Gauge Theory Gravity with GeometricCalculus

David Hestenes1

Department of Physics and AstronomyArizona State University, Tempe, Arizona 85287-1504

A new gauge theory of gravity on at spacetime has recently been developedby Lasenby, Doran, and Gull. Einsteins principles of equivalence and generalrelativity are replaced by gauge principles asserting, respectively, local rota-tion and global displacement gauge invariance. A new unitary formulation ofEinsteins tensor illuminates long-standing problems with energy-momentumconservation in general relativity. Geometric calculus provides many simpli-cations and fresh insights in theoretical formulation and physical applicationsof the theory.

I. Introduction

More than a decade before the advent of Einsteins general theory of relativity(GR), and after a lengthy and profound analysis of the relation between physicsand geometry [1], Henri Poincare concluded that:

One geometry cannot be more true than another; it can onlybe more convenient. Now, Euclidean geometry is and will remain,the most convenient. . . . What we call a straight line in astronomyis simply the path of a ray of light. If, therefore, we were to discovernegative parallaxes, or to prove that all parallaxes are higher than acertain limit, we should have a choice between two conclusions: wecould give up Euclidean geometry, or modify the laws of optics, andsuppose that light is not rigorously propagated in a straight line. Itis needless to add that every one would look upon this solution asthe more advantageous. [italics added]

Applied to GR, this amounts to claiming that any curved space formulation ofphysics can be replaced by an equivalent and simpler at space formulation.Ironically, the curved space formulation has been preferred by nearly everyonesince the inception of GR, and many attempts at alternative at space formula-tions have failed to exhibit the simplicity anticipated by Poincare. One wondersif the trend might have been dierent if Poincare were still alive to promote hisview when GR made its spectacular appearance on the scene.

A dramatic new twist on the physics-geometry connection has been intro-duced by Cambridge physicists Lasenby, Doran, and Gull with their at space

1electronic mail:hestenes@asu.edu

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alternative to GR called gauge theory gravity (GTG) [2, 3]. With geometric cal-culus (GC) as an essential tool, they clarify the foundations of GR and providemany examples of computational simplications in the at space gauge theory.All this amounts to compelling evidence that Poincare was right in the rstplace [4].

The chief innovation of GTG is a displacement gauge principle that assertsglobal homogeneity of spacetime. This is a new kind of gauge principle thatclaries and revitalizes Einsteins general relativity principle by cleanly sep-arating arbitrary coordinate transformations from physically signicant eldtransformations. GTG also posits a rotation gauge principle of standard typein nonAbelian gauge theory. The present account is unique in interpretingthis principle as a realization of Einsteins Equivalence Principle asserting localisotropy of spacetime. With rotation gauge equivalence and displacement gaugeequivalence combined, GTG synthesizes Einsteins principles of equivalence andgeneral relativity into a new general principle of gauge equivalence.

A unique consequence of GTG, announced here for the rst time, is theexistence of a displacement gauge invariant energy-momentum conservation law.

Many previous attempts to formulate a at space alternative to GR are awk-ward and unconvincing. The most noteworthy alternative is the self-consistenteld theory of a massless spin-2 particle (the graviton) [5]. That is a gaugetheory with local Lorentz rotations as gauge transformations. However, it lacksthe simplicity brought by the Displacement Gauge Principle in GTG.

GTG is also unique in its use of geometric calculus (GC). It is, of course,possible to reformulate GTG in terms of more standard mathematics, but at theloss of much of the theorys elegance and simplicity. Indeed, GC is so uniquelysuited to GTG it is doubtful that the theory would have developed without it.That should be evident in the details that follow.

This is the third in a series of articles promoting geometric algebra (GA)as a unied mathematical language for physics [6, 7]. Here, GA is extendedto a geometric calculus (GC) that includes the tools of dierential geometryneeded for Einsteins theory of general relativity (GR) on at spacetime. Mypurpose is to demonstrate the unique geometrical insight and computationalpower that GC brings to GR, and to introduce mathematical tools that areready for use in teaching and research. This provides the last essential piece fora comprehensive geometric algebra and calculus expressly designed to serve thepurposes of theoretical physics [8, 9].

The preceding article [7] (hereafter referred to as GA2) is a preferred prereq-uisite, but to make the present article reasonably self-contained, a summary ofessential concepts, notations and results from GA2 is included. Of course, priorfamiliarity with standard treatments of GR will be helpful as well. However, forstudents who have mastered GA2, the present article may serve as a suitableentree to GR. As a comprehensive treatment of GR is not possible here, it willbe necessary for students to coordinate study of this article with one of themany ne textbooks on GR. In a course for undergraduates, the textbook bydInverno [10] would be a good choice for that purpose. In my experience, thechallenge of reformulating GR in terms of GC is a great stimulus to student

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learning.As emphasized in GA2, one great advantage of adopting GC as a unied

language for physics is that it eliminates unnecessary language and conceptualbarriers between classical, quantum and relativistic physics. I submit that thesimplications introduced by GC are essential to tting an adequate treatmentof GR into the undergraduate physics curriculum. It is about time for blackholes and cosmology to be incorporated into the standard curriculum. Althoughspace limitations preclude addressing such topics here, the mathematical toolsintroduced are sucient to treat any topic with GC, and more details are givenin a recent book [2].

Recognition that GR should be formulated as a gauge theory has been along time coming, and, though it is often discussed in the research literature[11], it is still relegated to a subtopic in most GR textbooks, in part becausethe standard covariant tensor formalism is not well suited to gauge theory.Still less is it recognized that there is a connection between gravitational gaugetransformations and Einsteins Principle of Equivalence. Gauge theory is theone strong conceptual link between GR and quantum mechanics, if only becauseit is essential for incorporating the Dirac equation into GR. This is sucientreason to bring gauge theory to the fore in the formulation of GR.

This article demonstrates that GC is conceptually and computationally idealfor a gauge theory approach to GR conceptually ideal, because concepts ofvector and spinor are integrated by the geometric product in its mathematicalfoundations computationally ideal, because computations can be done with-out coordinates. Much of this article is devoted to demonstrating the eciencyof GC in computations. The GC approach is pedagogically ecient as well, asit develops GR by a straightforward generalization of Special Relativity usingmathematical tools already well developed in GA2.

Essential mathematical tools introduced in GA2 are summarized in SectionII, though mastery of GA2 may be necessary to understand the more subtleaspects of the theory. Section III extends the tool kit to unique mathematicaltools for linear algebra and induced transformations on manifolds. These toolsare indispensible for GTG and useful throughout the rest of physics.

Section IV introduces the gauge principles of GTG and shows how they gen-erate an induced geometry on spacetime that is mathematically equivalent tothe Riemannian geometry of GR. Some facility with GC is needed to appreci-ate how it streamlines the formulation, analysis and application of dierentialgeometry, so the more subtle derivations have been relegated to appendices.

Section V discusses the formulation of eld equations and equations of mo-tion in GTG. Besides standard results of GR, it includes a straightforwardextension of spinor methods in GA2 to treat gravitational precession and grav-itational interactions in the Dirac equation.

Section VI discusses simplications that GTG brings to the formulation andanalysis of solutions to the gravitational eld equations, in particular, motionin the eld of a black hole.

Section VII introduces a new split of the Einstein tensor to produce a newdenition for the energy-momentum tensor and a general energy-momentum

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conservation law.Section VIII summarizes the basic principles of GTG and discusses their

status in comparison with Newtons Laws as universal principles for physics.

II. Spacetime Algebra Background

This section reviews and extends concepts, notations and results from GA2 thatwill be applied and generalized in this paper. Our primary mathematical toolis the the real Spacetime Algebra (STA). One great advantage of STA is thatit enables coordinate-free representation and computation of physical systemsand processes. Another is that it incorporates the spinors of quantum me-chanics along with the tensors (in coordinate-free form) of classical eld theory.Some results from GA2 are summarized below without proof, and the readeris encouraged to consult GA2 for more details. Other results from GA2 willbe recalled as needed. The reader should note that most of GA2 carries overwithout change; the essential dierences all come from a generalized gauge con-cept of dierentiation. We show in subsequent sections that the gauge covariantderivative accommodates curved-space geometry in at space to achieve a at-space gauge theory of gravity. However, the accomodation is subtle, so it wasdiscovered only recently.

A. STA elements and operations

For physicists familiar with the Dirac matrix algebra, the quickest approachto STA is by reinterpreting the Dirac matrices as an orthonormal basis {; =0, 1, 2, 3} for a 4D real Minkowski vector space V4 with signature specied bythe rules:

20 = 1 and 21 =

22 =

23 = 1 . (1)

Note that the scalar 1 in these equations would be replaced by the identitymatrix if the were Dirac matrices. Thus, (1) is no mere shorthand for matrixequations but a dening relation of vectors to scalars that encodes spacetimesignature in algebraic form.

The frame {} generates an associative geometric algebra that is isomorphicto the Dirac algebra. The product of two vectors is called the geometricproduct. The usual inner product of vectors is dened by

12 ( + ) = , (2)where = 2 is the signature indicator. The outer product

12 ( ) = , (3)denes a new entity called a bivector (or 2-vector), which can be interpreted asa directed plane segment representing the plane containing the two vectors.

STA is the geometric algebra G4 = G(V4) generated by V4. A full basis forthe algebra is given by the set:

4

1 {} { } {i} i1 scalar 4 vectors 6 bivectors 4 trivectors 1 pseudoscalargrade 0 grade 1 grade 2 grade 3 grade 4

where the unit pseudoscalar

i 0123 (4)

squares to 1, anticommutes with all odd grade elements and commutes witheven grade elements. Thus, G4 is a linear space of dimension 1+4+6+4+1 =24 = 16.

A generic element of G4 is called a multivector. Any multivector can beexpressed as a linear combination of the basis elements. For example, a bivectorF has the expansion

F = 12F , (5)

with its scalar components F in the usual tensorial form. For components,we use the usual tensor algebra conventions for raising and lowering indices andsumming over repeated upper and lower index pairs.

Any multivector M can be written in the expanded forms

M = + a + F + bi + i =4

k=0

Mk, (6)

where = M0 and are scalars, a = M1 and b are vectors, and F = M2is a bivector, while bi = M3 is a trivector (or pseudovector) and i = M4 isa pseudoscalar. It is often convenient to drop the subscript on the scalar partso M = M0. The scalar part behaves like the trace in matrix algebra; forexample, MN = NM for arbitrary multivectors M and N . A multivectoris said to be even if the grades of its nonvanishing components are all even. Theeven multivectors compose the even subalgebra of G4, which is, of course, closedunder the geometric product.

Coordinate-free computations are facilitated by various denitions. The op-eration of reversion reverses the order in a product of vectors, so for vectorsa, b, c it is dened by

(abc)= cba . (7)It follows for any multivector M in the expanded form (6) that the reverse Mis given by

M = + a F bi + i . (8)

Computations are also facilitated by dening various products in terms of thefundamental geometric product.

5

The inner and outer products of vectors, (2) and (3) can be generalized toarbitrary multivectors as follows. For A = Ar and B = Bs of grades r, s 0,inner and outer products can be dened by

A B AB|rs|, A B ABr+s . (9)Coordinate-free manipulations are facilitated by a system of identities involvinginner and outer products [8]. These identities generalize to arbitrary dimen-sions the well-known identities for dot and cross products in ordinary 3D vectoralgebra. Only a few of the most commonly used identities are listed here. Fora vector a, the geometric product is related to inner and outer products by

aB = a B + a B . (10)For vectors a, b, c the most commonly used identity is

a (b c) = (a b)c (a c)b = a b c a c b . (11)This is a special case of

a (b B) = a bB b (a B) (12)for grade(B) 2, and we have a related identity

a (b B) = (a b) B . (13)We also need the commutator product

AB 12 (AB BA). (14)

This is especially useful when A is a bivector. Then we have the identity

AB = A B + AB + A B, (15)which should be compared with (10). For s = grade(B), it is easy to prove thatthe three terms on the right side of (15) have grades s 2, s, s+ 2 respectively.

Note the dropping of parentheses on the right hand side of (11). To reducethe number of parentheses in an expression, we sometimes use a precedence con-vention, which allows that inner, outer and commutator products take prece-dence over the geometric product in ambiguous expressions.

Finally, we note that the concept of duality, which appears throughout math-ematics, has a very simple realization in GA. The dual of any multivector issimply obtained by multiplying it by the pseudoscalar i (or, sometimes, by ascalar multiple thereof). Thus, in (6) the trivector bi is the dual of the vectorb. Inner and outer products are related by the duality identities

a (Bi) = (a B)i , a (Bi) = (a B)i . (16)

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B. Lorentz rotations and rotors

A complete treatment of Lorentz transformations is given in GA2. We areconcerned here only with transformations continuously connected to the identitycalled Lorentz rotations.

Every Lorentz rotation R has an explicit algebraic representation in thecanonical form

R(a) = RaR , (17)

where R is an even multivector called a rotor, which is subject to the normal-ization condition

RR = 1 . (18)

The rotors form a multiplicative group called the rotor group, which is a double-valued representation of the Lorentz rotation group, also called the spin groupor SU(2). The underbar serves to indicate that R is a linear operator. Anoverbar is used to designate its adjoint R . For a rotation, the adjoint is alsothe inverse; thus,

R (a) = R aR . (19)

A rotor is a special kind of spinor as useful in classical physics as in quantummechanics.

For a particle history parametrized by proper time , the particle velocityv = v() is a unit vector, so it can be obtained at any time from the xedunit timelike vector 0 by a Lorentz rotation

v = R0R , (20)

where R = R() is a one-parameter family of rotors. It follows from (18) thatR must satisfy the rotor equation of motion

R = 12R , (21)

where the overdot indicates dierentiation with respect to proper time and = () is a bivector-valued function, which can be interpreted as a generalizedrotational velocity. It follows by dierentiating (20) that the equation of motionfor the particle velocity must have the form

v = v . (22)The dynamics of particle motion is therefore completely determined by ().For example, for a classical particle with mass m and charge q in an electro-magnetic eld F , the dynamics is specied by

=q

mF , (23)

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which gives the standard Lorentz Force when inserted into (22). One of ourobjectives in this paper is to ascertain what should be for a particle subjectto a gravitational force.

One major advantage of the rotor equation (20) for the velocity is that itgeneralizes immediately to an orthonormal frame comoving with the velocity:

e = RR , (24)

where, of course, v = e0. Solution of the rotor equation of motion (21) thereforegives precession of the comoving frame along with the velocity vector. In SectionVII we adapt this approach to gyroscope precession in a gravitational eld.

C. Vector derivatives, dierential forms and eld equations

Geometric calculus is the extension of a geometric algebra like STA to in-clude dierentiation and integration. The fundamental dierential operator ingeometric calculus is the vector derivative. Although the vector derivative canbe dened in a coordinate-free way [9, 8], the quickest approach is to use thereaders prior knowledge about partial dierentiation.

For a vector variable a = a dened on V4, the vector derivative can begiven the operator denition

a =

a. (25)

This can be used to evaluate the following specic derivatives where K = Kkis a constant k-vector:

a a K = kK , (26)

a a K = (n k)K , (27)

a Ka = K = (1)k(n 2k)K , (28)where n is the dimension of the vector space (n = 4 for spacetime).

In special relativity the location of an event is designated by a vector x inV4. For the derivative with respect to a spacetime point x we usually use thespecial symbol

x . (29)This agrees with the standard symbol for the gradient of a scalar eld = (x). Moreover, the same symbol is used for the vector derivative of anymultivector eld. It is most helpful to consider a specic example.

Let F = F (x) be an electromagnetic eld. This is a bivector eld, withstandard tensor components given by (5). The vector derivative enables us toexpress Maxwells equation in the compact coordinate-free form

F = J , (30)

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where the vector eld J = J(x) is the charge current density.The vector derivative can be separated into two parts by using the the dif-

ferential identity

F = F + F , (31)which is an easy consequence of the identity (10). The terms on the right-handside of (29) are called, respectively, the divergence and curl of F . Since F isa vector and F is a trivector, we can separate vector and trivector parts of(29) to get two Maxwells equations:

F = J , (32)

F = 0 . (33)Although these separate equations have distinct applications, the single equation(30) is easier to solve for given source J and boundary conditions [9].

Something needs to be said here about dierential forms, though we shallnot need them in this paper, because we will not be doing surface integrals.Dierential forms have become increasingly popular in theoretical physics andespecially in general relativity since they were strongly advocated by Misner,Thorne and Wheeler [12]. The following brief discussion is intended to convincethe reader that all the aordances of dierential forms are present in geometriccalculus and to enable translation from one language to the other.

A dierential k-form is a scalar-valued linear function, which can be giventhe explicit form

= dkx K , (34)where K = K(x) is a k-vector eld and dkx is a k-vector-valued volume element.In this representation, Cartans exterior derivative d is given by

d = dk+1x (K) . (35)Thus, the exterior derivative of a k-form is equivalent to the curl of a k-vector.Dierential forms are often used to cast the Fundamental Theorem of IntegralCalculus (also known as the generalized Stokes Theorem) in the form

Rd =

R

. (36)

By virtue of (34) and (35), this theorem applies to a dierentiable k-vector eldK = K(x) dened on a (k + 1)-dimensional surface R with a k-dimensionalboundary R.

Now it is easy to translate Maxwells equations from STA into dierentialforms. The electromagnetic eld F and current J become a 2-form and a 1-form:

= d2x F , = J dx . (37)

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Dual forms can be dened by

d2x (Fi) , d3x (Ji) . (38)The duality identity (16) implies the duality of divergence and curl:

( F )i = (Fi). (39)Consequently, the Maxwell equations (32) and (33) are equivalent to the equa-tions

d = , (40)

d = 0 . (41)

The standard formalism of dierential forms does not allow us to combine thesetwo equations into a single equation like F = J . This is related to a more se-rious limitation of the dierential forms formalism: It is not suitable for dealingwith spinors.

As we are interested to see how quantum mechanics generalizes to curvedspacetime, we record without proof the essential results from GA2, which shouldbe consulted for a detailed explanation. To be specic, we consider the electronas the quintessential fermion.

The electron wave function is a real spinor eld = (x), an even multi-vector in the real STA. For the electron with charge e and mass m, the eldequation for is the real Dirac equation

21h eA = m0 , (42)

where A = A is the electromagnetic vector potential. This coordinate-freeversion of the Dirac equation has a number of remarkable properties, beginningwith the fact that it is formulated entirely in the real STA and so implies thatcomplex numbers in the standard matrix version of the Dirac equation have ageometric interpretation. Indeed, the unit imaginary is explicitly identied in(38) with the spacelike bivector 21, which does square to minus one. Thevector derivative = will be recognized as the famous dierential oper-ator introduced by Dirac, except that the are vectors rather than matrices.Our reformulation of Diracs operator as a vector derivative shows that it is thefundamental dierential operator for all of spacetime physics, not just quan-tum physics. The present paper shows that adapting this operator is the mainproblem in adapting quantum mechanics to curved spacetime.

Physical interpretation of the Dirac equation depends on the specicationof observables, which are bilinear functions of the wave function. The Diracwave function determines an orthonormal frame eld of local observables

= e , (43)

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where = (x) is a scalar probability density and

e = RR , (44)

where R = R(x) is a rotor eld. The vector eld 0 = e0 is the Diracprobability current, which doubles as a charge current when multiplied by thecharge e. The vector eld e3 = R3R species the local direction of electronspin. The vector elds e1 and e2 specify the local phase of the electron, ande2e1 = R21R relates the unit imaginary in the Dirac equation to electronspin. A full understanding of this point requires more explanation than can begiven here. Details are given in GA2.

Finally, note the strong similarity between the spinor frame eld (40) andthe comoving frame (24) for a classical particle. This provides a common groundfor describing spin precession in both classical and quantum mechanics.

III. Mathematical Tools

This section introduces powerful mathematical tools and theorems of generalutility throughout physics, though the treatment here is limited to results neededfor gauge gravity theory. It is particularly noteworthy that these tools enablelinear algebra and dierential transformations without matrices or coordinates.

A. Linear Algebra

Within geometric calculus (GC), linear algebra is the theory of linear vector-valued functions of a vector variable. GC makes it possible to perform coordinate-free computations in linear algebra without resorting to matrices, as demon-strated in the basic concepts, notations and theorems reviewed below. Linearalgebra is a large subject, so we restrict our attention to the essentials neededfor gravitation theory. A more extensive treatment of linear algebra with GC isgiven elsewhere [13, 8, 14] as well as [2].

Though our approach works for vector spaces of any dimension, we will beconcerned only with linear transformations of Minkowski space. To begin, weneed a notation that clearly distinguishes linear operators and their productsfrom vectors and their products. Accordingly, we distinguish symbols represent-ing a linear transformation (or operator) by axing them with an underbar (oroverbar). Thus, for a linear operator f acting on a vector a, we write

fa = f(a) . (45)

As usual in linear algebra, the parenthesis around the argument of f can beincluded or omitted, either for emphasis or to remove ambiguity.

Every linear transformation f on Minkowski space has a unique extensionto a linear function on the whole STA, called the outermorphism of f becauseit preserves outer products. It is convenient to use the same notation f for theoutermorphism and the operator that induces it, distinguishing them when

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necessary by their arguments. The outermorphism is dened by the property

f(A B) = (fA) (fB) (46)

for arbitrary multivectors A, B, and

f = (47)

for any scalar . It follows that, for any factoring A = a1 a2 . . . ar of anr-vector A into vectors,

f(A) = (fa1) (fa2) . . . (far) . (48)

This relation can be used to compute the outermorphism directly from theinducing linear operator.

Since the outermorphism preserves the outer product, it is grade preserving,that is

fM k = fM k (49)

for any multivector M . This implies that f alters the pseudoscalar i only by ascalar multiple. Indeed

f(i) = (det f)i or det f = if(i) , (50)

which denes the determinant of f . Note that the outermorphism makes itpossible to dene (and evaluate) the determinant without introducing a basisor matrices.

The product of two linear transformations, expressed by

h = gf (51)

applies also to their outermorphisms. In other words, the outermorphism ofa product equals the product of outermorphisms. It follows immediately from(51) that

det( gf) = (det g)(det f) , (52)

from which many other properties of the determinant follow, such as

det(f1) = (det f)1 (53)

whenever f1 exists.Every linear transformation f has an adjoint transformation f which can

be extended to an outermorphism denoted by the same symbol. The adjointoutermorphism can be dened in terms of f by

MfN = N fM , (54)

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where M and N are arbitrary multivectors and the bracket, as usual, indicatesscalar part. For vectors a, b this can be written

b f (a) = a f(b) . (55)Dierentiating with respect to b we obtain [15],

f (a) = ba f(b) . (56)

This is the most useful formula for obtaining f from f . Indeed, it might wellbe taken as the preferred denition of f .

Unlike the outer product, the inner product is not generally preserved byoutermorphisms. However, it obeys the fundamental transformation law

f [f(A) B] = A f (B) (57)for (grade A) (grade B). Of course, this law also holds with an interchangeof overbar and underbar. If f is invertible, it can be written in the form

f [A B] = f1(A) f (B) . (58)For B = i, since A i = Ai, this immediately gives the general formula for theinverse outermorphism:

f1A = [f (Ai)](f i)1 = (det f)1f (Ai)i1 . (59)

This relation shows explicitly the double duality inherent in computing the in-verse of a linear transformation, but not at all obvious in the matrix formulation.

B. Transformations and Covariants

This section describes the apparatus of geometric calculus [8] for handlingtransformations of spacetime and corresponding induced transformations of mul-tivector elds on spacetime. We concentrate on transformations (or mappings)of 4-dimensional regions, including the whole of spacetime, but our apparatusapplies with only minor adjustments to mapping vector manifolds of any di-mension including submanifolds in spacetime. It also applies to curved as wellas at manifolds and allows the target of a mapping to be a dierent manifold.As a matter of course, we assume whatever dierentiability is required for per-forming indicated operations. Accordingly, all transformations are presumed tobe smooth and invertible unless otherwise indicated.

In the next section we model spacetime as a vector manifoldM4 = {x} withtangent space V4(x) at each spacetime point x. Tangent vectors in V4(x) aredierentiably attached to M4 in two distinct ways: rst, as tangent to a curve;second as gradient of a scalar. Thus, a smooth curve x = x() through a pointx has tangent

dx

d= lim

01

{x( + ) x()}. (60)

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A scalar eld = (x) has a gradient determined by the vector derivative

= x(x) = x(x). (61)

The two kinds of derivative are related by the chain rule:

d(x())d

=dx

dx(x). (62)

By the way, we usually write = x for the derivative with respect to aspacetime point and a for the derivative with repect to a vector variable ain V4(x). The vector-valued gradient is commonly called a covector todistinguish its kind from the vector dx/d. The dierence in kind is manifestin the dierent ways they transform, as we see next.

Let f be a smooth mapping (i.e. dieomorphism) that transforms each pointx in some region of spacetime into another point x, as expressed by

f : x x = f(x) . (63)

The mapping (63) induces a linear transformation of tangent vectors at x totangent vectors at x, given by the dierential

f : a a = f(a) = a f . (64)More explicitly, it determines the transformation of a vector eld a = a(x) intoa vector eld

a = a(x) f [a(x); x] = f [a(f1(x)); f1(x) ] . (65)

The outermorphism of f determines an induced transformation of speciedmultivector elds. In particular, the induced transformation of the pseudoscalargives

f(i) = Jf i, where Jf = det f = ifi (66)

is known as the Jacobian of f .The transformation f also induces an adjoint transformation f which takes

covectors at x back to covectors at x, as dened by

f : b b = f (b) `f` b = xf(x) b . (67)More explicitly, for covector elds

f : b(x) b(x) = f [ b(x); x ] = f [ b(f(x)); x ] . (68)

As in (55), the dierential is related to the adjoint by

b f(a) = a f (b) . (69)

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Since the induced transformations for vectors and covectors are linear trans-formations, they generalize as outermorphisms to dierential and adjoint trans-formations of the entire tangent algebra at each point of the manifold. Accordingto (59), therefore, the outermorphism f determines the inverse transformation

f1(a) = f (ai)(Jf i)1 = a . (70)

Also, however,

f1(a) = a xf1(x) . (71)Hence, the inverse of the dierential equals the dierential of the inverse.

Since the adjoint maps backward instead of forward, it is often conve-nient to deal with its inverse

f1 : b(x) b(x) = f1 [ b(f1(x)) ] . (72)

This has the advantage of being directly comparable to f . Note that it is notnecessary to distinguish between f1 and f 1.

f

xa

x' = f (x)f_

f_

f (a) = a'_

f (b' ) = b_

.

.

b'

Fig. 1. A dierentiable transformation f of points induces a dier-ential f and its adjoint f transforming tangent vectors between thepoints.

To summarize, we have two kinds of induced transformations for multivectorelds: the dierential f and the adjoint f , as shown in Fig. 1. Multivector eldstransformed by f are commonly said to be contravariant or to transform like avector, while elds transformed by f are said to be covariant or to transformlike a covector. The term vector is thus associated with the dierential whilecovector is associated with the adjoint. This linking of the vector concept toa transformation law is axiomatic in ordinary tensor calculus, because vectorsare dened in terms of coordinates. In geometric calculus, however, the two

15

concepts are separate, because the algebraic concept of vector is determinedby the axioms of geometric algebra without reference to any coordinates ortransformations.

As we have just seen, one way to assign a transformation law to a vector ora vector eld is by relating it to dierentiation. Another way is by the rule ofdirect substitution: A eld F = F (x) is transformed to F (f1(x)) = F (x) or,equivalently, to

F (x) F (f(x)) = F (x) . (73)

Thus, the values of the eld are unchanged although they are associated withdierent points by changing the functional form of the eld.

Directional derivatives of the two dierent functions in (73) are related bythe chain rule:

a F = a xF (f(x)) = (a xf(x)) xF (x) = f(a) F = a F .(74)

The chain rule is more simply expressed as an operator identity

a = a f () = f(a) = a . (75)Dierentiation with a yields the general transformation law for the vectorderivative:

= f () or = f 1() . (76)

This is the chain rule for the vector derivative, the most basic form of the chainrule for dierentiation on vector manifolds. All properties of induced transfor-mations are essentially implications of this rule, including the transformationlaw for the dierential, as (75) shows.

Sometimes it is convenient to use a subscript notation for the dierential:

fa f(a) = a f. (77)Then the second dierential of the transformation (63) can be written

fab b `a f` = b [f(a)] [f(b a)] , (78)and we note that

fab = fba. (79)

This symmetry is equivalent to the fact that the adjoint function has vanishingcurl. Thus, using = aa we prove

` f` (a) = b fb(a) = b cfcb a = f a = 0 . (80)

16

The transformation rule for the curl of a covector eld a = f (a) is therefore

a = f (a) = f ( a) . (81)

To extend this to multivector elds of any grade, note that the dierential of anoutermorphism is not itself an outermorphism; rather it satises the productrule

fb(A B) = fb(A) f (B) + f (A) fb(B) . (82)

Therefore, it follows from (80) that the curl of the adjoint outermorphism van-ishes, and (81) generalizes to

A = f ( A) or A = f 1(A) , (83)

where A = f (A). Thus, the adjoint outermorphism of the curl is the curl of anoutermorphism.

The transformation rule for the divergence is more complex, but it can bederived from that of the curl by exploiting the duality of inner and outer prod-ucts

a (Ai) = (a A)i (84)and the transformation law (58) relating them. Thus,

f ( (Ai)) = f [ ( A)i ] = f1( A)f (i) .Then, using (83) and (66) we obtain

f (Ai) = [ f1(A)f (i) ] = (JfA)i ,where A = f(A). For the divergence, therefore, we have the transformationrule

A = f(A) = Jf1 f [ (JfA) ] = f [ A + ( lnJf ) A ] , (85)This formula can be separated into two parts:

` f` (A) = f [ ( ln Jf ) A ] = ( ln Jf ) f(A) , (86)

` f(A`) = f( A) . (87)The whole can be recovered from the parts by using the following generalizationof (75) (which can also be derived from (58)):

f(A) = f(A ) . (88)

17

IV. Induced Geometry on Flat Spacetime

Every real entity has a denite location in space and time this is the funda-mental criterion for existence assumed by every scientic theory. In Einsteinsrelativity theories the spacetime of real physical entities is a 4-dimensional con-tinuum modeled mathematically by a 4D dierentiable manifoldM4. The stan-dard formulation of general relativity (GR) employs a curved space model ofspacetime, which places points and vector elds in dierent spaces. In contrast,our at space model of spacetime identies the spacetime manifold M4 withthe Minkowski vector space of special relativity. A spacetime map describes aspatiotemporal partial ordering of physical events by representing the events aspoints (vectors) in M4 = {x}. All other properties of physical entities are rep-resented by elds on spacetime with values in the tangent algebra V4(x) and thegeometric algebra G(V4) that it generates. For this reason it is worthwhile tomaintain a distinction between V4 and M4 even though they are algebraicallyidentical vector spaces.

To incorporate gravity into at space theory we represent metrical relationsamong events by elds in V4 rather than intrinsic properties ofM4 as in curvedspace theory. The following subsections explain how to do that in a systematicway that is easily compared to Einsteins curved space theory.

A. Displacement Gauge Invariance

A given ordering of events can be represented by a map in many dierentways, just as the surface of the earth can be represented by Mercator projection,stereographic projection or many other equivalent maps. As the physical worldis independent of the way we construct our maps, we seek a physical theorywhich is equally independent. The Cambridge group has formulated this ideaas a new kind of gauge principle, which can be expressed as follows:

Displacement Gauge Principle (DGP): The equations of physicsmust be invariant under arbitrary smooth remappings of events ontospacetime.

To give this principle a precise mathematical formulation, from here on weinterpret the transformation (63) as a smooth remapping of at spacetime M4onto itself. It will be convenient to cast the direct substitution transformation(73) of a eld F = F (x) in the alternative form

F (x) F (x) = F (f(x)) . (89)

This simple transformation law, assumed to hold for all physical elds and eldequations, is the mathematical formulation of the DGP. The descriptive termdisplacement is justied by the fact that (89) describes displacement of theeld from point x to point x without changing its values. The DGP shouldbe recognized as a vast generalization of translational invariance in specialrelativity, so it has a comparable physical interpretation. Accordingly, the DGPcan be interpreted as asserting that spacetime is globally homogeneous. In

18

other words, with respect to the equations of physics all spacetime points areequivalent. Thus, the DGP has implications for measurement, as it establishesa means for comparing eld congurations at dierent places and times.

The Cambridge group had the brilliant insight that enforcement of the trans-formation law (89) requires introduction of a new physical eld that can be iden-tied with the gravitational eld. Properties of this eld, called the gauge eldor gauge tensor, derive from the transformation laws of the preceding section.The most essential step is dening a gauge invariant derivative. To that end,consider a gauge invariant scalar eld = (x) transformed by (89); accordingto (76) the transformation of its gradient is given by

(x) = (f(x)) = f [(x)] . (90)To make this equation gauge invariant, we introduce an invertible tensor eldh dened on covectors so that

h[((x)] = h[((x)] . (91)Comparison with (90) shows that h must be a linear operator on covectors withthe transformation law

h = hf 1 . (92)

More explicitly, when operating on a covector b = the rule ish(b;x) = h(f 1(b;x);x) . (93)

We usually suppress the position dependence and write

h(b) = h(f 1(b)) . (94)

Applying this rule to the vector derivative with its transformation rule (76),we can dene a position gauge invariant derivative by

h() . (95)The symbol is a convenient abuse of notation to remind us that the linearoperator h is involved in its denition.

From the operator we obtain a position gauge invariant directional deriva-tive

a = a h() = (ha) , (96)where h is the adjoint of h and a is a free vector, which is to say that it canbe regarded as constant or as a vector transforming by the direct substitutionrule (89). This is claried by a specic example.

Let x = x() be a timelike curve representing a particle history. Accordingto (64), the dieomorphism (63) induces the transformation

x =dx

d x = f(x.) . (97)

19

Thus the description of particle velocity by x. is contravariant under spacetimedieomorphisms. Comparing (62) with (96), it is evident that an invariantvelocity vector v = v(x()) can be introduced by writing

x = h(v) , (98)

where h obeys the adjoint of the gauge transformation rule (92), that is

f1h = h or h = f h . (99)

Accordingly, (97) implies that

x = h(v) , (100)

where v has the same value as in (98), but it is taken as a function of x()instead of x(). To distinguish the x from the invariant velocity

v = h1(x) = h1(x) , (101)

one could refer to x as the map velocity. Otherwise the term velocity desig-nates v. The invariant normalization

v2 = 1 (102)

xes the scale on the parameter , which can therefore be interpreted as propertime.

For any eld F = F (x()) dened on a particle history x(), the chain rulegives the operator relation

d

d= x = h(v) = v h() = v , (103)

so that

dF

d= v F . (104)

For F (x) = x, this gives

dx

d= v x = (hv) x = hv , (105)

recovering (98).Since h and h are mutually determined, we can refer to either as the gauge

tensor or, simply, the gauge on spacetime. Actually, the term gauge ismore appropriate here than elsewhere in physics, because h does indeed deter-mine the gauging of a metric on spacetime. To see that, use (97) in (101)to derive the following expression for the invariant line element on a timelikeparticle history:

d2 = [ h1(dx) ]2 = dx g(dx) (106)

20

where

g = h1h1 (107)

is a symmetric metric tensor. This formulation suggests that gauge is a morefundamental geometric entity than metric, and that view is conrmed by devel-opments below. Einstein has taught us to interpret the metric tensor physicallyas a gravitational potential, making it easy and natural to transfer this inter-pretation to the gauge tensor. Some readers will recognize h as equivalent to atetrad eld, which has been proposed before to represent gravitational eldson at spacetime [16]. However, the spacetime calculus makes all the dierencein turning the tetrad into a practical tool.

To verify that (106) is equivalent to the standard invariant line element ingeneral relativity, coordinates must be introduced. Let x = x(x0, x1, x2, x3)be a parametrization of the points, in some spacetime region, by an arbitraryset of coordinates {x}. Partial derivatives then give tangent vectors to thecoordinate curves x, which, in direct analogy to (98) and (101), determine aset of displacement gauge invariant vector elds {g} according to the equation

e x = xx

= h(g) , (108)

or, equivalently, by

g = h1(e) . (109)

The components for this coordinate system are then given by

g = g g = e g(e) . (110)Therefore, with dx = dxe, the line element (106) can be put in the form

d2 = gdxdx , (111)

which is familiar from the tensor formulation of GR.To complete the introduction of coordinates into the at space gauge theory,

we introduce coordinate functions x = x(x) and the coordinate frame vectors

e = x (112)

reciprocal to the frame {e} dened by (108); whence

e e = e x = x = . (113)The displacement gauge invariant frame {g} reciprocal to (109) is then givenby

g = h(e) = x . (114)

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So the gauge invariant derivative is given by

= h() = g . (115)

The at space and curved space theories can now be compared throughtheir use of coordinates. The vector elds g encode all the information in themetric tensor g = g g of GR, as is evident in (110). However, equations(111) and (114) decouple coordinate dependence of the metric, expressed bye = x, from the physical dependence, expressed by the gauge tensor h or itsadjointh. In other words, the remapping of events in spacetime is completelydecoupled from changes in coordinates in the gauge theory, whereas the curvedspace theory has no means to separate passive coordinate changes from shiftsin physical congurations. This crucial fact is the reason why the DisplacementGauge Principle has physical consequences, whereas Einsteins general relativityprinciple does not.

The precise theoretical status of Einsteins general relativity principle (GRP),also known as the general covariance principle, has been a matter of perennialconfusion and dispute that some say originated in confusion on Einsteins part[17]. One of Einsteins main motivations in formulating it was epistemologi-cal: to extend the special relativity principle, requiring physical equivalence ofall inertial frames, to physical equivalence of arbitrary reference frames. Heformulated the GRP as requiring covariance of the equations of physics un-der the group of arbitrary coordinate transformations (also known as generalcovariance). While acknowledging criticisms that the requirement of generalcovariance is devoid of physical content, Einstein continued to believe in theGRP as a cornerstone of his gravitation theory.

Gauge theory throws new light on the GRP, which may be regarded as vin-dicating Einsteins obstinate stance. The problem with Einsteins GRP is thatit is not a true symmetry principle [17]. For a transformation group to be aphysical symmetry group, there must be a well dened geometric object thatthe group leaves invariant. Thus the displacement group of the DGP is a sym-metry group, because it leaves the at spacetime background invariant. That iswhy it can have the physical consequences described above. There is no com-parable symmetry group for curved spacetime, because each mapping producesa new spacetime, so there is no geometric object to be left invariant. Moreover,general covariance does not discriminate between passive coordinate transfor-mations and active physical transformations. Those distinctions are generallymade on an ad hoc basis in applications. The intuitive idea of physically equiv-alent situations was surely at the back of Einsteins mind when he insisted onheuristic signicance of the GPR over the objections of Kretschmann and otherson its lack of physical content. I submit that the DGP provides, at last, a pre-cise mathematical formulation for the kind of GRP that Einstein was lookingfor. Once again I am impressed by Einsteins profound physical insight, whichserved him so well in assessing the signicance of mathematical equations inphysics. Of course, his conclusions depended critically on the mathematics athis disposal, and displacement gauge theory was not an option available to him.

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B. Rotation Gauge Covariance

Gauge theory gravity requires one other gauge principle, which we formulateas follows:

Rotation Gauge Principle (RGP): The equations of physics mustbe covariant under local Lorentz rotations.

Generalizing the treatment in Section IIIB, we characterize a local Lorentz ro-tation by a position dependent rotor eld L = L(x) with LL = 1. This enablesus to dene two kinds of covariant elds, a multivector eld M = M(x) and aspinor eld = (x), with respective transformation laws:

L : M M = L M LML , (116)

L : = L . (117)Spinors and multivectors are related by the fact that the spinor eld deter-mines a frame of multivector observables {}, where {} is the standardorthonormal frame of constant vectors introduced in equation (1). Frame inde-pendence of the gauge tensor is ensured by (116), which implies that the gaugetensor satises the operator transformation law

L : h h = L h . (118)From (114) it follows that this transformation leaves the metric tensor g =g g invariant.

To construct eld equations that satisfy the RGP, we dene a gauge co-variant derivative or coderivative operator D as follows. With respect to acoordinate frame {g = h1(e)}, components D = g D of the coderivativeare dened for spinors and multivectors respectively by

D = ( + 12) , (119)

DM = M + M , (120)where the connexion for the derivative

= (g) (121)

is a bivector-valued tensor evaluated on the frame {g}. From Section IA,we know that the bivector property of implies that M is gradepreserving. Hence D is a scalar dierential operator in the sense thatgrade(DM) =grade(M). Note that D is not grade-preserving on the spinorin (119).

To ensure the covariant transformations for the derivatives

L : D L(D) = D = ( + 12) , (122)

23

L : DM L (DM) = DM = M + M , (123)

the connexion must obey the familiar gauge theory transformation law

= = LL 2(L)L , (124)

where it should be noted that the operator = g is rotation gauge in-variant. The full vector coderivative D can now be dened by the operatorequation

D = gD , (125)

for which gauge covariance follows from (118) and the denition of D. SinceD preserves the grade of M in (120), the coderivative is a vector dierentialoperator, so we can decompose it in the same way that we decomposed thevector derivative into divergence and curl in equation (31). Thus, for anarbitrary covariant multivector eld M = M(x), we can write

DM = D M + D M , (126)where D M and D M are respectively the codivergence and the cocurl.

Combining (118) with (92), we see that the most general transformation ofthe gauge tensor is given by

h = L hf 1, h = f hL . (127)

In other words, every gauge transformation on spacetime is a combination ofdisplacement and local rotation. This operator equation enables us to turn anymultivector eld on spacetime into a gauge covariant eld, as shown explicitlyin the next section for the electromagnetic eld. Finally, it should be notedthat all the above rotation covariant quantities, including the connexion , aredisplacement gauge invariant as well.

In special relativity, Lorentz transformations are passive rotations express-ing equivalence of physics with respect to dierent inertial reference frames.Here, however, covariance under active rotations expresses local physical equiv-alence of dierent directions in spacetime. In other words, the rotation gaugeprinciple asserts that spacetime is locally isotropic. Thus, passive equivalenceis an equivalence of observers, while active equivalence is an equivalence ofstates. This distinction generalizes to the physical interpretation of any sym-metry group principle: Active transformations relate equivalent physical states;passive transformations relate equivalent observers.

To establish the connection of GTG to GR, we need to relate the coderivativeto the usual covariant derivative in GR. Like the directional derivative =g , the directional coderivative D = g D is a scalar dierential operatorthat maps vectors into vectors. Accordingly, we can write

Dg = Lg , (128)

24

which merely expresses the derivative as a linear combination of basis vectors.This denes the so-called coecients of connexion L for the frame {g}. Bydierentiating g g = , we nd the complementary equation

Dg = Lg . (129)

When the coecients of connexion are known functions, the coderivative of anymultivector eld is determined.

Thus, for any vector eld a = ag we have

Da = (Da)g + a(Dg).

Then, since the a are scalars, we get

Da = (a + aL)g . (130)

Note that the coecient in parenthesis on the right is the standard expressionfor a covariant derivative in tensor calculus.

Further properties of the connection are obtained by contracting (129) withg to obtain

D g = g gL . (131)This bivector-valued quantity is called torsion. In the Riemannian geometry ofGR torsion vanishes, so we restrict our attention to that case. Considering theantisymmetry of the outer product on the right side of (131), we see that thetorsion vanishes if and only if

L = L . (132)

This can be related to the metric tensor by considering

Dg = g = (Dg) g + g (Dg) ,whence

g = gL + gL . (133)

Combining three copies of this equation with permuted free indices, we solve for

L =12g

(g + g g) . (134)This is the classical Christoel formula for a Riemannian connexion, so the de-sired relation of the coderivative to the covariant derivative in GR is established.

C. Curvature and Coderivative

In standard tensor analysis the curvature tensor is obtained from the commu-tator of covariant derivatives. Likewise, we obtain it here from the commutatorof coderivatives. Dierentiating (129) we get

[D, D ]g = Rg , (135)

25

where the operator commutator has the usual denition

[D, D ] DD DD , (136)

and

R = L L + LL LL , (137)

is the usual tensor expression for the Riemannian curvature of the manifold.This suces to establish mathematical equivalence of our at space gauge theoryto the standard tensor formulation of Riemannian geometry and hence to GR.Now we can condently turn to the full gauge theory treatment of curvatureand gravitation to see what advantages it has over standard GR.

The commutator of coderivatives dened by (121) gives us

[D, D ]M = R(g g)M , (138)

where

R(g g) = + = D D

(139)

is the curvature tensor expressed as a bivector-valued function of a bivectorvariable. This is the fundamental form for curvature in GTG. It follows from(123) that the curvature bivector has the covariance property

L : R(g g) R(g g) = L R(L (g g)), (140)

where L = L1.For the gravity eld variables g, equations (138) and (135) give us

[D, D ]g = g = R g ; (141)whence

R = (g g). (142)Coordinate frame elds {g} and {g} have been used to introduce the

coderivative and curvature in order to make connection with the standard ten-sor formalism as direct as possible. However, the vector coderivative and thecurvature bivector can be dened and all their properties can be derived ina completely coordinate-free way. Derivations are given in Appendix B, andresults are summarized in Table 1.

26

Curvature

R(a b) a (b) b (a) + (a) (b) ([a, b])[a D`, b D` ]M` = R(a b)M = [a D, b D ]M ([a, b])[a, b] a b b a = a Db b DaR(g g) = + [D, D ]g = R(g g) g = R gR(a b) = abR(g g) A R(B) = B R(A)Curvature Contractions and Coderivative Identities

R(a) b R(b a) = bR(b a) a R(a b) = 0R a R(a) = aR(a) a R(a) = 0D Da = (D D) a = R(a) (D D) M = D (D M) = 0D D M = 0 D DM = (D D)MBianchi Identity and its Contractions

D` R`(a b) = 0 a D`R`(b c) + b D`R`(c a) + c D`R`(a b) = 0R`(D` b) = D` R`(b) R`(D`) = 12DRG(a) R(a) 12aR G`(D`) = 0

Table 1. Coderivative and Curvature. All derivatives are rotationgauge covariant if the elds are. However, coderivatives are well-denedeven for quantities that are not covariant. The elds a and b are vector-valued, while A and B are bivectors, and the multivector M must havegrade(M) 2 in the double codivergence identity. Note that accents areemployed in some equations to indicate which quantities are dierentiated.

V. Field Equations and Particle Motion

With the dierential geometry of gauge elds well in hand, we are ready toapply it to gravity theory.

A. Einsteins Equation

Einsteins gravitational eld equation can be created from contractions ofthe curvature tensor in the usual way. Contraction of the curvature bivector(139) gives the usual Ricci tensor expressed as a vector-valued function of avector variable.

R(g) = g R(g g) , (143)

27

With help from the identity (13), a second contraction gives the usual scalarcurvature

R g R(g) = (g g) R(g g) . (144)The gravitational eld equation can thus be written in the form

G(g) R(g) 12g R = T (g) , (145)where = 8 in natural units with the gravitational constant set equal tounity. The vector-valued function of a vector variable G(a) is Einsteins tensor,and its source T (a) is the total energy-momentum tensor for material particlesand non-gravitational elds. Of course, decomposition into tensor componentsG = g G(g) and T = g T (g) yields Einsteins equation in its standardtensor component form.

The spacetime algebra reviewed in Section I enables us to express Einsteinstensor G = G(g) in the new unitary form:

G = 12 (g g g) = 12 (g ) (g g) . (146)

The last equality is a consequence of the well-known symmetry of the curvaturetensor,

(g g) R(g g) = (g g) R(g g) . (147)Identities (12) and (13) can be used to expand the inner product in (146) to get

G = g g 12g(g g) , (148)which will be recognized as equivalent to the standard form (145) for the Einsteintensor. Thus, expansion of the inner product has split the unitary Einsteintensor into two parts. Let us refer to this as the Ricci split of the Einsteintensor.

As is well known, Einstein originally chose the particular combination ofRicci tensor and scalar curvature in (145) because, as shown in Appendix B andTable 1, its codivergence G`(D`) vanishes in consequence of the Bianchi identity.Using D(gg) = 0 from appendix A, we can write this version of the Bianchiidentity in the form

D[gG(g)] = gG`(D`) = 0 . (149)

It follows, then, from Einsteins eld equation (145) that

D[gT (g)] = gT` (D`) = 0 . (150)

Initially, Einstein wanted to interpret this as a general energy-momentum con-servation law. However, that interpretation is not so straightforward, because atrue conservation law requires vanishing of an ordinary divergence rather thana codivergence. Instead, we shall see in Section VC that (150) determines the

28

equation of motion for matter. Then in Section VII we see that an alternativeto the Ricci split is more suitable for deriving the general energy-momentumconservation law that Einstein wanted.

B. Electrodynamics with Gravity

GTG diers from GR in providing explicit equations for the inuence ofgravity on physical elds. This is best illustrated by electrodynamics. Maxwellsequation F = J with F = A is not gravitation gauge covariant. To makethe vector eld A gauge covariant and incorporate the eect of gravity, we simplywrite [15]

A = h(A) , (151)

and gauge covariance is assured by (127). In fact, the inuence of gravity onthe electromagnetic eld is fully characterized by the gauge tensor h.

Now applying (240) from Appendix A to (151), we immediately obtain agauge covariant expression for the electromagnetic eld strength:

F = h(F ) = D A = h(A ) . (152)

Similarly, applying the cocurl again and using Table 1, we obtain

D F = D D A = h(A ) = 0 . (153)

Thus, Maxwells equation F = 0 applies irrespective of the gravitationaleld. Now it is obvious that the gauge covariant form of Maxwells equationmust be

DF = D F + D F = J . (154)From the vector part of this equation, we have, using the double codivergenceidentity in Table 1,

D (D F ) = 0 = D J , (155)A little more analysis is needed to relate the right side of this equation to thestandard charge current conservation law.

Explicit dependence of the codivergence D F on the gauge tensor can bederived from (152) by exploiting duality. The derivation is essentially the sameas that for (85) and yields the result

D F = hh1[ (h1h(F ))] , (156)

where h det(h). For the vector eld J the result is a scalar, and we obtainthe simpler equation

D J = h (h1h(J )) , (157)

29

so comparison with (155) identies J = h1h(J ) with the conserved chargecurrent. Combining this with (156), we can write D F = J in the equivalentform

(h1hh(F )) = J . (158)This enables us to interpret the operator h1hh = h1 g1 as a kind of dielectrictensor or generalized index of refraction for the gravitational medium [18]. Ofcourse, equation (158) can be used to compute refraction of light by the sun,with a result in agreement with alternative GR calculations.

The standard Maxwell energy-momentum tensor (as given in GA2) is easilygeneralized to the gauge covariant form:

TEM (a) = 12F aF (159)Using (153) to compute its coderivative , we obtain

T`EM (D`) = 12 (F` D`F + F DF ) = 12 (J F F J ) . (160)In other words,

D[gTEM (g)] = gT`EM (D`) = gF J . (161)One way to assure mutual consistency of Einsteins eld equation with Maxwellsequation is to derive them from a common Lagrangian. This has been doneby the Cambridge group in a novel way with GC, and the method has beenextended to spinor elds with provocative new results. The reader is referredto the literature for details [2].

C. Equations of Motion from Einsteins Equation

In classical electrodynamics Maxwells eld equation and Lorentzs equationof motion for a charged particle must be postulated separately. It came asquite a surprise, therefore, to discover that in GR the equation of motion fora material particle can be derived from Einsteins eld equation, provided it isassumed to hold everywhere, including at any singularities in the gravitationaleld. This discovery was made independently by several people, though mostof the credit is unfairly given to Einstein [19]. Here we sketch essential elementsin one especially simple approach to deriving equations of motion to illustrateadvantages of using GA. The form of the equation of motion is determined bythe form of the energy-momentum tensor. The idea is to describe a localizedmaterial system as a point particle by a multipole expansion of the mass distri-bution. Then the problem is to reformulate the result as a dierential equationfor the particle.

We consider only the simplest case of a structureless point particle. For auid of non-interacting material particles, the energy-momentum tensor can beput in the form

TM (a) = m(a v)v, (162)

30

where is the proper particle density, v is the proper particle velocity and m isthe mass per particle. Its codivergence is

T`M (D`) = T`M (`) + T (g) = mvD (v) + mv Dv. (163)As shown in (157) for a charge current, the codivergence D (v) can be ex-pressed as a true divergence, so its vanishing expresses particle conservation,which we assume holds here. For a single particle, is a -function on the par-ticle history, and it can be eliminated by integration, but we dont need thatstep here.

For a charged particle the total energy-momentum tensor is

T (a) = TM (a) + TEM (a) = ma vv 12F aF . (164)According to (161) and (163), its codivergence is

D[g(TM (g) + TEM (g)] = gmv Dv + gJ F . (165)

For a particle with charge q the current density is J = qv, so the vanishing of(165) gives us

mv Dv qF v = 0. (166)This has the desired form for the equation of motion of a point charge. ForF = 0 it reduces to the geodesic equation, as described in the next Section.

The electromagnetic eld in (166) can be expressed as a sum F = F self +F ext, where F self is the particles own eld and F ext is the eld of other parti-cles. Evaluated on the particle history, F self describes reaction of the particleto radiation it emits. This process has been much studied in the literature [19],but the analysis is too complicated to discuss here. If we neglect radiation, wecan write F = F ext in (166), and qF v is the usual Lorentz force on a testcharge.

D. Particle Motion and Parallel Transfer

The equation for a particle history x = x() generated by a timelike velocityv = v(x()) in the presence of an ambient gauge tensor h is x. = h(v). For anycovariant multivector eld M = M(x()), eqn. (120) gives us the directionalcoderivative

v DM = dd

M + (v)M , (167)

where the gauge invariant proper time derivative is given by (104). Obviously,all this applies to arbitrary dierentiable curves in spacetime if the requirementthat v be timelike is dropped.

The equation for a geodesic can now be written

v Dv = v. + (v) v = 0 (168)

31

With (v) specied by the ambient geometry, this equation can be integratedfor v = h1(x.) and then a second integration gives the geodesic curve x().The solution is facilitated by noting that (168) implies

v v. = 12dv2

d= v (v) v = (v) (v v) = 0 . (169)

Therefore v2 is constant on the curve and, as noted in GA2, the value of v atany point on the curve can be obtained from a timelike reference value v0 by aLorentz rotation. Thus,

v = R v0 = Rv0R , (170)

where R = R(x()) is a rotor eld on the curve satisfying the dierential equa-tion

v DR = dRd

+ 12(v)R = 0 . (171)

This is equivalent to the spinor derivative (119), so the rotor R can be regardedas a special kind of spinor.

More generally, the parallel transport of any xed multivector M0 to a eldM = M(x()) dened on the whole curve is given by

M = RM0 = RM0R . (172)

Of course, it satises the dierential equation

v DM = M + (v)M = 0 . (173)The formal similarity of this formulation for parallel transport to a directionalgauge transformation is obvious, but the two should not be confused.

With the spinor coderivative dened by (171) (and not necessarily vanish-ing), we can rewrite (167) as an operator equation

v D = R dd

R, (174)

or equivalently, as

Rv D = v R = dd

R . (175)

This shows that a coderivative can be locally transformed to an ordinary deriva-tive by a suitable gauge transformation.

According to (166), for a particle with unit mass and charge q in an elec-tromagnetic eld F = h(F ) = D A = h(A), the geodesic equation (168)generalizes to

v. = v (v) + qF v , (176)

32

The rst term on the right can be interpreted as a gravitational force, thoughthe standard GR interpretation regards it as part of force-free geodesic mo-tion. From (246) and (242) in Appendix A, we see that v (v H) = (vv) H =0, so

v (v) = v H(v) = v h [` h`1(v)] , (177)where the accent indicates which quantity is dierentiated. Thus, the gravi-tational force is determined by the bivector-valued function H(v), and we canwrite

v H(v) + qF v = h[ (` h`1(v) +A) ] h1(x.)= h[ (` h`1(v) +A) x. ].

(178)

Finally, on expanding the last term and introducing the metric tensor g denedby (107), we can put the equation of motion (176) in the form

d

d[ g(x.) + qA] = `[ 12x

. g`(x.) + qA` x. ] . (179)

Except for the presence of the metric tensor g, this is identical to the equationof motion for a point charge in Special Relativity, so we can use familiar tech-niques from that domain to solve it (details in Section VII). Ignoring the vectorpotential and solving for

..x , it can be put in the form

..x = g1

[12`x. g`(x.) g(x.)] . (180)

This is equivalent to the standard formulation for the geodesic equation in termsof the Christoel connection. Note how the metric tensor appears when theequation of motion is formulated in terms of a spacetime map rather than incovariant form.

E. Gravitational Precession and Dirac Equation

The solution of equation (171) gives more than the velocity. In perfect anal-ogy with the treatment of electromagnetic spin precession in GA2, it gives usa general method for evaluating gravitational eects on the motion and preces-sion of a spacecraft or satellite, and thus a means for testing gravitation theory.To the particle worldline we attach a (comoving orthonormal frame) or mobile{e = e(x()) = e(); = 0, 1, 2, 3}. The mobile is tied to the velocity byrequiring v = e0. Rotation of the mobile with respect to a xed orthonormalframe {} is described by

e = RR , (181)

where R = R(x()) is a unimodular rotor. This has the same form as equation(24) for a mobile in at spacetime because the background spacetime is at.

33

Applying (167) to (181) we obtain the coderivative of the mobile:

v De = e + (v) e . (182)The coderivative here includes a gauge invariant description of gravitationalforces on the mobile. In accordance with (21) and (23), eects of any nongrav-itational forces can be incorporated by writing

v De = e + (v) e = e , (183)where = (x) is a bivector such as an external electromagnetic eld actingon the mobile. The four equations (183) include the equation of motion

dv

d= ( (v)) v (184)

for the particle, and are equivalent to the single rotor equation

dR

d= 12 ( (v))R . (185)

For = 0 the particle equation becomes the equation for a geodesic, and therotor equation describes parallel transfer of the mobile along the geodesic. Thisequation has been applied to gravitational precession in [20], so there is no needto elaborate here.

Our rotor treatment of classical gravitational precession generalizes directlyto gravitational interactions in quantum mechanics. We saw in (40) that a realDirac spinor eld = (x) determines an orthonormal frame of vector eldse = e(x) = RR . Generalization of the real Dirac equation (38) to includegravitational interaction is obtained simply by replacing the partial derivative by the coderivative D dened by equation (119). Thus, we obtain

gD21h = g( + 12)21h = eA + m0 , (186)

where A = h(A) is the gauge covariant vector potential. This is equivalent tothe standard matrix form of the Dirac equation with gravitational interaction,but it is obviously much simpler in formulation and application. Comparisonof the spinor coderivative (119) with the rotor coderivative (185) tells us imme-diately that gravitational eects on electron motion, including spin precession,are exactly the same as on classical rigid body motion. The Cambridge grouphas studied solutions of the real Dirac equation (186) in the eld of a black holeextensively [21].

VI. Solutions of Einsteins Equation

Though GC facilitates curvature calculations, the nonlinearity of Einsteinsequation still makes it dicult to solve. Gauge theory gravity introduces simpli-cations both in calculation and physical interpretation by cleanly separating

34

the functional form of the gauge tensor from arbitrary choices of coordinatesystem. Using results in the Appendices and Table 1, gravitational curvaturecan be calculated from a given gauge tensor by the straightforward sequence ofsteps:

h(a) ` h`1

(a) (a) R(a b) . (187)

To solve Einsteins equation for a given physical situation, symmetry consider-ations are used to restrict the functional form of the gauge tensor, from whicha functional form for the curvature tensor is derived by (187). This, in turn, isused to simplify the form of Einsteins tensor and reduce Einsteins equation tosolvable form. Finally, gauge freedom is used to analyze alternative forms forthe solution and ascertain the simplest for a given application. Details, framedsomewhat dierently, are given in [2] and other publications by the Cambridgegroup, so they need not be repeated here. We conne our attention to a reviewof results that showcase advantages of the GC formulation and analysis.

A. Static Black Hole

The most fundamental result of GR is the gravitational eld of a point particleat rest called a black hole. To construct a spacetime map of this eld and themotion of particles within it, we represent the velocity of the hole by the constantunit vector 0 and locate the origin on its history. As explained in GA2, thisdenes a preferred reference frame and a spacetime split of each spacetime pointx into designation by a time t = x 0 and a relative position vector x = x 0,and a split of the vector derivative t = 0 and = 0 , as expressed by

x0 = t + x, 0 = t +. (188)

It will also be convenient to write r = |x 0| = |x| so that x = rx, and tointroduce the unit radius vector

er = rx = r = x0. (189)

The Cambridge group has shown that the curvature tensor for a black holecan be written in the wonderfully compact form [2, 3]

R(B) = M2r3

(B + 3xBx), (190)

where M is the mass of the black hole. Considering the local Lorentz rotation ofthe curvature (140), it is evident at once that the curvature tensor is invariantunder a boost in the x = er0 plane, as expressed by

L x = LxL = x. (191)

This simple symmetry of the curvature tensor has long remained unrecognizedin standard tensor formulations of GR.

35

Using the vector derivative identity (28), it is not dicult to verify by directcalculation that

R(b) = `aR(a` b) = 0, (192)so Einsteins tensor G(a) vanishes when r = 0. As is evident in (190), thecurvature is singular at r = 0; this can be shown to arise from a delta-functionin the energy-momentum tensor:

T (a) = 43(x)(a 0)0 . (193)This simple expression for the singularity contrasts with standard GR where theblack hole singularity is not so well dened, and it is often claimed that suchentities as wormholes are allowed [12]. Unfortunately for science ction bus,GTG rules out wormholes by requiring global solutions of Einsteins equation,as explained below.

The standard Schwarzschild solution of Einsteins eld equation for a blackhole is equivalent to the symmetric gauge tensor

hS(a) = (1a 0 + a 0)0, =(1 2M

r

) 12. (194)

Alternative solutions can be generated by gauge transformations that leave thecurvature (190) invariant. Allowed transformations are determined by threeparameters. One parameter is xed by boundary conditions at innity, a sec-ond generates displacements, and the third generates boosts dened by (191).The Cambridge group [3] has found boost-displacements that transform theSchwarzschild gauge (194) into a Kerr-Schild gauge

hKS(a) = a +M

ra (0 er)(0 er), (195)

and a Newtonian gauge

h(a) = a (a 0)er, =(2Mr

) 12. (196)

The former corresponds to such well-known solutions as the Eddington-Finkelsteinmetric, while the latter has hardly been noticed in the GR literature, despite itsconsiderable virtues, which we now expose. It should be noted that the Newto-nian gauge (196) is well-dened for every r = 0, while the Schwarzschild gauge(194) has a well-known singularity at r = 2M . Therefore, the former is a moregeneral solution to Einsteins equation than the latter.

To interpret results of physical measurements in the gravitational eld, wefollow the usual practice of setting up reference systems of imaginary observers.First, consider a stationary observer at a xed distance r from the black hole,so his world line is dened by the map velocity

x. = t

.0, (197)

36

so x = 0. As only gauge covariant quantities are physically signicant, we mustconsider the observers gauge velocity determined by (196):

v = h1x. = t.(0 + er). (198)

Whence the constraint v2 = 1 implies

t.=(1 2M

r

) 12. (199)

This tells us that the stationary observers clock (reading proper time ) runsslow compared to the proper time t of the black hole. But how can we read aclock on the black hole? That is answered by considering a freely-falling observerwith a constant gauge velocity v = 0 equal to that of the black hole, so hismap velocity is

x. = h10 = 0 + er, (200)

which we solve immediately for

t.= x. 0, r. = x. er = =

(2Mr

) 12. (201)

This tells us that the clock of a freely falling observer has the same rate as theproper time of the black hole, and his velocity r. is identical to the infall velocityin Newtonian theory. This justies the name Newtonian gauge introducedabove and suggests referring to time t as Newtonian time.

As our discussion of observers suggests, the Newtonian gauge simplies phys-ical interpretation and analysis of motion in the gravitational eld of a pointparticle. It has the great advantage of showing precisely how GR diers fromNewtonian theory in this case. To compare with standard GR, we need theNewtonian gauge tensor, which from (196) is

g a = h1h1a = a + (a 0 er + a er0) 2a 00. (202)This determines a line element

ds2 = dx g dx = dt2 (dr + dt)2 r2d2. (203)This line element does not appear in any GR textbook. Only Ron Gautreauhas expounded its pedagogical benets in a series of AJP articles [22]. All hisanalysis can be neatly reformulated in Newtonian gauge theory.

Now, to analyze the general motion of a point charge in the Newtoniangauge, we multiply equation (179) by 0 to do a space-time split. Consideringrst the scalar part of the split, for static elds t g = 0 and tA = 0 we ndimmediately the constant of motion

E = ( gx. + qA) 0 = v u + V , (204)

37

where V = qA 0 is the usual electric potential andv u = 0 gx = h1(0) h1(x.) = t. (1 2) r. . (205)

We recognize this as a gauge invariant generalization of energy conservation inspecial relativity. The time derivative can be eliminated from this expressionwith the constraint

1 = v2 = (x. + t.er)2 = (1 2)t.2 2t.r. x2 . (206)

As our analysis to this point is sucient to show how electromagnetic interac-tions can be included, let us omit them from the rest of our analysis to con-centrate on the distinctive features of the gravitational interaction. Introducingthe angular momentum (per unit mass)

L = x x = r2x x (207)so we can write

x2 = r.2 +L2

r2(208)

with L2 = L2, we eliminate t. between (204) and (206) to get an equationgoverning radial motion of the particle:

E2 = (1 2)(1 +

L2

r2

)+ r.2. (209)

A pedagogically excellent analysis of the physical implications this equation andrelated issues in elementary black hole physics is given by Taylor and Wheeler[23]. Their treatment could be further simplied by adopting the Newtoniangauge.

To complete our analysis of motion in the Newtonian gauge, we return tothe space-time split of equation (179) to consider the bivector part, which givesus the spatial equation of motion

g = ` 12x. g`(x.) , (210)

and, from (202), we get the more explicit expressions

g ( gx.) 0 = x+ t. x (211)

` 12x. g`(x.) = t.

[r. +

r(x r.x) + 12 t

.2]. (212)

Inserting these expressions into (210), we get the equation of motion in theform

x = x , (213)

38

where is a complicated expression that is dicult to simplify. However, allwe need is to recognize that the right side of (213) is a central force, so angularmomentum is conserved. Then, with the help of (208), we obtain

x =(

..r L

2

r3

)x , (214)

and, dierentiating the radial equation (209) to eliminate..r , we get the equation

of motion in the simple explicit form

x = Mr2

x 3ML2

r3x . (215)

Thus, the deviation of GR from the Newtonian gravitational force has beenreduced to the single term on the right side of this equation.

To solve the equation of motion (215), we rst neglect the GR correction, sothe equation has the familiar Newtonian form, although the parameter is propertime rather than Newtonian time. This equation is most elegantly and easilysolved by using (207) to put it in the form

Lx = Mr2

Lx = M x , (216)

which reveals at once the constant of motion

M = LxM x . (217)The solution is now completely determined by the two constants of motion andL, and GA facilitates analysis of all its properties, as demonstrated exhaustivelyelsewhere [24]. The solution is a conic section with eccentricity = ||, and isthe direction of its major axis. For 0 < 1 the solution is an ellipse. Changingthe independent variable from proper time to Newtonian time has the eect ofmaking the ellipse precess. That is a purely kinematic eect of special relativity.

To ascertain the GR eect on orbital precession, we use (217) to write (215)in the form

d

d= 3L3

x Lr3

. (218)

Averaging over an orbital period, we get [2]

=6M

a(1 2)L , (219)

where a is the semimajor axis of the ellipse. For Mercury, this gives the famousGR perihelion advance of 43 arcseconds per century.

B. Other Solutions

The Cambridge group has applied GTG to most of the domains where GRis applied, with many new simplications, insights and results. For example,

39

anyone who has studied the Kerr solution for a rotating black hole should bepleased, if not astounded, to learn that the Kerr curvature tensor can be castin the compact form

R(B) = M2(r iL cos )3 (B + 3xBx), (220)

where L is a constant and is the angle with respect to the axis of rotation. Ithas been known for a long time that the Kerr curvature can be obtained fromthe Schwarzschild curvature by complexifying the coordinates, but (220) showsthat it can be reduced to a substitution for the radius in (190), and the unitimaginary is properly identied as the unit pseudoscalar, so the substitutioncan be interpreted as a duality rotation-scaling of the Schwarzschild curvature.GTG provides other insights into Kerr geometry, but so far they have not beensucient to solve the important problem of matching the Kerr solution to theboundary of a rotating star.

The Cambridge group, including a number of graduate students and post-docs, is actively applying GTG to a number of outstanding problems in GR,particularly in cosmology and black hole quantum theory. As this work is con-tinually progressing, interested readers are referred to the Cambridge website[21] for the latest news.

VII. Energy-momentum Conservation

Shortly after publishing his general theory of relativity, Einstein endeavored todene an energy-momentum tensor for the gravitational eld so he could inter-pret his gravitational eld equation as a general equation for energy-momentumconservation [25]. His proposal was immediately confronted with severe di-culties that have remained unresolved to this day despite considerable eortsof many physicists. The general consensus today is that a unique energy-momentum tensor cannot be dened, so only global energy-momentum con-servation is allowed in GR [26]. However, there are dissenters who maintainthat local energy-momentum conservation is a fundamental experimental fact,so it must be and can be incorporated in GR [27]. In this section we showhow GTG provides a new approach to the problem of local energy-momentumconservation in GR.

An alternative to the Ricci split (148) is obtained by substituting the curva-ture bivector (139) into the new unitary form for Einsteins tensor (146) to getEinsteins equation in the form

gG(g) = g(g g g) (D 12 ) = gT (g) = T , (221)

where g = |det(g g)|12 = |det(g)|

12 is themetric density, so T gT (g)

is a tensor density. Next, we dene an energy-momentum superpotential by

U g(g g g) , (222)

40

and, using the following identity derived from an identity at the end of AppendixA

D[g(g g g)] = 0 , (223)

we write

g(g g g) D = DU = U + U . (224)Inserting this into (221) we obtain a split of Einsteins equation in the followingform:

1gG(g) = T t = T , (225)

where

t U + 121g(g g g) ( ) (226)is identied as the energy-momentum tensor (density) for the graviational eld,so

T U , (227)

can be identied as the total energy-momentum tensor (density). It followsimmediately that we have the true energy-momentum conservation law:

T = (t + T ) = 0 , (228)

because U = U = 0. Because of its evident importance ingiving us a general conservation law, let us refer to (225) with the associateddenitions (226) and (227) as the canonical energy-momentum split.

The most striking and perhaps the most profound feature of the canonicalsplit is the simple linear relation (222) between the the superpotential, whichdetermines energy-momentum density, and the connexion, which determinesgravitational force. It is a kind of duality between a vector-valued function of abivector variable U and a bivector-valued function of a vector variable .The duality relation can be inverted by expanding the right side of (222) to get

U = g(g g g) = 12(g g) I + g(g g) g , (229)where

I 12 (g g) U = g(g U) = 21gg (230)is a frame-independent vector eld. Solving (229) for the connexion bivectorwe obtain

1g = 12 (g g g) U + 12I g . (231)

41

This surprising dual equivalence of the connexion to the superpotential U

can be regarded as a hitherto unsuspected relation of gravitational force toenergy-momentum tensor. Accordingly, it can be expected to provide newphysical insights, but there is insucient space to explore the possibilities inthis paper. A more complete analysis of the canonical split, including its re-lation to alternative splits by Einstein and others, will be published elsewhere[28].

Physically, the most important question about the canonical split is theuniqueness of the energy-momentum tensor dened by (227). In other words,to what degree is energy-momentum localizable? We have already noted thatthe connexion = (g) is invariant under the position gauge transformations.Hence, the superpotential and all other elements of the canonical split are gaugeinvariant. This completely solves Einsteins problem of dening a coordinateindependent energy-momentum tensor.

In contrast to the Ricci split (148), however, the canonical split is not ro-tation gauge covariant. According to (124), the connexion transforms as apseudotensor under local rotations. Since the superpotential U is directlyproportional to and the energy-momentum tensor is a gradient thereof, thesequantities depend on the choice of the local rotation gauge, which is not uniquelydened by the theory. However, local energy-momentum conservation is notnecessarily destroyed by this gauge variability, because the local gravitationalforce retains its duality to local energy-momentum density. To make the pointexplicit, consider the geodesic equation for a particle with invariant velocityv = h1(x):

v Dv = v + (v) v = 0. (232)For a given gauge, the term (v) v can be regarded as the gravitational forceon the particle. This force changes with a change in gauge, but the superpoten-tial and energy-momentum tensor change in the same way. Therefore, the localrelation of force to energy-momentum is preserved. In other words, with respectto local energy-momentum exchange, all gauges are equivalent. In this sense,energy-momentum is localizable whatever gauge is chosen. However, much moreanalysis will be needed to conrm this line of reasoning.

VIII. Spacetime Modeling Games

Every scientic theory is characterized by general principles, often ill-denedor unstated, that dene the domain and structure of the theory. Accordingly,an essential step in evaluating a theory is articulating its dening principles.Einstein taught us that models of spacetime underlie every physical theory,so that is where we begin. To facilitate comparison with alternative theories,especially Einsteins general relativity (GR), I have outlined the basic principlesof gauge theory gravity (GTG) as I see it in Table 2. Some commentary isneeded to understand the design and intent of the table.

42

In the early stages of GTG development the Cambridge group was keen toshow how it diers from GR, particularly in the treatment of black holes [18].I cannot speak for their current opinions, but my own view is that GTG is notfundamentally dierent from GR. I see it, rather, as simplifying and clarifyingthe structure of GR in a sense, as perfecting the foundations and completingthe development of GR. As a bonus, GTG brings the full power of geometriccalculus to the analysis and solution of problems in the domain of GR, and itgoes beyond GR when torsion is included [29].

As Newtonian mechanics is the original prototype for all physical theories,I have followed Newtons example in formulating a set of Universal Laws thatdene GTG. The Laws are universal in the sense that they apply to all modelsand explanations consistent with the theory, and they are intended to embraceall of physics. This is not to say that the Laws cannot be modied, improvedor extended.

I have argued elsewhere that it takes six Universal Laws to dene Newtonstheory completely [30, 31], so I have constructed analogs of those six Laws forGTG in Table 2. This stimulates an enlightening comparison of Newtonian the-ory with GR and GTG, giving us a new