Time-Independent Massive EKG Solution in a Schwarzschild Exterior

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Time-Independent Massive EKG Solution irf aSchwarzschild Exterior Background

Vincent Mamo, Heather Miller, Chris WoodApril 23, 2003

The purpose of this project was to attempt to find solutions to the Einstein-Klein-Gordon equation on an exterior Schwarzschild background. Many solu-tions were attempted with the massless equation dependent on radius and timeand finally was refined into a massive, radius only dependent equation.

We begin with a general metric of th$bliowing form:ds 2 =&'dt2 - e?'dr2 - r2d92 - r2sin29dq2

(where both ji and A were dependent upon rand t), and a Lagrangian ofL =

in 2 c2+ __w=0

Using the equation:02 WVaVb'I'=Dx9x' - ab3: (4 )We then assumed that the metric that we were using was a Schwarzschild.

This would imply several things:=2 - 2MG".

1!i 2MG

'V1/ 0 o\! 0 00 001

0 0 00)

(1)

(2)

(3 )

1


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1'ab

( e"'zi' AA A '= 2o 0o 0

o oo 0

_re_? 0o -resn2O

Other Christoffel symbols are obviously non-zero, but for the purposes of ourcalculations, the Christoffel symbol associated with rand t were only applicable.

So it can be seen that A = -ii, therefore A = -l1. Upon first attempts wedid not use the massive term in the equation. The massless equation was usedsimply for simplicity, to see if there were any solutions that could be obtained.We also assumed that JJ = RT and the differential equation could be separatedobtaining the following:

82I,=e" [w " + (v' - A')W]Dividing through by 'I', we obtain:

1 2T - e"' [R" + (ii' - A')R'] -R -

Since either side of the equation is dependent upon a different variable, itcan be said that either side of the equation is equal to the constant

However, since the exterior Schwarzschild metric coefficients are only depen-dent upon r (and not t), we may simplify the Christoffel symbols as follows:

0 0O-,o_ - 000ab 0 0 0 00000

!eUAvF 0 0 00 0 00 0 re 00 0 0 _re_Asin2O

With the above Christoffel symbols (and definition for the Laplacian incurved space-time), we obtain:

= evA [(iii' + w") + - (7 )A nice check for this would be to assume that this is in fiat space. Therefore,

the above equation reduces to:2 = w ' + w "r

(5 )

(6 )

i-1ab-

2


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which is the correct wave equation in flat space.At this point, we attempted a separation of variables. Since we also assumed

that the time function can be represented as T(t) = Ac i, the variable 2mustbe equal toUsing this, we obtain the equation for R(r):

(2MG'\22 1R'+R"- 2MG 1 21 c2r ) [r c2r - 2MG ] R=

This turns out to be a situation in which the solution is not solvable by theteammates at hand. In attempts to rectify our situation, at this point, we aban-doned the time-dependent massless to a time-independent massive equation:

We short-cut the problem by assuming that the Einstein 's equation is notnecessary to solve. This brings us straight to:

gabVaVb'P + E2'P=0.Using the same Schwarzschild exterior background and rules for differentia-

tion, we obtained the following equation for 'P(r):(1_)W+(_-)P"--W=0 (9 )

i- . _2MG iwuerea--p- anu 7 -The solution for 'P was intially assumed to be a power series in r and this

led to the need for a double power series. Finally after more analysis, it wasassumed that 'P = and placing the equation into Maple, it simplified to thefollowing:

1 a ,, 2a,, 2a, a a - 0 10r +3J +3 r -A "dsolve" function was then used (a differential equation solver), and thisproduced the following result:

f=e' [clF ([- - + 1], [2- a], 2,v'(a - r)) +C2F ({- + , [a', 2/(a - r)) (11)where the function F is defined as the Barnes's extended hypergeometric

function

(8 )

and 1'(z)=' e_ttz_ldt.

1+'F=E11r(d+k) k!r(d)3


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The beginning attempts at a massless equation dependent upon the radiusand time led to the revision of the equation to f inally a massive, radius onlydependent equation. With the assumed solution of 'I', the differential equationwas expanded and then solved using Maple. Maple outputted a solutTon inhypergeometry with an exponential term, which was expected, as the seriesshould decay as the radius goes to infinity. The next step in this project will beto look for eigenvalues of field-created masses.

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Time-Independent Massive EKG Solution in a Schwarzschild Exterior