A note on Perng's unified field theory

A NNALES DE L ’I. H. P., SECTION A

S. C. C ^HANG A ^LLEN I. J ^ANIS

A note on Perng’s unified field theory

Annales de l’I. H. P., section A, tome 17, n

^o

3 (1972), p. 291-293

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© Gauthier-Villars, 1972, tous droits réservés.

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p. 291

A note

^on

Perng’s ^{unified field} theory*

S. C. CHANG and ALLEN I. JANIS University ^of Pittsburgh,

Pittsburgh, Pennsylvania, ^{U. S. A.}

Ann. Inst. Henri Poincaré, Vol. XVII, ^n° 3, 1972,

Section A : .-.

Physique théorique.

ABSTRACT. - Based on

Perng’s assumptions

^{and a} fundamental

property

^{of a}

symmetric

^affine

connection,

^a

simple argument

^shows

the

inconsistency

^of

Perng’s theory.

Mr.

Jeng

^J.

Perng recently published

^a paper with the title A

general-

lization

o f

^{metric tensor and its}

application

^{in this}

periodical [1].

^Some

of the basic

assumptions

of this paper have been found ^to be

inconsistent,

or even worse, lead to the fantastic conclusion that there is no electro-

magnetic

^field. The details will be shown below.

One of

Perng’s assumptions

is that the tensor

Aik

satisfies the

equation

By interchanging indices,

^{we have}

and

If we subtract

(1)

^{from the sum of}

(2)

^and

(3),

^{and use}

Perng’s

assump- tion that the affine connection is

symmetric,

^we

immediately get

* Supported ⁱⁿ part by ^a grant ^{from the} National Science Foundation.

ANN. INST. POINCARÉ, ^A-XVII-3 20

292 S. C. CHANG AND ALLEN I. JANIS

.and

then, by interchanging indices,

From

(4)

^and

(5),

^it is esay ^{to see} that

If the tensor (/an in

Peng’s

paper is defined

by [later

^{we will}

^show,

based ^on

Perng’s assumption,

^{that this must be this case}

!] :

then from

(6)

^and

(7)

^we conclude that

[remember

^{~W must be}

symmetric according

^to

(7)] :

Perng specifies { ( c. ) ^}

^{as the} Christoffel

symbol

constructed

by

Therefore it is safe to conclude that

Now

according

^to

assumption (i)

^of

Perng’s

paper, the

symmetric

affine connection

r {k

^{is defined as}

From

(9)

^and

(10)

^we conclu de that

If ^{se =}

0,

^then

Perng’s

^whole

theory

^is

empty.

^If

0,

^then

Eik sis Cf$ ⁼

0,

^{and because det}

(E~S) ~

⁰

[sap

sp, ⁼

dû ^!]

^{we must conclude}

cps ⁼

0,

^{i. e. there is no}

electromagnetic

^{field 1}

Next we want to prove that if the Christoffel

symbol ~ ^is

^defined

as

with eap any second rank contravariant tensor, and if is defined as

in

Equation (10),

^{then gap} Ep, ⁼

3~.

A NOTE ON PERNG’S UNIFIED FIELD THEORY 293

Proof. 2014

^{Under an}

arbitrary

coordinate

transformation, {03C303BD}

^as

defined in

(12)

will transform as

Since 2 x s,v ^gGS cps is ^a third rank mixed

tensor, r~v

^as defined in

(10)

will transform as

Because is ^a

symmetric

^affine

connection,

^{it should transform}

as

Subtracting (14)

^from

(13)

^{we have}

or

If we

define{03C3 03BD ⁼

^-}- 2014 ~03BD ,

p]

^{then we} will conclude

s,çi ⁼

ôfi

and because s,,i is

symmetric,

^{then must be}

symmetric

and thus we also

get

^{ssi =}

We thus see that the

symmetric

affine connection is

totally

^deter-

mined

by

^{the tensor ~ik if ~ik is}

nonsingular

^{and if we}

postulate

^{= 0.}

Also,

^{as we have}

shown,

^an

attempt

^to

modify

the définition of the tensor s~ but

keep

the form of the

symmetric

affine connection

proposed by Perng

is doomed to be a failure.

[1] ^{JENG J.} PERNG, Ann. Inst. H. Poincaré, ^t. XIII, ^n° 4, 1970, p. 287.

(Manuscrit reçu le 1er ^mars 1971.)

20.