Polarizable Vacuum (pv) And The Schwarzschild Solution

PACS numbers: 03.03.+p, 03.50.De, 04.20.Cv, 04.25.Dm, 04.40.Nr, 04.50.+h

Polarizable Vacuum and the Schwarzchild Solution Joseph G. Depp April 14, 2005

The PV model is examined for a solution that is mathematically equivalent to the Schwarzchild solution of Einstein’s equations for general relativity. A simple solution is found and the resulting Lagrangian density and equation of motion are presented for further investigation.

Introduction In 1957 Dicke published an oft-cited paper [1] that examined the possibility that general relativity could be derived from field theory. The form of the Lagrangian density for the index of refraction was taken as,

 2  ∂K  2  Ld = −λF ( K )(∇K ) 2 − K     c 2  ∂t  

(1)

K is the index of refraction, l is an arbitrary constant, and F(K) is an arbitrary scalar function. The function F(K) was initially taken by Dicke to be 1/K. More recently, Puthoff [2] investigated the case for which F(K) is 1/K2. In this paper we follow the notation of Puthoff. We investigate the mathematically allowable forms of F(K) to find a form that is compatible with general relativity. Schwarzchild Solution and the Index of Refraction We are interested in the spherically symmetric, time-independent solution to Einstein’s equations that gives rise to the Schwarzchild metric. The development of the Schwarzchild metric can be found in most introductory texts to general relativity. Here we follow the development and notation as found in [3]. The most general spherically symmetric, timeindependent metric can be written as, ds 2 = Ac 2 dt 2 − Bdr 2 − r 2 (dθ 2 + sin 2 θ dϕ 2 )

(2)

A and B are intrinsically positive functions of r only. Einstein’s equations are usually solved in this metric, obtaining the solution, A = B −1 = (1 − 2m / r )

(3)

The constant m is GM / c 2 with G the gravitational constant and M the gravitating mass. As noted by Desiato [4] and others it is this metric that should be used for comparison with the index of refraction (see appendix for more details) so that,

K (r ) = (1 − 2m / r ) −1

(4)

We take (4) as given and ask what form must F(K) take if the Lagrangian density is to produce an identical index of refraction. Equation of Motion for the Index of Refraction We begin by obtaining the equation of motion for the index of refraction with an arbitrary scalar function, F(K).

 K 2 ∂2K 1 F ' (K ) K 2 F ' ( K )  1  ∂K  2 ∇ K− 2 = − ( ∇ K ) + K +    2 F (K ) 2 F ( K )  c 2  ∂t  c ∂t 2  2

2

(5)

For F(K) = 1/K in (5), we obtain K 2 ∂2K 1 1 ∇ K− 2 = (∇K )2 + 1 K2  ∂K  2 2K 2 c  ∂t  c ∂t

2

2

(6)

in agreement with [1, eq.67]. For F(K) = 1/K2, (5) yields

∇2K −

K 2 ∂2K 1 2 = (∇K ) 2 2 K c ∂t

(7)

in agreement with [2, eq.34]. Since we are interested in a spherically symmetric, time-independent solution (5) reduces to d 2 K 2 dK 1 F ' ( K )  dK  + =−   2 r dr 2 F ( K )  dr  dr

2

(8)

Substituting (4) into (8) yields

F ' (K ) 4 =− F (K ) K

(9)

Solving for F(K) gives F (k ) =

1 K4

(10)

The constant of integration is taken to be zero since there is already an arbitrary constant, l, in (1). The resulting Lagrangian density is

λ 

2  ∂K  Ld = − 4 (∇K ) 2 − K   2  ∂t  K  c 

2   

(11)

and the equation of motion for the index of refraction becomes K 2 ∂2K 2 K  ∂K  2 ∇ K− 2 = (∇K ) − 2   2 K c ∂t c  ∂t  2

2

(12)

Equation (11) differs from the Lagrangian densities used by Dicke and Puthoff. However, it has the benefit that it produces an index of refraction, K, which is in exact agreement with the Schwarzchild solution of Einstein’s equations for general relativity. The assertion in equation (4) implies that the coordinates (r ,θ , ϕ , t ) in equations (11) and (12) are equivalent to those in the metric in equation (2). Conclusion It is possible to find a Lagrangian density for the index of refraction such that the index of refraction is mathematically compatible with Schwarzchild solution for Einstein’s equations of general relativity. The resulting Lagrangian density is presented for further investigations into its physical implications. Acknowledgements I would like to thank H. Puthoff and T. Desiato for several helpful discussions during the preparation and review of this manuscript.

Appendix The most general radially symmetric, time-independent line element is given by ds 2 = Ac 2 dt 2 − Bdσ 2 − Cσ 2 (dθ 2 + sin 2 θ dϕ 2 )

(A1)

A , B , and C are intrinsically positive functions of the radial coordinate, σ . However, a radial transformation

r = σ C1/ 2

(A2)

brings (A1) into the form ds 2 = Ac 2 dt 2 − Bdr 2 − r 2 (dθ 2 + sin 2 θ dϕ 2 )

(A3)

where A and B are positive definite functions of the radial coordinate, r. The solution to Einstein’s equations of general relativity for the metric in (A3) [3,eq.6.48] is ds 2 = (1 − 2m / r )c 2 dt 2 − (1 − 2m / r ) −1 dr 2 − r 2 (dθ 2 + sin 2 θ dϕ 2 )

(A4)

We assert that it is this form that should be compared with the index of refraction, K, such that ds 2 = K −1c 2 dt 2 − Kdr 2 − r 2 (dθ 2 + sin 2 θ dϕ 2 )

(A5)

The comparison yields K (r ) = (1 − 2m / r ) −1

(A6)

It is customary to perform one more radial transformation on the line element to bring it into isotropic form. The transformation [3, eq.6.60]

 m  r = ρ 1 +  2ρ 

2

(A7)

yields for the isotropic line element ds

2

2 ( 1 − m / 2ρ ) 2 2 4 = c dt − (1 + m / 2 ρ ) (dρ 2 + ρ 2 dθ 2 + ρ 2 sin 2 θ dϕ 2 ) 2 (1 + m / 2 ρ )

(A8)

Under the same transformation the index of refraction, K, becomes, in isotropic coordinates 2 ( 1 + m / 2ρ ) K (ρ ) = (1 − m / 2 ρ )2

(A9)

The assertion in (A5) implies that the coordinates (r ,θ , ϕ , t ) in (11) and (12) are equivalent to those in the metric (A4). The physical interpretation of (A9) is that the index of refraction approaches unity very far from the gravitating mass but becomes infinite on the event horizon. Clearly, any test of the theory of relativity performed with the index of refraction, K, will produce the same result as that obtained from the Schwarzchild solution since they are identical by definition.

REFERENCES 1. R. H. Dicke, ‘‘Gravitation without a principle of equivalence’’, Rev. Mod. Phys. 29,363–376 (1957). 2. H. E. Puthoff, “Polarizable-Vacuum (PV) approach to general relativity”, Found. Phys. 32, 927943 (2002). 3. R. Adler, M. Bazin, M. Schiffer, “Introduction to General Relativity”, McGraw-Hill 1994 Ch 6, 164-173. 4. T. J. Desiato, R. C. Storti, “Event horizons in the PV model”, E-print 2003