WASHINGTON UNIVERSITY

Department of Physics

Dissertation Examination Committee:

Cliﬁord M. Will, Chair

Mark Alford

Ramanath Cowsick

Renato Feres

Barry Spielman

Wai-Mo Suen

˜EXTENSIONS OF THE EINSTEIN-SCHRODINGER NON-SYMMETRIC

THEORY OF GRAVITY

by

James A. Shi†ett

A dissertation presented to the

Graduate School of Arts and Sciences

of Washington University in

partial fulﬂllment of the

requirements for the degree

of Doctor of Philosophy

May 2008

Saint Louis, MissouriAcknowledgements

Thanks to Cliﬁord Will for his help and support. Thanks also to my mother Betsey

Shi†ett for her encouragement during my graduate studies, and to my late father

John Shi†ett for his encouragement long ago. This work was funded in part by the

National Science Foundation under grants PHY 03-53180 and PHY 06-52448.

iiContents

Acknowledgements ii

Abstract v

1 Introduction 1

2 Extension of the Einstein-Schr˜odinger theory for Abelian ﬂelds 7

2.1 The Lagrangian density . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Maxwell’s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 The connection equations . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Exact Solutions 28

3.1 An exact electric monopole solution . . . . . . . . . . . . . . . . . . . 28

3.2 An exact electromagnetic plane-wave solution . . . . . . . . . . . . . 32

4 The equations of motion 34

4.1 The Lorentz force equation . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Equations of motion of the electric monopole solution . . . . . . . . . 36

4.3 The Einstein-Infeld-Hoﬁmann equations of motion . . . . . . . . . . . 41

5 Observational consequences 51

5.1 Pericenter advance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 De ection and time delay of light . . . . . . . . . . . . . . . . . . . . 60

5.3 Shift in Hydrogen atom energy levels . . . . . . . . . . . . . . . . . . 69

6 Application of Newman-Penrose methods 71

6.1 Newman-Penrose methods applied to the exact ﬂeld equations . . . . 71

26.2 asymptotically at O(1=r ) expansion of the ﬂeld

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7 Extension of the Einstein-Schr˜odinger theory for non-Abelian ﬂelds107

7.1 The Lagrangian density . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 Invariance properties of the Lagrangian density . . . . . . . . . . . . 112

7.3 The ﬂeld equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8 Conclusions 122

iiiA A divergence identity 123

¾~B Variational derivatives for ﬂelds with the symmetry ¡ =0 124

[„ ]

C Approximate solution for N in terms of g and f 126”„ ”„ ”„

ﬁ~D Approximate solution for ¡ in terms of g and f 129”„ ”„”„

E Derivation of the generalized contracted Bianchi identity 134

F Validation of the EIH method to post-Coulombian order 139

G Application of point-particle post-Newtonian methods 143

H Alternative derivation of the Lorentz force equation 147

¡1Ie derivation of the O(⁄ ) ﬂeld equations 151b

J A weak ﬂeld Lagrangian density 153

K Proca-waves as Pauli-Villars ghosts? 161

„L L , T , j and kinetic equations for spin-0 and spin-1/2 sources 165m „”

M Alternative ways to derive the Einstein-Schr˜odinger theory 172

N Derivation of the electric monopole solution 179

^O The function V(r) in the electric monopole solution 185

P The electric monopole solution in alternative coordinates 190

Q The electromagnetic plane-wave solution in alternative coordinates192

R Some properties of the non-symmetric Ricci tensor 197

S Calculation of the Ricci tensor in tetrad form 201

T Proof of a nonsymmetric matrix decomposition theorem 204

aU Calculation of the exact ¤ in Newman-Penrose form 206bc

aV Check of the approximate ¤ inenrose form 210bc

WKursunoglu’s theory with sources and non-Abelian ﬂelds 214

X Possible extension of the theory to non-Abelian symmetric ﬂelds 220

Bibliography 247

ivAbstract

We modify the Einstein-Schr˜odinger theory to include a cosmological constant ⁄z

which multiplies the symmetric metric. The cosmological constant ⁄ is assumed toz

be nearly cancelled by Schr˜odinger’s cosmological constant ⁄ which multiplies theb

nonsymmetric fundamental tensor, such that the total ⁄=⁄ +⁄ matches measure-z b

ment. The resulting theory becomes exactly Einstein-Maxwell theory in the limit

2as j⁄ j!1. For j⁄ j»1=(Planck length) the ﬂeld equations match the ordinaryz z

¡16Einstein and Maxwell equations except for extra terms which are < 10 of the

usual terms for worst-case ﬂeld strengths and rates-of-change accessible to measure-

ment. Additional ﬂelds can be included in the Lagrangian, and these ﬂelds may

couple to the symmetric metric and the electromagnetic vector potential, just as

in Einstein-Maxwell theory. The ordinary Lorentz force equation is obtained by

taking the divergence of the Einstein equations when sources are included. The

Einstein-Infeld-Hoﬁmann (EIH) equations of motion match the equations of motion

for Einstein-Maxwell theory to Newtonian/Coulombian order, which proves the exis-

tenceofaLorentzforcewithoutrequiringsources. Anexactchargedsolutionmatches

¡66the Reissner-Nordstr˜om solution except for additional terms which are»10 of the

usualtermsforworst-caseradiiaccessibletomeasurement. Anexactelectromagnetic

vplane-wave solution is identical to its counterpart in Einstein-Maxwell theory. Peri-

center advance, de ection of light and time delay of light have a fractional diﬁerence

¡56of <10 compared to Einstein-Maxwell theory for worst-case parameters. When

a spin-1/2 ﬂeld is included in the Lagrangian, the theory gives the ordinary Dirac

¡50equation, and the charged solution results in fractional shifts of <10 in Hydrogen

atomenergy levels. Newman-Penrose methods are used to derivean exact solution of

theconnectionequations, andtoshowthatthechargedsolutionisPetrovtype-Dlike

2the Reissner-Nordstr˜om solution. The Newman-Penrose asymptotically at O(1=r )

expansion of the ﬂeld equations is shown to match Einstein-Maxwell theory. Finally

we generalize the theory to non-Abelian ﬂelds, and show that a special case of the

resulting theory closely approximates Einstein-Weinberg-Salam theory.

viChapter 1

Introduction

Einstein-Maxwell theory is the standard theory which couples general relativity with

electrodynamics. In this theory, space-time geometry and gravity are described by

a metric g which is symmetric (g = g ), and the electromagnetic ﬂeld F is„” „” ”„ „”

antisymmetric (F =¡F ). The fact that these two ﬂelds could be combined to-„” ”„

gether into one second rank tensor was noticed long ago by researchers looking for

a more uniﬂed description of the physical laws. The Einstein-Schr˜odinger theory is

a generalization of vacuum general relativity which allows a nonsymmetric ﬂeld N„”

in place of the symmetric g . The theory without a cosmological constant was ﬂrst„”

proposed by Einstein and Straus[1, 2, 3, 4, 5]. Schr˜odinger later showed that it could

be derived from a very simple Lagrangian density if a cosmological constant ⁄ wasb

included[6, 7, 8]. Einstein and Schr˜odinger suspected that the theory might include

electrodynamics, where the nonsymmetric \fundamental tensor" N contained both„”

the metric and electromagnetic ﬂeld. However, this hope was dashed when it was

foundthatthetheorydidnotpredictaLorentzforcebetweenchargedparticles[9,10].

1In this dissertation we describe a simple modiﬂcation of the Einstein-Schr˜odinger

theory[11,12,13,14]whichcloselyapproximatesEinstein-Maxwelltheory,andwhere

the Lorentz force does occur. The modiﬂcation involves the addition of a second cos-

mological term ⁄ g to the ﬂeld equations, where g is the symmetric metric. Wez „” „”

assumethistermisnearlycanceledbySchr˜odinger’s\bare"cosmologicalterm⁄ N ,b „”

where N is the nonsymmetric fundamental tensor. The total \physical" cosmolog-„”

ical constant ⁄=⁄ +⁄ can then be made to match cosmological measurements ofb z

the accelerating universe.

The origin of our ⁄ is unknown. One possibility is that ⁄ could arise fromz z

vacuum uctuations, an idea discussed by many authors[15, 16, 17, 18]. Zero-point

uctuations are essential to both quantum electrodynamics and the Standard Model,

and are thought to be the cause of the Casimir force[16] and other eﬁects. With

this interpretation, the ﬂne tuning of cosmological constants is not so objectionable

because it resembles mass/charge/ﬂeld-strength renormalization in quantum electro-

dynamics. For example, to cancel electron self-energy in quantum electrodynamics,

the \bare" electron mass becomes large for a cutoﬁ frequency ! »1=(Planck length),c

and inﬂnite if ! !1, but the total \physical" mass remains small. In a similarc

manner, to cancel zero-point energy in our theory, the \bare" cosmological constant

4 2⁄ »! £ (Planck length) becomes large if ! »1=(Planck length), and inﬂnite ifb cc

! !1, but the total \physical" ⁄ remains small. There are other possible ori-c

gins of ⁄ . For example ⁄ could arise dynamically, related to the minimum of az z

potential of some additional ﬂeld in the theory. Apart from the discussion above,

speculation about the origin of ⁄ is outside the scope of this dissertation. Our mainz

26

goalistoshowthatthetheorycloselyapproximatesEinstein-Maxwelltheory, andfor

non-Abelian ﬂelds the Einstein-Weinberg-Salam theory (general relativity coupled to

electro-weak theory).

LikeEinstein-Maxwelltheory, ourtheorycanbecoupledtoadditionalﬂeldsusing

a symmetric metric g and vector potential A , and it is invariant under a U(1)„” „

gauge transformation. The theory does not enlarge the invariance group. When

coupledtotheStandardModel,thecombinedLagrangianisinvariantundertheusual

„”U(1)›SU(2)›SU(3)gaugegroup. TheusualU(1)gaugetermF F isincorporated„”

togetherwiththegeometry, andisnotexplicitlyintheLagrangian. Thenon-Abelian

version of the theory can also be coupled to the Standard Model, in which case both

the U(1) and SU(2) gauge terms are incorporated together with the geometry. This

is done much as it is done in [19, 20] with Bonnor’s theory. Whether the SU(3) gauge

term of the Standard model could also be incorporated with a larger gauge group, or

by using higher space-time dimensions, is beyond the scope of this dissertation.

The Abelian version of our theory is similar to [21, 22] but with the opposite sign

of⁄ and⁄ . BecauseofthisdiﬁerenceourtheoryinvolvesHermitianﬂeldsinsteadofb z

realﬂelds,andthesphericallysymmetricsolutionshavemuchdiﬁerentpropertiesnear

theoriginanddonotcomeinaninﬂniteset. TheAbelianversionofourtheoryisalso

roughly the electromagnetic dual of another theory[23, 24, 25, 26]. Compared to all

of these other theories, our theory also allows coupling to additional ﬂelds (sources),

and it allows ⁄=0, and it is derived from a Lagrangian density which incorporates a

new type of non-symmetric Ricci tensor with diﬁerent invariance properties.

Many other modiﬂcations of the Einstein-Schr˜odinger theory have been consid-

3ered. For example in Bonnor’s theory[27, 28] the antisymmetric part of the funda-

mental tensor N or its dual is taken to be the electromagnetic ﬂeld, and a Lorentz[¿‰]

p

a[‰¿]force is derived, but only because a ¡NN N term is appended onto the usual[¿‰]

Lagrangian density. Other theories include an assortment of additional terms in the

Lagrangiandensity[29,30]. Suchtheorieslackthemathematicalsimplicityoftheorig-

inal Einstein-Schr˜odinger theory, and for that reason they seem unsatisfying. This

criticism seems less applicable to our theory because there are such good motivations

p

for including a ⁄ ¡g term in the Lagrangian density.z

Some previous work[31, 32, 33] shows that the original Einstein-Schr˜odinger the-

oryhasproblemswithnegativeenergy\ghosts". Aswillbeseeninx2.4, thisproblem

is avoided in our theory in an unusual way. In [31, 32, 33] referenced above, the elec-

tromagnetic ﬂeld is assumed to be an independent ﬂeld added onto the Lagrangian,

and it is unrelated to N . Because of the coupling of N to the electromagnetic[”„] ”„

ﬂeld in such theories, there would be observable violations[34, 35, 36] of the principle

of equivalence for values of N which occur in the theory. Such problems do not[”„]

applyinourtheory,mainlybecauseweassumeasymmetricmetricwhichisdeﬂnedin

terms of N , and it is this symmetric metric which appears in Maxwell’s equations,”„

and any coupling to additional ﬂelds. Such problems are also avoided in our theory

partly because of the small values of N which occur.[”„]

InmostpreviousworkontheoriginalEinstein-Schr˜odingertheory,theelectromag-

netic ﬂeld is assumed to be the dual of N . Even though this is the same deﬂnition[¿‰]

usedin[9,10]toshowthereisnoLorentzforce, severalauthorsclaim thataLorentz-

like force can be demonstrated[37, 38, 39]. However, the solutions[40, 41, 42] that

4