aa r X i v : . [ g r- q c ] D ec Gravitational Field Tensor
Stephen M. Barnett ∗ School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK (Dated: September 22, 2018)We present a tensorial relative of the familiar affine connection and argue that it should beregarded as the gravitational field tensor. Remarkably, the Lagrangian density expressed in termsof this tensor has a simple form, which depends only on the metric and its first derivatives and,moreover, is a true scalar quantity. The geodesic equation, moreover, shows that our tensor plays arole that is strongly reminiscent of the gravitational field in Newtonian mechanics and this, togetherwith other evidence, which we present, leads us to identify it as the gravitational field tensor.We calculate the gravitational field tensor for the Schwarzschild metric. We suggest some of theadvantages to be gained from applying our tensor to the study of gravitational waves.
PACS numbers: 04.20.Cv, 04.30.-w
That physical quantities should be represented as ten-sors is one of the fundamental ideas in the general theoryof relativity. It is this way that we arrive naturally ata theory that applies in any coordinate system and forany state of motion. It comes as something of a surprise,therefore, to discover in one’s first course on the subjectthat the affine connection Γ λµν , or Christoffel symbol ofthe second kind, is not a tensor. Despite this shortcom-ing, it appears naturally in the geodesic equation for themotion of a particle playing a role akin to the electro-magnetic field tensor, F µν , in the equation of motion fora charged particle. In this sense, at least, the connectionis a gravitational analogue of the electromagnetic field.The geodesic equation illustrates clearly, however, thatthe affine connection cannot be a tensor, for were it tobe a tensor then the existence of a local inertial frame,in which the affine connection vanishes would necessar-ily require the tensor itself to be zero in all coordinatesystems.It is well-known that although the affine connectionis not a tensor, the difference between any two affineconnections is a tensor [1]. Consider the tensor quantitydefined to be the difference between the two connections,Γ λµν and ˜Γ λµν , expressed in terms of a common set ofcoordinates: ∆ λµν = Γ λµν − ˜Γ λµν . (1)Each connection is derived from a metric, g µν , ˜ g µν , inthe usual way [2]. We choose g µν to be the usual metrictensor for our space–time, and thus Γ λµν is the true con-nection. We leave unspecified, for the present, the preciseform of ˜ g µν and thus of ˜Γ λµν , other than to state that weshall choose it to correspond to a metric for a flat spaceso that the Ricci tensor vanishes ( ˜ R µν = 0) [3]. Our taskwill be to investigate the role of ∆ λµν within the gen-eral theory of relativity. We work throughout with thenatural system of units in which Newton’s gravitationalconstant, G , and the speed of light, c , are both set tounity.We note, first, that the difference between the Riemann curvature and that for a space–time with connection ˜Γ λµν has a natural and simple expression in terms of ∆ λµν [1]: R βνρσ − ˜ R βνρσ = ∆ βνσ ; ρ − ∆ βνρ ; σ +∆ ανσ ∆ βαρ − ∆ ανρ ∆ βασ , (2)where ; denotes covariant differentiation and we follow,in this manuscript, the conventions adopted by Dirac [2].This formula is reminiscent of the expression for the Rie-mann tensor written in terms of derivatives and productsof the connection but has the important feature that eachterm in it is a tensor. This follows directly from the factthat ∆ λµν is a tensor. A correspondingly simple expres-sion for the difference in the Ricci tensor follows directlyfrom that for the difference in the Riemann tensor: R νρ − ˜ R νρ = ∆ βνβ ; ρ − ∆ βνρ ; β +∆ ανβ ∆ βαρ − ∆ ανρ ∆ βαβ . (3)As we have selected ˜Γ λµν such that ˜ R µν = 0, this is alsoan expression for the Ricci tensor, R νρ , alone. A furthercontraction using the inverse of the metric, g νρ , gives,naturally enough, the curvature scalar.As a first application of the above ideas, we considerthe action for the gravitational field, which is usuallygiven in the Einstein-Hilbert form [2, 4, 5]: I EH = 116 π Z d x √− g R , (4)where R = g µν R µν is the curvature scalar. Variationof this action leads, of course, to the free-field Einsteinequation, but the task of extracting this is complicatedby the fact that the Lagrangian density depends on themetric, its first derivatives and also second derivatives.It is possible to use integration by parts to remove thesecond derivatives and thereby simplify the derivation.This widely adopted procedure leads to the Lagrangiandensity [2, 6–9]: L D = 116 π g µν (cid:0) Γ σµν Γ ρσρ − Γ ρµσ Γ σνρ (cid:1) . (5)This is not a scalar quality, however, by virtue of thefact that the affine connection is not a tensor [2, 6, 9] butwas, nevertheless, used as the staring point in Dirac’sHamiltonian formulation of gravitation [10]. If we use(3) in place of the familiar expression for the Ricci tensor,however, then a similar integration by parts leads to theLagrangian density [11]: L ∆ = 116 π g µν (cid:0) ∆ αλα ∆ λµν − ∆ λαµ ∆ αλν (cid:1) . (6)That this is a scalar follows directly from the fact that∆ λµν is a tensor. It is also clear that this form of theLagrangian density depends only on the metric and itsfirst derivatives. The highly desirable situation of havingonly first order derivatives, and therefore straightforwardEuler-Lagrange equations, has motivated the search foralternative Lagrangians, yet none of the existing rivalswe are aware of have the simplicity of L ∆ [12, 13]. Wenote, in particular, that it is bilinear in the tensor ∆ λµν ,much as the electromagnetic Lagrangian density is bilin-ear in the electromagnetic field tensor. There is a longhistory of attempts to formulate an action principle onthe basis of bilinear combinations of the curvature , as op-posed to the linear form embodied in the Einstein-Hilbertaction [14, 15], and also to express other physically sig-nificant properties in this way [16–19]. Our form for theLagrangian density, however, emphasises the role of thetensor ∆ λµν rather than the curvature. We elaborate onthis point further below, but consider first an example ofthe explicit the form of our tensor and of its applicationto the study of gravitational waves.Working with ∆ λµν rather than the connection or themetric itself may present significant calculational advan-tage, by making a suitable choice of flat-space metric˜ g µν . As a simple illustration of this we consider theSchwarzschild metric corresponding to a mass m localisedat the origin, with the tilde space corresponding to theabsence of this mass. If we employ the natural sphericalpolar coordinates then there are 9 non-vanishing distinctcomponents of the connection [2] but only 5 non-zero dis-tinct components of ∆ λµν :∆ = mr (cid:18) − mr (cid:19) ∆ = − mr (cid:18) − mr (cid:19) − ∆ = 2 m ∆ = 2 m sin θ ∆ = mr (cid:18) − mr (cid:19) − , (7)where the coordinates are numbered as (0 , , ,
3) =( t, r, θ, φ ). The corresponding Ricci tensor is zero (awayfrom the origin) as is the curvature scalar. Proving this using our tensor is far simpler than using the affine con-nection as 16 π L ∆ , ∆ λµλ and ∆ λµν ; λ g µν , which combineto give R , are each separately zero.The fact that the tensor ∆ λµν is the difference betweenconnections suggests, in particular, application to situa-tions in which it is appropriate to linearize about a back-ground metric and hence to the theory of gravitationalwaves [4, 20]. The linearization usually leads to a waveequation for the small deviation of the metric associatedwith the wave, but its derivation requires the choice of asuitable gauge choice (the transverse traceless gauge) ina particular coordinate system [20]. Gravitational wavespropagating in a region of free space satisfy R µν = 0.Use of the tensors ∆ λµν and working to first order thenleads directly to the equation∆ αµα ; ν − ∆ αµν ; α = 0 . (8)If we rewrite this in terms of the small correction to themetric in a Minkowski background and choose the trans-verse traceless gauge then this becomes the familiar grav-itational wave-equation. It should be emphasised, how-ever, that this equation is a tensor equation and henceapplies in any coordinate system. The tensor ∆ λµν is,moreover, gauge-invariant at least to this lowest order.To see this we need only note that a gauge transforma-tion corresponds to a local coordinate transformation [21]and that we must, therefore transform both g µν and ˜ g µν : g µν → g µν − ξ µ,ν − ξ ν,µ ˜ g µν → ˜ g µν − ξ µ,ν − ξ ν,µ (9)where the , denotes differentiation as usual [2]. Hence ourgravitational wave equation (8) is both a tensor equation,and hence holds in any coordinate system, and it is alsogauge-invariant. In this sense it should be viewed as thenatural analogue of the free-field Maxwell equations forelectromagnetic waves [22]. The symmetries and conser-vation laws for gravitational waves may be lifted from thesymmetries of the background metric, ˜ g µν , by applyingNoether’s theorem [23] to the action I = 116 π Z d x p − ˜ g ˜ g µν (cid:0) ∆ αλα ∆ λµν − ∆ λαµ ∆ αλν (cid:1) , (10)where the tensors ∆ λµν are restricted to first order in thedifference between ˜ g µν and g µν .It remains to make explicit the case for referring to thetensor ∆ λµν as the gravitational field tensor. There arethree strong indications of this and the combination ofthese is compelling. These are (i) the form of the geodesicequation for the motion of a test particle, (ii) comparisonwith corresponding quantities in electromagnetism and(iii) the existence of an analogy with Yang-Mills theories.Let us take each of these in turn. (i) The geodesic equation . The geodesic equation forthe motion of a test particle is [2] d x λ dτ = − Γ λµν dx µ dτ dx ν dτ = − ∆ λµν dx µ dτ dx ν dτ − ˜Γ λµν dx µ dτ dx ν dτ . (11)In this equation of motion the term containing ˜Γ λµν de-scribes the motion as it would be in the absence of thebody or bodies responsible for the curvature. The termcontaining the tensor ∆ λµν provides the modification ofthis motion due to the presence of the gravitating bodies,that is the gravitational field. In this sense it providesthe natural analog of the Newtonian gravitational force,as encapsulated in Newton’s first law of motion [24], inthat it induces the deviation from the uniform motionassociated with ˜Γ λµν acting alone. (ii) Comparison with electromagnetism . Analogies be-tween electromagnetism and gravitation have often beenapplied as an aid to understanding and teaching phenom-ena within general relativity [9, 22]. This analogy en-hances the case for identifying ∆ λµν as the gravitationalfield tensor. To see this we recall that in electromag-netism we introduce the four-potential A µ from whichwe can form the field F µν by differentiation. Finally thefields are coupled to charged matter through Maxwell’sequations in which the derivatives of the fields appear.All of these quantities, the four-potential, the field andits derivatives are tensor quantities [25].A strongly analogous scheme for gravitational fieldstreats the metric, g µν , as a potential. From this we ob-tain the connection, Γ λµν , by differentiation and thence,via further differentiation, the Riemann tensor and theequation of motion expressed in terms of the Ricci tensor.As with electromagnetism, each of these are tensors withthe exception of the connection. But for this, it would benatural in this scheme to associate the connection withthe gravitational field, in analogy with the electromag-netic field tensor, F µν . We can complete the analogywith electromagnetism by replacing the affine connectionwith the gravitational field tensor ∆ λµν . Like the con-nection, it is obtained from the metric by differentiationand further differentiation of it leads to the Riemann ten-sor and to the Ricci tensor, which is coupled to mattersources in the Einstein field equation. The relationshipsbetween these quantities are depicted in the tables.In table I we present the electromagnetic potential andfield together with the governing Maxwell equation thatcouples the fields to the material sources. In the sec-ond line we have the analogous quantities for gravity.Note the appearance of the connection, which is the onlyquantity in the table that is not a tensor. In table II theconnection is replaced by our gravitational field tensor,so that every quantity in the table is a tensor. (iii) Analogy with other field theories . Finally, andmost speculatively, we note that the Lagrangian densitywhen expressed in terms of the tensor ∆ λµν is bilinear. Potential Field? Field Equation A µ F µν F µν ; ν = j µ g µν Γ λµν G µν = − πT µν TABLE I. The familiar quantities in electromagnetism and ingravitation. In both cases we progress from potential to fieldto field equation by differentiation.Potential Field Tensor Field Equation A µ F µν F µν ; ν = j µ g µν ∆ λµν G µν = − πT µν TABLE II. A more natural assignment in which the non-tensorial connection is replaced by the tensor ∆ λµν . In this sense it is reminiscent of the Lagrangian densityfor Yang-Mills theories and electromagnetism [26, 27]: L Y M = − F αβ · F αβ . (12)Although our Lagrangian density, L ∆ , is not explicitly ofthis form, it is bilinear in the gravitational field tensor.The formal similarity with Yang-Mills theories can bemade yet stronger if we add to Eq. (12) the zero-valuedquantity F λλ · F µµ . It is possible that this analogy(or similarity in form) between L ∆ and the Yang-MillsLagrangian density might suggest new directions in thestudy of quantum effects in gravity.I am grateful to John Cameron, Rob Cameron, JimCresser, Sarah Croke, Claire Gilson, Norman Gray, Mar-tin Hendry, Fiona Speirits, Graham Woan and AlisonYao for helpful comments and suggestions. ∗ [email protected][1] L. P. Eisenhardt, Riemannian Geometry (Princeton Uni-versity Press, Princeton NJ, 1926).[2] P. A. M. Dirac,
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