Weyl Geometry, This Scale Invariance Becomes Local

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General relativity is naturally invariant under global changes of units. By formulating general relativity in an integrable Weyl geometry, this scale invariance becomes local. We refer to this generalization as scale invariant general relativity. As soon as we make a definition of a fundamental standard of length – for example, as the distance light travels in one second1 – scale invariant general relativity reduces to general relativity. 2. In [7], Ehlers, Pirani and Schild make the following argument. First, the paths of light pulses may be used to determine a conformal connection on spacetime. Then, a projective connection is found by tracing trajectories of massive test particles (“dust”). Finally, requiring the two connections to approach one another in the limit of near-light velocities reduces the possible connection to that of a Weyl geometry. When this program is carried out with mathematical precision [8], the resulting geometry is an integrable Weyl geometry. 3. Deeper physical interest in Weyl geometry also arises in higher symmetry approaches to gravity. Gravi- tational theories based on the full conformal group ([1],[9]-[19]) often yield general relativity formulated on an integrable Weyl geometry rather than on a Riemannian one and are therefore equivalent to general relativity while providing additional natural structures. For these reasons, it is useful to recognize the typical forms and meaning of the connection and curvature of Weyl geometry. Here we use the techniques of modern gauge theory [20, 21] to develop the properties of Weyl geometry, beginning in the next section with a review of the conformal properties of Riemannian spacetimes. Decompo- sition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime, and generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, in the final section, we enlarge the symmetry from Poincaré to Weyl to develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor, and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We conclude with a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that its vacuum solutions are conformal equivalence classes of Ricci-flat metrics in an integrable Weyl geometry. This theory is invariant under dilatations, but not the full conformal group.

2 Conformal transformations in Riemannian geometry 2.1 Structure equations for Riemannian geometry The Cartan structure equations of a Riemannian geometry are

Ra = dαa αc αa (1) b b − b ∧ c 0 = dea eb αa (2) − ∧ b ea ad µ a where the solder form, = eµ x , provides an orthonormal basis, α b is the spin connection 1-form, and Ra 1 a ec ed the curvature 2-form is b = 2 R bcd . Diﬀerential forms are written in boldface. The structure equations satisfy integrability∧ conditions, the Bianchi identities, found by applying the

1The second is currently defined as the duration of 9, 192, 631, 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom [physics.nist.gov]. The metre is defined as the 1 length of the path travelled by light in a vacuum in 299792458 second [17th General Conference on Weights and Measures (1983), Resolution 1].

2 Poincaré lemma, d2 0: ≡ 0 d2αa = d (αc αa + Ra ) ≡ b b ∧ c b = dαc αa αc dαa + dRa b ∧ c − b ∧ c b = (αe αc + Rc ) αa αc (αe αa + Ra )+ dRa b ∧ e b ∧ c − b ∧ c ∧ e c b = dRa + Rc αa αc Ra b b ∧ c − b ∧ c DRa ≡ b 0 d2ea = d eb αa ≡ ∧ b = deb αa eb dαa ∧ b − ∧ b = (ec αc ) αa eb (αc αa + Ra ) ∧ c ∧ b − ∧ b ∧ c b = eb Ra − ∧ b In components, these take the familiar forms

a R b[cd;e] = 0 a R [bcd] = 0 Here we use Greek and Latin indices to distinguish diﬀerent vector bases. Use of the covariantly constant a coeﬃcient matrix of the solder form, eµ , allows us to convert freely between orthonormal components α α b c d a (Latin indices) and coordinate components (Greek indices), R βµν = ea eβ eµ eν R bcd.

2.2 Conformal transformation of the metric, solder form, and connection A conformal transformation of the metric is the transformation

2φ gµν g˜µν = e gµν (3) → This is not an invariance of Riemannian geometry, but it is an invariance of Weyl geometry. If we make a change of this type in a Riemannian geometry, the solder form changes by

ea e˜a = eφea (4) → since the solder form and metric are related via the orthonormal metric, ηab = diag ( 1, 1, 1, 1) by − a b gµν = eµ eν ηab The corresponding structure equation then gives the altered form of the metric compatible connection,

deã = e˜b αã ∧ b d eφea = eφeb αã ∧ b eφ (dea + dφ ea) = eφeb αã ∧ ∧ b so to find the new spin connection we must solve

dea = eb α˜a dφ ea (5) ∧ b − ∧ a ad c Since the spin connection is antisymmetric, α˜ = η ηbcα˜ , this is solved by setting b − d ˜a a ac µ ed α b = α b + 2∆dbec ∂µφ (6)

µ a ac where ec is inverse to eµ . The convenient symbol ∆db is deﬁned as

ac 1 a c ac ∆ (δ δ η ηbd) (7) db ≡ 2 d b −

3 1 This is meant to act on any tensor according to 1

ac d 1 a c ac d ∆ T (δ δ η ηbd) T db c ≡ 2 d b − c 1 a ac d = T η ηbdT 2 b − c 1 ac = η (Tcb Tbc) 2 − 0 1 which is just an antisymmetric tensor with the ﬁrst index raised to give a tensor. It is a 2 1 projection since it is idempotent,

ac ed 1 a c ac e d ed ∆ ∆ = (δ δ η ηbd) δ δ η ηcf db cf 4 d b − c f − 1 e a ae ea a e = δ δ η ηbf η ηbf + δ δ 4 b f − − f b ae = ∆fb Returning to check eq.(6) in eq.(5), we have, dea eb ˜a d ea = α b φ b ∧ a− ∧ac µ d a = e α + 2∆ e ∂µφ e dφ e ∧ b db c − ∧ eb a a c ac µ eb ed d ea = α b + (δd δb η ηbd) ec ∂µφ φ b ∧ a µ − b a a ∧ − ∧ = e α + e ∂µφ e e dφ e ∧ b b ∧ − ∧ = eb αa + dφ ea dφ ea ∧ b ∧ − ∧ = eb αa ∧ b as required. Since the spin connection is uniquely determined (up to local Lorentz transformations) by the structure equation, this is the unique solution.

2.3 Transformation of the curvature Now compute the new curvature tensor. For this longer calculation it is convenient to deﬁne the orthonormal b component of dφ = φbe , µ φa e ∂µφ. ≡ a ˜a a ac ed Then α b = α b + 2∆dbφc and the conformal transformation changes the curvature to:

1 R˜a ec ed = dα˜ a α˜ c α˜ a 2 bcd ∧ b − b ∧ c a c a ac d c ae d = dα α α + d 2∆ φce α 2∆ φee b − b ∧ c db − b ∧ dc ce d a ce d ag f 2∆ φee α 2∆ φee 2∆ φge − db ∧ c − db ∧ fc 1 a c d ac d ce d ag f = R e e + D 2∆ φce 2∆ φee 2∆ φge 2 bcd ∧ db − db ∧ fc Since Dea = dea ec αa =0 − ∧ c this reduces to

1 a c d 1 a c d ac d ce d ag f R˜ e e = R e e + 2∆ Dφce 2∆ φee 2∆ φge 2 bcd ∧ 2 bcd ∧ db − db ∧ fc 4 as is easily checked by expanding the ∆s. We need only simplify the ﬁnal term:

ce ed ag ef ce ed ag ef 4∆dbφe ∆fcφg = 2∆dbφe 2∆fcφg ∧ ∧ c ce d a ag f = φbe η ηbdφee φce η ηfcφge − ∧ − c a ag c f ce d a = φcφbe e η φgφbηfce e ηbdη φcφee e ag ∧ ce− d f ∧ − ∧ +η ηfcφgη ηbdφee e f a c ∧ d a ag f d = φbφf e e ηbdφ φce e η ηbdφgφf e e ∧ − ∧ − ∧ a e ae f d c d a = (δ δ φeφf η ηbdφeφf ) e e ηbdφ φce e d b − ∧ − ∧ a c ac e d 1 a f c d 1 a f c d = (δ δ η ηbd) φcφee e δ ηbcφ φf e e + δ ηbdφ φf e e d b − ∧ − 2 d ∧ 2 c ∧

ac 1 a c f 1 ac f e d = 2∆ φcφe δ δ ηceφ φf + η ηbdηceφ φf e e db − 2 d b 2 ∧

ac 1 f e d = 2∆ φcφe ηceφ φf e e db − 2 ∧

(α) d (x) d Now setting D φc = φc;de and d φ = φde . the ﬁnal result is,

a a ac 1 f e d R˜ = R + 2∆ φc e φeφc + φ φf ηce e e (8) b b db ; − 2 ∧ In components, eq.(8) becomes

a a ae 1 f ae 1 f R˜ = R + 2∆ φe c φcφe + φ φf ηec 2∆ φe d φdφe + φ φf ηed bcd bcd db ; − 2 − cb ; − 2 The Ricci tensor and scalar are

c c R˜bd = Rbd (n 2) φb d ηbdφ + (n 2) φbφd + (n 2) ηbdφ φc − − ; − ;c − − and

ab R˜ =g ˜ Rãb −2φ ab ce c = e g (Rbd (n 2) φb;d η φe;cηbd + (n 2) φdφb (n 2) φ φcηbd) −2φ − − c − c − c − − = e R 2 (n 1) φ + (n 2) φ φc n (n 2) φ φc − − ;c − − − −2φ c c = e R 2 (n 1) φ (n 1) (n 2) φ φc − − ;c − − − 2.4 Invariance of the Weyl curvature tensor a In general, we may split the Riemann curvature R bcd into its trace, the Ricci tensor, and its traceless part, called the Weyl curvature. This decomposition is most concisely expressed if we first define the Schouten tensor, 1 1 bd Rbd ηdbR (9) R ≡ (n 2) − 2 (n 1) − − where Rab is the Ricci tensor, c Rab R ≡ acb b The Schouten tensor often arises as a 1-form, Ra = abe . Except in 2-dimensions, it is equivalent to the Ricci tensor, since we may invert eq.(9) to write R

Rbd = (n 2) bd + ηdb (10) − R R R = 2 (n 1) (11) − R

5 In terms of ab, the Weyl curvature 2-form is deﬁned as R a a ae d C R + 2∆ Re e (12) b ≡ b db ∧ a Ca Expanding to ﬁnd the components, C bcd, of b, a a ae ae C bcd = R bcd + 2∆db ec 2∆cb ed a a eR ae− R a e ae = R bcd + (δd δb η ηbd) ec (δc δb η ηcb) ed a a − a R a− −a R = R + δ bc ηbd δ bd + ηbc bcd d R − R c − c R R d a 1 a a a a R a a = R (δ Rbd δ Rbc R ηbc + R ηbd)+ (δ ηbd δ ηbc) (13) bcd − (n 2) c − d − d c (n 1) (n 2) c − d − − − we readily verify its tracelessness,

c 1 R C = Rbd (nRbd Rbd Rbd + ηbdR)+ (n 1) ηbd bcd − n 2 − − (n 1) (n 2) − − − − = 0 with all other nontrivial traces equivalent to this one. By contrast, the second term in eq.(12) is equivalent ae a ae ae to knowing the Ricci or Schouten tensor, since the components of the ∆cb term, D bcd ∆db ec ∆cb ed in eq.(12) give ≡ R − R

c 2 a 1 fg a bd = δ D + η D ηbd R a −n 2 bcd (n 1) (n 2) fcg − − − To check this we expand,

c 2 a 1 fg a bd = δ D + η D ηbd R a −n 2 bcd (n 1) (n 2) fcg − − − c 2 ae ae 1 fg ae fg ae = δ (∆ ec ∆ ed)+ η ∆ ec η ∆ eg ηbd a −n 2 dbR − cb R (n 1) (n 2) fgR − cf R − − − 2 1 a e ae 1 = (δ δ η ηdb) ea (n 1) bd −n 2 2 d b − R − 2 − R − 1 1 fg a e ae 1 + η δ δ η ηfg ea (n 1) ηbd (n 1) (n 2) 2 f g − R − 2 − R − − 1 1 = ( bd ηbd (n 1) bd)+ (1 n (n 1)) ηbd −n 2 R − R− − R 2 (n 1) (n 2) − − − R − − − 1 = [ bd + (n 1) bd + ηbd ηbd] n 2 −R − R R − R − = bd R We now have the decomposition of the Riemann curvature into traceless and trace parts,

a a ae d R = C 2∆ Re e (14) b b − db ∧ After a conformal transformation, the new Riemann curvature 2-form may also be decomposed in the same way, a a ae d R˜ = C˜ 2∆ R˜ e e˜ b b − db ∧ Combining this with eq.(8) we have

a ae d a ae d ae 1 2 c d C˜ 2∆ R˜ e e˜ = C 2∆ Re e + 2∆ φe c φeφc + φ ηce e e b − db ∧ b − db ∧ db ; − 2 ∧

a ae 1 2 c d = C 2∆ ec φe c + φeφc φ ηce e e b − db R − ; − 2 ∧

6 Equality of the traceless and trace parts shows immediately that both

C˜ a Ca b = b

−φ 1 2 c R˜ e = e ec φe c + φeφc φ ηce ˜e R − ; − 2 so the Weyl curvature 2-form is conformally invariant. The components of each part transform as

˜a −2φ a C bcd = e C bcd (15)

−2φ 1 2 ˜ab = e ab φa b + φaφb φ ηab (16) R R − ; − 2

−2φ a 1 a ˜ = e φ (n 2) φ φa (17) R R− ;a − 2 − where the factor of e−2φ comes from replacing e˜a = eφea. This proves that the Weyl curvature tensor is covariant with weight 2 under a conformal transformation of the metric, and yields the expression for the change in the Schouten− (and therefore, Ricci) tensor under conformal transformation.

2.5 Conditions for conformal Ricci flatness Next, we find the condition required for the metric of a Riemannian geometry to be conformally related to the metric of a Ricci-flat spacetime. This follows as a pair of integrability conditions for φ when we set eq.(16) equal to zero. First, we rewrite eq.(16) as a 1-form equation,

−φ 1 ab e R˜ c = e Rc Dφc + φcdφ η φaφbηcee − − 2 D d e µ where φc = φc φeω c. Using the vector ﬁeld φc ec ∂µφ, we deﬁne the corresponding 1-form c − ≡ χ φce = dφ and ask for the conditions under which R˜ c = 0 has a solution for φ. This may be written as≡ a pair of equations,

e 1 ab e dφc = Rc + φeω + φcχ η φaφbηcee (18) c − 2 dφ = χ (19)

The integrability conditions follow from the Poincaré lemma, d2 0, ≡ 2 0 d φc ≡ e e ab e 1 2 e = dRc + dφe ω + φedω + dφc χ η φadφb ηcee φ ηcede (20) ∧ c c ∧ − ∧ − 2 0 d2φ ≡ = dχ (21)

The second condition, eq.(21), is identically satisﬁed by the deﬁnition of χ. Substituting the original equation

7 for dφc, eq.(18), into the ﬁrst integrability conditon, eq.(20),

d 1 2 d e e 0 = dRc + Re + φdω + φeχ φ ηede ω + φedω e − 2 ∧ c c

e 1 2 e + Rc + φeω + φcχ φ ηcee χ c − 2 ∧

ab d 1 2 d e 1 2 e η φa Rb + φdω + φbχ φ ηbde ηcee φ ηcede − b − 2 ∧ − 2 e ab e e d e = (dRc + Re ω )+ Rc χ η φaRb ηcee + φedω + φdω ω ∧ c ∧ − ∧ c e ∧ c 1 2 e d e 1 2 e ab d e φ ηcede + ηede ω φ ηcee χ η φaφdω ηcee −2 ∧ c − 2 ∧ − b ∧ 2 e 1 2 a e φ χ ηcee + φ φae ηcee − ∧ 2 ∧ b a ab e e 1 2 e d e = DRc + φa δ δ η ηce Rb e + φeR φ ηec De e ω c e − ∧ c − 2 − ∧ d 2 1 e e 1 e da e +φ ηceχ e ηceχ e + ηceχ e φaφdω ηcee 2 ∧ − ∧ 2 ∧ − ∧ a ab e = DRc + φa R + 2∆ Rb e c ec ∧ which we see from eq.(14) becomes D Ca 0= Rc + φa c (22)

Though this well-known condition still depends on the gradient of the conformal factor, φa, Szekeres has shown using spinor techniques that it can be broken down into two integrability conditions depending only on the curvature [22].

2.6 Conditions for conformal Einstein equation with matter We may apply the same approach to the Einstein equation with conformal matter. Let the matter be of definite conformal weight, Ψ˜ ekφΨ for a generic field Ψ. Then the covariant form of the stress-energy tensor will be of conformal weight→ 2, − −2φ Tãb = e Tab to have the correct weight for the Einstein equation. The Einstein tensor, of course, is not of definite −2φ conformal weight, but it acquires an overall factor of e . We assume that Tab is of definite weight. 1 Then, writing the Einstein equation, Rab 2 ηabR = Tab in terms of the Schouten tensor using eqs.(10) and (11), gives − 1 ab ηab = Tab R − R n 2 − Now define, for arbitrary curvatures, not necessarily solutions, 1 Eab ab ηab Tab ≡ R − R− n 2 − We would like to know when there exists a conformal transformation, Eab Eãb such that Eãb = 0. The → calculation is simpler if we notice that Eab =0 if and only if 1 1 1 Eab Eηab = ab Tab T ηab =0 − n 1 R − n 2 − n 1 − − − Defining 1 1 ab Tab T ηab (23) T ≡ n 2 − n 1 − −

8 ˜ 1 ˜ we ask for a conformal gauge in which Eab n−1 Eηab = ab ab =0. We establish clearly that this is equivalent− to the EinsteinR equation.− T The essential question is the condition for a conformal transformation such that Eãb = 0. Substituting the conformally transformed fields to find Eãb, 1 Eãb = ab ηab Tab R − R− n 2 − −2φ 1 2 −2φ c 1 c 1 −2φ = e ab φa b + φaφb φ ηab ηabe φ (n 2) φ φc e Tab R − ; − 2 − R− ;c − 2 − − n 2 − so we examine integrability of

1 2 c 1 c 1 0 = ab φa b + φaφb φ ηab ηab φ (n 2) φ φc Tab (24) R − ; − 2 − R− ;c − 2 − − n 2 − c However, by solving the trace of this equation for φ ;c,

a a 1 c c 1 c 1 0 = φ + φ φa nφ φc n nφ n (n 2) φ φc T R− ;a − 2 − R− ;c − 2 − − n 2 − c 1 c 1 = (n 1) + (n 1) φ + (n 1) (n 2) φ φc T − − R − ;c 2 − − − n 2 − c 1 c 1 φ = (n 2) φ φc + T (25) ;c R− 2 − (n 1) (n 2) − − and substituting eq.(25) back into eq.(24) we ﬁnd the simpler form,

1 2 0 = ab φa b + φaφb φ ηab ab (26) R − ; − 2 − T ˜ 1 ˜ and this is just Eab n−1 Eηab =0. Conversely, the trace of eq.(26) reproduces the trace condition, eq.(25). − 1 Therefore, conformal vanishing of Eãb is equivalent to conformal vanishing of Eãb Eη˜ ab. − n−1 T eb d ˜ 1 ˜ Returning to the find the condition, we set a ab and χ = φ, then write Eab n−1 Eηab =0 as a 1-form equation, ≡ T −

b 1 2 b 0 = Ra dφa + φbω + φaχ φ ηabe T a − a − 2 − We therefore require

b 1 2 b dφa = Ra + φbω + φaχ φ ηabe T a a − 2 − dχ = d2φ 0 ≡ The trace relation of eq.(25) also holds. The integrability condition is: 2 0 d φa ≡ b 1 2 b = d Ra T a + φbω + φaχ φ ηabe − a − 2

b 1 2 b = d (Ra T a)+ φbdω φ ηabde − a − 2 c 1 2 c b + Rb T b + φcω + φbχ φ ηbce ω − b − 2 ∧ a

b 1 2 b + Ra T a + φbω + φaχ φ ηabe χ − a − 2 ∧

c b 1 2 b d φ Rc T c + φbω + φcχ φ ηcbe ηade − − c − 2 ∧

9 Distributing and collecting like terms,

b c d 0 = d (Ra T a) + (Rb T b) ω + (Ra T a) dφ φ (Rc T c) ηade − − ∧ a − ∧ − − ∧ b c b 1 2 b c b +φb dω ω ω φ ηab de e ω a − a ∧ c − 2 − ∧ c b b 1 c c 1 c d + φbχ ω + φbω χ + φ φc φ φc + φ φc dφ ηade ∧ a a ∧ 2 − 2 ∧ b c b d bc d = D (Ra T a)+ φbR + φbδ δ (Rc T c) e φbη ηad (Rc T c) e − a a d − ∧ − − ∧ D Rb cb ed = (Ra T a)+ φb a +2φb∆ad (Rc T c) − b cb −d ∧ = DRa + φbC DT a 2φb∆ T c e a − − ad ∧ leaving us with D Cb D bc ed Ra + φb a = T a + φb2∆daT c (27) This is a new result. When eq.(26) is written using the Riemann tensor instead of the Weyl tensor,

b cb d D (Ra T a)+ φbR +2φb∆ (Rc T c) e = 0 − a ad − ∧ we recognize the same condition as that for Ricci ﬂatness, but with the Schouten tensor replaced by Ra T a. − 3 Weyl geometry

A simple extension of the Poincaré symmetry underlying Riemannian geometry leads to the Cartan structure equations for the Weyl group: Ra = dωa ωc ωa (28) b b − b ∧ c Ta = dea eb ωa ω ea (29) − ∧ b − ∧ Ω = dω (30) Ta 1 a eb ec where the most general case includes both the torsion, = 2 T bc , and the dilatational curvature, Ω 1 eb ec ∧ = 2 Ωab . In our treatment of a Dirac-like theory, we will not assume vanishing torsion. A conformal∧ transformation of the metric, eq.(3), now transforms both the solder form and the Weyl vector, according to eã = eφea ω˜ = ω + dφ The final structure equation, eq.(30) then remains unchanged, since dω˜ = dω The basis equation transforms as T˜ a = deã e˜b ω˜ a ω˜ eã − ∧ b − ∧ = eφdφ ea + eφdea eφeb ω˜ a (ω + dφ) eφea ∧ − ∧ b − ∧ = eφ dφ ea + (Ta +ec ωa + ω ea) eb ω˜ a ω ea dφ ea ∧ ∧ c ∧ − ∧ b − ∧ − ∧ = eφT a + eφeb (ωa ω˜ a ) ∧ b − b We conclude that it is sufficient to take the spin connection to be conformally invariant, and the torsion a weight-1 conformal tensor:

a a ω˜ b = ω b T˜ a = eφTa

10 These inferences are correct, as may be shown directly from the gauge transformation properties of the Cartan connection. Since the spin connection is invariant, the Lorentz curvature 2-form is also invariant, R˜ a Ra b = b. We again use the Poincaré lemma, d2 0 to ﬁnd the integrability conditions: ≡ DRa b = 0 (31) DTa = eb Ra Ω ea (32) ∧ b − ∧ DΩ = 0 (33) where the covariant derivatives are given by

DRa dRa + Rc ωa Ra ωc b ≡ b b ∧ c − c ∧ b DTa dTa + Tb ωa ω Ta ≡ ∧ b − ∧ DΩ dΩ ≡ When the torsion vanishes, we have a pair of algebraic identities since the Weyl-Ricci tensor may have an antisymmetric part. From

eb Ra = Ω ea ∧ b ∧ Ra a [bcd] = δ[bΩcd] (34)

we ﬁnd the symmetric and antisymmetric parts, Ra Ra Ra a a a bcd + cdb + dbc = δb Ωcd + δc Ωdb + δd Ωbc Rbd Rdb = (n 2)Ωbd − − − Ra R While the Lorentz curvature 2-form is conformally invariant, the components bcd, ab and Ωab all have conformal weight 2. − 3.1 The connection with the Weyl vector and torsion As with Riemannian geometry, the structure equation for the solder form, eq.(29). allows us to solve for the connection.

3.1.1 Weyl connection with torsion Look at the solder form equation, dea = eb ωa + ω ea + Ta ∧ b ∧ Notice that when Ta =0 this has exactly the same form as the conformally transformed solder form structure equation of a Riemannian geometry, eq.(5), with dφ replaced by ω. Thus, the solution for the Weyl spin connection is completely analogous to the eﬀect of a dilatation on− the connection of a Riemannian geometry, with the Weyl vector Wc replacing the negative of the gradient of the scale change, φc in eq.(6). Taking a a a a a − a advantage of this observation, let ω b = α b + β b + γ b where α b is the compatible connection and β b is the required Weyl vector piece,

dea eb a = α b a ∧ ac d β = 2∆ Wce b − db

11 a ac d and each term has the same antisymmetry of indices as the full spin connection, i.e., ω b = η ηbdω c. Then −

dea = eb (αa + βa + γa )+ ω ea + Ta ∧ b b b ∧ eb a eb a eb a ea Ta = α b + β b + γ b + ω + ∧ ac b ∧ d ∧ a b∧ a a 0 = 2∆ Wce e + ω e + e γ + T − db ∧ ∧ ∧ b = eb γa + Ta ∧ b 1 = γa + T a eb ec bc 2 bc ∧

e The ﬁnal equation must involve antisymmetric ηaeγ bc = γabc = γbac. Lowering indices in the remaining condition and cycling, −

0 = γabc γacb + Tabc − 0 = γbca γbac + Tbca − 0 = γcab γcba + Tcab − we combine with the usual sum-sum-diﬀerence and solve.

0 = γabc γacb + γbca γbac γcab + γcba + (Tabc Tcab + Tbca) − − − − = (γabc γbac) (γacb + γcab) + (γbca + γcba) + (Tabc Tcab + Tbca) − − − = 2γabc + (Tabc Tcab + Tbca) − 1 γabc = (Tabc Tcab + Tbca) −2 − Therefore, 1 γa = (T a + T a + T a) bc −2 bc cb bc and the spin connection is given by

a a ac d a c ω = α 2∆ Wce C e (35) b b − db − bc where we define the contorsion tensor to be 1 Ca (T a + T a + T a) (36) bc ≡ 2 bc cb bc Now check, dea eb a ea Ta = ω b + ω + b ∧ a ∧ ac d a c a a = e α 2∆ Wce C e + ω e + T ∧ b − db − bc ∧ b a a c ac b d a b c a a = e α (δ δ η ηbd) Wce e C e e + ω e + T ∧ b − d b − ∧ − bc ∧ ∧ b a b a a 1 a a a b c 1 a b c = e α Wbe e + ω e (T + T + T ) e e + T e e ∧ b − ∧ ∧ − 2 bc cb bc ∧ 2 bc ∧ 1 1 = eb αa T a eb ec + T a eb ec ∧ b − 2 bc ∧ 2 bc ∧ = eb αa ∧ b 3.1.2 The covariant derivative of Weyl geometry in a coordinate basis When a tensor transforms linearly and homogeneously with a power λ of the conformal transformation that applies to the solder form eã = eφea, TÃ = eλφT A

12 it is a conformal tensor of weight λ. When differentiating a conformal tensor of weight λ the covariant derivative in Weyl geometry is not just the partial derivative, but includes the weight of the field times the field, times the Weyl vector. For example, for a scalar field we have

Dµϕ = ∂µϕ λWµϕ − This means that metric compatibility gives a diﬀerent expression for the connection.

0 = Dµgαβ ν ν = ∂µgαβ gνβΓ˜ gαν Γˆ 2Wµgαβ − αµ − βµ − = ∂µgαβ Γˆβαµ Γˆαβµ 2Wµgαβ − − − The derivative of a contravariant vector of weight λ is given by

α α β α α Dµv = ∂µv + v Γˆ λv Wµ (37) βµ − and, checking the tranformed derivative,

α λφ α D˜ µv˜ = D˜µ e v λφ α λφ β α λφ α = ∂µ e v + e v Γ˜ λe v (Wµ + ∂µφ) βµ − λφ α β ˜α α = e ∂µv + v Γ βµ λv Wµ − λφ α = e Dµv and is therefore properly covariant. Notice that the Weyl connection is invariant under a conformal trans- ˜α α formation, Γ βµ =Γ βµ.

3.1.3 Weyl connection with torsion in a coordinate basis The corresponding expression in a coordinate basis starts with the deﬁnition of torsion as the antisymmetric part of the connection,

Tαµβ Γˆαµβ Γˆαβµ (38) ≡ − Then, starting from metric compatibility,

0 = Dµgαβ ν ν = ∂µgαβ gνβΓ˜ gαν Γˆ 2Wµgαβ − αµ − βµ − = ∂µgαβ Γˆβαµ Γˆαβµ 2Wµgαβ − − − we cycle the expression in the usual way

Γˆβαµ + Γˆαβµ = ∂µgαβ 2Wµgαβ − Γˆαµβ + Γˆµαβ = ∂βgµα 2Wβgµα − Γˆµβα + Γˆβµα = ∂αgβµ 2Wαgβµ − Each of these three expressions is a conformal tensor since

2φ 2φ ∂µg˜αβ 2W˜ µg˜αβ = ∂µ e gαβ 2 (Wµ + ∂µφ) e gαβ − − 2φ = e (∂µgαβ 2Wµgαβ) −

13 Adding the ﬁrst two and subtracting the third we no longer assume the connection is symmetric,

0 = Γˆβαµ + Γˆαβµ + Γˆαµβ + Γˆµαβ Γˆµβα Γˆβµα − − ∂µgαβ +2Wµgαβ ∂βgµα +2Wβgµα + ∂αgβµ 2Wαgβµ − − − = 2Γˆαβµ + Γˆαµβ Γˆαβµ + Γˆβαµ Γˆβµα + Γˆµαβ Γˆµβα − − − (∂µgαβ + ∂βgµα ∂αgβµ)+2(Wµgαβ + Wβgµα Wαgβµ) − − − = 2Γˆαβµ + (Tαµβ + Tβαµ + Tµαβ)

(∂µgαβ + ∂βgµα ∂αgβµ)+2(Wµgαβ + Wβgµα Wαgβµ) − − − we ﬁnd 1 Γˆαβµ = Γαβµ (Wµgαβ + Wβgµα Wαgβµ) (Tαµβ + Tβαµ + Tµαβ) − − − 2 Now, if we raise the ﬁrst index,

α α α α α 1 α α α Γˆ =Γ δ Wµ + δ Wβ W gβµ T + T + T (39) βµ βµ − β µ − − 2 µβ β µ µ β we arrive at the coordinate form of the connection.

3.2 The Weyl-Schouten tensor R˜ a Ra The invariance of the full curvature, b = b, means that not only is the Weyl curvature of a Weyl geometry conformally covariant, but so is the Weyl-Schouten tensor, Ra. By the Weyl-Schouten tensor, we mean the conformally covariant Schouten tensor of a Weyl geometry. To compute it for a torsion-free Weyl geometry, we must expand the curvature, eq.(28) using the Weyl connection,

a a ac d ω = α 2∆ Wce (40) b b − db a Ra with α b still the metric-compatible spin connection. The diﬀerence is that now all of b will be conformally covariant because the connection is scale invariant,

de˜a = e˜b ω˜ a + ω˜ e˜a ∧ b ∧ d eφea = eφeb ω˜ a + (ω + dφ) eφea ∧ b ∧ eφdφ ea +eφdea = eφeb ω˜ a + (ω + dφ) eφea ∧ ∧ b ∧ eb ωa = eb ω˜ a ∧ b ∧ b a a and therefore ω˜ b = ω b. It follows from eq.(28) that the full Weyl curvature tensor is scale invariant, R˜ a Ra b = b. Substituting into the curvature, the algebra is identical to that leading up to eq.(8), with φa replaced by Wa. This results in −

a a ac 1 2 e d R = R 2∆ Wc e + WeWc W ηce e e b b − db ; − 2 ∧ which decomposes into three parts when we separate the symmetric and antisymmetric parts of the trace term. With

Ω = dω

Ωab = W[b;a]

14 we have

a a ac 1 2 e d ac e d R = C 2∆ ce + W + WeWc W ηce e e 2∆ Ωece e (41) b b − db R (c;e) − 2 ∧ − db ∧ Ra d a c a where b = α b α b α c is the Riemannian part of the curvature. Carrying out the decomposition of the curvature into trace− and∧ trace-free parts, we ﬁnd the relationship between the Weyl and Schouten tensors of the Weyl and Riemannian geometries. In addition, the asymmetry of the Ricci tensor gives rise to a third independent component, the dilatational curvature:

Ca Ca b = b

1 2 c Ra = Ra + W WaWc + W ηac e (a;c) − 2 ∧ Ω = W ea eb [b;a] ∧ or in components, 1 2 Rab = ab + W WaWb + W ηab (42) R (a;b) − 2

We define the Weyl-Schouten tensor Rab to be this symmetric part only. In an integrable Weyl geometry, defined as one in which the dilatational curvature, Ω, vanishes, there exists a conformal transformation which makes the Weyl vector vanish, Wa = 0. In this gauge, the Weyl- Ca Schouten tensor reduces to the Schouten tensor. The gravitational field, b, is the same in all gauges. The modification of eq.(41) in the presence of torsion follows immediately since it only changes the Weyl connection of eq.(40) by the contorsion tensor,

a a ac d a c a a ωˆ = α 2∆ Wce C e = ω C b b − db − bc b − b This changes the curvature to

a Rˆ = Ra DCa + Cc Ca b b − b b ∧ c 4 Scale invariant gravity

We now turn to the formulation of a scale invariant gravity theory, based in a Weyl geometry. For this we must construct a Lorentz- and dilatation-invariant action functional from the curvature and any other available tensors. As we have noted, the conformal weight of the curvature components in an orthonormal a b basis is 2. Since gµν = eµ eν ηab, the Minkowski metric has conformal weight zero, making the conformal weight of− the Weyl-Ricci scalar equal to 2 as expected. This introduces a diﬃculty in writing a scale invariant action in dimensions greater than− two, since the volume element has weight +n in n-dimensions.

4.1 Actions nonlinear in the curvature In 2n-dimensions, we may use n-products of the curvature:

ab cd ef S = R R R Qabcd···ef ˆ ∧ ∧···∧ where Qabcd···ef is a rank-n invariant tensor. In 4-dim the most general such action is curvature-quadratic action, and takes the form

abcd ab 2 4 S = αR Rabcd + βR Rab + γR √ gd x ˆ −

15 However, this may be simpliﬁed using the invariance of the Euler character. Variation of the Gauss-Bonnet 1 Rab Rcd combination for the Euler character χ = 32π2 εabcd, gives − ´ ∧ ab cd δχ = δ R R εabcd ˆ ∧ ab eb a ae b cd = 2 d δω δω ω (δω ) ω R εabcd ˆ − ∧ e − ∧ e ab cd = 2 D δω R εabcd ˆ ∧ ab cd ab cd = 2 D δω R εabcd + δω DR εabcd ˆ ∧ ∧ and this vanishes identically when we use the second Bianchi identity, DRcd 0, and let the variation vanish on the boundary, ≡

ab cd δχ = 2 D δω R εabcd ˆ ∧ V

ab cd = 2 d δω R εabcd ˆ ∧ V ab cd = 2 δω R εabcd ∧ δV

= 0 The additon of any multiple of the Euler character to the action therefore makes no contribution to the field equations. We expand the Euler character as follows. Define a convenient volume element as the dual of unity, Φ ∗1. Then: ≡ ∗ Φ 1 ≡ 1 a b c d = εabcde e e e 4! ∧ ∧ ∧ ∗ ∗ 1 a b c d Φ = εabcde e e e 4! ∧ ∧ ∧ 1 = ε εabcd 4! abcd = 1 − Φ 1 d µ d ν d α d β In a coordinate basis, = 4! √ gεµναβ x x x x , and if we ignore orientation this is simply √ gd4x. From the definition of−Φ it follows∧ that ∧ ∧ −

abcd 1 abcd e f g h ε Φ = ε εefghe e e e 4! ∧ ∧ ∧ 1 = 4!δabcd ee ef eg eh 4! − efgh ∧ ∧ ∧ = ea eb ec ec − ∧ ∧ ∧

16 2 abcd abcd ea eb ec ed abcdΦ where ε εefgh = 4!δefgh. Therefore, substituting = ε into the expression for the Euler character, − ∧ ∧ ∧ −

1 ab cd χ = R R εabcd −32π2 ˆ ∧

1 ab cd e f g h = R R e e e e εabcd −128π2 ˆ ef gh ∧ ∧ ∧ 1 = Rab Rcd εefghε Φ 128π2 ˆ ef gh abcd 1 = Rab Rcd 4!δefghΦ 128π2 ˆ ef gh abcd

efgh Expanding the antisymmetric δabcd ,

4!δefgh δe δf δgδh δhδg + δg δhδf δf δh + δh δf δg δgδf abcd ≡ a b c d − c d b c d − c d b c d − c d δf δe δgδh δhδg + δg δhδe δeδh + δh (δeδg δgδe) − a b c d − c d b c d − c d b c d − c d δg δf δeδh δhδe + δe δhδf δf δh + δh δf δe δeδf − a b c d − c d b c d − c d b c d − c d h f g e e g g e f f e e f g g f δa δb (δc δd δc δd)+ δb δc δd δc δd + δb δc δd δc δd − − − − we explicitly write out the integrand of the Euler character in the Gauss-Bonnet form3,

ab cd 1 ab cd efgh R R εabcd = R R 4!δ Φ ∧ −4 ef gh abcd 2 b d abcd = R 4R R + R Rabcd − − d b 2We check the normalization by contracting all pairs of indices, (ae) , (bf) , (cg) , (dh): a b c d − d c c d b − b d d b c − c b 4! = δa δb δc δd δc δd + δb δc δd δcδd + δb δcδd δcδd − b a c d − d c c d a − e d d a c − c a δa δb δc δd δc δd + δb δc δd δc δd + δb (δc δd δc δd ) − b a c d − d c c d a − e d d a c − c a δa δb δc δd δc δd + δb δc δd δc δd + δb (δc δd δc δd ) − c b a d − d a a d b − f d d b a − a b δa δb δc δd δc δd + δb δc δd δc δd + δb δcδd δc δd − d b c a − a c c a b − b a a b c − c b δa δb (δcδd δc δd)+ δb δc δd δc δd + δb δcδd δcδd = 4 (4 (16 − 4) + 4 − 16 + 4 − 16) − (64 − 16 + 4 − 16 + 4 − 16) − (64 − 16 + 4 − 16 + 4 − 16) − (64 − 16 + 4 − 16 + 4 − 16) = 24

Rab Rcd δefgh Rab Rcd δe δf δg δh − δhδg δg δhδf − δf δh δh δf δg − δg δf 4! ef gh abcd = ef gh a b c d c d + b c d c d + b c d c d −Rab Rcd δf δe δg δh − δhδg + δg δhδe − δeδh + δh δeδg − δg δe ef gh a b c d c d b c d c d b c d c d −Rab Rcd δg δf δeδh − δhδe δe δhδf − δf δh δh δf δe − δeδf ef gh a b c d c d + b c d c d + b c d c d −Rab Rcd δh δf δg δe − δeδg + δg δeδf − δf δe + δe δf δg − δg δf ef gh a b c d c d b c d c d b c d c d Rab Rcd Rab Rcd Rab Rcd − Rab Rcd Rab Rcd Rab Rcd = 2 ab cd + ad bc + ac bd 2 ba cd + da bc + ca db − Rab Rcd Rab Rcd Rab Rcd − Rab Rcd Rab Rcd Rab Rcd 2 cb ad + bd ac + dc ab 2 db ca + bc da + cd ba R2 − Rb Rd RabcdR = 4 4 d b + abcd

17 so that

1 2 b d abcd χ = R 4R R + R Rabcd Φ 32π2 ˆ − d b The invariance of χ allows us to replace

abcd 2 2 b d R RabcdΦ = 32π χ R 4R R Φ ˆ − ˆ − d b Dropping the invariant ﬁrst term leaves the most general curvature-quadratic action in the form

2 ab 4 S = aR + bR Rab √ gd x (43) ˆ − for constants a,b. Quadratic gravity theories, especially the b =0 case, have often been studied because the scale invariance allows the theory to be renormalizable. However, fourth order field equations such as those resulting from eq.(43) are sometimes found to introduce ghosts in the quantum theory. The Einstein-Hilbert term may be included as well, but while this has desirable effects, it breaks the scale invariance we examine here. For further discussion and references on quadratic gravity, see [24]. The particular case of Weyl (conformal) gravity deserves mention. As first shown by Bach [9], the fully conformally action for Weyl gravity may be written as

abcd 4 SW = α C Cabcd√ gd x ˆ −

abcd bd 1 2 4 = α R Rabcd 2R Rbd + R √ gd x ˆ − 3 −

2 2 b d 4 bd 1 2 4 = 32π αχ α R 4R R √ gd x + α 2R Rbd + R √ gd x − ˆ − d b − ˆ − 3 − 1 = 32π2αχ +2α Rb Rd R2 √ gd4x (44) ˆ d b − 3 − Naturally, fourth order ﬁeld equations result from metric variation of eq.(44). However, it is shown in [25] that varying the full conformal connection in the action 44 gives the additional integrability condition to reduce the fourth order equations to the Einstein equation. Quadratic gravity applies only in four dimensions, with dimension 2n theories having correspondingly more complicated ﬁeld equations. Instead, we consider Weyl invariant theories linear in the curvature.

4.2 Actions linear in the curvature There are two classes of gravitational theories with Lorentz and dilatational symmetry with actions linear in the curvatures. Both may be written in any dimension.

4.2.1 Biconformal gravity Based in geometries ﬁrst developed by Ivanov and Niederle [16, 17] and Wheeler [18], what is now called biconformal gravity takes place in a 2n-dimensional symplectic manifold. The canonical conjucacy of ths space makes the volume element dimensionless, allowing for a curvature-linear action in any dimension [19] of the form

a1 a1 a1 a2 an b1...bn S = (αΩ + βδ Ω + γe fb ) e e fb fb ε εa ...a (45) ˆ b1 b1 ∧ 1 ∧ ∧···∧ ∧ 1 ∧···∧ 1 1 n The resulting theory describes n-dimensional gravity on an n-dimensional Lagrangian submanifold. Arising as the quotient of the conformal group by the Weyl group, the theory leads to scale invariant general relativity. Because the underlying structure is the full conformal group instead of the Weyl group, resulting in Kähler geometry instead of Weyl geometry, we will not go into further detail here.

18 4.2.2 The Dirac theory An alternative approach to scale invariant gravity was developed by Dirac in an attempt to give rigor to his Large Numbers Hypothesis [26], the idea that extremely large dimensionless numbers in the description of nature should be related to one another. In [27], Dirac presents a scale invariant gravity theory in which the gravitational constant varies with time in such a way that the large dimensionless magnitude constructed from the fundamental charge e and G is related to the age of the universe. The result follows from a single, simple solution to the scale invariant theory. Here we examine the scale invariant theory without further discussion of the Large Numbers Hypothesis. In the Dirac theory, scale invariance is achieved by including a gravitationally coupled scalar field in addition to the curvature. There is a wide literature on scalar fields coupled to gravity. Fierz [29] and Jordan [28] showed that the energy-momentum tensor of scalar theories may sometimes be unphysical. This flaw is corrected by the Brans-Dicke scalar-tensor theory [30], and discussion contiues. In more recent work, [31], Romero, Fonseca-Neto, and Pucheu study the relationship between Brans-Dicke theory and integrable Weyl geometry is studied. See also the history by Brans [32]. Dirac begins with an action which in our notation takes the form

1 µν 2 µν 4 n SD = Ω Ωµν ϕ R +6g DµϕDν ϕ + cϕ √ gd x (46) ˆ 4 − − where Ωµν = Wµ,ν Wµ,ν is the dilatational curvature. Dropping the dilatation, and allowing an arbitrary constant multiplying− the kinetic term gives the Brans-Dicke scalar-tensor theory, usually written as

ω0 µν n SD = ϕR + g DµϕDν ϕ + m √ gd x ˆ ϕ L − where m is a matter Lagrangian. In theL next Section, we take a similar but slightly different approach, allowing torsion and a mass term but no quadratic dilatational term. Our treatment also differs by our use of a Palatini variation, so the metric and connection are regarded as independent variables. Finally, we consider arbitrary spacetime dimension. We find a locally scale invariant theory that exactly reproduces general relativity as soon as a suitable definition of the unit of length is made.

5 Curvature linear, scale invariant gravity in any dimension

Beginning with the action for a Klein-Gordon scalar ﬁeld, ϕ, in curved, n-dimensional Weyl geometry, we add a non-quadratic term analogous to Dirac’s ϕ4 potential,

2 2 µν m c 2 s n Sϕ = g DµϕDν ϕ + ϕ + βϕ √ gd x (47) ˆ ~2 − where the covariant derivative of ϕ is Dµϕ = ∂µϕ λWµ, we include a gravitational term of the formwhere ∗ 1 a b − Φ ··· e e 1= n! εa b in n-dim and the power k will be chosen to make the full action scale invariant. There≡ is no scalar we∧···∧ can form which is linear in the dilatational curvature. We choose units ~ = c =1. To include gravity, we multiply a power of the scalar ﬁeld times the scalar curvature. Thus, we arrive at

k µν 2 2 s n S = αϕ R + g DµϕDν ϕ + m ϕ + βϕ √ gd x (48) ˆ −

19 If the scalar ﬁeld, metric, curvature and geometric mass scale as

ϕ eλφϕ → 2φ gµν e gµν → g e2nφg → R e−2φR 2 2 → m c − m2 = e 2φm2 ~2 → α,β dimensionless then S scales as

k µν 2 2 s n S˜ = αϕ˜ R˜ +˜g DµϕD˜ ν ϕ˜ + m ϕ˜ + βϕ˜ gd˜ x ˆ − p (λk−2)φ k −2φ 2λφ µν −2φ 2λφ 2 2 sλφ s nφ n = αe ϕ R + e e g DµϕDν ϕ + e e m ϕ + e βϕ e √ gd x ˆ − (λk−2+n)φ k (n−2+2λ)φ µν 2 2 (n+sλ)φ s n = e αϕ R + e g DµϕDν ϕ + m ϕ + e βϕ √ gd x ˆ − h i and is therefore locally scale invariant if n 2 λ = − − 2 k = 2 2n s = n 2 − We may make these assignments in any dimension greater than two, resulting in

2n 2 µν 2 2 − n S = αϕ R + g DµϕDν ϕ + m ϕ + βϕ n 2 √ gd x (49) ˆ − 5.1 Variation of the action We consider the Palatini variation of S, varying the solder form, spin connection, Weyl vector and scalar ﬁeld, ea a ( , ω b, Wa, ϕ), independently. The variation of the solder form is most easily accomplished by varying the metric. These are equivalent, since

a b δgαβ = 2ηabeα δeβ and conversely

a ab β δeα = 2η eb δgαβ

a The connection variation is easiest using diﬀerential forms and varying ω b directly.

5.1.1 Metric variation R µν R R 1 µν Writing = g µν where µν depends only on the spin connection, and noting that δ√ g = 2 √ ggµν δg , the metric variation is − − −

2n µν 2 1 2 µν 2 2 − n 0= δg αϕ Rµν + DµϕDν ϕ gµν αϕ R + g DµϕDν ϕ + m ϕ + βϕ n 2 √ gd x ˆ − 2 −

20 so we immediately get the ﬁeld equation,

2n 2 1 1 α 2 2 − αϕ Rµν gµν R = DµϕDν ϕ + gµν D ϕDαϕ + m ϕ + βϕ n 2 − 2 − 2 This takes the form of the scale covariant Einstein equation

4 1 1 1 1 1 α 2 − Rµν gµν R = DµϕDν ϕ + gµν D ϕDαϕ + m + βϕ n 2 (50) − 2 α −ϕ2 2 ϕ2

5.1.2 Scalar ﬁeld variation The scalar ﬁeld variation is

n n+2 µν 2 2 − n 0 = 2αδϕϕR +2g DµδϕDν ϕ +2m ϕδϕ + βδϕϕ n 2 √ gd x ˆ n 2 − − n n+2 µν 2 2 − n = δϕ 2αϕR 2Dµ (g Dν ϕ)+2m ϕ + βϕ n 2 √ gd x ˆ − n 2 − − and therefore we ﬁnd a nonlinear wave equation coupled to the scalar curvature,

n n+2 a 2 − D Daϕ m ϕ αRϕ βϕ n 2 =0 (51) − − − n 2 − 5.1.3 Weyl vector variation

The Weyl vector only appears in the kinetic term for the scalar ﬁeld, where Dµϕ = ∂µϕ λWµϕ. Varying − Wµ, µν n 0= (2g λδWµϕDν ϕ) √ gd x ˆ − and therefore,

Dαϕ = 0 (52)

This may immediately be solved for the Weyl vector

dϕ λωϕ = 0 − 1 ω = d ln ϕ λ 2 1 = dϕ −n 2 ϕ − which implies an integrable Weyl geometry and the existence of a gauge in which the Weyl vector vanishes. We easily ﬁnd the gauge transformation φ required to remove the Weyl vector.

ω˜ = ω + dφ 1 0 = d ln ϕ + dφ λ 1 ϕ φ = ln −λ ϕ0 With this transformation, we have ω˜ =0. This now remains the case for arbitrary global scale transformations.

21 The same transformation changes the scalar ﬁeld according to

ϕ˜ = ϕeλφ λ − 1 ln ϕ = ϕe λ ϕ0

= ϕ0 so the transformation that removes the Weyl vector simultaneously makes the scalar ﬁeld constant. In an arbitrary gauge φ, the Weyl vector and scalar ﬁeld become

Wµ = ∂µφ λφ ϕ = ϕ0e

but the physical content of the theory remains the same. ∗ If we were to allow curvature squared terms in the action, it would involve Ω Ω where Ωµν = Wµ,ν Wν,µ. Such a dilatational curvature would lead, in general, to a nonintegrable Weyl geometry. However, the physical− constraints against such a geometry are extremely strong – we do not experience changes of relative physical size. ∗ A quadratic term, Ω Ω, is a kinetic term for a nonintegrable Weyl vector. Except in dimension n = 4, scale invariance of such a kinetic term would also require a factor of the scalar ﬁeld,

2(n−4) 1 − µα νβ n ϕ n 2 g g Ωµν Ωαβ√ gd x 4 ˆ − This would lead to a ﬁeld equation of the form

− 2(n 4) µν µ ϕ n−2 Ω =2λD ϕ ;ν which allows nontrivial ϕbut also a nontrivial dilatation, Ωµν . The dilatational curvature would also act as a source to the wave equation for ϕ,

− µ 2 n n+2 n 4 n 6 αβ D D ϕ + m ϕ + βϕ n−2 + αϕR + − ϕ n−2 Ω Ω =0 µ n 2 n 2 αβ − − and as an energy source for the Einstein equation in the form

2(n−4) − β 1 αβ Tµν = ϕ n 2 Ω Ωνβ gµν Ω Ωαβ µ − 4 We continue without the kinetic term, because of the unphysical nature of dilatations.

5.1.4 Connection variation The variation of the spin connection is much easier to work with than the variation of the Christoﬀel connection. Since only the curvature depends on the connection, we need only the ﬁrst term in the action and the equivalence

α 2 ab c d α 2 1 ab e f c d ϕ R e e εabc···d = ϕ R e e e e εabc···d (n 2)! ∧ ∧···∧ (n 2)! 2 ef ∧ ∧ ∧···∧ − − α 2 1 ab efc···d = ϕ R ε εabc···dΦ −(n 2)! 2 ef − α 1 = ϕ2 Rab (n 2)! δeδf δf δe Φ (n 2)! 2 ef − a b − a b − = αϕ2RΦ

22 k n α 2 ab c d a Therefore, replacing αϕ R√ gd x with ϕ δ a R e e ε ··· , we vary ω , (n−2)! ωb abc d b ´ − ´ ∧ ∧···∧ α 2 ab c d δωab S = ϕ δωa R e e εabc···d ˆ (n 2)! b ∧ ∧···∧ − α 2 ab eb af c d = ϕ δ ab dω ηef ω ω e e εabc···d ˆ (n 2)! ω − ∧ ∧ ∧···∧ − α 2 ab eb a ae b c d = ϕ dδω δω ω δω ω e e εabc···d ˆ (n 2)! − ∧ e − ∧ e ∧ ∧···∧ − α 2 ab c d = ϕ D δω e e εabc···d ˆ (n 2)! ∧ ∧···∧ − α 2 ab c d α a c d = D ϕ δω e e εabc···d 2ϕ (Dϕ) δω e e εabc···d ˆ (n 2)! ∧ ∧···∧ − ˆ (n 2)! b ∧ ∧···∧ − − α 2 ab c d + (n 2) ϕ δω De e εabc···d ˆ (n 2)! − ∧ ∧···∧ − ab ab ek Dec The ﬁrst term is a total divergence which we discard. Then, writing δω = δω k and noticing that is the torsion 2-form, Tc

α ab k c d 2 c d e 0 = δω e 2ϕ (Dϕ) e e εabc···d + (n 2) ϕ T e e εabcd···e ˆ (n 2)! k ∧ ∧ ∧···∧ − ∧ ∧···∧ − and therefore

2 1 k c d c k d e 0 = Dϕ e e e εabc···d + T e e e εabcd···e −n 2 ϕ ∧ ∧ ∧···∧ ∧ ∧ ∧···∧ − 2 1 m k c d 1 c m n k d e = Dmϕe e e e εabc···d + T e e e e e εabcd···e −n 2 ϕ ∧ ∧ ∧···∧ 2 mn ∧ ∧ ∧ ∧···∧ − 2 1 mkc···d 1 c mnkd···e = Dmϕε εabc···d T ε εabcd···e Φ n 2 ϕ − 2 mn − Taking the dual eliminates the volume form. Then, resolving the pairs of Levi-Civita tensors,

2 1 m k k m 0 = Dmϕ (n 2)! δ δ δ δ −n 2 ϕ − a b − a b − 1 + (n 3)!T c δm δnδk δkδn + δn δkδm δmδk + δk (δmδn δnδm) 2 − mn a b c − b c a b c − b c a b c − b c 1 k k n m k k k c 2 k k 0 = T δ T + T δ T +2δ T δ Daϕ δ Dbϕ 2 ab − b an ma b − ba a bc − ϕ b − a k k m k c 2 k k = T + δ T δ T δ Daϕ δ Dbϕ ab b ma − a cb − ϕ b − a The trace gives

m 2 0 = (n 2) T (n 1) Daϕ − ma − ϕ − 2 n 1 T m = − D ϕ ma ϕ n 2 a − so that

k k c k m 2 k k T = δ T δ T + δ Daϕ δ Dbϕ ab a cb − b ma ϕ b − a 2 n 1 k n 1 k k k = − δ Dbϕ − δ Daϕ + δ Daϕ δ Dbϕ ϕ n 2 a − n 2 b b − a − − 2 n 1 k n 1 k = − 1 δ Dbϕ − 1 δ Daϕ ϕ n 2 − a − n 2 − b − −

23 and we arrive at a solution for the torsion,

c 1 2 c c T = (δ Dbϕ δ Daϕ) (53) ab n 2 ϕ a − b − 5.2 Collected ﬁeld equations Collecting the variational ﬁeld equations, eqs.(50-53), and restoring the orthonormal basis,

4 1 1 1 1 1 c 2 − Rab ηabR = DaϕDbϕ + ηab D ϕDcϕ + m + βϕ n 2 (54) − 2 α −ϕ2 2 ϕ2

c 2 1 c c T = (δ Dbϕ δ Daϕ) (55) ab n 2 ϕ a − b − Daϕ = 0 (56) n n+2 a 2 − D Daϕ m ϕ αRϕ βϕ n 2 = 0 (57) − − − n 2 − It is interesting to note that the form of the torsion in eq.(55) is that of Einstein’s lambda transformation, which is from the most general transformation of the connection leaving the curvature unchanged. However, substituting eq.(56) into eq.(55), we see that the torsion vanishes,

c T ab = 0 Again using eq.(56) in eq.(57) get a relationship between the mass, the scalar ﬁeld and the scalar curvature,

2 m n β 4 R = ϕ n−2 − α − n 2 α − A similar but diﬀerent relation arises when we look at the trace of eq.(54) with ϕa =0,

1 1 2 2 2n R Rη η m ϕ βϕ n−2 ab ab = 2 ab + − 2 2αϕ 1 n 2 4 R = m + βϕ n−2 −α n 2 − Comparing the two,

2 m n β 4 1 n 2 n β 4 ϕ n−2 = m ϕ n−2 − α − n 2 α −α n 2 − n 2 α − 2 − − m2 = 0 n 2 − 4 β n−2 so the mass of ϕ must vanish in any dimension. However, there is no constraint on α ϕ , which in the Riemannian gauge leads to the usual Einstein equation with cosmological constant. As note following eq.(52), Dϕ = 0 requires that there exist a local gauge in which the Weyl vector vanishes. In this gauge, the Weyl-Riemann tensor reduces to the usual Riemann curvature, Ra Ra b W =0,T a =0 = b | a bc and the Einstein tensor takes the usual Riemannian form. The system has reduced to exactly the vacuum Einstein equation with cosmological constant Λ in a Riemannian geometry, 1 Rab ηabR ηabΛ = 0 − 2 − c T ab = 0

ϕ = ϕ0

Wa = 0

24 where

β 4 Λ ϕ n−2 ≡ 2α 0 the only diﬀerence being that in this formulation we retain local scale invariance with an integrable Weyl vector. In a general gauge φ, the solution takes the form: 1 Rab Rηab = 0 − 2 a T bc = 0 Wa = ∂aφ − − n 2 φ ϕ = ϕ0e 2

where this form of the scalar ﬁeld ϕ insures that Daϕ = 0. The form in a general gauge describes exactly the same physical situation, but in a set of units which may vary from place to place.

6 Conclusion

We developed the properties of Weyl geometry using the Cartan formalism for gauge theories, including enough details of the calculations to illustrate the techniques and show the advantages of the Cartan approach. Beginning with a review of the conformal properties of Riemannian spacetimes, we present an efficient form of the decomposition of the Riemann curvature into trace and traceless parts. This allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors produced by a conformal transformation. By writing the change in the Schouten tensor as a system of differential equations for the conformal factor, we reproduce the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime as a pair of integrability conditions. Continuing with the streamlined approach, we generalize this condition to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. The inclusion of matter sources is a new result. Next, enlarging the symmetry from Poincaré to Weyl, we develop the Cartan structure equations of Cartan-Weyl geometry without assuming vanishing torsion. We find the form of the curvature tensor and its relationship to the Riemann curvature of the corresponding Riemannian geometry, and show that the spin connection reproduces the expected coordinate form of Weyl connection plus contorsion tensor. We then look at possible gravity theories. We use the Gauss-Bonnet form of the Euler character to write the general form of quadratic-curvature action in terms of the Ricci tensor and scalar, then turn to a detailed description of a modified form of the Dirac scalar-tensor action. Our approacg differs from that of either Dirac or Brans-Dicke in three ways: we allow nonvanishing torsion, we vary the solder form and spin connection independently, and we work in arbitrary dimension. We find that the torsion and gradient of the scalar field must both vanish, exactly reducing the system to locally scale-covariant general relativity with cosmological constant.

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Appendix: Bianchi identities

The Cartan structure equations,

dωa = ωc ωa + Ra b b ∧ c b dea = eb ωa + ω ea + Ta ∧ b ∧ dω = Ω ν µ 1 µ ν Wµ,ν dx dx = Ωµν dx dx ∧ 2 ∧ Wµ,ν Wν,µ = Ωµν − have integrability conditions, which for gravity theories are called Bianchi identities. These follow from the Poincaré lemma, d2 0: ≡ dRa = d2ωa dωc ωa + ωc dωa b b − b ∧ c b ∧ c 0 = dRa + (Rc + ωe ωc ) ωa ωc (Ra + ωe ωa ) b b b ∧ e ∧ c − b ∧ c c ∧ e = dRa + Rc ωa Ra ωc b b ∧ c − c ∧ b DRa b = 0 d2ea = deb ωa eb dωa + dω ea ω dea + dTa ∧ b − ∧ b ∧ − ∧ 0 = ec ωb + ω eb + Tb ωa eb (ωc ωa + Ra )+ Ω ea ω eb ωa + ω ea + Ta + dTa ∧ c ∧ ∧ b − ∧ b ∧ c b ∧ − ∧ ∧ b ∧ 0 = eb Ra + Ω ea + dT a + Tb ωa ω Ta − ∧ b ∧ ∧ b − ∧ DTa = eb Ra Ω ea ∧ b − ∧ dΩ = 0

Summary:

DRa dRa + Rc ωa Ra ωc 0 b ≡ b b ∧ c − c ∧ b ≡ DTa = eb Ra Ω ea ∧ b − ∧ DΩ = 0 where

DRa dRa + Rc ωa Ra ωc b ≡ b b ∧ c − c ∧ b DTa dTa + Tb ωa ω Ta ≡ ∧ b − ∧ DΩ dΩ ≡ When the torsion vanishes, we have an algebraic identity,

ebRa = Ω ea b ∧ Ra a [bcd] = δ[bΩcd] Ra Ra Ra a a a bcd + cdb + dbc = δb Ωcd + δc Ωdb + δd Ωbc Rbd Rbd = (n 2)Ωbd − − −