# Novel properties of bound states of KleinâGordon equation in ...

Novel properties of bound states of KleinâGordon equation in ...

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**Novel** **properties** **of** **bound** **states** **of** Kle**in**–Gordon **equation** **in**... 91studied until now, because for the standard Hilbert form **of** the Schwarzschild solution”the bare rest-mass density is never even **in**troduced” [8] correctly. The dependence **of**the first four energy levels ε 2 0 l,1 l,2 l,3 lon the mass ratio ϱ is shown **in** Figure 5 for l = 2.A typical steep-like dependence **of** the discrete eigenvalues ε 2 n l on the variable ϱ2 is seen.For each value **of** the pr**in**ciple quantum number n the number **of** the jumps **of** ε 2 n l (ϱ2 )depends on the number **of** the maxima **of** the correspond**in**g wave function which aremoved from the **in**ner well to the exterior one dur**in**g the transition regime, which developswith the **in**crease **of** the values **of** the mass ratio ϱ.It is clear that such nontrivial behavior **of** the eigenvalues ε 2 n l (ϱ2 ) **of** the Kle**in**-Gordon**equation** **in** a gravitational field **of** a po**in**t particle is due to the existence **of** two f**in**itepotential wells: 1) An **in**ner one which has a size **of** order **of** the Schwarzschild radiusand **in** general case **of** macroscopic orbital momenta is very deep; and 2) An exterior one,which is extremely shallow **in** comparison with the **in**ner well. The normal world withalmost Newton gravity “lives” **in** the exterior well. The both wells are separated typicallyby a huge potential barrier which looks like a centrifugal barrier from outside, and as asuspensory barrier **of** the type **of** a spherical potential wall from **in**side. Our calculationsdescribe the quantum penetration under this barrier. The quantum result depends stronglyon the mass defect **of** the po**in**t source.0Cases: µ = 1, l = 2, n = 0, 1, 2, 3, 4−0.01Eigenvalues−0.02−0.03ε 42ε 32ε 22ε 12ε 02−0.041Ε−007 1Ε−005 0.001 0.1ϲ 2Figure 6: The attraction and the repulsion **of** the quantum levels ε n l for different values **of** thegravitational mass defect **of** the po**in**t source.The obta**in**ed **in** this article behavior **of** a quantum test particles **in** the gravitationalfield **of** po**in**t source **of** gravity seems to us to be much more physical then the one **in**the wide spread models **of** black holes. Clearly, **in** contrast to such space-time holeswith nonphysical **in**f**in**itely deep well **in** them, **in** our case the f**in**ite **in**ner well plays therole **of** a trap for the test particles. It is not excluded that this way one may be able toconstruct a model **of** very compact matter objects with an arbitrary large mass and a size**of** order **of** Schwarzschild radius. It’s possible that such type **of** objects may expla**in** theobserved **in** the Nature compact dark objects. They may be the f**in**al product **of** the stelarevolution, **in**stead **of** the quite formal and sophisticated constructions like black holes

92 P.P. Fiziev, T.L. Boyadjiev, D.A. Georgievawhich correspond to empty space solutions **of** E**in**ste**in** **equation**s. Of course, at presentthese possibilities are an open problem which deserves further careful study.It is easy to observe one more astonish**in**g phenomena **in** the discrete spectrum **of** atest quantum particle **in** gravitational field **of** po**in**t source. It is related, too, with themass defect **of** the source. If one puts **in** the same figure the functional dependence **of**all discrete eigenvalues ε n l on the squared mass ratio ϱ 2 = g tt | r=0 (see **in** [1]), one canobserve a repulsion and an attraction (up to a quasi-cross**in**g) **of** these levels, as shown**in** Figure 6. Such type **of** behavior **of** quantum discrete levels is well known **in** the laserphysics, **in** the neutr**in**o oscillations and **in** other branches **of** quantum physics. To the best**of** our knowledge we are observ**in**g such behavior **of** quantum levels **in** the fundamentalgravitational physics for the first time.4. ConclusionsIn the present-days traditional approach to the motion **of** test particles **in** the Schwarzschildgravitational field, **in**side the event horizon there exist a nonphysical **in**f**in**itely deep potentialwell. Everyth**in**g which somehow can be placed **in**side this well will fall unavoidableto its center ρ = 0 for a f**in**ite time. It is well known that this center is a physically**in**admissible geometrical s**in**gularity. To hide such undesired s**in**gularity one must usea unprovable mathematical hypothesis like the cosmic censorship one. Today we haveenough examples which show that such hypotheses can not be correct at least **in** its orig**in**alformulation, see, for example [9] and the references there**in**.In the article [1] the pr**in**cipal role **of** the matter po**in**t source **of** gravity was stressed.In GR the matter po**in**t source presents a natural cutt**in**g factor for the physical values **of**the lum**in**osity variable ρ ∈ [ρ 0 , ∞), where ρ 0 > ρ G (5). This is because the **in**f**in**ite massdensity **of** the matter po**in**t changes drastically the geometry **of** the space-time around it.In full accord with Dirac’s suggestion [6] this cutt**in**g places all nonphysical phenomena,together with the very event horizon **in** the nonphysical doma**in** **of** the variables and yieldsa large number **of** new **in**terest**in**g phenomena.In particular, as shown **in** the present article, the described cutt**in**g yields an **in**terest**in**gnovel discrete spectra for Kle**in**-Gordon test particles **in** the gravitational field **of** massivepo**in**t. Mathematically this is caused by the universal change **of** the **bound**ary conditionsfor the correspond**in**g wave **equation**, due to the presence **of** the matter source **of** gravityand depends on its mass defect.A real discrete spectrum does not exist **in** the case **of** black hole solution, just because**of** the presence **of** a hole **in** the space-time, which absorbs everyth**in**g around it. Ourconsideration gives a unique possibility for a direct experimental test **of** the existence **of**space-time holes. If one will f**in**d a discrete spectra **of** correspond**in**g stationary waves,propagat**in**g around some compact dark object, one will have **in**disputable evidence thatthere is no space-time hole **in**side such object. If, **in** contrast, one will f**in**d only a decay**in**gquasi-normal modes with complex eigenvalues (see [5] and the references here**in**), thiswill **in**dicate that we are observ**in**g **in**deed a black hole.We have obta**in**ed similar results for electromagnetic and gravitational waves **in** thegravitational field **of** po**in**t particle[10]. Their further study is an important physical problemand will give us more practical physical criteria for an experimental dist**in**guish**in**g **of**the two complete different hypothetical objects: 1) the space-time black holes and 2) thepossible new very compact dark objects made **of** real matter. Present days astrophysical

**Novel** **properties** **of** **bound** **states** **of** Kle**in**–Gordon **equation** **in**... 93observations are still not able do make a difference between them. The only real observationalfact is that we are seen a very compact and very massive objects, which showup only due to their strong gravitational field. An actual theoretical problem is to f**in**d aconv**in**c**in**g model for description **of** these observed compact dark objects and the criteriafor the experimental justification **of** such model. We hope that the present article is a newstep **in** this direction.Acknowledgments. One **of** the authors (PPF) is grateful to the High Energy PhysicsDivision, ICTP, Trieste, for the hospitality and for the nice work**in**g conditions dur**in**g hisvisit **in** the autumn **of** 2003. There the idea **of** the present article was created. PPF isgrateful to the JINR, Dubna, too, for the priority f**in**ancial support **of** the present article.All **of** the authors are deeply **in**debted to the JINR, Dubna for the hospitality and for theexcellent work**in**g conditions dur**in**g their three months stay there at the end **of** 2003 andat the beg**in**n**in**g **of** 2004, when the most **of** the work was done. We wish to acknowledge,too, the participants **in** the scientific sem**in**ars **of** the BLTF and LIT **of** JINR for thestimulat**in**g discussions **of** the basic ideas, methods and results **of** the present article.References[1] Fiziev, P., Gravitational Field **of** Massive P**in**t Particle **in** General Relativity, gr-qc/0306088;Abdus Salam ICTP, Trieste, Italy: prepr**in**t IC/2003/122; On the Solutions **of** E**in**ste**in** Equationswith Massive Po**in**t Source, gr-qc/0407088; Fiziev, P., Dimitrov, S., Po**in**t Electric Charge**in** General Relativity, hep-th/0406077.[2] Hilbert, D. (1917) Nach. Ges. Wiss. Göt**in**gen, Math. Phys. Kl. 53; Droste, J. (1917) Proc. K.Ned. Acad. Wet., Ser. A 19 197; Weyl, H. (1917) Ann. Phys (Leipzig) 54 117.[3] Feshbach, H., Villars, F. (1958) Rev. Mod. Phys. 30 24.[4] Jacobson, T., Introduction to Quantum Fields **in** Curved Spacetime and Hawk**in**g Effect, grqc/0308048.[5] Regge, T., Wheeler, J. A. (1957) Phys. Rev. 108 1063; Chandrasekhar, S. (1983) The MathematicalTheory **of** Black Holes, Oxford University Press, Oxford; Kokkotas, K.D., Schmidt,B.G. (1999) Liv**in**g Rev. Relativity 2; Abdalla, E., Mol**in**a, C., Saa, A., Field propagation **in**the Schwarzschild–de Sitter black hole, gr-qc/0309078; Das, S., Shankaranarayanan, S., Highfrequences quasi-normal modes for black-holes with generic syngularities, gr-qc/0410209;Szpak, N., Quas**in**ormal mode expansion and the exact solution **of** the Cauchi problem forwave **equation**s, hep-th/0411050; Konoplya, R. A., Zhidenko, A. V., Decay **of** massive scalarfield **in** a Schwarzschild background, gr-qc/0411059.[6] Dirac, P. A. M. (1962) Proc. Roy Soc. (London), A270 354; (1964) **in** Conference **in** Warszawaand Jablonna, L. Infeld (Ed), pp. 163–175, Gauthier-Villars, Paris.[7] Puzyn**in**, I.V. et al. (1999) Physics **of** Elementary Particles and Atomic Nuclei (AIP), vol. 30(1)97.[8] Arnowitt, R., Deser, S., Misner, C.W. (1960) Phys. Rev. Lett. 4(7) 375.[9] Joshi, P. S. (1993) Global Aspects **in** Gravitation and Cosmology, Clarendon Press, Oxford;S**in**gh, T. P., Joshi, P. S. (1996) Class. Quant. Grav. 13 559; Hamade, R. S., Stewart, J. M.(1996) Class. Quant. Grav. 13 497; Giambo, R., Giannoni, F., Magli, G., Piccone, P. (2003)

94 P.P. Fiziev, T.L. Boyadjiev, D.A. GeorgievaClass. Quant. Grav. 20 L75; New Solutions **of** E**in**ste**in** Equations **in** Spherical Symmetry: TheCosmic Censor to the Court, gr-qc/0204030.[10] Fiziev, P. P., Boyadjiev, T.L., Georgieva, D.A. (2004) **Novel** Stationary States **of** Scalar, Electromagneticand Gravitational Waves **in** Gravitational Field **of** Massive Po**in**t (**in** preparation).