Second-Order Biconf ormally Invariant Scalar-Tensor F ield Theories in a F our-Dim ensional Space

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Second-Order, Biconformally Invariant

Scalar-Tensor Field Theories

in a Four-Dimensional Space

Gregory W. Horndeski

Adjunct Associate Professor of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada

N2L 3G1

email:

ghorndeski@uwaterloo.ca

horndeskimath@gmail.com

October 9, 2022

ABSTRACT

In this paper I shall consider field theories in a space of four-dimensions which have field

variables consisting of the components of a metric tensor and scalar field. The field equations of

these scalar-tensor theories will be derived from a variational principle using a Lagrange scalar

density which is a concomitant of the field variables and their derivatives of arbitrary, but finite,

order. I shall consider biconformal transformations of the field variables, which are conformal

transformations which affect both the metric tensor and the scalar field. A necessary and sufficient

condition will be developed to determine when the Euler-Lagrange tensor densities are biconformally

invariant. This condition will be employed to construct all of the second-order biconformally

invariant scalar-tensor field theories in a space of four-dimensions. It turns out that the field

equations of these theories can be derived from a linear combination of (at most) two second-order

Lagrangians, with the coefficients in that linear combination being real constants.

SECTION 1: INTRODUCTION

In June, 2022 Prof. Paul J.Steinhardt contacted me with a problem. He wanted to know what

all of the conformally invariant second-order scalar-tensor field theories were, since that would assist

him and his students in their endeavors to construct models of bouncing universes. Well, in Ref. 1,

I constructed all such field theories of arbitrary differential order, and I directed him to that result.

His response indicated that he was not interested in conformal transformations that only affected the

metric, but wanted the transformation to also affect the scalar field. I did not know the answer to that

question, and the purpose of this paper is to provide the answer when the field tensor densities are

of second-order.

To begin, let me introduce some terminology.

When I refer to a theory as a scalar-tensor field theory that means that the field variables are

the local components of a pseudo-Riemannian metric tensor, gab, and a scalar field ö, and that the

field equations are derivable from a Lagrangian which is a scalar density tensorial concomitant of

the form

L = L(gab; gab,c;. . . ; ö; ö,c; . . . ) Eq.1.1

which is of finite differential order and a comma is used to denote partial differentiation with respect

to the local coordinates employed to compute the metric tensor’s components. The associated Euler-

Lagrange tensor densities are defined by

Eab(L) / äL := ML ! d ML + . . . Eq.1.2a

ägab Mgab dxc Mgab,c

E(L) / äL := ML ! d ML + . . . . Eq.1.2b

äö Mö dxc Mö,c

In Ref. 2, I show that in a space of any dimension the Euler-Lagrange tensor densities are not

independent but are related by the identity

b(L)|b = ½ö,aE(L) Eq.1.3

where the vertical bar denotes covariant differentiation with respect to the Levi-Civita connection

determined by the metric tensor, and repeated indices are summed. The definitions of Eab(L) and

E(L) given above differ from those I used in Refs. 2 and 3, by a minus sign, but that will not prove

to be an obstacle in what is done here. In addition the geometric quantities that I shall use here are

defined as they were in Refs. 2 and 3, with Vi

|jk ! Vi

|kj := VhRh

jk , where Vi is an arbitrary vector

field, Rhj:=Rh

ji and R:= ghjRhj . I shall also let öa:=ö,a ; öab:= ö|ab , Gö:= gaböab and g:= |det(gab)|.

For scalar-tensor field theories I define a biconformal transformation of degree k (k0ú) by:

gab÷g'ab := e2ógab and ö ÷ ö':= ekóö Eq.1.4

where ó is an arbitrary scalar field.

If L is a Lagrangian of the form given in Eq.1.1, then the transformed Lagrangian L' is

obtained from L by simply replacing gab, ö and their derivatives by g'ab, ö' and their derivatives. L

is said to be conformlly invariant if

L'(g'ab; g'ab,c; . . . ; ö'; ö',c; . . .) = L(gab; gab,c; . . . ;ö; ö,c; . . . ) . Eq.1.5

This means that if you take L as in Eq.1.1 then

L(e2ógab; (e2ógab),c; . . . ; ekóö; (ekóö),c; . . . ) = L(gab; gab,c; . . . ; ö; ö,c; . . . ) . Eq.1.6

When I say that a Lagrangian L of the form given in Eq.1.1 is biconformally invariant up to a

divergence I mean that there exists a vector density concomitant Vi built from gab, ö, ó and their

derivatives which is such that Eq.1.5 is replaced by

L' = L + d Vi . Eq.1.7

dxi

A well-known property of the Euler-Lagrange operator is that it annihilates divergences. (For a direct

proof of this fact see Lovelock [4].) In this case we can use Eq.1.7 to write

ä L' = ä (L + d Vi ) = ä (L + d Vi) ägrs . Eq.1.8

äg'ab äg'ab dxi ägrs dxi äg'ab

Since

ägrs = Mgrs = ½ e!2ó(äa

räb

s + äa

säb

äg'ab Mg'ab

(for those unfamiliar with the process of differentiating tensor concomitants please see Appendix A

in [1]) Eq.1.8 tells us that

Eab(L)' / äL' = e!2ó äL = e!2ó Eab(L) . Eq.1.9

äg'ab ägab

Similarly, Eq.1.7 tells us that when L is biconformally invariant up to a divergence then

E(L)' / äL' = e!kó äL = e!kó E(L) . Eq.1.10

äö' äö

Eqs. 1.9 and 1.10 allow us to deduce that when a scalar-tensor field theory is such that its Lagrangain

is biconformally invariant of degree k up to a divergence then

b(L)' = g'bcEac(L)' = gbcEac(L) = Ea

b(L) Eq.1.11

and

ö'E(L)' = öE(L) . Eq.1.12

When I say that the field tensor densities of a scalar-tensor field theory are biconformally

invariant I mean that they satisfy Eqs. 1.11 and 1.12.

We require an easy way to determine when the field tensor densities of a scalar-tensor field

theory are biconformally invariant of degree k. This is provided by the following:

Proposition 1: Let Eab(L) and E(L) be the Euler-Lagrange tensor densities of a scalar-tensor field

theory. This theory will be biconformally invariant of degree k if and only if

2gabEab(L) + köE(L) = 0 . Eq.1.13

If Eab(L) and E(L) satisfy Eq.1.13 then L is biconformally invariant of degree k up to a divergence.

Proof: The proof is modeled on the proof of Proposition 2.1 in Ref. 1.

Y The Euler-Lagrange tensor densities Ea

b(L)' and E(L)' satisfy the primed version of Eq.1.3; viz.,

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Second-Order,BiconformallyInvariantScalar-TensorFieldTheoriesinaFour-DimensionalSpacebyGregoryW.HorndeskiAdjunctAssociateProfessorofAppliedMathematicsUniversityofWaterlooWaterloo,OntarioCanadaN2L3G1email:ghorndeski@uwaterloo.caorhorndeskimath@gmail.comOctober9,20221ABSTRACTInthispaperIshallconsiderf...

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Second-Order Biconf ormally Invariant Scalar-Tensor F ield Theories in a F our-Dim ensional Space

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