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New solutions of Einstein’s equations utilizing Poincar´e spaces I. Geometry Lawrence R. Mead † Dept. of Physics and Astronomy University of Southern Mississippi Hattiesburg MS. 39406 Harry I. Ringermacher General Electric Global Research Center Schenectady, NY 12309 Sungwook Lee Dept. of Mathematics University of Southern Mississippi Hattiesburg, MS 39406 November 23, 2005 Abstract We describe new N+1-dimensional curved space-time solutions to the Einstein equations. The spatial metric is scale invariant and based on a generalization of the classical two-dimensional Poincar´e metric to arbitrary dimension. A special case is the well-known RobertsonWalker space of standard cosmology. The solutions have remarkable properties. We prove that a certain axisymmetric subset of the solutions is Riemann flat when a single parameter takes on a special value. In this case, the N-space remains curved, thus exhibiting the phenomenon of “induced matter.”

PACS: 04.20.Jb, 04.20-q, 02.40.Ky †

Communications to [email protected]

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1

Introduction

Standard cosmological models begin with the Friedman-Robertson-Walker space (FRW) with line element given by, dr 2 (1) + r 2 dΩ2 . 1 − kr 2 In this description of a homogeneous and isotropic universe, k is the curvature index taking values 0, ±1, and dΩ2 = sin2 θdφ2 +dθ 2 is the usual solid angle in spherical coordinates. a(t) is the evolution factor determined by the relative amounts of mass-energy present through the Einstein-Friedman equations. Current measurements indicate an accelerating expansion of a virtually flat space [1]. An often seen statement in cosmology is to the effect that “the universe expands but galaxies do not” [2]. As the Hubble expansion takes place, galaxies presumably reach a kind of equilibrium where the mutual gravitational attraction of the stars counterbalances the expansion of space itself. Indeed, one might ask whether or not a simple modification of the FRW metric could hold within galaxies or clusters of galaxies; that is, does there exist a region of space-time in which the expansion is negated? We attempt to answer this question, at least in part. In this paper, we derive a new solution of Einstein’s equations for a uniform fluid which is intimately related to the metric in Eq. (1). The present work is also related to and expands upon recently introduced extensions of the concept of “induced matter” [3, 4, 5]. In the sections which follow, we motivate and describe some of the properties of the new solution and its relation to known metric spaces and prove a theorem which generalizes the new space to arbitrary dimension. dS 2 = c2 dt2 − a(t)2

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CRW space

Einstein’s equations are, 8πG 1 Rµν − Rgµν = − 4 Tµν , 2 c where the matter-energy tensor is that of a perfect fluid, Tµν = ρ uµ uν + P (uµ uν − gµν ), 2

(2)

(3)

with uµ uµ = 1, P is pressure and ρ is energy density. Throughout, we use the conventions of Adler [6]. Let us now construct a space whose line-element is of the form, 2

2



2

2

2

dS = f (t)dt − a(t) g(x, y) dx + dy + dz

2



.

(4)

Eq.(4) is a modification of the standard (flat, k = 0) FRW space Eq.(1) in three spatial dimensions. We will later generalize to arbitrary dimension and include nonzero curvature. The form is chosen to break the standard cosmological isotropy as might be found within an axisymmetric galaxy. We require that the functions f (t) and g(x, y) be chosen such that the metric components of Eq.(4) satisfy the vacuum Einstein equations (2), Tµν = 0, analogous to the empty space surrounding a gravitating point source. This choice is tantamount to positing that geometry alone (possibly associated with galaxies) and not traditional matter can compensate for the expansion. Inserting these into Eq.(2) yields the following equations for the two unknown functions: f (t) = a0 (t)2 , (5) ∇2 g = 6g 2 2ggxx = 3gx2 2ggyy = 3gy2 2ggxy = 3gx gy ,

(6)

where the over-prime denotes differentiation in t, and subscripts denote partial differentiation. Except for trivial scalings of the coordinates, Eqs.(6) have a unique solution, 2 . (7) g(x, y) = (x + y)2 Our N = 3 “Cancel Robertson-Walker” (CRW) line element is thus, 2a(t)2 dS = a (t) dt − dx2 + dy 2 + dz 2 . (x + y)2 2

0

2



2



(8)

Again, this is a vacuum solution of Eq.(2). Calculating the full curvature tensor Rµνσρ , we find that it vanishes identically [7]: that is, the full fourspace given in Eq.(8) is Riemann flat (we show explicit Minkowski coordinates 3

in the appendix). Note that the first term a0 (t)2 dt2 is just da2 which allows the line element to be written as, 2a2 dx2 + dy 2 + dz 2 . dS = da − 2 (x + y) 2



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(9)

In this form, coordinate time t has disappeared and the dependent variable a, originally the FRW evolution factor, now plays the role of the time variable. The expansion of the space has been “canceled” by removing all matterenergy from the space and measuring “time” via the evolution function a itself. In addition, the space defined in Eqs.(8), (9) is scale invariant: if (t, x, y, z) → (λt, λx, λy, λz), the line element remains independent of the scaling factor λ. In the next section, we will detail some important properties of the space of Eq.(9) and generalize to arbitrary dimension and curvature.

3 3.1

CRW spaces No singularity

Because there is an apparent (plane) singularity at y = −x, it is natural to ask whether there is some sort of singular distribution of matter in the space (9) analogous to the point mass of the standard Schwarzschild solution. This question is answered in the negative by simply adopting the coordinate system in the appendix; neither the metric nor the curvature tensor has any singular components at any point in space.

3.2

Relation to Poincar´ e spaces

Poincar´e spaces are two-dimensional spaces defined by the metric pair below: dx2 + dy 2 , y2 dρ2 + ρ2 dφ2 = , (1 + 14 kρ2 )2

dSh2 = λ2

Half plane

(10)

dSd2

Disc

(11)

and the latter is expressed in polar coordinates (ρ, φ). The first metric is defined only for y ≥ 0, the upper half plane; its Gaussian curvature is −1/λ2 . 4

The second describes a full disc-space when the Gaussian curvature is positive, but serves as a model of hyperbolic space when k = −λ2 . These spaces have played a historical role in the development of hyperbolic geometry, a subject of current interest in relation to cosmology and string theory. We see immediately that our CRW 3-space is the symmetrized (rotated) Poincar´e half-plane, Eq.(10), while the Poincar´e space on the disk, Eq.(11), is the isotropic form of the FRW 3-space under the standard coordinate transformation, ρ . r= 1 + 14 kρ2 Clearly, the CRW and FRW metrics are intimately related and we do not think that these connections are too well known in the physics literature. Indeed, the Poincar´e metrics Eqs. (10,11), have been generalized by Laugwitz [8], who considers the metric, dS 2 =

a2 (dx2 + dy 2 + dz 2 ) . [A + B(x + y) + C(x2 + y 2 + z 2 )]2

(12)

The spatial FRW metric follows from the choices A = 1, B = 0 and C = k/4; the CRW metric from the choices A = C = 0 and B = 1. Even the timedependence of these spaces can be included. From the work of Stephani [9] we find the following 4-space: dS 2 =

1 Q0 (t) dt2 − (dx2 + dy 2 + dz 2 ), η(t)Q(t) Q(t)2

(13)

where

F (t) + η(t) [A + B(x + y) + C(x2 + y 2 + z 2 )]2 , (14) 2 and where F (t) and η(t) are arbitrary functions of time. Now, with the same choices above for the set of constants A, B, C, the time-dependences of the FRW and CRW metrics can be produced by choosing, Q(t) ≡

η(t) = F (t) = 1/a(t) 0

η(t) = a (t)/a(t),

CRW η + F = 2/a(t)

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(15) FRW.

(16)

3.3

Arbitrary curvature and dimension

The CRW metric can be immediately generalized to arbitrary dimension and curvature. We can introduce the basic idea starting from the metric in Eq.(9). Let us replace the factor of two in front of the space part by the real number D:   Da2 2 2 2 2 2 (17) dS = da − dx + dy + dz . (x + y)2 The Ricci scalar, R, for this space is R ≡ Rµµ =

6(D − 2) . Da2

(18)

Hence, the curvature can be positive and negative as well as zero (at D = 2) depending on D. This space remains an exact solution of Einstein’s equation but with nonzero Tµν . Taking the uniform fluid form for the energy tensor as in Eq.(3) and in the rest frame of the fluid, one finds simple relations between the pressure and density which imply that for all time t (or a) [10], 1 P = − ρ. 3

(19)

The negative pressure implies an expansion of the space in which the (positive definite) matter ρ is embedded, albeit a power law expansion as opposed to an exponential expansion of the universe. Thus, one might consider this region as comparatively rigid. Next, change the term in the spatial denominator from (x + y)2 to (x + y + z)2 . The metric now reads, Da2 dS = da − dx2 + dy 2 + dz 2 . 2 (x + y + z) 2



2



(20)

This metric is still a solution of Einstein’s equations, but now becomes Riemann flat at D = 3 rather than at D = 2; indeed, the curvature scalar is proportional to D − 3. These observations suggest a general theorem which we now state and prove. Theorem 1 The metric dS 2 = da2 − D a2

dx21 + dx22 + · · · + dx2N , (x1 + x1 + · · · + xM )2 6

N ≥M ≥1

(21)

is Riemann flat (all of the components of the Riemann tensor vanish identically) whenever D = M . We prove this theorem in stages. Consider the Riemannian manifold defined by dS 2 = g˜ij dxi dxj . (22) This space is said to be maximally symmetric if and only if it has constant sectional curvature κ = κ(i, j), for any 1 ≤ i 6= j ≤ N . In the plane spanned by the basis vectors (ˆ ei , eˆj ) the sectional curvature is defined by, j κ(i, j) ≡ g˜ii Riji .

(23)

j For a maximally symmetric space Riji = κ˜ gii , j 6= i. Standard examples are the hyperbolic 3-space (κ = −1), Euclidean 3-space (κ = 0) or the 3-sphere (κ = +1). For such a maximally symmetric space,

Rii = −κ(N − 1)˜ gii .

(24)

The first step is to show that the Riemannian metric dx21 + dx22 + · · · + dx2N (x1 + x2 + · · · + xM )2 1 = δi (x1 + x2 + · · · xN )2 j = (x1 + x2 + · · · xN )2 δ ij

dS 2 = g˜ij g˜ij

1 ≤ M ≤ N,

(25)

has sectional curvature κ = −M . This result is demonstrated from the definition and by direct calculation. One finds that the non-vanishing components of the connections Γijk are all ±(x1 + x2 + · · · + xM )−1 , from which it follows that M 2 . (26) R121 =− (x1 + x2 + · · · + xM )2 2 Thus, by symmetry, κ(i, j) = κ(1, 2) = g˜11 R121 = −M . That is, the metric has constant sectional curvature so that the result of Eq.(24) holds for this space. Next, examine the metric,

dS 2 = da2 − Da2 g˜ij dxi dxj , 7

(27)

where g˜ij denotes a maximally symmetric space of curvature κ, such as in Eq. (25). In a previous paper [5] we proved that given a line element of the form Eq.(27), the corresponding space will be Riemann flat if and only if D = −κ. Thus, D = M and the proof is complete.

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Summary

We have described a new scale-invariant N + 1 dimensional solution to the Einstein equations, CRW space (Eq. 21), which no longer has the inherent isotropy of its FRW cousin. The spatial part of the metric has constant negative sectional curvature while the full CRW space has Riemann curvature proportional to D − M . The CRW space is thus curved according to the sign of this factor, being Riemann flat at precisely the value D = M . The CRW space with nonzero curvature contains a fluid of matter satisfying P = − 31 ρ, and is thus relatively rigid compared to exponential expansion. We have shown that the CRW space and its FRW cousin of standard cosmology are intimately related to the two traditional models of hyperbolic geometry: the Poincar´e metrics on the half-space and disk, respectively. These connections seem not to be well-known. In the second paper following, we will further explore the geometric properties of the CRW spaces, and in addition, the remarkable dynamics of particle motion in the space.

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References [1] W.L. Freedman and M.S. Turner, “Measuring and understanding the universe,” Rev. Mod. Phys., 75, 1433-1447 (2003). [2] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman and Co., 1973, §27.5, p. 719. [3] T. Liko and P.S. Wesson, J. Math. Phys.46, 062504-1 (2005). ¨ Delice, Phys. Rev. D68, [4] M. Arik, A. Baykal, M.C.C ¸ alik, D. C ¸ iftci and O. 123503 (2003). [5] H.I. Ringermacher and L.R. Mead, J. Math. Phys.46, 102501 (2005). [6] R. Adler, M. Bazin and M. Schiffer, Introduction to General Relativity, McGraw-Hill, 2nd Ed., 1975. [7] All tensor calculations have been verified by MAPLE(tm) and/or Mathematica(tm). [8] D. Laugwitz, Differential and Riemannian geometry, Academic Press, 1965. [9] H. Stephani, Comm. Math. Phys. 4, 137 (1967). [10] Einstein’s equations for the function a(t) are the same as the Friedman equations of FRW cosmology with the specific a(t) = t; thus, the second derivative terms are missing and Eq.(19) then follows. The same equation of state follows for the most general metric we consider.

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A

Flattening transformation

In general, finding coordinates which explicitly put a flat space into standard Minkowski form is not easy. After some labor, we find that the following coordinate change accomplishes that goal for our 3+1 dimensional CRW metric of Eq.(9): az x+y a(x2 + y 2 ) U = x+y a(x − y) V = x+y a W = . x+y T =

In these coordinates, the CRW line element is dS 2 = dT 2 − dU 2 − dV 2 − dW 2 .

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