Mean Field Theory and Astrophysical Black Holes C. Chevalier and F. Debbasch ERGA-LERMA Université Paris VI UMR 8112, 3 rue Galilée, 94200 Ivry, France Abstract. We review the basics of a newly developed mean field theory of relativistic gravitation. A particularly simple coarse graining of the Schwarzschild space-time is presented as an example. This example is then used to discuss current and near future observations of Sgr A*. Keywords: General Relativity, Black Holes, Relativistic Statistical Physics PACS: 04.20.-q, 05.20.-y, 04.70.Bw, 98.35.Jk

INTRODUCTION Developing a mean field theory for general relativity has long been the subject of active research ([10, 11, 17, 12, 26, 3, 4]). This problem is of undeniable theoretical interest, but it is also of real practical importance because finite precision effects in astrophysical observations of relativistic objects and in observational cosmology can only be modeled properly through a field theory of relativistic gravitation ([9]). The last three years have witnessed the construction of the first general mean field theory for Einstein gravitation ([5, 6]). The aim of the present contribution is to review the basic elements of this theory and to present a simple application of astrophysical interest. The material is organized as follows. Section 2 is devoted to the mean field theory itself. We introduce statistical ensembles of space-times and show how to define in a mathematically consistent and physically meaningful way a mean or coarse grained space-time associated with such an ensemble. Section 3 presents a particular statistical ensemble which can be interpreted as modeling finite precision observations of a single Schwarzschild black hole. The mean space-time corresponding to this ensemble is found to be also a black hole, but of different horizon radius. Further investigation of this space-time is presented in Section 4. Contrary to the Schwarzschild black hole, the mean black hole is surrounded by a matter which describes the net effect of the averaged degrees of freedom of the gravitational field on the coarse grained field. This matter is characterized in Section 4.1 by its energy density and pressures. We then show in Section 4.2 that the total mass of the space-time is not modified by the averaging. The averaging thus only redistributes the mass in space-time. The temperature of the mean space-time is finally evaluated in Section 4.3. It is, at fixed total mass of the black hole, an increasing function of the coarse graining. Our results are discussed in Section 5. In particular, the conclusions of the theoretical calculations presented here can be used to interpret, at least qualitatively, the results of current and near future observations of Sgr A*, which is the black hole candidate closest to us. We argue that current observations, despite the very poor relative precision (of order 400), probably deliver a correct estimate of the mass M of Sgr A*. As far as near future observations are concerned, we argue that they will probably detect a shadow, but that the relative difference between the size of this shadow and the theoretical estimate 5 × 2M will probably be of order unity. The size of the observed shadow should therefore not furnish a direct, unbiased estimate of the mass of Sgr A*.

A MEAN FIELD THEORY FOR GENERAL RELATIVITY Suppose that a certain characteristic ω of the physical space-time is not accessible to current observations; for example, the parameter ω can represent a finite spatial cutoff which takes into account the finite angular resolution of astrophysical and/or cosmological observations. The natural way to model the results of these observations is to introduce a statistical ensemble of ω -dependent space-times and to perform an averaging over ω to generate from this ensemble a single coarse grained space-time. How to define precisely such a coarse grained space-time in a physically and mathematically meaningful way has been an open problem for many years ([10, 11, 17, 12, 26, 3, 4]). The aim of this Section is to provide a self contained introduction to a recently developed ([5, 6]), rather general solution to this problem.

Let M be a fixed ‘base’ manifold and let Ω be an arbitrary probability space. Let g(ω ) be an ω -dependent Lorentzian metric defined on M . Each ω -dependent couple S (ω ) = (M , g(ω )) represents a physical space-time and the collection of all couples defines a statistical ensemble Σ of space-times. Each member of Σ is naturally equipped with the Levi-Civita connection Γ(ω ) associated with g(ω ) ([25]) and associated with a stress-energy tensor T (ω ) related to the metric g(ω ) and Γ(ω ) through Einstein equation. It has been shown in ([5]) that the statistical ensemble Σ of space-times can be used to define a single, mean or coarse grained space-time S¯ = (M , g) ¯ where the metric g¯ is simply the average of the metrics g(ω ) over ω ; one thus has, for all points P of M : (1) g(P) ¯ = hg(P, ω )i, where the brackets on the right-hand side indicate an average over the probability space Ω. The connection of the mean space-time S¯ is simply the Levi-Civita connection associated to the metric g¯ and will be conveniently called the mean or coarse grained connection. Since the relations linking the coordinate basis components g µν of an arbitrary metric g to the Christoffel symbols Γαµν of its Levi-Civita connection are non-linear, the Christoffel symbols of the mean connection are not identical to the averages of the Christoffel symbols associated to the various space-times S (ω ). That definition (1) of the mean metric is physically quite natural is perhaps best understood by averaging a statistical ensemble of ω -dependent geodesics. Let P be a point of M and let U be a (contravariant) 4-vector tangent to M at P. Given a coordinate system x on M around P, the geodesic of the metric g(ω ) passing through P with tangent U at P obeys the relation: dVµ 1 (ω ) = ∂α gµβ (xP , ω )U α U β , (2) ds |P 2 where s is an affine parameter along the geodesic and dsµ |P (ω ) stands for the acceleration of the geodesic at point P. Note that this acceleration is defined as an element of the space cotangent to M at P i.e. it is a covariant vector at P. This definition is natural because, for time-like geodesics, the acceleration can be interpreted as the s-derivative of the momentum of a point particle following the geodesic and the momentum of a particle, being opposite of the x-derivative of a Hamiltonian ([1, 16]), is by definition a cotangent vector. Relation (2) can be averaged over ω to deliver: dV

<

dVµ (ω ) > = ds |P =

1 ∂α < gµβ (xP , ω ) > U α U β 2 1 ∂α g¯ µβ (xP )U α U β . 2

(3)

Equation (3) means that the collection CP,U of ω -dependent geodesics passing through P with tangent U at P can be averaged locally into a single geodesic and that this averaged or coarse grained geodesic is the geodesic of the metric g¯ (see equation (1)) passing through P with tangent U at P. Reference ([6]) provides a thorough discussion of how the time-like, space-like or null caracters of geodesics may be altered by coarse graining. The local nature of the averaging is also discussed in ([6]), allbeit in the cosmological context only. The more general case where test matter evolving in space-time is not represented by point particles following geodesics but by classical fields is adressed in ([5]). Now on to the so-called ‘source’ of the averaged or coarse grained field. Because the Einstein tensor depends ¯ g) non-linearly on the metric, the Einstein tensor E¯ = E (∇(Γ), ¯ associated to g¯ and Γ¯ does not generally coincide with the average of the Einstein tensors E (∇(Γ(ω )), g(ω )). The tensor E¯ is nevertheless the Einstein tensor of the defines, via Einstein equation, stress-energy tensor T¯ for the mean space-time. Since mean space-time. It therefore



aαβ αβ ¯ Eµν (g) ¯ 6= Eµν (g(ω )) , T is generally different from T (ω ) . The additional, generally non vanishing tensor field ∆T = T¯ − hT (ω )i, can be interpreted as the stress-energy tensor of an ‘apparent matter’. This apparent matter simply describes the cumulative effects of the averaged upon (small scale) fluctuations of the gravitational field on the (large scale) behaviour of the coarse grained field. In particular, the vanishing of T (ω ) for all ω does not necessarily imply the vanishing of T¯ . The mean stress-energy tensor T¯ can therefore be non vanishing in regions where the unavera ged stress-energy tensor actually vanishes. A general discussion of this and other perhaps unexpected consequences of definition (1) can be found in ([5]); possible cosmological applications are also discussed in ([6, 7]); finally, related issues are addressed, albeit in a less general framework, by ([18, 15]. Let us finally mention that the averaging scheme just presented is the only one which ensures that the motions in a mean field can actually be interpreted, at least locally, as the averages of ‘real’ unaveraged motions. This very important point is fully developed in ([6]).

EXAMPLE: DESCRIPTION OF A SINGLE SCHWARZSCHILD BLACK HOLE OBSERVED WITH FINITE PRECISION Definition of the statistical ensemble and determination of the mean metric The statistical ensemble considered in ([7]) describes a single Schwarzschild black hole observed with finite precision measurements of the three spatial Kerr-Schild coordinates. More precisely, this ensemble is defined as an ensemble of space-times S (ω ), each member of the ensemble being equipped with the metric g(t, r, ω ) given by:   2dt 2M 1 2 2 2 2 2 dt − (dr · (r − ω )) . dr · (r − ω ) + (4) dsω = dt − dr − |r − ω | |r − ω | (r − ω )2

The parameter M represents the mass of the black hole and r stands for the set of three ‘spatial’ coordinates x, y, z. The set Ω of possible values for ω is taken to be the Euclidean 3-ball of radius a: Ω = {ω ∈ R3 ; ω 2 6 a2 }. The probability measure on Ω is defined by its density p(ω ) with respect to the Lebesgue measure d 3 ω and we take this density to be uniform i.e. p(ω ) = 1/Va, where Va = 4π a3/3. In (4), u.v and u2 are short-hand notations for the usual Euclidean scalar products of elements (u, v) ∈ R3 and we also use the notation u for the Euclidean norm |u| = (u · u)1/2 of an element of R3 . The exact expression of the mean metric g¯ corresponding to this ensemble can be obtained for every a < r. The mean metric expressed in Kerr-Schild coordinates is given by:      2 

2 2M 2M 6a2 M  r 2a2 M 2 ds = 1 − dt − dr2 − · dr − 1 + r r 5r3 r 5r3 (5)     r 3M r+a 3M 3Mr 2 2 2 − 2 + 3 2 a − r ln · drdt. + − 2r 2a 4a r r−a r

One can construct a new set of coordinates (τ , ρ , θ , φ ) which makes the static and spherically symmetric character of the mean space-time apparent. These coordinates are called the Schwarzschild coordinates for the mean space-time. The Schwarzschild radial coordinate ρ reads: r 2a2 M ρ (r) = r 1 + (6) 5r3 and the Schwarzschild time coordinate τ is defined by:

d τ = dt − α (ρ )d ρ , with

α (ρ ) =

   15Mrρ r+a 3 3 4 2 2 4 −2(a r + ar ) + (a − 2a r + r ) ln . 8a3 (2M − r)(a2 M − 5r3) r−a

The metric then takes the form: with

2 ds = F(ρ )d τ 2 − G(ρ )d ρ 2 − ρ 2 dΓ2 F(ρ ) = 1 −

2M r

(7)

(8) (9) (10)

and G(ρ ) =

h  −2a2M − 5r3 4 64a8M(2M − r) + 90a4M 2 r4 + 45a2M 2 r6 64a6 (2M − r)(a2 M − 5r3 )2    r+a 6 2 2 4 2 2 2 2 2 2 +a (−275M r + 80r ) − 180aM r(a − r ) (a + r ) ln r−a 2 #  r+a . +45M 2 (a2 − r2 )4 ln r−a

In (9), dΓ2 stands for the usual volume element on the unit sphere S2 .

(11)

The coarse grained space-time describes a black hole for a < 2M Let us introduce the adimensionalized coarse graining parameter x = a/M. For all values of x inferior to 2, the firstpsingularity of G encountered when coming from infinity in ρ -space is located at r = r+ = 2M i.e. ρ = ρ+ = 2M 1 + x2/20. This singularity of G is also a zero of F; this p shows that the coarse grained space-time is a spherically symmetric black hole with horizon radius ρH = ρ+ = 2M 1 + x2/20. Kruskal coordinates for this black hole can be constructed in the usual way. One first expands the components of g¯ given by (9) in the neighbourhood of ρ = ρ+ and gets:

2 1 ds = (ρ − ρ+)F ′ (ρ+ )d τ 2 − d ρ 2 − ρ 2dΓ2 , (12) (ρ − ρ+)P′ (ρ+ )

where P(ρ ) = 1/G(ρ ). One then considers the class1 of coordinate systems (τK , XK , θ , φ ) which verify, for ρ = ρ+ + 0+ : p ρ XK + τK − 1; ln ( ) = τ F ′ (ρ+ )P′ (ρ+ ). XK2 − τK2 = (13) ρ+ XK − τK By extension of the usual terminology, any of these coordinate systems can be called a Kruskal coordinate system for S¯. Indeed, when ρ approaches ρ+ by positive values, the components of g¯ in any of these coordinate systems read

2  4ρ+ (14) ds = ′ d τK2 − dXK2 − ρ 2 (XK , τK )dΓ2 , P (ρ+ )  with ρ (XK , τK ) = ρ+ + ρ+ XK2 − τK2 . In this form, the metric g¯ can be extended through the surface ρ = ρ+ to values of ρ inferior to ρ+ . Equation (13) and (14) also make it clear that the surface ρ = ρ+ , over which the Killing field ∂τ is null, is a bifurcate Killing horizon. The coarse grained space-time thus describes a black hole and its geometry around the horizon ρ = ρH is completely similar to the geometry of the original, unaveraged Schwarzschild solution around ρ = 2M.

SOME PROPERTIES OF THE COARSE GRAINED SPACE-TIME Stress-energy tensor It is interesting to further investigate the properties of this coarse grained black hole by determining the stressenergy tensor in the region lying ‘outside’ the horizon, where the Killing field ∂τ is time-like. The exact expressions for the Schwarzschild components of the mean stress-energy tensor are too complicated to warrant reproduction in this review. We just recall the approximate expressions of these components already given in ([7]), which are valid when a ≪ r: 6a2 M 2 8π T¯00 = ε ≈ − ; 5ρ 6 12a2M 2 8π T¯22 = −p2 ≈ ; 5ρ 6

6a2 M 2 8π T¯11 = −p1 ≈ − ; 5ρ 6 12a2 M 2 8π T¯33 = −p3 ≈ . 5ρ 6

(15)

This shows that the coarse graining procedure endows the original vacuum space-time with a non vanishing stressenergy tensor T¯ . This tensor describes how the averaged upon (small scale) degrees of freedom of the Schwarzschild gravitational field can be viewed as an apparent matter which acts as the effective ‘source’ of the coarse grained (large scale) field. The apparent matter is characterized by a negative energy density and an anisotropic pressure tensor. Note that all energy conditions (i.e. the weak, strong and dominant energy conditions ([25])) are violated by the mean stress energy tensor T¯ . Finally, by taking the trace of Einstein’s equation, the scalar curvature R¯ of the mean space-time outside the horizon can be obtained directly from the exact components of T¯ ; one finds, at second order in a/ρ : 2

2

12a M µ . R¯ = −8π T¯µ ≈ − 5ρ 6

1

More precisely, (13) defines a jet of coordinate systems

(16)

The coarse graining thus endows the space-time with a negative scalar curvature. This inevitably evokes the recent observations ([24]) of a positive, non vanishing cosmological constant Λ, which also endows vacuum regions of space-time with a negative scalar curvature ([21]) RΛ = −4Λ. The similarity and differences between the coarse grained space-time constructed here and space-times of cosmological interest are further explored in ([7]).

Mass of the coarse grained space-time The far field regime of the mean gravitational field is best studied by expanding the exact expression (9) in powers of a/ρ around infinity (in ρ ). This gives, at the first non vanishing order in a/ρ [7]:

2 ds = F(ρ )d τ 2 −

with

F(ρ ) = 1 −

1 d ρ 2 − ρ 2dΓ2 F(ρ )

2M 2a2 M 2 . − ρ 5ρ 4

(17)

(18)

This makes apparent that the mean space-time is asymptotically flat and has, therefore, a finite total mass. This mass can be evaluated by recalling [19] that any asymptotically flat space-time S admits a class C∞ (S ) of coordinate systems (τ , R) in which the metric takes the asymptotic form:     2µ 2µ 2 3µ 2 µ3 µ3 2µ 2 2 ds = 1− + 2 + O( 3 ) d τ − 1 + + 2 dR2 + O( 3 )d τ dRi R R R R R R +

(gravitational radiation terms) dRi dR j

(19)

around infinity (in R space); the coefficient µ is common to all coordinate systems in C∞ (S ) and is the mass of the space-time under consideration. Let now S be the usual, unaveraged Schwarzschild space-time of mass M; a particularly simple coordinate system in C∞ (S ) is the so-called isotropic coordinate system [19, 25]. Its time τ coincides with the time of the Schwarzschild coordinate system, and the spatial coordinates R = (R sin θR cos φR , R sin θR sin φR , R cos θR ) are linked to the three Schwarzschild spatial coordinates (ρ , θ , φ ) by θR = θ , φR = φ and   M 2 ρ = R 1+ . 2R

(20)

By equations (17) and (18), the modifications induced on the metric of space-time by the finite coarse graining involves terms which decrease at infinity like 1/ρ 4 or faster; in particular, a does not contribute to the 1/ρ factor in (18). It is therefore tempting to use simply equation (20) to define a new ‘radial’ coordinate R for the mean spacetime and to investigate the asymptotic form of the mean metric in the new coordinate system (τ , R, θ , φ ). A direct calculation proves that, in this coordinate system, the mean metric takes indeed the standard form (19) with µ = M, the mass of the original, unaveraged Schwarzschild black hole. Thus, the total mass of the space-time is unchanged by the averaging procedure. Note that this result is valid for any finite value of the coarse graining parameter a.

Temperature of the coarse grained black hole The thermal properties of the usual Schwarzschild black hole are perhaps most simply derived by studying the natural topology of the Euclidean Schwarzschild space time ([25]). Let us therefore introduce ‘outside the horizon’ q ρ −ρ H the so-called imaginary time ψ = iτ and the new radial coordinate: U(ρ ) = 2 P′ (ρH ) which tends towards 0+ as one approaches the horizon. Near the horizon, the metric g¯ (see equation (12)) takes the simple form:   !2

2 ψ p ds = − U 2 d + dU 2 − ρ 2(U)dΓ2 (21) 2/ F ′ (ρH )P′ (ρH )

with ρ (U) = ρH + U 2 P′ (ρH )/4. In these coordinates, the apparent singularity of the (ψ ,U) part of the metric on the horizon at U = 0 is clearly similar to the singularity displayed by the 2-D Euclidean metric in polar coordinates at the origin. It is therefore natural to consider p  the Euclidean coarse grained space-time as periodic in the imaginary time ψ , F ′ (ρH )P′ (ρH ) . This periodicity in imaginary time is characteristic of a thermal density matrix of period β = 4π / ([25]) of temperature 1 p ′ F (ρH )P′ (ρH ), T= (22) 4π The temperature T can be expressed in terms of M and x: T (x, M) = − where g(x) =

1 g(x) 12π M

x3      . 2 2 2 −x 1 + x4 + 1 − x4 ln 1+x/2 1−x/2

(23)

(24)

Expanding the above expression at order two in x, one finds: T (x, M) =

  1 x2 1+ . 8π M 20

(25)

This confirms that T (0, M) coincides with the Hawking temperature 1/8π M ([25]) of the Schwarzschild black hole.

Calorific capacity at constant x The calorific capacity at constant x, Cx , is defined by: Cx =



∂M ∂T



.

(26)

x

Equation (23) can be rewritten as 1 g(x). 12π T

(27)

1 g(x), 12π T 2

(28)

M=− Deriving (27) with respect to T at fixed x yields: Cx = or, equivalently, in terms of x and M:

Cx = 12π M 2 g−1 (x).

(29)

Equation (29) can be expanded at second order in x:   x2 Cx ≈ −8π M 1 − . 20 2

(30)

The calorific capacity at constant x, Cx , can be evaluated in the Schwarzschild limit, which simply corresponds to the case x = 0; one finds: 1 Cx |x=0 = − ; (31) 8π T 2 As it should, this T -dependent quantity coincides with what is usually called [25] the calorific capacity of the Schwarzschild black hole.

DISCUSSION We would like now to discuss what the results presented so far suggest about the interpretation of current or near future observations of astrophysical black holes. To make what follows specific, we will focus on the super massive black hole candidate Sgr A*, which is located at the Galactic Center, i.e. at approximately D = 8 kpc from the Sun ([23]). Let us first address infra-red observations. These study stellar motions in the immediate vicinity of the Galactic Center and they have led to an estimation of the central dark mass of roughly 4.106 M⊙ ([22, 13, 14]). The associated length is given by LM = GM/c2 = 5 µ as at D = 8 kpc. The stars that have been studied so far are located within a region of radius 0.01 pc (250 mas at 8 kpc) around the Galactic Center and some approach 10−4 pc from the Galactic Center. In these observations, the relative positions of the stars with respect to the cen tral mass are determined with an angular resolution δ of order 2 mas ([13]). In the context of the example presented in this contribution, the finite angular resolution δ translates, as a first approximation, into a finite resolution ∆ = δ D in position measurements around Sgr A*. The realistic order of magnitude of an adimensionalized coarse graining a/M corresponding to these observations is therefore Ma ∼ δLMD . The numerical values given above for δ and LM /D lead to: a 2 · 10−3 ≃ 400. ≃ M 5 · 10−6

(32)

Since r & 10−4 pc and a ≃ 10−5 pc, the condition r ≫ LM is approximately realized in these observations. This means all these observations concern the motions of test objects situated ‘at infinity’ with respect to Sgr A*. The mass estimate derived from the study of these motions is therefore a mass estimate ‘at infinity’. The result presented in Section 4.2 strongly supports the fact that this estimate is correct and free of any systematic bias, whatever the finite value of the angular resolution δ may be. Thus, 4.106 M⊙ should be a realistic estimate of the true mass of Sgr A*, even though the angular resolution of the observations from which this result is deduced is actually quite poor from a theoretical point of view (the corresponding value of the adimensionalized coarse graining parameter a/M being approximately 400). Let us now address high resolution observations of Sgr A* using Very Large Baseline Interferometry (VLBI). The current resolution of these observations is 20 µ as, i.e. 2 Schwarzschild radii 2M at 7 mm wavelength ([2]). VLBI imaging resolution is however expected to be narrowed down to 20 µ as at 1.3 mm wavelength during the next decade ([8] ). In the context of the example presented here, this resolution corresponds roughly to: a δD ≃ 4. ∼ M LM

(33)

Thus, a coarse graining parameter a taking into account the typical finite resolution of the near future observations would be of the same order of magnitude as the Schwarzschild radius 2M of Sgr A*. The material developed in Section 3.2 supports the fact that a black hole observed with a finite precision a smaller or comparable to its horizon radius should still appear as a black hole. In particular, near future VLBI observations, which will have a resolution comparable to the theoretical Schwarzschild radius 2M of Sgr A*, should be able to detect a shadow ([8]). But the apparent size of the shadow should be compared with a theoretical estimate which takes into account the finite resolution of the observations by using a mean or coarse grained gravitational field to compute the motion of stars and photons orbiting the black hole. Typically, one can expect the size of the apparent shadow to be comparable to a few ρH . In the context of the example presented in Section 3.2, ρH depends on both the mass M and on the coarse graining parameter a tracing the finite resolution of the observations. Since the projected observational situation corresponds to a value of x = a/M of order unity, the difference between ρH and the ‘theoretical’ horizon radius 2M of the unaveraged black hole cannot be expected to be small, and will probably be of order unity. So will also be the difference between the observed size of the shadow (∼ 5 × ρH ) and the ‘theoretical’ estimate 5 × 2M. Thus, contrary to what is usually expected ([20]), the apparent size of the shadow will probably not provide a direct, unbiased estimate for the mass of Sgr A*. Let us conclude by mentioning a few extensions of the work presented in this contribution. On the astrophysical side, one should now systematically consider ensembles of black holes which do take into account the real (current and future) observational conditions and evaluate, from each of these ensembles, a realistic mean or coarse grained black hole. Naturally, for a given observational procedure, one should not only construct ensembles of Schwarzschild black holes, but also ensembles of Kerr black holes if one wants to extract from the data information about the angular momentum of the observed object.

Possible cosmological implications of the mean field theory introduced in ([5]) should also be explored systematically, in both perturbative ([18]) and fully non linear regimes. On the purely theoretical side, the fact that a purely classical averaging modifies the temperature of a black hole suggests a link between the mean field theory developed in Section 2 and string theory; this link obviously demands investigation. A first step would be the computation of the total entropy of coarse grained black holes; in particular, does the apparent matter surrounding the mean black hole carry a non vanishing entropy?

REFERENCES 1. Barut, A.O., Electrodynamics and classical theory of fields and particles The Macmillan Company (1964) 2. Bower, G., 2004, Science 304, p.704–708. 3. Buchert, T., 2000, On average properties of inhomogeneous fluids in general relativity: dust cosmologies. Gen. Rel. Grav., 32:105–26. 4. Buchert, T., 2001, On average properties of inhomogeneous fluids in general relativity: perfect fluid cosmologies. Gen. Rel. Grav., 33:1381–405. 5. Debbasch, F., 2004, What is a mean gravitational field? Eur. Phys. J. B 37(2), p.257–270. 6. Debbasch, F., 2005 Mean field theory and geodesics in General Relativity Eur. Phys. J. B 43(1), p.143–154. 7. Debbasch, F., & Ollivier, Y., 2005, Observing a Schwarzschild black hole with finite precision Astron. Astrophys. 433(2), p.397–404. 8. Doeleman, S., & Bower, G., 2004, GCNews 18, p.6–12. 9. Ellis, G. F. R., & van Elst, H., 1998, Cosmological Models in Cargese Lecture 1998, arXiv:gr-qc/9812046 10. Futamase, T., 1991, Prog. Theor. Phys., 86:389–99. 11. Futamase, T., 1993, Prog. Theor. Phys., 581–97. 12. Futamase, T., 1996, Averaging of a locally inhomogeneous realistic universe. Phys. Rev. D, 53:681–9. 13. Ghez et al., 2003, Astron. Nachr. Suppl. 324(1), p.527–533. 14. Ghez et al., 2003, ApJ 586(2), p.L127–L131. 15. Ishibashi, A. & Wald, R. M., Can the acceleration of our Universe be explained by the effects of inhomogeneities? arXiv:gr-qc/0509108 16. Israel, W., 1987, Covariant fluid mechanics and thermodynamics: An introduction. In A. Anile and Y. Choquet-Bruhat, editors, Relativistic Fluid Dynamics, volume 1385 of Lecture Notes in Mathematics, Berlin, Springer-Verlag. 17. Kasai, M., 1992, Construction of Inhomogeneous universes which are Friedmann-Lemaitre-Robertson-Walker on average. Phys. Rev. D, 69:2330–2. 18. Kolb, E. W., Matarese, S., & Riotto, A., 2005, On cosmic acceleration without dark energy. arXiv:astro-ph/056534v1. 19. Misner, C. W., Thorne, K. S., & Wheeler, J. A., 1973, Gravitation (W.H. Freeman and Co., New-York). 20. Miyoshi, M., 2004, An approach detecting the horizon of Sgr A* in Proceedings of the 7th European VLBI Network Symposium (Toledo). 21. Peebles, P.J.E., 1993, Princeton Series in Physics (Princeton University Press, Princeton) 22. Schödel et al., 2003, ApJ 596(2), p.1015–1034. 23. Schödel et al., 2003, ApJ 597(2), p.L121–L124. 24. D.N. Spergel et al., 2003 First Year Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. Astrophys. J. Suppl., 148:175. 25. Wald, R. M., 1984, General Relativity (The University of Chicago Press, Chicago). 26. Zalaletdinov, R., 1997 Averaging problem in general relativity, macroscopic gravity and using Einstein’s equation in cosmology. Bull. Astron. Soc. India, 25:401–416.

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call. The first models were endowed with symmetric connexion weights which induced relaxation dynamics ... are presented in the present conference [7]. The nature of the ... duce the activation variable xi(t) of the neuron at time t. For BF and ...

heat exchanger mean temperature differences for ... - Semantic Scholar
This creates a temperature glide during phase change (at which point the con- ... In the analysis of heat exchangers, the Log Mean Tempera- ture Difference .... Software Tools ... Comparison of the UAs: Development of Error Scales. Once the ...

heat exchanger mean temperature differences for ... - Semantic Scholar
compositions, systems employing zeotropic refrigerant mixtures must be liquid ... a heat exchanger, both of which complicate traditional heat ex- changer ...

Cognitive Psychology Meets Psychometric Theory - Semantic Scholar
This article analyzes latent variable models from a cognitive psychology perspective. We start by discussing work by Tuerlinckx and De Boeck (2005), who proved that a diffusion model for 2-choice response processes entails a. 2-parameter logistic ite

Belief Revision in Probability Theory - Semantic Scholar
Center for Research on Concepts and Cognition. Indiana University ... and PC(A2) > 0. To prevent confusion, I call A2 the ..... If we test a proposition n times, and the results are the same ... Pearl said our confidence in the assessment of BEL(E).

Cognitive Psychology Meets Psychometric Theory - Semantic Scholar
sentence “John has value k on the ability measured by this test” derives its meaning exclusively from the relation between John and other test takers, real or imagined (Borsboom, Kievit, Cer- vone ...... London, England: Chapman & Hall/CRC Press.

Theory Research at Google - Semantic Scholar
28 Jun 2008 - platform (MapReduce), parallel data analysis language (Sawzall), and data storage system. (BigTable). Thus computer scientists find many research challenges in the systems where they can .... In theory, the data stream model facilitates

Theory of Communication Networks - Semantic Scholar
Jun 16, 2008 - services requests from many other hosts, called clients. ... most popular and traffic-intensive applications such as file distribution (e.g., BitTorrent), file searching (e.g., ... tralized searching and sharing of data and resources.

Economic game theory for mutualism and ... - Semantic Scholar
In fact, the principal can achieve separation of high and low-quality types without ever observing their ...... homogeneous viscous populations. J. Theor. Biol., 252 ...

true motion estimation — theory, application, and ... - Semantic Scholar
From an application perspective, the TMT successfully captured true motion vectors .... 6 Effective System Design and Implementation of True Motion Tracker.

A Theory of Credit Scoring and Competitive Pricing ... - Semantic Scholar
Chatterjee and Corbae also wish to thank the FRB Chicago for hosting them as ...... defines the feasible action set B) and Lemma 2.1, we know that the budget ...

Evaluation methods and decision theory for ... - Semantic Scholar
ral dependence. We formulate the decision theory for streaming data classification with tem- poral dependence and develop a new evaluation methodology for data stream classification that takes ...... Kappa statistic. 8 Conclusion. As researchers, we

A Theory of Credit Scoring and Competitive Pricing ... - Semantic Scholar
Chatterjee and Corbae also wish to thank the FRB Chicago for hosting them as visitors. ... removal of a bankruptcy flag; (2) for households with medium and high credit ratings, their ... single company, the Fair Isaac and Company, and are known as FI

true motion estimation — theory, application, and ... - Semantic Scholar
5 Application in Motion Analysis and Understanding: Object-Motion Estima- ...... data, we extend the basic TMT to an integration of the matching-based technique ...

Bucketing Coding and Information Theory for the ... - Semantic Scholar
mate nearest neighbors is considered, when the data is generated ... A large natural class of .... such that x0,i ⊕bi = j and x0,i ⊕x1,i = k where ⊕ is the xor.