Polarizable Vacuum (pv) And The Reissner-nordstrom 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 Reissner-Nordstrom Solution Joseph G. Depp May 6, 2005

In a previous publication the author showed that the PV model of Dicke with a modified Lagrangian density produced a solution that is mathematically equivalent to the Schwarzchild solution of Einstein’s equations for general relativity. In this paper it is shown that the same PV model with modified Lagrangian also gives an exact solution to the charged mass point metric of Reissner and Nordstrom.

Introduction In 1957 Dicke (1) published an oft-cited paper 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  K 2 Ld = −λF ( K )(∇K ) −     c 2  ∂t  

(1)

K is the index of refraction, l is an arbitrary constant, and F(K) is an arbitrary scalar function. In a previous publication (2) the author showed that the above Lagrangian density exactly reproduced the Schwarzschild solution of Einstein’s equations for general relativity when F (k ) = 1 / K 4 . In this paper we show that the Reissner-Nordstrom metric can be derived from the same Lagrangian density if we include the Lagrangian density for the electromagnetic field in the usual fashion and we obtain a value for the arbitrary constant lambda that agrees with the value obtained by Dicke through heuristic arguments. We follow the notation for the PV model as given by Puthoff (3).

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Reissner-Nordstrom and the Index of Refraction The Lagrangian density for the index of refraction and an electromagnetic field is given by

λ 

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

2  1  B2   − Kε 0 E 2   −    2  Kµ 0

(2)

For a static point charge the field is given in MKS units by

E=

q K 4πε 0 r 2

and B = 0

(3)

It is readily apparent that we must retain a factor of K in the denominator to account for the charged mass that is imbedded in the dielectric. The equation of motion obtained from (2) above is 2 2 2λ  2 2 1 q2 1  ∂K   2 2 ∂ K − 4 ∇ K − K − ( ∇ K ) + K + =0    2 2 2 4 K ∂ (ct ) K   ∂ct   2 K (4π ) ε 0 r

(4)

We seek an index of refraction that is time-independent and radially symmetric. Under these assumptions (4) reduces to

2λ − 4 K

 d 2 K 2 dK 2  dK  2  1 q2 1 + − + =0    2  2 2 r dr K  dr   2 K (4π ) ε 0 r 4  dr

(5)

We can rewrite (5) as follows 2  d 2 K 2 dK 2  dK  2  2 α −   =K 4  2 + r dr K  dr   r  dr

(6)

We define the constant α for the sake of brevity as

α2 =

q2

(7)

4(4π ) ε 0 λ 2

We now seek a solution of the form K (r ) = (1 − b / r + a 2 / r 2 ) −1

(8)

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The algebra of substituting (8) into (6) is easily done and the result shows that (8) is a solution of (6) if

a2 = α 2 / 2

(9)

Now we examine the metric of Reissner and Nordstrom. ds 2 = (1 − 2m / r + Q 2 / r 2 )c 2 dt 2 − (1 − 2m / r + Q 2 / r 2 ) −1 dr 2 − r 2 (dθ 2 + sin 2 θ dϕ 2 )

(10)

The constant, m, has the usual definition m = GM / c 2

(11)

G is the gravitational constant and M is the mass of the gravitating body. The constant Q written in MKS units is Q2 =

q2 G 4πε 0 c 4

(12)

Following Desiato (4) and others we assert that the metric can also be written as ds 2 = K −1c 2 dt 2 − Kdr 2 − r 2 (dθ 2 + sin 2 θ dϕ 2 )

(13)

We can make this assertion if we identify the constant b in (8) with the constant 2m in (10) and if we identify the constant a 2 in (8) with the constant Q 2 in (10). This leads to a definition for the previously undefined constant, λ , that appears in (1), (2), (4), (5) and (7).

λ=

c4 32πG

(14)

Equation (14) is identical to the value obtained by Dicke through heuristic arguments. Conclusion We have shown that the modified PV Lagrangian density presented previously by the author will give the Reissner-Nordstrom metric and, in the process, we have established a value for the previously unspecified constant in the Lagrangian density. The value is in agreement with the value suggested by Dicke based on heuristic arguments.

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REFERENCES 1. R. H. Dicke, ‘‘Gravitation without a principle of equivalence’’, Rev. Mod. Phys. 29,363–376 (1957). 2. J. G. Depp, “Polarizable Vacuum and the Schwarzschild Solution”, online, April 2005 3. H. E. Puthoff, “Polarizable-Vacuum (PV) approach to general relativity”, Found. Phys. 32, 927-943 (2002). 4. T. J. Desiato, R. C. Storti, “Event horizons in the PV model”, E-print 2003

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