MaxPlanckInstitut

fur Mathematik

in den Naturwissenschaften

Leipzig

Electromagnetic resolution of curvature

and gravitational

by

N K Dadhich K B Marathe and G Martucci

PreprintNr

IL NUOVO CIMENTO VolN

Decemb er

c

So cieta Italiana di Fisica

N K DADHICH ETC

Electromagnetic Resolution of Curvature and Gravitational In

stantons

N K Dadhich K B Marathe andG Martucci

InterUniversity Center for Astronomy Astrophysics

Pune India



City University of New York Brooklyn Col lege Department of Mathematics Brooklyn NY

USA



Dipartimento di Fisica Universita di Firenze Largo E Fermi Firenze Italy

ricevuto approvato

Summary We study the electromagnetic resolution of the Riemann curvature on

a spacetime manifold M with metric g into its electric and magnetic parts relative

to a unit timelikevector with resp ect to g There exists a duality transformation

between the active and passive electric parts whichleaves invariant a sub class of eld

equations which corresp ond to the generalized gravitational equations

We also discuss various geometric formulations of the equations of gravitational

instantons and their generalization which includes as a sp ecial case the classical

Einstein equations Some sp ecial solutions and their physical signicance

are also considered

PACS Classical general Relativity

PACS Dierential geometry Riemannian geometry

emailnkdiucaaernetin

emailkbmscibro oklyncunyedu

emailmartucciinfnit

ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS

Intro duction

It is well known since the b eginning of this century that the electric and magnetic

elds can b e combined in a natural way to obtain the electromagnetic eld tensor over

the Minkowski space and that Maxwells equations can b e expressed as dierential equa

tions for the comp onents of this tensor Conversely the electric and magnetic parts of

the eld tensor can be recovered once a co ordinate system is xed However the fact

that the electromagnetic eld tensor can b e regarded as the curvature form or a gauge

eld of a connection or a gauge p otential in a principal bundle with gauge group U

and that Maxwells equations can be expressed as gauge eld equations b ecame clear

only recently This identication of curvature and gauge eld extends to bundles over

arbitrary Riemannian manifolds The study of the space of solutions of gauge eld equa

tions has led to new results and a b etter understanding of the top ology and geometry

of low dimensional manifolds For background material on geometry and top ology with

applications to gauge theory see for example Marathe and Martucci and It

is natural to ask ab out the signicance of electric and magnetic parts of an arbitrary

gauge eld regarded as the curvature of a gauge p otential There do es not seem to b e

any discussion of this asp ect of gauge elds in the literature However a decomp osition

of the Weyl and the Riemann curvature of the LeviCivita connection on a spacetime

manifold is known see for example and references therein In section we con

sider the decomp osition of the full Riemann curvature of a spacetime manifold M g

into its electric and magnetic parts relative to a unit timelikevector with resp ect to g

We shall refer to this decomp osition as the electromagnetic resolution of the curvature

In Dadhich has dened a duality transformation which exchanges the active and

passive electric parts and leaves the antisymmetric magnetic part unchanged This du

ality transformation implies interchanging the Ricci and the Einstein tensors Duality

transformation leaves the Einstein vacuum equations invariantbut changes the sign of

the Duality transformation would then imply that the mass M of the

Schwarzschild solution go es over to M Furthermore it turns out that the Einstein

vacuum equations can b e mo died so that the mo died equations are not dualityinvari

ant and yet admit the same solution while the dual set of equations admit a solution

distinct from the Schwarzschild solution Similarly we can construct spacetimes dual

to the ReissnerNordstrom NUT and the Kerr solutions The dual spacetimes

describ e the original spacetimes with global monop ole charge and global texture The

duality so dened is thus intimately related to the top ological defects

Weshow that the gravitational instanton Lagrangian R and the instanton equations

are invariant under the duality transformation The main aim of this pap er is to study the

role played by the duality transformation in the case of gravitational instantons with or

without their coupling to matter elds In section we consider a gravitational instanton

coupled to classical elds The resulting eld equations include the usual gravitational

equations with source term and with or without the Its relation

to duality is considered in section In the concluding section wemention some op en

problems

Electromagnetic Resolution of Curvature

Let M g b e a spacetime manifold with metric g of signature We

identify the Riemann curvature with the symmetric op erator it induces on M for

x

each x M and denote the Ho dge star op erator on M by J Let u b e a unit timelike

a x

N K DADHICH ETC

vector We dene the electromagnetic resolution of the Riemann curvature relativeto

the unit timelikevector u as follows

a

b d b d

E R u u E JRJ u u

ac abcd ac abcd

b d b d

H JR u u H H H RJ u u

ac abcd abcd

ac ac ac

where

b d e b d

H JC u u and H R u u

abcd abce

ac ac

d

are the symmetric and the skew symmetric parts of the tensor H Here C is the

ac abcd

Weyl conformal curvature and is the dimensional volume element The pair of

abcd

tensors E E constitutes the electric part of curvature The tensor E is called the

ac ac ac

active part and the tensor E is called the passivepart The tensor H is called the

ac ac

magnetic part of the curvature H H and hence is dropp ed from further considera

ac ca

a a a

tion WewriteE E H H and E E The active and the passive electric parts

a a a

are symmetric tensors The electric and the magnetic parts have the following additional

prop erties

b b d

a E u E R u u

ab bd

b b d

E u E R b Rg u u

ab bd bd

b

c H u H

ab

We note that the electric and magnetic parts are orthogonal to the timelike vector u

a

and thus b ehave at least lo cally as tensors The electric parts are symmetric and

account for comp onents while the magnetic part is tracefree and accounts for

comp onents of the Riemann curvature Thus the entire gravitational eld is decomp osed

into electromagnetic parts The resolution is not unique and dep ends on the choice of

a unit timelike vector Using the ab ove decomp osition Dadhich has dened the

duality transformation as follows

E E H H H H

ab ab

ab ab ab ab

Thus the duality transformation interchanges the activeandpassive electric parts and

changes the sign of the symmetric magnetic part while leaving the antisymmetric mag

netic part unchanged From equation it is clear that the duality transformation would

map the Ricci tensor into the Einstein tensor and viceversa This is b ecause contraction

of Riemann is Ricci while that of its double dual is Einstein The Ricci tensor can b e

written in terms of the electric and magnetic parts as follows

mn c

H u u u R E E E E u u Eg

acmn b bcmn a ab ab ab a b ab

The scalar curvature R is given by

R E E

ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS

In section weshow that the vacuum Einstein equations with the cosmological constant

are equivalent to the equations

b b

E E and H

ab

a a

This together with equation imply that the vacuum equation R is in general

ab

equivalentto

E E H and E or equivalently E

ab ab

ab

Thus the vacuum equation is symmetric in E and E and hence is invariant under

ab ab

the duality transformation

For obtaining the Schwarzschild solution wefollow and consider the spherically

symmetric metric

ds C rtdt A rtdr r d sin d

The natural choice for the resolving vector is in this case p ointing along the tline

whichishyp ersurface orthogonal From equation H and E E leadto

ab

AC for this no b oundary condition of asymptotic atness is needed Now E

gives A Mr which is the Schwarzschild solution Note that we did not

need to use the remaining equation E E which is hence free and is implied bythe

rest Without aecting the Schwarzschild solution we can intro duce some distribution

in the direction

We consider the following equation which is not invariant under the duality transfor

mation

H E E E w w

ab ab a b

ab

where is a scalar and w u ju j is a spacelikeunitvector parallel to the acceleration

a a a

It is clear that it will also admit the Schwarzschild solution as the general solution and

it determines For spherical symmetry the ab ove alternate equation also

characterizes vacuum b ecause the Schwarzschild solution is unique Now applying the

duality transformation to the ab ove equation we obtain the equation

H E E E w w

ab ab a b

ab

Again we shall have AC and E will then yield C k Mr and

kr Thus the general solution of equation for the metric is given by

M

C A k

r

This is the BarriolaVilenkin solution for the Schwarzschild particle with global

monop ole charge This has nonzero stresses given by

k

T T

r

N K DADHICH ETC

A global monop ole is describ ed by a triplet of scalars

a a a a

r f r x r where x x r

Through the usual Lagrangian it generates the energymomentum distribution at large

distance from the core precisey of the form given ab ovein LiketheSchwarzschild

solution the monop ole solution is also the unique solution of equation

On the other hand at spacetime could also b e characterized in an alternativeform

by

E H E w w

ab ab a b

ab

leading to C A and implying Its dual will b e

E H E w w

ab ab a b

ab

yielding the general solution

 

C A C A const k

which is nonat and represents a global monop ole of zero mass as it follows from the

solution when M

In general the duality transformation for nonvacuum spacetime maps a Lorentzian

spacetime into another Lorentzian spacetime with Ricci and Einstein tensors inter

changed The vacuum equation is however invariant Note that under the duality trans

formation the Weyl tensor do es change sign and this implies that the mass angular

momentum and the NUT parameter change sign In the case of the Kerr and the

NUT solutions H changes sign We can however have distinct solutions only when the

ac

vacuum equation is suitably mo died without disturbing the original vacuum solution

so that it is not dualityinvariant This would happ en when there exists a free equation

whichplays no role in determining the original vacuum solution Then it is p ossible to

write the alternate equation which though duality noninvariant givesthesamevacuum

solution Note that as in equation the distribution is intro duced along the di

rection of the acceleration vector which is radial for spherical symmetry Now the dual

equation would in general give a solution dierent from the original solution This is how

we can construct solutions dual to vacuum or at spacetime

Finally the duality transformation essentially means interchange of roles of the New

tonian active part and the relativistic passive part asp ects of the eld The two

asp ects are b ound together by the Einstein eld equations As argued and demonstrated

ab ove duality seems to indicate minimal departure in physical features from the orig

inal spacetime and hence the covariant dualatness condition could b e considered as

dening minimally curved spacetime

The most general dualityinvariant expression consisting of the Ricci and the metric

R

a a

The corresp onding vacuum equation is g is R

b b

R

a a

g R

b b

ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS

By taking traces of b oth sides of the ab ove equation we get and hence it reduces

to the equation for gravitational instanton

R

a a

g R

b b

We note that over a dimensional Riemannian manifold M g the gravitational

instantons are critical p oints of the quadratic Riemannian functional or action A g

dened by

Z

A g R dv

g

M

The action A g is conformally invariant From equation it follows that it is also in

variant under the duality transformation A generalization of this action is considered

in section Furthermore the standard Hilb ertEinstein action

Z

A g Rdv

g

M

also leads to the instanton equations when the variation of the action is restricted to

metrics of volume

There are several dierences b etween the Riemannian functionals used in theories of

gravitation and the YangMills functional used to study gauge eld theories The most

imp ortant dierence is that the Riemannian functionals are dep endent on the bundle of

frames of M or its reductions while the YangMills functional can be dened on any

principal bundle over M However wehave the following interesting theorem

Theorem Let M g be a compact dimensional Riemannian manifold Let

M denote the bund le of selfdual forms on M with induced metric G Then

the LeviCivita connection on M satises the gravitational instanton equations if and

g

only if the LeviCivita connection on M satises the YangMil ls instanton equa

G

tions

The instanton action and the corresp onding eld equations are dualityinvariant

They are also conformally invariantas well As a matter of fact conformal invariance

singles out the R instanton action That means the conformal invariance includes the

dualityinvariance while the dualityinvariance of the Palatini action with the condition

that the resulting equation b e valid for all values of R would lead to the conformal invari

ance The simplest and wellknown instanton solution is the time

Here the duality only leads to the antide Sitter spacetime

Coupling matter and other elds to gravitational instanton leads naturally to the

notion of generalized gravitational instantons It was intro duced in We discuss this

in the next section and then consider the relation b etween duality and instantons

Generalized Gravitational Instantons

We b egin with a summary of the results obtained in earlier pap ers which

are relevant to the presentpaper Let M b e a spacetime manifold The space M

x

of forms at x M is a real sixdimensional vector space The Ho dge star op erator J

on M denes a complex structure on it ie J I is the identity transformation x

N K DADHICH ETC

of M Using the complex structure J M can b e made into a complex three

x x

dimensional vector space The space of complex linear transformations of this complex

vector space can be identied with the subspace of real transformations S of the real

vector space dened by fS j SJ JSg

Wenow dene a tensor of curvature typ e

Denition Let C be a tensor of type on M We can regard C as a quadrilinear

mapping pointwise so that for each x M C can be identied with a multilinear map

x

   

C T M T M T M T M R

x

x x x x

where R is the real number eld We say that the tensor C is of curvature type if C

x

satises the algebraic properties of the RiemannChristoel curvature tensor ie if C

x



satises the fol lowing conditions for each x M and for al l T M

x

C C

x x

C C

x x

C C C

x x x

From the ab ove denition it follows that a tensor C of curvature typ e also satises the

following condition

C C x M

x x

Weobserve that if C is of curvature typ e then C can b e identied with a selfadjoint

x

linear transformation of M for each x M where M is the space of second

x x

order dierential forms at x We now dene the curvature pro duct of two symmetric

tensors of typ e on M It was intro duced in and used in to obtain a

geometric formulation of Einsteins equations

Denition Let g and T be two symmetric tensors of type on M The curva

ture pro duct of g and T denotedbyg T is a tensor of type denedby

c

g T g T g T

c x

g T g T



for al l x M and T M

x

In the following prop osition we collect together some imp ortant prop erties of the curva

ture pro duct and tensors of curvature typ e

Proposition Let g and T be two symmetric tensors of type on M and let C be

a tensor of curvaturetypeonM Then we have the fol lowing

g T T g

c c

g T is a tensor of curvaturetype

c

Let G denote the tensor g g Then G induces a pseudoinner product on

c x

M x M x

ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS

C induces a symmetric linear transformation of M x M

x

x

When g is the fundamental tensor of M and T is the energymomentum tensor we

call g T T g the interaction tensor b etween the eld and the source energy

c c

momentum tensor It plays an essential role in denition of the new eld equations

given here We denote the Ho dge star op erator on M by J The fact that M is

x

x

aLorentz manifold implies that J denes a complex structure on M x M

x

x

Using this complex structure we can give a natural structure of a complex vector space

to M Wenow dene the gravitational tensor W of curvature typ e which includes

g

x

the source term

Denition Let M beaspacetime manifold with fundamental tensor g and let T be

a symmetric tensor of type on M Then the gravitational tensor W is dened

g

by

W R g T

g c

where R is the RiemannChristoel curvature tensor of type

We are now in a p osition to give a geometric formulation of the generalized eld equations

of gravitation

Theorem Let W denote the gravitational tensor dened by with source tensor

g

T We also denote by W the linear transformation of M inducedby W Then the

g g

x

fol lowing statements areequivalent

g satises the generalized eld equations of gravitation

R Rg T Tg

W commutes with J ie

g

W J

g

W induces a complex linear transformation of the complex vector space M for

g

x

each x M

Let G denote the inner product on M induced by g Dene the submanifolds

x

D and D of M by

x

M jGv v and v v g D fv



x

Dene a real valued function f on D D cal led the gravitational sectional

curvatureby

f v sv GW v v

g

where sv if v D and sv ifv D Then

f f J

N K DADHICH ETC

We shall call the triple M g T a generalized gravitational eld if any one of the

conditions of the ab ove theorem is satised Generalized gravitational eld equations

were intro duced by Marathe in In the sp ecial case when the energymomentum

tensor T is zero the results of the ab ove theorem were obtained in Solutions of

Marathes generalized gravitational eld equations which are not solutions of Einsteins

equations have b een discussed in

We note that the ab ove theorem and the last condition in prop osition can be

used to discuss the Petrov classication of gravitational elds see Petrov We

observethat the generalized eld equations of gravitation contain Einsteins equations

with or without the cosmological constant as sp ecial cases Solutions of the sourcefree

generalized eld equations are called gravitational instantons If the base manifold

is Riemannian then gravitational instantons corresp ond to Einstein spaces A detailed

discussion of the structure of Einstein spaces and their mo duli spaces maybefoundin

Besse

If one wishes to retain the coupling constant say k in Einsteins equations one need

only intro duce it in the denition of W as follows

W R kg T

c

The equations are then replaced by

ij ij ij ij

R Rg k T Tg

In particular by taking R kT in equations we obtain

ij ij ij

Rg kT R

which are Einsteins equations with the coupling constant k

We observe that the equation do es not lead to any relation b etween the scalar

curvature and the trace of the source tensor since b oth sides of equation are trace

free Taking divergence of b oth sides of equation and using the Bianchi identities

we obtain the generalized conservation condition

ij ij

r T g

i i

k

where r is the covariant derivative with resp ect to the vector

i i

x

kT R

and Using the function dened by equation the eld equations can

i i

x

b e written as

ij ij ij ij

R Rg g kT

In this form the new eld equations app ear as Einsteins eld equations with the cosmo

logical constant replaced by the function whichwemay call the cosmological function

ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS

The cosmological function is intimately connected with the classical conservation condi

tion expressing the divergencefree nature of the energymomentumtensorasisshown

by the following prop osition

Proposition The energymomentum tensor satises the classical conservation condi

tion

ij

r T

i

if and only if the cosmological function is a constant Moreover in this case the

generalized eld equations reduce to Einsteins eld equations with cosmological constant

Equation is called the dierential or lo cal law of conservation of energy and

momentum To obtain global conservation laws one needs to integrate equations

However such integration is p ossible only if the spacetime manifold admits Killing

vectors In general a manifold do es not admit any Killing vectors Thus equations

have no clear physical meaning An interesting discussion of this p ointisgiven in Sachs

and Wu We note that if the energymomentum tensor is nonzero but is lo calized

in the sense that it is negligible away from a given region then the scalar curvature acts

as a measure of the cosmological constant

Duality and Instanton Equations

By setting the energymomentum tensor to zero in Theorem we obtain various

characterizations of the usual gravitational instanton Thus equation reduces to the

instanton equation and equation b ecomes

R J ie RJ JR which is equivalentto R JRJ

Using equations and we obtain

b b b mn c b b b b b

R Rg g H u u E u E E E u u

acmn a a

a a a cmn a a

Equations and imply that

b b

E E and hence E E

a a

Equations and imply that

H

ab

Thus the instanton equations are equivalent to the pair of equations and We

note that the instanton equations are equivalenttothevacuum Einstein equations with

the cosmological term These well known results are sp ecial cases of statementsand

of Theorem and are characteristic of Einstein spaces Equations and

contain as a sp ecial case the vacuum Einstein equations when E or E is zero and

are therefore dualityinvariant On the other hand Einsteins equations with nonzero

energymomentum tensor are not dualityinvariant Thus to include matter elds wenow

consider the generalized instanton equation with the coupling constant k Let

N K DADHICH ETC

us consider the case of p erfect uid In this case the energymomentum tensor is given

by

b b b

T pu u pg

a

a a

Its trace T is then given by

T p p p

Thus equation b ecomes

b b b b

R Rg pu u g

a

a a a

Note that equation reduces to the instanton equation if p We do not need

and p to vanish separately

In general equation can give only the three dierences T T T T T T

between the comp onents of the energymomentum tensor For a p erfect uid this will

determine p and an equation of state p will be necessary to nd and p

separately For example the condition of pressure isotropy will require

E E E E E E

In equation if weput p and R a constant then it reduces to

Einsteins equation with the uid distribution replaced by dust with density p

and acts as the cosmological constant Wenow obtain a solution of equation in

the sp ecial case of the static spherically symmetric metric

ds C r dt A r dr r d sin d

The natural choice for the resolving vector in this case is along the tdirection A direct

calculation leads to the following two equations for the three unknowns A C and

    

A C A A C C

C AC r r A C

 

A C

rA

A C

Given as a function of r we can eliminate C from equations and to obtain

the following nonlinear dierential equation for A

 

A A A

 

rAA r A rA

A A r

A particular solution is given by

constant A r and C

ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS

Here the constant remains undetermined and weget

R and hence R

The particular solution given by corresp onds to dust of uniform density with the

We note that for this solution E and hence the cosmological constant

ab

gravitational eld is entirely supp orted by the passive electric part of the curvature This

is a remarkable prop erty of this spacetime which is an Einstein space The constantpa

rameter of the Einstein metric remains undetermined Thus in the generalized equations

a p erfect uid with the equation of state p constant corresp onds to a uniform

density dust with a cosmological constant

Some Op en problems

It is intriguing and remarkable that the dualityisintimately related to the top ological

defects ie it simply incorp orates a top ological defect global monop ole into the original

eld Thus duality transformation seems to b e a top ological defect generating pro cess In

this context it is imp ortant to note that this pro cess generates zero gravitational charge

a

T T and hence it would continue to share the basic physical prop erties at

a

the Newtonian level with the original vacuum or at spacetime This raises a couple

of interesting questions i Is the top ological defect pro ducing prop erty general or is it

sp ecic only to spherical and axial symmetries ii Would a spacetime with top ological

defect b e always dual to some vacuum including electrovac with cosmological term or

at spacetime ie is it always obtainable as a solution of equations dual to the vacuum

or at spacetime equations This is the case for the Schwarzschild Reissner Nordstrom

Kerr and the NUT solutions

We now discuss a natural generalization of the instanton equations by considering

the metric and connection as indep endentvariables that usually go es under the name of

the Palatini principle Note that the connection is not on an arbitrary principal bundle

but on the bundle of frames LM ofM g

In the Palatini principle has b een used to study the eld equations obtained from

the variation of the action

Z

p

n

S g f gd x

M

where M gisanndimensional pseudoRiemannian manifold with metric g andisa

torsionfree connection on LM The function f is taken as an analytic function of one

variable which is the scalar curvature determined by the pair g If we require the

Lagrangian function f to b e a nonconstant function then it follows that the pair g

satisfy the equations

g

n

These equations must b e considered together with the consistency condition

n



f f

N K DADHICH ETC

We note that the consistency condition is identically satised by the conformally invariant

Lagrangian

n



f

In particular when n and is the LeviCivita connection we obtain the well known

quadratic Lagrangian f R R considered in section The equations contain as a

very sp ecial case the vacuum Einstein equations but are much more general The Palatini

principle has b een used with quadratic Lagrangians f as well as for other Lagrangians

in Following the denition given in we call the pair g an Einstein

pair if it satises the equations The equations can b e generalized further as

follows

g

where is a real scalar eld on M The equations generalize the gravitational

instanton equations which in turn generalize the vacuum Einstein equations

Thus the universal equations obtained from the nonconstant Lagrangian f by

the Palatini principle are not the Einstein equations but the generalized gravitational

instanton equations The equations and as well as equations can b e

regarded as deformations of the usual gravitational instanton equations A discussion of

some prop erties and solutions of these equations is given in It is imp ortanttonote

that the tensor app earing in the equations is not the usual Einstein tensor but is

the tracefree part of the Ricci tensor Denition of the electromagnetic resolution

is not applicable in this case since the Riemann tensor cannot b e considered as a form

with values in the space of forms It would b e interesting to nd a general denition

of the electromagnetic resolution and then dene a generalized duality transformation

Work on these problems is in progress

Acknowledgements

NKD would like to thank Jose Senovilla L K Patel and Ramesh Tikekar for useful

discussions The work of KBM was supp orted in part by the Istituto Nazionale di Fisica

Nucleare sezione Firenze and MaxPlanckInstitute for Mathematics in the Sciences

Leipzig and was carried out during the Fall Fellowship leave from Bro oklyn College

of the City UniversityofNewYork He would like to thank the Dipartimento di Fisica

Universita di Firenze for providing a very pleasantworking environment He would also

like to thank IUCAA Pune for a visiting professorship in January when this work

was b egun

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