013) 73:2587

Fig. 4 Plot for the dimensionless proper time, Θ ≡ τ/�, as a functionof the dimensionless radial coordinate X ≡ r/�, where X0 ≡ R0/�.Here we present several curves for different values of the dynamicalexponent z with the same value of the constant of motion E. Noticethat the Schwarzschild case is recuperated when z = 1

3 A note on the proper system

Since the metric of a Lifshitz space-time is static, the La-grangian associated is independent of the temporal coordi-nate t , and thus, its corresponding canonical conjugate mo-mentum is a conserved quantity. So, considering the La-grangian (4) we obtain

Πt ≡ ∂L∂t

= − r2z

�2zf (r)t = −√

E. (22)

Therefore, by combining Eq. (5) with Eq. (22) we obtain thequadrature

dr

dτ= ±�z−1

rz−1

√E, (23)

and its dimensionless solution is given by

Θ(X) = ±Xz − Xz0√

E, (24)

where Θ ≡ τ/� is the dimensionless proper time, see Fig. 4.The first observation of the solution (24) is that the propersystem no detects the shape of the lapse function f (r), i.e.,if photons are in the region r+ < r < ∞ (X+ < X < ∞),then they cannot determine the presence of black hole,which is in agreement with General Relativity, moreover, theSchwarzschild case is recuperated when z = 1. In this sense,massless particles presents the same asymptotic behavior asin Einstein’s space-times:

1. They cross the event horizon in a finite (dimensionless)proper time Θ+ given by

Θ+ ≡ Θ(X+) = Xz0 − Xz+√

E, (25)

and, eventually, arrives at the origin in a finite (dimen-sionless) proper time given by

Θ0 ≡ Θ(X = 0) = Xz0√E

. (26)

2. They require an infinite (dimensionless) proper time toarrive at the spatial infinite, i.e., τ → ∞ when r → ∞.

4 Summary

We obtained a generalization for the behavior of masslessparticles in a Lifshitz space-time reported in previous works[9–11]. This generalization is independent of the dynami-cal exponent z, and the dimension of space-time. We foundthat space-time looks similar to space-times of general rel-ativity in the sense that an external observer measures aninfinite coordinate time to photons traveling to the eventhorizon (or to the origin, if a Lifshitz vacua space-time isconsidered). On the other hand, by using of a simple cri-terion of convergence, we found a general condition overthe measure of coordinate time employed by photons toarrive at spatial infinite. Our result is that an external ob-server will see that photons arrive at the spatial infinite ina finite coordinate time, and the condition is that the lapsefunction, f (r), tends to one when r → ∞. Obviously, thiscondition always satisfies the asymptotic Lifshitz space-times, and therefore, our novel result becomes general forall space-times that satisfy the above condition. Addition-ally, we obtained the result that, from the point of view ofthe proper system, massless particles present the same be-havior as Einstein’s space-times, because the proper time isindependent of the lapse function f (r), therefore, the pho-ton cannot determine the presence of the black hole in theregion r+ < r < ∞.

Acknowledgements YV is supported by FONDECYT grant11121148; JRV thanks the UFRO for their hospitality and dedicatesthis work to JGDVB on his 10th birthday.

References

1. S. Kachru, X. Liu, M. Mulligan, Gravity duals of Lifshitz-likefixed points. Phys. Rev. D 78, 106005 (2008). arXiv:0808.1725

2. S.A. Hartnoll, J. Polchinski, E. Silverstein, D. Tong, Towardsstrange metallic holography. J. High Energy Phys. 1004, 120(2010)

3. K. Balasubramanian, J. McGreevy, An analytic Lifshitz blackhole. Phys. Rev. D 80, 104039 (2009). arXiv:0909.0263

4. M. Taylor, Non-relativistic holography. arXiv:0812.05305. R.B. Mann, Lifshitz topological black holes. J. High Energy Phys.

06, 075 (2009). arXiv:0905.11366. G. Bertoldi, B.A. Burrington, A. Peet, Black Holes in asymptot-

ically Lifshitz space-times with arbitrary critical exponent. Phys.Rev. D 80, 126003 (2009). arXiv:0905.3183

7. E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, Lifshitzblack hole in three dimensions. Phys. Rev. D 80, 104029 (2009).arXiv:0909.1347

8. M.H. Dehghani, R.B. Mann, Lovelock-Lifshitz black holes.J. High Energy Phys. 1007, 019 (2010). arXiv:1004.4397

9. N. Cruz, M. Olivares, J.R. Villanueva, Geodesic structure of theLifshitz black hole in 2 + 1 dimensions. Eur. Phys. J. C 73, 7(2013). arXiv:1305.2133

Eur. Phys. J. C (2013) 73:2587 Page 5 of 5

10. M. Olivares, G. Rojas, Y. Vásquez, J.R. Villanueva, Particles mo-tion on topological Lifshitz black holes in 3 + 1 dimensions. As-trophys. Space Sci. 347, 83–89 (2013). arXiv:1304.4297

11. M. Olivares, Y. Vásquez, J.R. Villanueva, F. Moncada, Motion ofparticles on a z = 2 Lifshitz black hole in 3+1 dimensions (2013).arXiv:1306.5285

12. N. Cruz, M. Olivares, J.R. Villanueva, The geodesic structure ofthe Schwarzschild anti-de sitter black hole. Class. Quantum Grav-ity 22, 1167–1190 (2005). arXiv:gr-qc/0408016

13. E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, Analytic Lif-shitz black holes in higher dimensions. J. High Energy Phys. 1004,030 (2010). arXiv:1001.2361