INDIAN WRITING IN ENGLISH: SOME LANGUAGE ISSUES AND TRANSLATION ...

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INDIAN WRITING IN ENGLISH: SOME LANGUAGE ISSUES AND TRANSLATION PROBLEMS Christopher Rollason, Ph.D (Metz, France) - This is the revised and updated text of a paper given by the author at Jawaharlal Nehru University (New Delhi) on 8 March 2006, as part of the event Writers' Meet. The theme of this paper requires that we establish the nature of the object of study: what precisely is the thing that we are used to calling Indian Writing in English, or IWE? I shall begin my discussion with some remarks from over three decades ago, by the late David
  • creation of literary works
  • iwe
  • depth model with the analysis of the contemporary translation scholar
  • high degree of cultural empathy
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  • urdu
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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