WORDS MD

Internat. J. Math. & Math. Sci.

VOL. 20 NO. (1997) 201-204 201

A

HYPERCONTINUOUS HYPERSMOOTH SCHWARZSCHILD LINE ELEMENT TRANSFORMATION

ROBERT A.HERRMANN Mathematics

Department

Uxfited

States

NavalAcademy 572HollowayRd.

Annapolis,MD21402-5002USA

(Received April 3, 1995 andin revised form June 21, 1995)

ABSTRACT. In

this paper,anewderivationforoneoftheblack holelineelementsisgivensince the basic derivation for this lineelement is flawed mathematically. This derivationpostulates a transformationprocedure that utilizesa transformationfunction that ismodeled by anideal nonstandard physical worldtransformation processthat yieldsaconnectionbetweenan exterior Schwarzschild lineelement and distinctlydifferent interior lineelement. Thetransformation isan idealtransformationinthatinthenatural world the transformationis conceivedofasoccurring atanunknownmoment intheevolutionofagravitationallycollapsing sphericalbodywith radius greater than butnearto the Schwarzsclfild radius.

An

ideal transformationmodels this trans- formation ina mannerindependent of the objects standard radius. Ityields predictedbehavior baseduponaNewtoniangravitational field priortothe transformation, predictedbehaviorafter the transformation forafield internal tothe Schwarzschildsurface and predictedbehavior with respectto field alteration processesduring thetransformation.

KEY WORDS AND PHRASES.

Eddington-Finkelsteintransformation,hypercontinuous,hyper- smooth,black hole metric, nonstandard analysis, nonstandard substratum.

1992

AMS SUBJECT CLASSIFICATION CODES.

83C57,03H10.

I.

INTRODUCTION.

In [I],

the linear effectlineelementisderivedand,in

[2],

^ageneralllne elementd$’ isderived byconsideringanon-reversible

P-process

emanating from the center ofasphericalconflguratlon andits interaction withthesubstratum. This interaction ismodeled by taking the Special Theory chronotopicintervalandmodifyingitssphericalcoordinate transformationbyatype of damping ofthe basiclight-clockcounts. Thisdampingischaracterizedbythetwoexpressions

(i)

^dR"

dR"

+

dT" and

(ii)

^{dT" gdR"}

+

^{dT’,where the}

,

and]9aretobe determined. Fromthese determinations, the followinggeneral^lineelementisderived.

aS

(cdt") ’ (1/A)(dR")’ (R)’(sin ’ ^0(db’) ’ ^{+ (dO’)’). (1.1)}

TheEddington-Finkelsteintransformationisthe least ad hoc andismorephysically justified than others.

But,

in

[5],

^{thederivationand}argumentfor using thethissimpletransformation

(1)

202 R.A. HERRMANN

dU"’ dr"

+ .fM(R’)dR"

^{to obtain ablack holeline elementisflawed. Thisflawiscausedby}

theusualad hoclogicalerrorsin "removinginfinities." Equation57.11 in

[5,

p.

157],

specifically requires that

R > 2GM/c ^2. However,

inarguing for the useof the transformed Schwarzschild lineelement

(1.1),

^{Lawdenassumesthatitis}possible for

R" 2GM/c .

^{But theassumedreal}

valuedfunction definedbyequation57.11 isnot defined for

R"

such that

R 2GM/c 2.

^Since

the derivation of the Schwarzschild line element in

[2]

^{does not} require the General Principle of Relativity and, indeed, assumes that there is a privileged observer within a substratum, ^a

new and rigorously correct procedureis necessary. This is accomplished by showing that

(1)

canbeconsideredasahypercontinuous andhypersmoothtransformationassociated witha new non-reversible

P-process

that yieldsan alteration tothe gravitationalfield inthevicinityof the Schwarzschildsurface during the process of gravitational collapse. This speculationis modeled by the expression

(1)

which is conceivedof as an alteration in the time measuring light-clock.

Further,thisalteration isconceptually thesameastheultrasmooth microeffects model for fractual behavior

[4].

^Thistransformationtakes theSchwarzschild lineelement,which appliesonlytothe casewhere

R" > 2GM/c2,

^{and yieldsan}NSP-world black holelineelement thatonlyapplies for the casewhere

R" < 2GM/c .

^Likeultrasmooth microeffects, the nonstandardtransformation process is consideredas anideal model of behavior that approximates the actual natural world process. Thuswehave twodistrict linedementsconnected by such a transformationand each appliestoaspecific

R ’

domain.

2.

THE

FUNCTION

f(R’).

Toestablishthataninternal function

fM(Rm)

exists withthe appropriate propertiesproceed

asfollows: let be the set of allnonsingletonintervals in

(at). ^Let "

^C

^{(at lit)}

^{be the set}

of all nonempty functional sets of ordered pairs.

For

each

I ,

^let

C(I, at)

^C

"

^{bethesetof all}

real valuedcontinuousfunctions

(end

pointsincludedas

necessary)

^definedon

I.

Foreacha

>

^0,

]f.

6

C((-oo, 0], at), (-oo,0]

fi2r, suchthat

Vz

6

(-oo,0], f.(z) 1/(z -a).

^Further,

^Bg.

^fi

C((0,2a], at), (0,2a] ,

^{such that}

^Vz (0,2a], g.(z) -z/(2a) + 7z2/(4a ) z/a 1/a.

Then

Bh. C((2a, +cx),at), (2a, +oo)

⁽⁵

’,

^{such that} Vz

(2a, +cx), h.(x)

0. Finally, it follows that

limz--.o- ft(x)

lim,--.o+

ga(x),

lim,_.2,-

ga(x)

limt.-.2,4

ha(x). ^Hence

’.(=);

n.(=) 9.(=);

(-oo, o]

z

(0,2a]

xfii

(2a, +co)

is continuousfor eachzE at and has the indicated properties.

Now H: ^(z)

^existsandis continuousforallxE at and

f:();

e (-oo, 0]

n:() :(); (0,2a]

:(); e (2., +oo)

All of the above can be easily expressed in afirst-order language and all the statements hold in our superstructure enlargement

[4]. Let

0

< tt(0).

^{Then there exists an internal}

hypercontinuoushypersmooth

H,:

*at^-, *t suchthat

Vz

q

*(-oo,0], H,(z) 1/(z- )

^and

Yz

^q

*(-oo,0)t3

at,

st(H,(z)) st(1/(z -)) 1/z;

and for z 0,

H,(0)

exists, although

t(H,(0))

^{does notexistasa}real number. Further,

Yz

^q

(2, +oo)t3at (0, +oo), st(H,(z))

O.

To

obtainthe hypercontinuoushypersmooth

fM,

simply let

c.fM ^H,,

^x

^{A, R"}

^*at.

SCHWARZSCHILD LINE ELEMENT TRANSFORMATION 203

3. MOTIVATION FOR FUNCTION

SELECTION.

Recall that a function

f

^{defined on interval I is} standardizable

(to F)

^on I if’x E Irl lit,

F(x) sv.(f(x)) r=

lit.

Now,

^considerthetransformation

(1)

inthe nonstandard formdU dt

+ fM(R)dR

whereinternal

fM(R’)

is afunction definedon A C *lit, and A

A(R’).

There are infinitelymany nonstandardfunctions that can be standardized to produce the line elementd

’2.

Inthis lineelement, ^considersubstituting forthe functionA

A(R m,

thefunction

*A-e.Thetransformedlineelementthenbecomes,prior tostandardizingthe coefficient functions

(i.e.

restricting them the the natural

world),

T

*A

e)c’((dU)

²

2fMdUdR + fw(dR) ’) (1/(

^*A

e))(dRm) ’-

(R")2(sin ’ ^{O(d’) + (dO") )}

*A

e)ce(dUm) ’ ²⁽

^*A

e)c’fMdUmdR+

(( "A e)c’ f

^w

^(1/(

^*A

e)))d ^:

^dR

"-

(R’)’(sin ’ ^0re(de’) ’ + (dO")’). (3.1)

Following theprocedureoutlined in

[4],

firstconsiderthe partition IR

(-o0, 0]

^U

(0, 2e]

^U

(2e, +),

whereeisapositive infinitesimal. Considerthe requiredconstraints.

(2)

^Asrequired,

for specific real intervals, allcoefficients ofthe termsof the transformedlineelement areto be standardizedand, hence,arestandard functions.

(3)

Since any lineelement transformation, prior tostandardization,^{shouldretain its}infinitesimalcharacterwithrespect toanappropriate interval

I,

then forany infinitesimaldR and for

ea

^value

R

E

I

terms such as

G(R)dR ,

^where

G(Rm)

^isacoefficientfunction,must be ofinfinitesimalvalue.

Forthe importantconstraint

(3),

Deilnition4.1.1,andtheorems 4.1.1,4.1.2 in

[3]

^implythat

forafixed infinitesimaldR in ordertohaveexpressionb infinitesimalas

R

varies, thecoefficient

h(R) "A-e)c’f

w

1/(*A-e)

^mustbe infinitesimalon asubset

A

ofanappropriateinterval

I

such that 0 E A. The simplest case would beto assumethat

A *(

-o,

0].

^{Let standard}

r

e A

^t3 IR.Then itfollows that

h(r)

C

p(0).

^Thus

st(h(r))

^0.

Inleed,

let x

e (tg{ft(r)

r

<

0, ^r

e lit})

^t9

(p(0)

f3

A).

^Then

st(h(x))

^{0. Sincewe are}seeking atransformationprocess that ishypercontinuous,at leaston

*(

^c,

0],

^{thislast statement}suggests the simplesttoconsider would be that on

*(- oo,0],

^{h 0. Thus the} basic constraint yields the basic requirement that on

*(- o,0]

the simplest function tochooseis

CfM(X) 1/(x --e).

Since standardizing is required on

*(- o,0)rl

lit, we have for each x

e "(- o,0)f3

^{IR, that}

st(cfM(x)) cst(fM(X)) st(1/(x- e)) 1/X.

^Thisleads to the assumption that on

(-o,0]

^thefunction

f(x) 1/(x a),

a

>

0,should be considered. After *-transferringand prior to standardizing, this selection would satisfy

(3)

for both of the coefficients in which

fM

appears and for the interval

I *(-

oo,

0].

^The function ga isarbitrarily selected to satisfy the hypercontinuous andhypersmooth property and, obviously,

ha

^isselected topreservethe originallineelement for the interval

(2e, +oo).

^Finally, it is necessary that the resultingnew coefficient functions, prior to standarizing, all satisfy

(3)

at least forafixeddR andavarying

R

E

*(

^-o,

0]

^{for the}

expression

(1).

It is not difficult toshow that

IH,(x)l < 2/e

^{for all x}

e

^*IR. Consequently, for e

(dRY)

^1/3 expression

(1)

^{is an}infinitesimalforall

R" e

^*IR.

20 R.A. HERRMANN

Let I

v2/c

^{2 A. For} the collapsescenario

R RM.

^If

2GM/(R’c 2) <

1, substituting

2GM/R

^v

,

^into

(1.1)

yields the so-called Schwarzschildline element. Withrespect tothe transformation,

(A)

^if

R" < 2GM/c ,

^{then for}

st(fM(R")) 1/(cA),

^1-

2GM/(R’’c);

^for

(B) ^R" > 2GM/c2; s’C(fM(R"))

0, and for the^casethat

(C) R" 2GM/c ,

the function

fM

isdefinedandequaltoaNSP-world value

fM(R’). But,

^forcase

(C), st(fM(R’))

^doesnotexist

asareal number.

Hence, (C)

^{hasnodirecteffectwithin the}natural worldwhen

R" 2GM/c2,

althoughthe fact that

fM(R")dR"

^{is an}infinitesimal implies that

s’c(fM(R’)dR’)

^{0. Using}

these NSP-worldfunctionsand

(3.1),

^cases

(A)

^and

(C)

^yield

dS[ $(cdU’)

2cdU"dR"-

(n-)(i= 0-(d-) + (dO’)). ^(3.2)

Butcase

(B),

^leadsto

(1.1).

^Thetwo constraintsaremetby

fM(R"),

^andindeedthestandardized

(A)

^{form for}

fM(R’)

^{isunique if}

(3)

^istobe satisfied foraspecificinterval.

Since this isanidealapproximatingmodel,ⁱⁿordertoapplythis idealmodeltothe natural

vorld,

one mostselect an appropriate real afor the real valuedfunction

Ha.

^{Finally, it is not}

assumed that the functionga isunique.

In

any solutionsfor hnedement

(3.2),

the dU"

[resp.

dR’]

referstothe timing

[resp. length]

infinitesimallight-clockcountsand doesrefertouniversal time

[resp. length]

alterations.

REFERENCES

1.

HERRMANN,

R.A.Anoperator equation andrelativisticalterations inthetime for radioactive decay,

Intern, J. Math.

^Math. Sci.,

(to appear).

2.

HERRMANN, R.A.

Constructing Logically Consistent Special and General Theoriesof .Relativity, Math.

Dept.,

U.S.NavalAcademy,Annapolis,

MD,

1993.

3.

HERRMANN,

R.A.

Some

application of nonstandard analysistoundergraduatemathematics:

infinitesimalmodeling and elementary_physics, Instructional Development Project, Mathematics

Department, U. S.

NavalAcademy,AnnapoliS,

MD,

21402-5002,1991.

4.

HER_RMANN, R.A.

Fractals and ultrasmoothmicroeffects,

J. Math.

^{Phys.,30(April}

1989),

805-808.

5.

LAWDEN, D.

F.

A_.n

IntroductiontoTensor Calculus,Relativity andCosmology,John Wiley

& Sons, New

York, 1982.