Remarks Concerning the Geometries of Gravity And

Home , Solder form

ct ys- tum avity e the bridge ysics. bia ometry. vitational en gr hrift some ara (Cam etwe w estsc een gauge elds timately related y somewhat un- e-time ge w eb esh onov-Bohm e e ction, and not the v ac y the fact that the enc har avitational phase for onne etom v , gauge elds, and quan y en the A t here for his F asic di er vit thega an and C. V. Vish able for an op en path, unlik etwe y of South Carolina, Colum .Ry h I had with Charles Misner on funda- gy b ates a b ersit terference, and their implications to ph h can b e visualized, and is in estigation. , Univ v y Abstract v-Bohm (AB) e ect and parallel transp ort on tum in e of the solder form for the sp e the solder form and the c y B. L. Hu, M. P y and gauge eld. Then I shall illustrate, in section one. It il lustr evious suggestion that the fundamental variables for een them that arises due to the existence of the solder b er the encouragemen UGE FIELDS vit w h as the geometry of gra edited b esapr e some basic remarks ab out the similarities and di erences ap ers in honor of Charles Misner. c y explain the tremendous usefulness of geometry in ph ort on a c AN ysics, suc elativity, einfor ysics and Astronom ol. 1, P ansp h came out of this in avitational eld ar al R , using the Aharono y t di erence b et e is further shown by the observability of the gr s. This ma . It therefore seems appropriate to presen vit View metadata,citationandsimilarpapersatcore.ac.uk al lel tr tofPh enc y from their e ects on quan ODUCTION ar In section 4, I shall further illustrate this di erence b x attempts to understand the similarities and di erences b et ortant limitation is shown in the analo vit 〉 VITY AND GA A ANAND aths. This r ations whic PostScript processed by the SLAC/DESY Libraries on 29 Mar 1995. . In particular, I remem een the geometries of gra

GR-QC-9503055 tal asp ects of ph ctions in Gener w vitational phase for a spinless particle is observ e en p n imp a cone. gra AB e ect. This implies that the translational gauge symmetry of the gra 3, an imp ortan form for gra In section 2, Ibet shall mak Geometry is a part ofto mathematics symmetrie whic observ ical geometry ortho do and gra theory metric. 1. INTR I recall with great pleasuremen the discussions whic This di er op quantizing the gr and gauge elds due to the existenc SC 29208, USA A and the p JEEV Departmen Univ. Press, 1993), V REMARKS CONCERNING THE GEOMETRIES OF GRA Dir provided byCERNDocumentServer brought toyouby CORE

eld is broken by the existence of the solder form. It is then argued that the solder

form and the connection are the prop er variables for quantizing the gravitational

eld.

2. LOCALITY OF GRAVITY AND GAUGE FIELDS

Something which Misner emphasized to me during a conversation was the fundamen-

tal role assigned to lo calityby the theory of relativity. Already in sp ecial relativity

lo cality is incorp orated in the fact that signals cannot travel faster than the sp eed of

light. But in general relativity, lo cality plays an even more fundamental role: The

principle of equivalence states that the laws of physics are lo cally Minkowskian. Also,

b ecause space-time is curved, there is no distant parallelism and vectors at two dif-

ferent p oints can only b e compared by parallel transp orting them to a common p oint

with resp ect to the gravitational connection.

These three asp ects of lo cality are also present in gauge elds which are now b eing

used to describ e the three remaining fundamental interactions in physics. The prin-

ciple of equivalence for gauge elds may b e stated as follows: Given any p oint p in

space-time, a gauge can b e chosen so that the corresp onding connection co ecients

or vector p otential vanishes at p for all elds interacting with the given gauge eld.

Also, there is no distant parallelism for vectors parallel transp orted using the gauge

eld connection if the curvature or the Yang-Mills eld strength is non vanishing.

Contrary to what is sometimes said, gravity do es not di er fundamentally from gauge

elds simply b ecause it is asso ciated with a metric. Because if the gauge group is

unitary then it leaves invariant a metric in the vector space at each space-time p oint

that consists of all p ossible values of any matter eld interacting with the gauge eld

at that p oint. The essential di erence is that the gravitational metric can b e used to

measure distances along any curve in space-time, unlike the gauge eld metric. But

I shall shownow, by means of physical arguments, that this fundmental di erence

between gravity and gauge elds exists even prior to intro ducing the metric.

3. AHARONOV-BOHM EFFECT AND PARALLEL TRANSPORTON

A CONE

It is an interesting fact that the phase shifts in quantum interference due to gravity

and gauge elds are obtained in a simple manner from the distance due to the gravita-

tional metric and parallel transp ort due to gravitational and gauge eld connections

along the interfering b eams . Conversely, the phase shifts in quantum interference

can b e used to de ne gauge elds and gravity . This is most easily shown for the

simplest gauge eld, namely the electromagnetic eld, by means of the AB e ect .

We recall that the magnetic AB e ect is the phase shift in the interference of two

coherent electron b eams which enclose a cylinder containing a magnetic ux. In the

interference region, the wave function may b e written as (r;t)+ (r;t), where

1 2 1

and are the wave functions corresp onding to the two b eams. The intro duction of

the magnetic eld inside the cylinder mo di es this wave function to

(r;t)= (r;t)+ F (r;t);

1 2

in an appropriate gauge, where

A dx ): (1) F = exp(

Here the integral is along the curve going around the cylinder, A is the electromag-

netic 4-vector p otential and e is the charge of the electron. Therefore the intensity

distribution j (r;t)j in the interference region is mo di ed in an apparantly non lo cal

wayby the magnetic ux via F even though the magnetic and electric eld strengths

vanish everywhere along the b eams.

But this phenomenon is not surprising when we realize the analogy with the geometry

of a cone . The cone may b e formed by taking a at sheet of pap er b ounded bytwo

straight lines making an angle and identifying the two straight lines (Fig. 1a); we

denote this cone by C .For 0 2 , this is what we do when we make a cone by

rolling this at sheet so that these two lines coincide to form one of the generators of

the cone. Since the pap er is not stretched or compressed during this pro cess, a cone

has no intrinsic curvature except at the ap ex, which can b e smo othed out so that the

curvature is nite there. In the multiply connected geometry around the ap ex, the

intrinsic curvature is zero everywhere, same as the at geometry of the sheet which

was rolled up to b e the cone. In particular, a vector is parallel transp orted likeon

the at sheet. But a vector V parallel transp orted around a closed curve drawn on

the curvature free region of the cone so as to enclose the ap ex undergo es a rotation

by the angle , which is the holonomy transformation asso ciated with this curve.

If the curvature at the ap ex is regarded as analogous to the magnetic eld in the

cylinder then the zero intrinsic curvature everywhere else corresp onds to the vanishing

of magnetic eld strength outside the cylinder. Then V moving in a curvature free

region is analogous to b eams traveling in a eld free region. The rotation by the

angle which relates V and V (Fig. 1a) is analogous to the phase di erence b etween

the two b eams due to F . This suggests that the electromagnetic eld maybea

connection for parallel transp orting the value of the wave function and the AB e ect

arises b ecause a wave function when parallel transp orted around the closed curve ,

gets multiplied by F . The electric and magnetic eld strengths at each space-time

Figure 1

a) Analogy between the Aharonov-Bohm e ect and paral lel transport on a cone. The

cone may be obtained by identifying the lines OA and OA on a at sheet. There-

fore, the vector V paral lel transportedfrom B around the cone would come back to B

0 0

(identi ed with B )asV rotated by the angle . This is analogous to the AB phase

shift with the magnetic eld corresponding to the curvature at the apex of the cone.

b) The limitation of this analogy when is changedto+2 by adding an extra sheet

of paper. The vector paral lel transported along the closed curve BCDEB rotates by

+2 with respect to the tangent vector to the curve. This enables one to distinguish

this cone from the earlier one. This is unlike the AB e ect which cannot distinguish

between two enclosed magnetic uxes that di er by one quantum of ux.

p oint then constitutes the curvature of this connection at this p oint. Thus the phase

factor (1) is the holonomy transformation asso ciated with for this connection. The

statement that the electromagnetic eld is a gauge eld is the same as saying that it

is a connection as describ ed ab ove.

The ab ove mentioned conical geometry describ es the gravitational eld in each section

normal to a long straight string , such as a cosmic string. A gravitational analog of

the AB e ect is obtained if weinterfere two coherent b eams of identical particles with

intrinsic spin around the string. The resulting phase shift due to the cosmic string

is a sp ecial case of the phase shift due to an arbitrary gravitational eld obtained

b efore . Basically, this phase shift consists of two parts, one due to the change in path

lengths of the interfering b eams, and the other due to the holonomy transformation,

which in this case is a rotation undergone by the wave function when it is parallel

transp orted around the interfering b eams. This change in path length and holonomy

transformation, and consequently the phase shifts, o ccur even though the space is

lo cally at.

Now if the AB phase

= A dx

is changed by2, which corresp onds to changing the magnetic ux inside the cylinder

by a \quantum of ux", then (1) is unchanged. Therefore the AB exp eriment or for

that matter any other exp eriment outside the cylinder cannot detect the di erence

between these two magnetic uxes. Hence, Wu and Yang stated that, b ecause of

the AB e ect, the electromagnetic eld strength F has to o little information,

has to o much information, and it is the phase factor or the holonomy transformation

F , for arbitrary closed curves , which has the right amount of information of the

electromagnetic eld. This has b een generalised to an arbitrary connection by the

theorem which states that from the holonomy transformations the connection can b e

reconstructed and it is then unique up to gauge transformations. A simple physical

system to illustrate the Wu- Yang statement is a sup erconducting ring enclosing a

magnetic ux. No exp eriment p erformed in the interior of the ring using Co op er

pairs can distinguish b etween a given enclosed ux and + n , where is the

0 0

quantum of ux for the Co op er pair and n is an integer. For example, if we measure

the ux by inserting a Josephson junction in the ring and observe the Josephson

current, wewould obtain the same current for b oth uxes. Because the AB phases

for the two uxes di er by2n and therefore (1) is the same for b oth uxes, with e

now b eing the charge of the Co op er pair.

An imp ortant and interesting limitation of the analogy of the AB e ect with the

cone emerges when we consider the meaning of increasing the ux of the curvature

in the ap ex region of the cone by \one quantum". The new ux may b e regarded

as corresp onding to the cone C which has one extra sheet of pap er compared

to C . (Toembed C into a three dimensional Euclidean space it needs to b e

twisted in some way but it is well de ned by the identi cation stated ab ove.) The

holonomy transformations are the same for C and C (Fig. 1b). Therefore the

ab ove mentioned theorem implies that the cones C and C are the same as

far as their connections are concerned. Here a connection is regarded simply as a

rule for parallel transp orting abstract vectors attached to p oints on the cone and

not regarded as tangentvectors. Physically, the phase shift arising from spin in an

interference exp eriment which is determined by the holonomy transformation will b e

the same for b oth cones for a b osonic particle. For a fermionic particle, there is a

di erence of between the phase shifts b ecause this phase is acquired by a fermion

when it is rotated by2 radians. Therefore for fermions, C is not equivalentto

C , but is equivalentto C , b ecause of the nature of the spinor connection.

+2 +4

A straightforward application of the Gauss-Bonnet theorem shows that the ux or

integral of the curvature at the smo othed out ap ex of the cone C is 2 . Therefore

this ux is negative when > 2. In the latter case, it follows via Einstein's eld

equations that if C represents the geometry around a cosmic string then the string

has negative mass. In particular, C and C represent geomtries around cosmic

+2 +4

strings with negative mass.

The two cones, which are the same as far as their linear connections are concerned,

are of course, di erent when we takeinto account their metrics. This gives rise to

the phase shift due to changes in the path lengths of the interfering b eams . But

even if we forget their metrics, there is a subtle di erence b etween the two cones. To

see this, for each cone, parallel transp ort a vector around a closed smo oth curve that

encloses the ap ex and do es not intersect itself. This vector rotates with resp ect to

the tangentvector to the curveby the angle for C and by the angle +2 for

C . This di erence, which can b e observed by means of lo cal measurements, arises

ultimately b ecause we identify the vectors b eing parallel transp orted with the tangent

vectors to the cone. The mathematical concept used to make this identi cation in an

arbitrary manifold is called the solder form, or the canonical 1- form, or the canonical

form .For the electromagnetic eld the vector (x;t) b elongs to an internal space

and cannot b e compared with a tangentvector. Therefore the Wu-Yang statement

is valid. But the gravitational eld connection is for parallel transp orting tangent

vectors. Hence there is such an identi cation. This is the most fundamental di erence

between gravity and gauge elds .

If C and C have only the connections or only the solder forms then they are

identical. Since the two connections in the the frame bundles over C and C

have the same holonomy transformations, there exists a b er bundle isomorphism

e e

f between the frame bundles which maps one connection into the other . This f

induces a unique di eomorphism f between the base manifolds C and C in the

obvious way. The di erential f is a map b etween tangentvectors. It determines a

b er bundle isomorphism f between the two frame bundles that maps one solder form

into the other. But f and f are top ologically di erent in the sense that one cannot

b e continuously deformed into the other. This is whywewere able to distinguish

between C and C when the connections and the solder forms are b oth present,

even when the metric is absent.

4. GRAVITATIONAL PHASE FACTOR

The phase shift in quantum interference due to an arbitrary gauge eld, which gen-

2;11

eralizes the AB e ect, is determined by the \phase factor"

F = P exp( A T dx ); (2)

where T generate the Lie algebra of the gauge group, A is the Yang-Mills gauge

p otential, P denotes path ordering, and is a closed curve through the interfering

b eams. Here, F is an element of the gauge group. Its eigenvalues can b e determined

byinterference exp eriments . This shows the real signi cance of (1) as an elementof

the U (1) group, which is a sp ecial case of the gauge group. When is an in nitesimal

closed curve spanning a surface element represented by d ,

F =1+ F T d (3)

2hc

k k k i j

where F = dA gC A ^ A is the Yang-Mills eld strength.

The phase shift in quantum interference of a particle due to the gravitational eld is

determined by

i 1

ab a

F = P exp[ M )dx ]; (4) (e P +

ab a

h 2

which is an element of the Poincare group that may b e asso ciated with any path in

space-time. Here, P and M ;a;b =0;1;2;3 are resp ectively the energy-momentum

a ab

and angular momentum op erators which generate the representation of the Poincare

group corresp onding to the given particle, e is dual to the frame e used by lo cal

observers:

a a

e e = ; (5)

b b

and are the connection co ecients with resp ect to this frame eld. If the lo cal

observers use orthonormal frames then

e e g = ; (6)

a b

where g is the space-time metric and are the co ecients of the Minkowski metric.

When the particle has non zero intrinsic spin then the values of the wave function are

what are observed by observers using the frame eld e . Then (4) implies that the

spinor eld is parallel transp orted in addition to a phase that it acquires due to its

energy-momentum. This is obtained in the WKB approximation, disregarding here

for simplicity a real factor which do es not contribute to the phase .

For the sp ecial case of a spinless particle, M = 0, the gravitational phase acquired

by a lo cally plane wave is, to a go o d approximation,

e p ; (7) =

where p are the eigenvalues of the energy-momentum op erators P and the integral

a a

. A remarkable feature of (7) is that it is observable is along the classical tra jectory

for an op en curve unlike the phase shifts for gauge elds which can b e observed

only for closed curves. For example, (7) may b e observed by the Josephson e ect

for a path across the Josephson junction ,orby the oscillation of strangeness in the

Kaon system for an op en time-like path along the Kaon b eam . Both these phases

dep end on the geometry of space- time as determined by the gravitational eld.

To understand this di erence b etween gravity and gauge elds note that the eld

a a

e plays three roles here: First, comparing (4) with (2) suggests that e is likea

connection or gauge p otential asso ciated with the translation group. Indeed, e

and may b e regarded as constituting the connection in the ane bundle. The

curvature of this connection is obtained byevaluating (4) for an in nitesimal closed

curve :

i 1

a ab

F =1+ (Q P + R M )d ; (8)

a ab

2h 2

a a a b ab

using the Poincare Lie algebra, where Q = de + ^e is the torsion and R =

ab a cb

d + ^ is the curvature. Eq. (8) is the analog of (3) for gravity. This is the most

physical way that I know to regard gravity as the gauge eld of the Poincare group.

Second, e represents the solder form referred to earlier. In geometrical language, it

in the bundle is the pullback of the solder form with resp ect to the lo cal section e

of frames, which follows from (5). The (Lie-algebra valued) 1-form e P acts on the

tangentvector to to give an element in the Lie algebra of the translational group,

which is also an observable in the Hilb ert space. When this observable acts on a WKB

wave function it gives as an approximate eigenvalue the rate of change of phase along

. When this is integrated along the phase (7) is obtained. Third, (5) and (6)

imply that e is like the square ro ot of the metric:

a b

e e = g : (9)

In the rst role, there is no restriction on the values of e at any given p oint in space-

time. Indeed e may b e made to vanish by an appropriate choice of gauge along any

di erentiable curve that do es not intersect itself. The \gravitational eld" then has

the full gauge symmetry of the ane group A(4;R), i. e. the group of inhomogeneous

linear transformations on a four dimensional real vector space. The holonomy group

is a subgroup of the Poincare group which enables only the generators of the Poincare

Lie algebra to o ccur in (4). The corresp onding \gravitational phase", like the AB

phase, would then b e meaningful only for a closed curve .

However, in the second role, the matrix e is restricted to b e non singular. The

gauge symmetry group is reduced to the general linear group GL(4;R) A(4;R),

with b eing the connection or gauge eld. It is the breaking of the translational

gauge symmetry that enables the phase (7) to b e observable. The solder form is

the canonical 1-form de ned on the frame bundle whose structure group is GL(4;R).

The discussion of the parallel transp ort of a vector around a cone in section 3 shows

the imp ortant role played by the solder form which makes this theory richer than a

gauge theory with GL(4;R) as the internal symmetry. The e now transforms as a

tensor, instead of a connection, under lo cal gauge transformations which corresp onds

to space-time dep endent transformations of the frame eld. Therefore the phase (7)

is invariant under these gauge transformations for an op en curve , as it should b e

b ecause it is observable.

Despite the breaking of the translational gauge invariance, the torsion which app ears

in (8) as the curvature corresp onding to this group nevertheless arises naturally from

aphysical p oint of view. Because the motion of the amplitude of a spinor wave

function provides an op erational de nition of the connection which is indep endent

2;15

of the Christo el connection that comes from the metric . Therefore, the connec-

tion, in a co ordinate basis, can b e non symmetric and the torsion is then twice the

antisymmetric part of this connection. Hence the burden of pro of is on gravitational

theories with zero torsion to justify this constraint and not on torsion theories to

justify intro ducing torsion, b ecause kinematically torsion arises naturally whenever

there are elds with intrinsic spin as seen ab ove. But it is not necessary to intro duce

a metric in the rst two roles of e discussed here.

In the third role, the sp eci cation of the metric, which is the same as sp ecifying

the orthonormal frame eld e , breaks the gauge symmetry further to the Lorentz

group O (3; 1;R) GL(4;R) that leaves this metric invariant. But from the observed

3;15

phases (7), the metric may b e constructed . Therefore it do es not app ear to b e as

fundamental a physical variable as the solder form or connection. From an op erational

p oint of view, the motion of a quantum system in a gravitational eld is in uenced

directly by the solder form and connection, and the metric seems to arise only as

a secondary construct. Therefore in the reaction of the quantum system on the

gravitational eld, which needs to b e describ ed by quantum gravity, the solder form

and the connection would b e the fundamental dynamical variables that are a ected.

It was therefore prop osed that in quantizing the gravitational eld the variables e

ab 15

and should b e quantized and not the metric . The arguments in this pap er which

show further the imp ortant role played by the solder form reinforce this view. The

imp ortant role assigned to the vector p otential by the AB e ect nds its counterpart

in quantum electro dynamics in which it is the vector p otential which is quantized.

Similarly, the quantum e ects discussed ab ove which dep end on the gravitational

a ab

phase factor (4) suggest that the variables e and should b e quantized in order

to obtain quantum gravito dynamics. It is noteworthy that (4) is an element of the

Poincare group, even though the curvature of space-time classically breaks Poincare

invariance. This is analogous to the phase factor (2) for gauge elds b eing an element

of the corresp onding gauge group. This role of groups in the fundamental interactions,

together with the general role of symmetry in quantum physics, whichismuch more

fundamental, substantive and determinative than in classical physics, suggest that

the way forward in physics at the present time should p erhaps b e guided by the

precept 'symmetry is destiny'.

Acknowledgements

I thank P. O. Mazur, R. Penrose and R. Howard for useful remarks. This work was

partially supp orted by NSF grant no. PHY-8807812.

REFERENCES

1. C. N. Yang and R. L. Mills, Phys. Rev. 96 (1954) 191.

2. J. Anandan, Nuov. Cim. 53A (1979) 221.

3. J. Anandan, Phys. Rev. D 33 (1986) 2280; J. Anandan in Topological Properties and

Global StructureofSpace-Time, eds. P. G. Bergmann and V. De Sabbata (Plenum

Press, NY 1985), p. 1-14.

4. Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485.

5. J. S. Dowker, Nuov. Cim. 52B (1967) 129.

6. L. Marder, Pro c. Roy. So c. A 252 (1959) 45; in Recent Devolopments in General

Relativity (Pergamon, New York 1962).

7. T. T. Wu and C. N. Yang, Phys. Rev. D, 33 (1986) 2280.

8. J. Anandan in Conference on Di erential Geometric Methods in Physics, edited by

G. Denardo and H. D. Do ebner (World Scienti c, Singap ore, 1983) p. 211.

9. S. Kobayashi and K. Nomizu, Foundations of Di erential Geometry (John Wiley,

New York 1963) p. 118.