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Gravitational Anomaly and Hawking Radiation of Apparent Horizon in FRW Universe

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Gravitational Anomaly and Hawking Radiation of Apparent Horizon in FRW Universe Eur. Phys. J. C (2009) 62: 455–458 DOI 10.1140/epjc/s10052-009-1081-4 Letter Gravitational anomaly and Hawking radiation of apparent horizon in FRW universe Ran Lia, Ji-Rong Renb, Shao-Wen Wei Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, Gansu, China Received: 27 February 2009 / Published online: 26 June 2009 © Springer-Verlag / Società Italiana di Fisica 2009 Abstract Motivated by the successful applications of the have been carried out [8–28]. In fact, the anomaly analy- anomaly cancellation method to derive Hawking radiation sis can be traced back to Christensen and Fulling’s early from various types of black hole spacetimes, we further work [29], in which they suggested that there exists a rela- extend the gravitational anomaly method to investigate the tion between the Hawking radiation and the anomalous trace Hawking radiation from the apparent horizon of a FRW uni- of the field under the condition that the covariant conser- verse by assuming that the gravitational anomaly also exists vation law is valid. Imposing boundary condition near the near the apparent horizon of the FRW universe. The result horizon, Wilczek et al. showed that Hawking radiation is shows that the radiation flux from the apparent horizon of just the cancel term of the gravitational anomaly of the co- the FRW universe measured by a Kodama observer is just variant conservation law and gauge invariance. Their basic the pure thermal flux. The result presented here will further idea is that, near the horizon, a quantum field in a black hole confirm the thermal properties of the apparent horizon in a background can be effectively described by an infinite col- FRW universe. lection of (1 + 1)-dimensional fields on (t, r) space, where r is the radial direction. One could treat the original higher di- PACS 04.62.+v · 04.70.Dy · 11.30-j mensional field as a collection of two-dimensional quantum fields using dimensional reduction. This is because, near the horizon, the property of the black hole metric, mass or Using the techniques of quantum field theory in a curved potential terms for quantum fields in it can be suppressed. spacetime background, Hawking [1, 2] firstly discovered In this two-dimensional reduction, because all the ingoing the thermal radiation of a collapsing black hole. Since the modes cannot classically affect physics outside the horizon, Hawking radiation relates the theory of general relativity the two-dimensional effective action in the exterior region with quantum field theory and statistical thermodynamics, it becomes anomalous with respect to gauge or general coordi- is generally believed that a deeper understanding of Hawk- nate symmetries. To cancel the anomaly, they found that the ing radiation may shed some light on seeking the under- Hawking flux is universally determined only by the value of lying quantum gravity. Due to the difficulties existing in anomalies at the horizon. extending the Hawking’s original method to more compli- In the present work, motivated by the successful applica- cated spacetime backgrounds, several derivations of Hawk- tions of the anomaly cancellation method to derive Hawk- ing radiation have been proposed in the literature, including ing radiation from various types of black hole spacetimes, the so-called Damour–Ruffini method [3], the trace anom- we further extend the gravitational anomaly method to in- aly method [4], the quantum tunneling method [5] and the vestigate the Hawking radiation from apparent horizon of gravitational anomaly method [6, 7]. FRW universe by assuming that the gravitational anomaly A recent proposal of deriving Hawking radiation via also exists near the apparent horizon of a FRW universe. gravitational and gauge anomalies proposed by Wilczek and The hypothesis that a gravitational anomaly exists near the his collaborators [6, 7] has attracted a lot of interest. This apparent horizon of a FRW universe is reasonable because rejuvenates the interest of investigation for Hawking radia- the apparent horizon of a FRW universe can be viewed as tion. For various types of black holes, these investigations a causal boundary and previous investigations show that the apparent horizon is thermal like the event horizon of a black a e-mail: [email protected] hole. Under this hypothesis one can derive the Hawking ra- b e-mail: [email protected] diation from the apparent horizon of a FRW universe us- 456 Eur. Phys. J. C (2009) 62: 455–458 2 ing the general gravitational anomaly formalism. The result where xa = (t, r), h = diag(−1, a ) and dΩ2 repre- ab 1−kr2 2 shows that the radiation flux from the apparent horizon of sents the line element of S2. The apparent horizon is defined FRW universe measured by a Kodama observer is just the by the equation pure thermal flux. The result presented here will further con- ab firm the thermal properties of the apparent horizon of the h ∂aR∂bR = 0, (3) FRW universe. Recently, the thermal properties of the apparent horizon which gives us the location of apparent horizon explicitly as of the FRW universe have attracted a lot of interest. In fact, = 1 the investigations can be traced back to Jacobson’s work RA , (4) H 2 + k/a2 [30], in which the Einstein equation can be derived from the proportionality of entropy and horizon area together with the with H =˙a/a being the Hubble parameter. The temperature = = |κ| fundamental Clausius relation δQ TdSconnecting heat, of the apparent horizon is defined by T 2π , where the entropy, and temperature. It has been proved that the above surface gravity κ is given by idea can also be applied in establishing the relationship be- √ = √1 − ab =− 1 − tween the Friedmann equations and the first law of thermo- κ ∂a hh ∂bR = (1 ), (5) − R RA dynamics in the framework of the Friedmann–Robertson– 2 h RA Walker (FRW) universe. Assuming that the apparent horizon ˙ with = RA . In this paper, we will adopt the approxima- of a FRW universe has temperature T = 1/2πr˜ and en- 2HRA A tion 1 to simplify the calculation in the following. This tropy S = A/4, where r˜ and A are the radius and the area A approximation is also supported by the investigations about of the apparent horizon, respectively, the Friedmann equa- the Hawking radiation from the apparent horizon in the tun- tions can be derived from the Clausius relation [32] and the neling approach. Under the approximation, the temperature first Friedmann equation can be cast into the form of the of apparent horizon is simply given by T = 1 . unified first law [31, 33]. For some related investigations see 2πRA For simplicity, we will use the (t, R) coordinates, in also [34–42]. which the FRW metric can be rewritten as However, whether there is a Hawking temperature asso- ciated with the apparent horizon of a FRW universe is still 1 − R2/R2 2HR ds2 =− A dt2 − dt dR an assumption. In a recent paper [43], the scalar particles’ 1 − kR2/a2 1 − kR2/a2 Hawking radiation from the apparent horizon of FRW uni- 1 verse was investigated by using the tunneling formalism. + dR2 + R2 dΩ2. (6) 1 − kR2/a2 2 There it was found that the Kodama observer inside the ap- parent horizon does see a thermal spectrum with tempera- One can diagonalize the metric by introducing the coordi- ture T = 1/2πr˜A, which is caused by particles tunneling nates transformation from outside the apparent horizon to inside the apparent HR horizon. Subsequently, we considered the fermions’ tunnel- dt = dt + dR. (7) 1 − R2/R2 ing from the apparent horizon in [44], where the radiation A spectrum and Hawking temperature are correctly recovered. The FRW metric in the new coordinates is of the form The results in the present paper together with that in [43] and 1 − R2/R2 1 [44] will fill in the gap existing in the literature investigat- ds2 =− A dt2 + dR2 + R2 dΩ2. − 2 2 − 2 2 2 ing the relationship between the first law of thermodynamics 1 kR /a 1 R /RA and Friedmann equations and confirm the thermal properties (8) of the apparent horizon. For convenience, we firstly review some results related In this paper, we want to investigate the thermal flux mea- to the FRW universe. The FRW universe is described by the sured by a Kodama observer. So we must introduce the time metric T measured by a Kodama observer. The relationship be- tween Kodama time T and time t is given by dr2 ds2 =−dt2 + a2(t) + r2 dθ2 + sin2θdφ2 , 1 1 − kr2 dT = dt. (9) − 2 2 (1) 1 kR /a = − Finally, the FRW metric reads where a(t) is the scale factor and k 1, 0 and 1 represent the closed, flat and open universe, respectively. Introducing 2 =− − 2 2 2 ds 1 R /RA dT R = ar, the FRW metric (1) can be rewritten as 1 + dR2 + R2 dΩ2. (10) 2 = a b + 2 2 − 2 2 2 ds hab dx dx R dΩ2 , (2) 1 R /RA Eur. Phys. J. C (2009) 62: 455–458 457 Therefore, after dimensional reduction, the physics near the The total flux of the energy–momentum tensor measured by apparent horizon can be effectively described by the (1+1)- the Kodama observer inside the apparent horizon is given by dimensional metric R ai = N (RA). (20) 1 T ds2 =− 1 − R2/R2 dT 2 + dR2. (11) A − 2 2 N R 1 R /RA From the effective two-dimensional metric (11), T is given by Now, by assuming that the gravitational anomaly also ex- ists near the apparent horizon of a FRW universe, we will ap- R 1 2 N (RA) = πT , (21) ply the anomaly method to calculate the radiation flux from T 12 the apparent horizon.
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