Greybody factor and quasinormal modes of Regular Black Holes
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Greybody factor and quasinormal modes of Regular BlackHoles
Ángel Rincón a,1 , Victor Santos b,2 Instituto de Física, Pontificia Universidad Católica de Valparaíso, Avenida Brasil 2950, Casilla 4059,Valparaíso, Chile Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP), Av. OliveiraPaiva, 941, Cidade dos Funcionários, 60822-130, Fortaleza, Ceará, BrazilReceived: date / Accepted: date
Abstract
In this work, we investigate the quasinormal frequencies of a class of regularblack hole solutions which generalize Bardeen and Hayward spacetimes. In particular, weanalyze scalar, vector and gravitational perturbations of the black hole both with the semian-alytic WKB method and their time-domain profiles. We analyze in detail the behavior of thespectrum depending on the parameter p / q of the black hole, the quantum number of angular momentum and the s number. In addition, we compare our results with the classical solutionvalid for p = q = Contents
One of the most striking predictions of General Relativity (GR) [1] is the existence of BlackHoles (BHs), objects which produce a region where not even light can escape. They can beformed in the extreme final stages of the gravitational collapse of stars. Such astrophysicalobjects are remarkably simple, being characterized by three parameters: mass, charge andangular momentum, by virtue of the so-called “no-hair” theorems [2–4]. Since gravitationalradiation emitted by an oscillating black hole can carry information about its inner proper-ties like mass and charge [5, 6], this enables to use BHs as a laboratory for studying gravityin strong regimes, where quantum phenomena might take uttermost importance. Also, afterHawking’s seminal papers where the radiation from the black hole horizon was explained[5, 6], black boles become an excellent laboratory to study and also enhance our comprehen-sion about quantum gravity. Naively, the Hawking radiation is taken as black body radiation a e-mail: victor_santos@fisica.ufc.br b e-mail: [email protected] a r X i v : . [ g r- q c ] S e p parametrized by the hawking temperature T H . However, the latter is just an approximatedpicture because emitted particles feel an effective potential barrier in the exterior region.Such barrier backscatters a percentage of the outgoing radiation back into the black hole [7].Thus, the spectrum of the Hawking radiation as seen by an asymptotic observer has not acomplete blackbody distribution: it is better described by a greybody distribution. The grey-body factor can obtained from the transmission amplitude as the field modes pass fromnear horizon region to an asymptotic observer through the effective potential induced by thespacetime geometry. Estimation of this greybody factor is usually a difficult task and oftenone has to resort to approximations, usually in low/high frequency limits. There are mon-odromy methods [8, 9] and computations in a variety of scenarios [10–13], where one canalso employ numerical approaches to estimate them [14]. In particular, it should be noticedthat a considerable part of the literature is dedicated to the case of black holes with singu-larities. In order to fix that problem, it is expected that quantum ingredients play a dominantrole. Thus, although a complete theory of quantum gravity is still under construction, reg-ular black hole solutions can be obtained utilizing additional matter, for instance, takingadvantage of non-linear electrodynamics, as was previously pointed out by Ayon-Beato andGarcia in [15]. Additionally, an interesting feature of the gravitational collapse is that dur-ing the formation phase there is the emission of gravitational radiation, strong enough to bedetected by the gravitational wave detectors [16]. This connection was strongly provided bythe historical direct detection of GWs by LIGO, that two binary merger objects can coalesce into a super-massive black hole [17]. Moreover, recently was reported the first image of thesuper-massive black hole at the center of the giant elliptical galaxy Messier 87 (M87) by theEvent Horizon Telescope [18, 19] which increased the interest on the physics behind thisclass of intriguing objects.In the framework of GR, gravitational radiation arises as perturbations of spacetime it-self. Due to the nonlinear nature of Einsteins equations, it is often very hard to find closedsolutions, and usually one has to resort to perturbation theory. such perturbations give riseto a set of damped vibrations called quasinormal modes [20], complex numbers whose realpart represents the actual frequency of the oscillation and the imaginary part representsthe damping. A simple example of such oscillations are the oscillations of stars which aredamped by internal friction [21]. It is known that in general relativity damping might oc-cur even in frictionless scenarios. This effect arises because energy may be radiated awaytowards infinity by gravitational waves. As it was previously pointed in Ref. [22], even lin-earized perturbations of black holes exhibit quasinormal modes. QNMs are also important inthe investigation of the post-merger remnant of a binary black hole (BBH), as it coalescencesettles to a Kerr black hole at a sufficiently late time after the merging phase. The stationarystate is reached when the perturbed BH remnant emits gravitational waves (GWs) during aprocess known as the ringdown (RD), and The GW corresponding to this phase is describedby the linear perturbation theory of a Kerr BH [23, 24].As was previously said, the study of the quasinormal frequencies is quite relevant, be-cause they encode information on how a black hole relaxes after it has been perturbed. Evenmore, such frequencies depend on i) the geometry and ii) the type of perturbations [25].Along the years, the study of QNM becomes more essential than ever, and certain semi- nal works have been performed up to now, for instance see [26–28] and more recent works [29–43].One issue of the classical description of gravity is the existence of spacetime singular-ities [44]. Although the known solutions of Schwarzschild, Kerr and Reissner-Nordstromhave singularities protected by an event horizon [45], the prediction that BHs emit radiationcause them to shrink until the singularity if reached. This is still a conundrum, as singular- ities are generally regarded as an indicative of the breakdown in the theory, requiring newPhysics for a proper description. It is a common belief that only a consistent quantum theoryof gravity could solve it properly [46]. Since there are no consistent models yet, phenomeno-logical models have been proposed, and most of them are based on the avoidance of thecentral singularity. Such non-singular solutions are called regular black holes [25, 47, 48].Given the increasing interest in gravitational wave astronomy and on QNMs of blackholes, it would be interesting to investigate the QNM spectra from regular BHs. In previousworks quasinormal modes of regular black holes were computed by several authors, seee.g. [49–52]. In this paper we propose to investigate the QNMs of a class of regular blackholes which generalize Bardeen and Hayward spacetimes. In particular, we analyze scalar,vector and gravitational perturbations of the black hole both with the semianalytic WKBmethod and compute the greybody factor. Furthermore, it is essential to note that there isa vast collection of works where QNMs are calculated using the WKB approximation (see[31, 35, 53–56] and references therein).The work is structured as follows. In section 2 we introduce the spacetime backgroundof regular black holes and how to describe the field perturbations. In section 3 we computethe quasinormal frequencies emplying the 6th order WKB approximation, and in section 4we compute the greybody factor. Section 5 is devoted to the conclusions. s = − f ( r ) d t + f ( r ) − d r + r d Ω , (1)where d Ω is the metric of the 2-sphere and the lapse function f ( r ) = − m ( r ) / r explicitlydepends on the matter distribution. The particular mass function [57] m ( r ) = M [ + ( r / r ) q ] p / q , p , q ∈ Z (2)ensures an asymptotically flat spacetime for positive p and q . M and r can be interpretedas mass and length parameters. The regular BH solutions proposed by Bardeen [58] andHayward [47] correspond to the choices ( p , q ) = ( , ) and ( p , q ) = ( , ) in equation (2).The limit of large r is m ( r ) ≈ M (cid:18) − p q (cid:18) r r (cid:19) q (cid:19) , (3)the only restriction on (3) to obtain an asymptotically Schwarzschild solution is to have q >
0. To compare our results with results found for Bardeen and Hayward solutions, wewill restrict ourselves to the case p = q >
0. The behaviour of some typical cases isshown in Fig. (1). Thus, should be noticed that when the parameter q increases the massfunction m ( r ) tends to its constant value M . Fig. 1
Mass function (2) with p = q >
0. The parameter r is typically assumed to be microscopic( r (cid:28) M ) and hence the exterior region of the BH can be very close to the Schwarzschild space-time. The q → ∞ case, as evidenced by the case q =
90, corresponds to the usual matching between de Sitter andSchwarzchild solutions in the interior region of the BH. The grey shaded region indicates the interior region r < M . In this graphic we have used r / M = − . − ∂ Ψ∂ t + ∂ Ψ∂ r ∗ + V ( r ∗ ) Ψ = , (4)where the tortoise coordinate r ∗ is defined according to the differential equationd r ∗ d r = f ( r ) . (5)and the effective potential is V ( r ) = f ( r ) (cid:18) (cid:96) ( (cid:96) + ) r + − s r f (cid:48) ( r ) (cid:19) , (6)where (cid:96) denotes the multipole number of the spherical harmonics decomposition and s = , , Fig. 2
Effective potential (6) for the mass function (2) for gravitational perturbations s =
2, multipole number (cid:96) = p = q >
0. The grey shaded region indicates the interior region r < M . Notice that in theexterior region the potential is always positive and peaked. showed in 2 we can see that in the exterior region r > M the effective potential does notvaries sensibly for q >
1. Therefore, one should not expect large differences in the perturba-tion oscillation frequencies in the limit of large q . This can be further inferred by observingthe behavior of the effective potential with respect to the tortoise coordinate (5), as shownin Fig. 3. Employing the stationary ansatz Ψ ( t , r ∗ ) ∼ e i ω t ψ ( r ∗ ) , equation (6) becomesd ψ d r ∗ + (cid:2) ω − V ( r ∗ ) (cid:3) ψ = , (7)which precisely takes the form of a Schrödinger-like equation. The frequencies of the tem-poral decomposition take the form ω = ω R + i ω I , where ω R is the oscillation frequency ofthe QNM and ω I is the damping time. Therefore, any mode with ω I < The QNMs can be computed by imposing proper boundary conditions on (7), where thefields are purely ingoing at the BH horizon and purely outgoing at the spatial infinity.With such boundary conditions, the resulting frequencies are complex. If the potential (6) ispeaked and falls to a constant in the asymptotic region, one can compute the QNM frequen-cies from (7) employing the WKB approximation described by Schutz, Iyer and Will [59–61] and posteriorly improved by Konoplya [62]. With this last improvement the QNMs canbe computed by i Q (cid:112) Q (cid:48)(cid:48) − p ∑ i = Λ i = n + , n = , , . . . (8) Fig. 3
Effective potential (6) for the mass function (2) for gravitational perturbations s =
2, multipole number (cid:96) = p = q > r ∗ . (cid:96) n q = q = q = q =
100 0 0 . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . Table 1
Quasinormal frequencies of scalar perturbations for p = q >
1. for the regular Black Holewhere r / M = . where the correction term Λ i can be obtained for different orders of approximation. n is theovertone number and Q ( i ) is the i -th derivative of Q = ω − V computed at the maximumof the potential. It is worth to mention that the accuracy of the WKB method is dependentof the multipole number and overtone: in general the approximation is appropriate for (cid:96) > n and it is not applicable for (cid:96) < n . The results for the quasinormal modes are presented in tables 1, 2 and 3.
We can notice that all QNMs possess a negative imaginary part, conferring thereforestability for the black hole. Also, both real and imaginary parts of the QNMs do not seem tobe wildly sensible to the parameter q in the limit q → ∞ , indicating that the damping timeis, with good approximation, independent of it. However, we can observe a change the case q ≤
1, when the BH is smaller damping time than the classical case. (cid:96) n q = q = q = q =
101 0 0 . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . Table 2
Quasinormal frequencies of electromagnetic perturbations for p = q >
1. for the regular BlackHole where r / M = . (cid:96) n q = q = q = q =
102 0 0 . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . Table 3
Quasinormal frequencies of gravitational perturbations for p = q >
1. for the regular BlackHole where r / M = . Another interesting aspect for investigate the field perturbations around a black hole space-time is the amount of plunging field which is absorbed by the black hole, the absorptioncross section. It embodies the likelihood of a particle to be scattered/deflected by the blackhole. Based on the quantum mechanics analogy, the cross-section can be written as σ s ( ω ) = ∞ ∑ (cid:96) = σ ( (cid:96) ) s ( ω ) , (9)where we defined the partial absorption cross section σ ( (cid:96) ) s ( ω ) = πω ( (cid:96) + ) T (cid:96), s ( ω ) , (10)where T (cid:96), s ( ω ) is the transmission coefficient. To compute the transmission coefficient weemploy the transfer matrix method, as described in [63].The partial absorption cross sections versus M ω for (cid:96) = (cid:96) = (cid:96) = σ (cid:96) / M decreases when the valueof (cid:96) increases, due to the fact that the effective potential peak increases for larger values of (cid:96) . Fig. 4
Real and imaginary parts of the quasinormal mode ( (cid:96), n ) = ( , ) of scalar perturbations for p = q >
0. The dashed lines correspond to the respective values of the real and imaginary parts of the ClassicalSchwarzschild black hole.
In the present paper we have investigated the quasinormal frequencies and the greybodyfactors of gravitational perturbations around a class of regular black hole solutions. To ob-tain the corresponding quasinormal frequencies, we perturbed the background and takingadvantage of the symmetry we are able to decompose the original function into sphericalharmonics, writing down the differential equation of the problem involved. Then we con-sider the sixth order WKB approximation to obtain the QN frequencies. Further, as can beobserved in tables, our solutions reveal that the family of black holes here analysed recoverthe classical behaviour for q → ∞ , and possess a stronger absorption section in the micro-scopic regime. That means that BH regularity amends the dissipative effect of the black holeon its neighborhood. Finally, as can be observed in tables, for q >
1, the imaginary part ofthe quasinormal frequencies are also negative. Thus, our results indicate that all modes arefound to be unstable.
The author V. S. would like to thank the Fundação Cearense de apoio ao DesenvolvimentoCientífico e Tecnológico (FUNCAP) for financial support. The author A. R. acknowledgesDI-VRIEA for financial support through Proyecto Postdoctorado 2019 VRIEA-PUCV.
Fig. 5
Real and imaginary parts of the quasinormal mode ( (cid:96), n ) = ( , ) of electromagnetic perturbations for p = q >
0. The dashed lines correspond to the respective values of the real and imaginary parts of theClassical Schwarzschild black hole.
References
1. A. Einstein, Annalen Phys. (7), 769 (1916). DOI 10.1002/andp.200590044,10.1002/andp.19163540702. [Annalen Phys.14,517(2005); ,65(1916); AnnalenPhys.354,no.7,769(1916)]2. W. Israel, Physical Review (5), 1776 (1967). DOI 10.1103/PhysRev.164.1776.URL https://link.aps.org/doi/10.1103/PhysRev.164.1776 . Publisher: Amer-ican Physical Society3. W. Israel, Communications in Mathematical Physics (3), 245 (1968). DOI 10.1007/BF01645859. URL https://doi.org/10.1007/BF01645859
4. B. Carter, Physical Review Letters (6), 331 (1971). DOI 10.1103/PhysRevLett.26.331. URL https://link.aps.org/doi/10.1103/PhysRevLett.26.331 . Publisher:American Physical Society5. S.W. Hawking, Nature , 30 (1974). DOI 10.1038/248030a06. S.W. Hawking, Commun. Math. Phys. , 199 (1975). DOI 10.1007/BF02345020,10.1007/BF01608497. [,167(1975)]7. P. Kanti, J. March-Russell, Phys. Rev. D , 024023 (2002). DOI 10.1103/PhysRevD.66.0240238. B. Carneiro da Cunha, F. Novaes, Phys. Rev. D , 024045 (2016). DOI 10.1103/PhysRevD.93.024045. URL https://link.aps.org/doi/10.1103/PhysRevD.93.024045
9. A. Castro, J.M. Lapan, A. Maloney, M.J. Rodriguez, Classical and Quantum Gravity (16), 165005 (2013). DOI 10.1088/0264-9381/30/16/165005. URL https://doi.org/10.1088%2F0264-9381%2F30%2F16%2F165005 Fig. 6
Real and imaginary parts of the quasinormal mode ( (cid:96), n ) = ( , ) of gravitational perturbations for p = q >
0. The dashed lines correspond to the respective values of the real and imaginary parts of theClassical Schwarzschild black hole.
10. H. Gürsel, I. Sakallı, Eur. Phys. J. C (3), 234 (2020). DOI 10.1140/epjc/s10052-020-7791-311. Y.H. Hyun, Y. Kim, S.C. Park, JHEP , 041 (2019). DOI 10.1007/JHEP06(2019)04112. S. Kanzi, I. Sakallı, Nucl. Phys. B , 114703 (2019). DOI 10.1016/j.nuclphysb.2019.11470313. A. Chowdhury, N. Banerjee, Phys. Lett. B , 135417 (2020). DOI 10.1016/j.physletb.2020.13541714. F. Gray, M. Visser, Universe (9), 93 (2018). DOI 10.3390/universe409009315. E. Ayon-Beato, A. Garcia, Phys. Rev. Lett. , 5056 (1998). DOI 10.1103/PhysRevLett.80.505616. K.D. Kokkotas, B.G. Schmidt, Living Reviews in Relativity (1), 2 (1999). DOI 10.12942/lrr-1999-217. B.P. Abbott, et al., Phys. Rev. Lett. (6), 061102 (2016). DOI 10.1103/PhysRevLett.116.06110218. K. Akiyama, et al., Astrophys. J. (1), L1 (2019). DOI 10.3847/2041-8213/ab0ec719. K. Akiyama, et al., Astrophys. J. (1), L6 (2019). DOI 10.3847/2041-8213/ab114120. R.A. Konoplya, A. Zhidenko, Reviews of Modern Physics (3), 793–836 (2011). DOI10.1103/RevModPhys.83.79321. K.D. Kokkotas, B.F. Schutz, Monthly Notices of the Royal Astronomical Society (1), 119–128 (1992). DOI 10.1093/mnras/255.1.11922. W.H. Press, apjl , L105 (1971). DOI 10.1086/18084923. C.V. Vishveshwara, Nature (52615261), 936–938 (1970). DOI 10.1038/227936a024. M. Giesler, M. Isi, M.A. Scheel, S.A. Teukolsky, Physical Review X (4), 041060(2019). DOI 10.1103/PhysRevX.9.041060 Fig. 7
The partial absorption cross section for a massless scalar wave impinging upon a Bardeen Black Hole( p = , q =
2) for the first six multipoles for r / M = .
25. A. Flachi, J.P. Lemos, Phys. Rev. D (2), 024034 (2013). DOI 10.1103/PhysRevD.87.02403426. K.D. Kokkotas, B.G. Schmidt, Living Rev. Rel. , 2 (1999). DOI 10.12942/lrr-1999-227. R. Konoplya, Phys. Rev. D , 024018 (2003). DOI 10.1103/PhysRevD.68.02401828. R. Konoplya, A. Zhidenko, Rev. Mod. Phys. , 793 (2011). DOI 10.1103/RevModPhys.83.79329. A. Barrau, K. Martineau, J. Martinon, F. Moulin, Phys. Lett. B , 346 (2019). DOI10.1016/j.physletb.2019.06.03330. R. Konoplya, A. Zhidenko, A. Zinhailo, Class. Quant. Grav. , 155002 (2019). DOI10.1088/1361-6382/ab2e2531. G. Panotopoulos, A. Rincóón, Int. J. Mod. Phys. D (03), 1850034 (2017). DOI10.1142/S021827181850034732. A. Rincón, G. Panotopoulos, Phys. Rev. D (2), 024027 (2018). DOI 10.1103/PhysRevD.97.02402733. K. Destounis, G. Panotopoulos, A. Rincón, Eur. Phys. J. C (2), 139 (2018). DOI10.1140/epjc/s10052-018-5576-834. A. Rincón, G. Panotopoulos, Eur. Phys. J. C (10), 858 (2018). DOI 10.1140/epjc/s10052-018-6352-535. G. Panotopoulos, A. Rincón, Eur. Phys. J. Plus (6), 300 (2019). DOI 10.1140/epjp/i2019-12686-x36. A. Rincón, G. Panotopoulos, Phys. Dark Univ. , 100639 (2020). DOI 10.1016/j.dark.2020.100639
37. A. Rincon, G. Panotopoulos. Quasinormal modes of black holes with a scalar hair inEinstein-Maxwell-dilaton theory (2020)38. G. Panotopoulos. Quasinormal modes of charged black holes in higher-dimensionalEinstein-power-Maxwell theory (2020). DOI 10.3390/axioms901003339. R. Oliveira, D. Dantas, V. Santos, C. Almeida, Class. Quant. Grav. (10), 105013(2019). DOI 10.1088/1361-6382/ab187340. V. Santos, R. Maluf, C. Almeida, Phys. Rev. D (8), 084047 (2016). DOI 10.1103/PhysRevD.93.08404741. S. Devi, R. Roy, S. Chakrabarti, Eur. Phys. J. C (8), 760 (2020). DOI 10.1140/epjc/s10052-020-8311-142. K. Jusufi, M. Amir, M.S. Ali, S.D. Maharaj. Quasinormal modes, shadow and greybodyfactors of 5D electrically charged Bardeen black holes (2020)43. C.F.B. Macedo, L.C.B. Crispino, E.S. de Oliveira, International Journal of ModernPhysics D (09), 1641008 (2016). DOI 10.1142/S021827181641008X. URL https://doi.org/10.1142/S021827181641008X
44. S.W. Hawking, G.F.R. Ellis,
The Large Scale Structure of Space-Time (Cambridge Monographs on Mathematical Physics) (Cam-bridge University Press, 1975). URL
45. R. Penrose, General Relativity and Gravitation (7), 1141–1165 (2002). DOI 10.1023/ A:101657840820446. V.P. Frolov, G. Vilkovisky, in
The Second Marcel Grossmann Meeting on the RecentDevelopments of General Relativity (In Honor of Albert Einstein) (1979), p. 045547. S.A. Hayward, Physical Review Letters (3), 031103 (2006). DOI 10.1103/PhysRevLett.96.03110348. B. Toshmatov, A. Abdujabbarov, Z.e. Stuchlík, B. Ahmedov, Phys. Rev. D (8),083008 (2015). DOI 10.1103/PhysRevD.91.08300849. S. Fernando, J. Correa, Physical Review D (6), 064039 (2012). DOI 10.1103/PhysRevD.86.06403950. A. Flachi, J.P.S. Lemos, Physical Review D (2), 024034 (2013). DOI 10.1103/PhysRevD.87.02403451. B. Toshmatov, A. Abdujabbarov, Z. Stuchlík, B. Ahmedov, Physical Review D (8),083008 (2015). DOI 10.1103/PhysRevD.91.08300852. B. Toshmatov, Z. Stuchlík, J. Schee, B. Ahmedov, Physical Review D (8), 084058(2018). DOI 10.1103/PhysRevD.97.08405853. G. Panotopoulos, A. Rincón, Phys. Rev. D (8), 085014 (2018). DOI 10.1103/PhysRevD.97.08501454. S. Dey, S. Chakrabarti, Eur. Phys. J. C (6), 504 (2019). DOI 10.1140/epjc/s10052-019-7004-055. C. Wu. Quasinormal modes of gravitational perturbation around some well-knownregular black holes (2017)56. D. Mahdavian Yekta, M. Karimabadi, S. Alavi. Quasinormal modes of Regular blackholes: Non-minimally coupled massive scalar field perturbations (2019)57. J.C.S. Neves, A. Saa, Physics Letters B , 44–48 (2014). DOI 10.1016/j.physletb.2014.05.02658. S. Ansoldi. Spherical black holes with regular center: a review of existing modelsincluding a recent realization with gaussian sources (2008)59. S. Iyer, Phys. Rev. D , 3632 (1987). DOI 10.1103/PhysRevD.35.3632. URL https://link.aps.org/doi/10.1103/PhysRevD.35.3632
60. B.F. Schutz, C.M. Will, Astrophysical Journal, Letters , L33 (1985). DOI 10.1086/18445361. S. Iyer, C.M. Will, Phys. Rev. D , 3621 (1987). DOI 10.1103/PhysRevD.35.3621.URL https://link.aps.org/doi/10.1103/PhysRevD.35.3621
62. R.A. Konoplya, Phys. Rev. D , 024018 (2003). DOI 10.1103/PhysRevD.68.024018.URL https://link.aps.org/doi/10.1103/PhysRevD.68.024018
63. M.L. Du, Communications in Theoretical Physics25