aa r X i v : . [ qu a n t - ph ] N ov Equivalence Principle and Field Quantization in Curved Spacetime
H. Kleinert
Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin,Arnimallee 14, D14195 Berlin, GermanyandICRANeT, Piazzale della Republica 1,10 -65122, Pescara, Italy
To comply with the equivalence principle, fields in curved spacetime can be quantized only inthe neighborhood of each point, where one can construct a freely falling
Minkowski frame with zero curvature. In each such frame, the geometric forces of gravity can be replaced by a selfinteractingspin-2 field, as proposed by Feynman in 1962. At any fixed distance R from a black hole, thevacuum in each freely falling volume element acts like a thermal bath of all particles with Unruhtemperature T U = ~ GM/ πcR . At the horizon R = 2 GM/c , the falling vacua show the Hawkingtemperature T H = ~ c / πGMk B . PACS numbers: 04.62.+v e αµ ( x ) which couples Einstein indices withLorentz indices α . These serve to define anholonomic coordinate differentials dx α in curved spacetime x µ : dx α = e αµ ( x ) dx µ , (1)which at any given point have a Minkowski metric: ds = η αβ dx α dx β , η αβ = − − − = η αβ . (2)With the help of the vierbein field one can write the action simply as [1] A = Z d x √− g ¯ ψ ( x )[ γ α e αµ ( x )( i∂ µ − Γ αβµ Σ αβ ) − m ] ψ ( x ) , g µν ( x ) ≡ η αβ e αµ ( x ) e β ν ( x ) , (3)where Σ αβ is the spin matrix, which is formed from the commutator of two Dirac matrices as i [ γ α , γ β ] /
4, andΓ αβµ ≡ e αν e βλ Γ µνλ is the spin connection, It is constructed from combination of the so-called objects of anholonomityΩ µν λ = [ e αλ ∂ µ e αν − ( µ ↔ ν )], by taking the sum Ω µνλ − Ω νλµ + Ω λµν and lowering two indices with the help of themetric g µν ( x ).The theory of quantum fields in curvilinear spacetime has been set up on the basis of this Lagrangian, or a simplerversion for bosons which we do not write down. The classical field equation is solved on the background metric g µν ( x )in the entire spacetime. The field is expanded into the solutions, and the coefficients are quantized by canonicalcommutation rules, after which they serve as creation and annihilation operators on some global vacuum of thequantum system.The purpose of this note is to make this this procedure compatible with the equivalence principle.2.) If one wants to quantize the theory in accordance with the equivalence principle one must introduce creationand annihilation operators of proper elementary particles. These, however, are defined as irreducible representationsof the Poincar´e group with a given mass and spin. The symmetry transformations of the Poincar´e group can beperformed only in a Minkowski spacetime. According to Einstein’s theory, and confirmed by Satellite experiment, wecan remove gravitational forces locally at one point. The neighborhood will still be subject to gravitational fields.For the definition of elementary particles we need only a small neighborhood. In it, the geometric forces can bereplaced by the forces coming from the spin-2 gauge field theory of gravitation, which was developed by R. Feynmanin his 1962 lectures at Caltech [2]. This can be rederived by expanding of the metric in powers its deviations from theflat Minkowski metric. We define a Minkowski frame x a around the point of zero gravity, and extend it to an entirefinite box without spacetime curvature. Inside this box, particle experiments can be performed and the transformationproperties under the Poincar´e group can be identified.Inside the box, the fields are governed by the flat-spacetime action A = Z d x √− g ¯ ψ ( x ) { γ a e ab ( i∂ b − Γ bca Σ bc ) − m } ψ ( x ) . (4)In this expression, e ab = δ ab + h ab ( x ). The metric and the spin connection are defined as above, exchanging theindices α, β, . . . by a, b, . . . . All quantities must be expanded in powers of h ab .Thus we have arrive at a standard local field theory in the freely falling Minkowski laboratory around the point ofzero gravity. This action is perfectly Lorentz invariant, and the Dirac field can now be quantized without problems,producing an irreducible representation of the Poincar´e group with states of definite momenta and spin orientation | p , s [ m, s ] i .The Lagrangian governing the dynamics of the field h ab ( x ) is well known from Feynman’s lecture [2]. If thelaboratory is sufficiently small, we may work with the Newton approximation: A h = − κ Z d x h ab ǫ cade ǫ cbfg ∂ d ∂ f h eg + . . . , κ = 8 πG/c , G = Newton constant , (5)where ǫ cade is the antisymmetric unit tensor. If the laboratory is larger, for instance, if it contains the orbit of theplanet mercury, we must include also the first post-Newtonian corrections.Thus, although the Feynman spin-2 theory is certainly not a valid replacement of general relativity, it is so in aneighborhood of any freely falling point.The vacuum of the Dirac field is, of course, not universal. Each point x µ has its own vacuum state restricted tothe associated freely falling Minkowski frame.3.) There is an immediate consequence of this quantum theory. If we consider a Dirac field in a black hole, and goto the neighborhood of any point, the quantization has to be performed in the freely falling Minkowski frame withsmooth forces. These are incapable of creating pairs. An observer at a fixed distance R from the center, however,sees the vacua of these Minkowski frames pass by with acceleration a = GM/R , where G is Newton’s constant. At agiven R , the frequency factor e iωt associated with the zero-point oscillations of each scalar particle wave of the worldwill be Doppler shifted to e iωe iat/c c/a , and this wave has frequencies distributed with a probabilty that behaves like1 / ( e π Ω c/a − Z ∞∞ dt e i Ω t e iωe iat/c c/a = e − πc/ a Γ( i Ω c/a ) e − i Ω c/a log( ωc/a ) ( c/a ) . (6)we see that the probability to find the frequency Ω is | e − πc/ a Γ( i Ω c/a ) c/a | , which is equal to 2 πc/ (Ω a ) times1 / ( e π Ω c/a − T U = ~ a/ πck B [4],where k B is the Boltzmann constant. The particles in this heat bath can be detected by suitable particle reactions asdescribed in Ref. [5].The Hawking temperature T H is equal to the Unruh temperature of the freely falling Minkowski vacua at thesurface of the black hole, which lies at the horizon R = r S ≡ GM/c . There the Unruh temperature is equal to T U | a = GM/R ,R =2 GM/c = ~ c / πGM k B = T H .Note that there exists a thermal bath of nonzero Unruh temperature T U ( R ) = T H r S /R at any distance R fromthe center — even on the surface of the earth, where the temperature is too small to be measurable. In the light ofthis it is surprising that the derivation of the thermal bath from semiclassical pair creation is based on a coordinatesingularity at the horizon [6].For decreasing R inside the horizon, the temperature rises to infinity, but this radiation cannot reach any outsideoberver.Acknowledgment:The author thanks V. Belinski pointing out the many papers on the semiclassical explanation of pair creation in ablack hole. [1] Our notation followsH. Kleinert, Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation , World Scientific, Singapore 2009,pp. 1–497 ( ).The only exception is that the vierbein field is here called e αµ rather than h αµ to have the notation h ab free for the smalldeviations of e ab from the flat limit δ ab . [2] R.P. Feynman, F.B. Moringo, D. Pines, Feynman Lectures on Gravitation (held in 1962 at Caltech), Frontiers in Physics,New York. 1962.[3] P.M. Alsing and P.W. Milonni, Am. J. Phys. , 1524 (2004) (arXiv:quant-ph/0401170).[4] W.G. Unruh, Phys. Rev. D , 870 (1976).[5] A. Higuchi, G. E. A. Matsas, and D. Sudarsky, Phys. Rev. D 45, R3308 (1992); 46, 3450 (1992); A. Higuchi, G.E.A. Matsas,D. Sudarsky, Phys. Rev. D R3308 (1992); ibid. , 3450 (1992); D.A.T. Vanzella and G.E.A. Matsas, Phys. Rev. D ,014010 (2001). J. Mod.Phys. D , 1573 (2002).[6] M.K. Parikh, F. Wilczek, Phys. Rev. Lett.85