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Nonsingular Black Hole in Two-Dimensional Asymptotically Flat Spacetime

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Nonsingular Black Hole in Two-Dimensional Asymptotically Flat Spacetime PHYSICAL REVIEW D 104, 044064 (2021) Nonsingular black hole in two-dimensional asymptotically flat spacetime Wen-Yuan Ai * Centre for Cosmology, Particle Physics and Phenomenology, Universit´e catholique de Louvain, Louvain-la-Neuve B-1348, Belgium (Received 12 January 2021; accepted 19 July 2021; published 26 August 2021) 1 pffiffiffiffiffiffi 2 2 We study a special two-dimensional dilaton gravity with Lagrangian L ¼ 2 −gðϕR þ λ sech ϕÞ where λ is a parameter of dimension mass. This theory describes two-dimensional spacetimes that are asymptotically flat. Very interestingly, it has an exact solution for the metric, ds2 ¼ −ðc þ tanh λxÞdt2þ dx2=ðc þ tanh λxÞ, parametrized by c. For c ∈ ð−1; 1Þ, the solution presents an event horizon but no singularity. Because of the kink profile for the metric components appearing in the solution, we refer to the spacetime described by the metric as a gravitational domain wall with the wall simply being the event horizon and separating two asymptotically Minkowskian spacetimes. The global causal structure for such an object is studied via coordinate extension and the thermodynamical quantities are computed. Only when c ∈ ð−1; 0 are the entropy and energy non-negative. DOI: 10.1103/PhysRevD.104.044064 I. INTRODUCTION or anti–de Sitter space. For convenience, one usually takes jΛj¼1 so that Λ ¼1.2 Equation (1) also contains Two-dimensional (2D) gravity theories are interesting other interesting models as its special cases. For instance, because they allow one to isolate some important features the Callan-Giddings-Harvery-Strominger model [23] cor- of the higher-dimensional gravity while being very simple. ϕ − 1 ϕ −4λ2ϕ They are therefore viewed as a theoretical laboratory for responds to Uð Þ¼ ϕ, Wð Þ¼ and the spheri- quantum gravity. In recent years, motivated by the cally reduced gravity [24–27] is given by Sachdev-Ye-Kitaev model [1–3] (see also Refs. [4–6]), Jackiw-Teitelboim (JT) gravity [7–9] has received a resur- ðD − 3Þ UðϕÞ¼− ; ð2aÞ gence of interest, see, e.g., Refs. [10–16]. JT gravity ðD − 2Þϕ belongs to the family of two-dimensional dilaton gravities 1 −4 that can be parametrized by the following general action 2 ðD Þ WðϕÞ¼−λ ðD − 2ÞðD − 3ÞϕðD−2Þ; ð2bÞ Z 1 ffiffiffiffiffiffi 2 p− ϕ − ϕ αβ ∂ ϕ ∂ ϕ ϕ S ¼ 2 d x g½ R Uð Þg ð α Þ β þ Wð Þ; ð1Þ where D>2 is an integer representing the dimension of the spacetime before the spherical reduction. In both cases λ is where gαβ is the 2D spacetime metric. JT gravity corre- a parameter of dimension one in mass. ϕ sponds to UðϕÞ¼0, WðϕÞ¼2Λϕ. In this case, ϕ is an Actually, with a Weyl transformation, the factor Uð Þ in Eq. (1) can always be set to zero and hence 2D dilaton auxiliary field and can be eliminated by its algebraic 3 equation of motion, giving R ¼ −2Λ. Depending on gravity can be generally described by [28] the sign of Λ, JT gravity thus describes a 2D de Sitter Z 1 ffiffiffiffiffiffi 2 p− ϕ ϕ S ¼ 2 d x g½ R þ Wð Þ: ð3Þ *[email protected] 1In general, the ϕ in the first term in the square brackets can be a general function ZðϕÞ. But for physical reasons, Z0ðϕÞ is Varying the action with respect to ϕ, one has nonzero everywhere and we can introduce a new field to put the action into the form of Eq. (1). Moreover, here we only dWðϕÞ consider torsionless gravity. For reviews of dilaton gravity, see R ¼ − : ð4Þ Refs. [17–20], and for recent studies, see Refs. [21,22]. dϕ Published by the American Physical Society under the terms of 2 the Creative Commons Attribution 4.0 International license. Since Λ is a dimensionful parameter, one shall restore it in Further distribution of this work must maintain attribution to physical quantities via the dimensional analysis. the author(s) and the published article’s title, journal citation, 3Note that when UðϕÞ is singular the two theories before and and DOI. Funded by SCOAP3. after the transformation are not necessarily equivalent [19]. 2470-0010=2021=104(4)=044064(6) 044064-1 Published by the American Physical Society WEN-YUAN AI PHYS. REV. D 104, 044064 (2021) In this paper, we consider a special model with and then substituting GðxÞ¼AðxÞ into the equation of motion for GðxÞ. From Eq. (7), one has the Ricci scalar WðϕÞ¼λ2sech2ϕ; ð5Þ A0G0 GA00 GA02 R ¼ − − þ ; ð8Þ where λ is a parameter of dimension mass. As done for 2A A 2A2 JT gravity, we will also set λ ¼ 1 and one should be able to restore it in physical quantities. As we shall see shortly, where a prime denotes the derivative with respect to x. this theory describes asymptotically flat spacetimes. We Substituting Eq. (8) into the action (3), one obtains will show that it permits a very interesting solution with a Z rffiffiffiffi rffiffiffiffi free parameter in which there is an event horizon but no sin- 1 d G dA A S ¼ dtdx −ϕ þ WðϕÞ ; ð9Þ gularity. The structure is similar to the domain wall of a 2 dx A dx G scalar field theory. Therefore we refer to this new gravi- tational object as a gravitational domain wall (or gravita- from which the algebraic equation of motion for GðxÞ reads tional kink) with the “wall” representing the event horizon. pffiffiffiffi We will then study its causal structure and thermodynam- ϕ0A0 A pffiffiffiffiffiffiffi − WðϕÞ¼0: ð10Þ ical properties. We shall see that for a particular choice of AG G3=2 the free parameter in the solution, the gravitational domain wall can have a nonzero temperature while its energy and Now substituting G ¼ A into the above equation, one entropy are vanishing. Unlike JT gravity, which is related to finally arrives at the near-horizon region of extremal black holes in higher- dimensional spacetime through spherical reduction, it is not ϕ0A0 − WðϕÞ¼0: ð11Þ clear how the potential (5) arises from a higher-dimensional theory. Nonetheless, given the interesting properties men- To obtain the other equation of motion, we let G ¼ A in the tioned above and its potential role in building holography action (9) and get for asymtotically flat two-dimensional spacetimes, it still Z deserves to be investigated. 1 −ϕ 00 ϕ The remainder of this article is organized as follows. In S ¼ 2 dtdxð A þ Wð ÞÞ: ð12Þ the next section we study the 2D dilaton gravity with the potential (5) and show that it has an exact solution in which Varying the action with respect to A,wehave the zero-zero metric component takes the kink profile when using the Schwarzschild-like ansatz. We then study ϕ00 ¼ 0: ð13Þ the global causal structure for the obtained solution via coordinate extension. In Sec. III, the thermodynamical From Eq. (13), one has ϕ ¼ ax þ b where a, b are properties for the gravitational domain wall are investi- constants. If a ¼ 0, then from Eq. (11) we have WðϕÞ¼0. gated. We conclude in Sec. IV. For the model we consider [cf. Eq. (5)], WðϕÞ vanishes only at infinity, ϕ ¼∞. And in this case, R ¼ −dWðϕÞ= ϕ ≡ 0 II. GRAVITATIONAL DOMAIN WALL IN d jϕ¼∞ . Therefore, this solution describes the trivial TWO-DIMENSIONAL DILATON GRAVITY Minkowski spacetime, AðxÞ¼constant ≠ 0. If a ≠ 0, then one can always let A. Equations of motion and solutions A general classical solution in the theory given by Eq. (3) ϕ ¼ x ð14Þ can be put in the form [29,30] after a transformation ðt; x; AÞ → ðt=a;˜ ax˜ þ b;˜ a˜ 2AÞ 2 2 1 2 ˜ ≠ 0 ds ¼ −AðxÞdt þ dx : ð6Þ (a ), which leaves the metric (6) invariant. In this case, AðxÞ Eq. (11) reduces to Note that whether or not one shall interpret t as time and x dAðxÞ 0 ¼ sech2x; ð15Þ as space depends on the sign of AðxÞ.IfAðxÞ > then dx Eq. (6) describes a static spacetime region while if AðxÞ < 0 it describes a time-dependent spatially homo- from which one obtains AðxÞ¼ðtanh xÞþc with c being a geneous region. Following Ref. [30], one can obtain one of constant. Substituting AðxÞ¼ðtanhxÞþc and AðxÞ¼GðxÞ the dynamic equations by first considering the metric into Eq. (8), one confirms that R ¼ 2ðsech2xÞ tanh x ¼ − 0 → ∞ 1 W ðxÞ, which vanishes at jxj . Thus the above ds2 ¼ −AðxÞdt2 þ dx2; ð7Þ solutions parametrized by c describe asymptotically flat GðxÞ spacetimes. The metric reads 044064-2 NONSINGULAR BLACK HOLE IN TWO-DIMENSIONAL … PHYS. REV. D 104, 044064 (2021) 1 t ds2 ¼ −ðc þ tanh xÞdt2 þ dx2: ð16Þ c þ tanh x If c>1 or c<−1, then there is no solution for AðxÞ¼0 and thus there is no horizon for the spacetime considered. For c ∈ ð−1; 1Þ, the horizon is located at a finite distance xh ¼ −artanhc; ð17Þ while for c ¼ −1 or c ¼ 1, the horizon is located at þ∞ or −∞, respectively. Below we shall consider the case x c ∈ ð−1; 1Þ. The coordinate singularity at x ¼ xh is an event horizon. The region x>xh corresponds to the static region out- side of the horizon while x<xh the region inside of the horizon which is, however, time dependent. Note that one shall interpret x as the time for x<xh. The metric has no essential singularity so that it describes a novel gravita- tional structure where there is an event horizon but no geometrical singularity. Because of the kink solution, c þ tanh x in Eq.
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