General relativity as an effective theory

3.1 General relativity as an effective theory

The low-energy degrees of freedom in this case are the metric $gμν$ of spacetime itself. As has been seen in previous sections, Einstein’s action for this theory should be considered to be just one term in a sum of all possible interactions which are consistent with the symmetries of the low-energy theory (which in this case are: general covariance and local Lorentz invariance). Organizing the resulting action into powers of derivatives of the metric leads to the following effective Lagrangian:

Here $R μνλρ$ is the metric’s Riemann tensor, $R μν$ is its Ricci tensor, and $R$ is the Ricci scalar, each of which involves precisely two derivatives of the metric. For brevity only one representative example of the many possible curvature-cubed terms is explicitly written. (We use here Weinberg’s curvature conventions [146

], which differ from those of Misner, Thorne, and Wheeler [121 ] by an overall sign.)

The first term in Equation (28 ) is the cosmological constant, which is dropped in what follows since the observed size of the universe implies $λ$ is extremely small. There is, of course, no real theoretical understanding why the cosmological constant should be this small (a comprehensive review of the cosmological constant problem is given in [151 ]; for a recent suggestion in the spirit of effective field theories see [3 , 29 ]). Once the cosmological term is dropped, the leading term in the derivative expansion is the one linear in $R$ , which is the usual Einstein–Hilbert action of general relativity. Its coefficient defines Newton’s constant (and so also the Planck mass, $M p−2= 8πG$ ).

The explicit mass scales, $m$ and $Mp$ , are introduced to ensure that the remaining constants $a$ , $b$ , $c$ , $d$ and $e$ appearing in Equation (28 ) are dimensionless. Since it appears in the denominator, the mass scale $m$ can be considered as the smallest microscopic scale to have been integrated out to obtain Equation (28 ). For definiteness we might take the electron mass, $m = 5 × 10 −4 GeV$ , for $m$ when considering applications at energies below the masses of all elementary particles. (Notice that contributions like $2 m R$ or $3 2 R ∕M p$ could also exist, but these are completely negligible compared to the terms displayed in Equation (28 ).)

3.1.1 Redundant interactions

As discussed in the previous section, some of the interactions in the Lagrangian (28 ) may be redundant, in the sense that they do not contribute independently to physical observables (like graviton scattering amplitudes about some fixed geometry, say). To eliminate these we are free to drop any terms which are either total derivatives or which vanish when evaluated at solutions to the lower-order equations of motion.

The freedom to drop total derivatives allows us to set the couplings $d$ and $e$ to zero. We can drop $e$ because $√ −-g□R = ∂μ[√ − g-∇ μR ]$ , and we can drop $d$ because the quantity

integrates to give a topological invariant in 4 dimensions. That is, in 4 dimensions $∫ √ -- χ (M ) = (1∕32π2) M gX d4x$ gives the Euler number of a compact manifold $M$ – and so $X$ is locally a total derivative. It is therefore always possible to replace, for example, $RμνλρR μνλρ$ in the effective Lagrangian with the linear combination $4R R μν − R2 μν$ , with no consequences for any observables which are insensitive to the overall topology of spacetime (such as the classical equations, or perturbative particle interactions). Any such observable therefore is unchanged under the replacement $(a,b,d) → (a′,b′,d′) = (a − d,b + 4d,0 )$ .

The freedom to perform field redefinitions allows further simplification (just as was found for the toy model in earlier sections). To see how this works, consider the infinitesimal field redefinition $δgμν = Yμν$ , under which the leading term in $ℒeff$ undergoes the variation

In particular, we may set the constants $a$ and $b$ to zero simply by choosing $2 M p Y μν = 2aR μν − (a + 2b )Rg μν$ . Since the variation of the lower-order terms in the action are always proportional to their equations of motion, quite generally any term in $ℒeff$ which vanishes on use of the lower-order equations of motion can be removed in this way (order by order in $1∕m$ and $1∕M p$ ).

Since the lowest-order equations of motion for pure gravity (without a cosmological constant) imply $R μν = 0$ , we see that all of the interactions beyond the Einstein–Hilbert term which are explicitly written in Equation (28 ) can be removed in one of these two ways. The first interaction which can have physical effects (for pure gravity with no cosmological constant) in this low-energy expansion is therefore proportional to the cube of the Riemann tensor.

This last conclusion changes if matter or a cosmological constant are present, however, since then the lowest-order field equations become $R = S μν μν$ for some nonzero tensor $S μν$ . Then terms like $R2$ or $μν R μνR$ no longer vanish when evaluated at the solutions to the equations of motion, but are instead equivalent to interactions of the form $(S μμ)2$ , $S μνRμν$ , or $SμνS μν$ . Since some of our later applications of $ℒeff$ are to the gravitational potential energy of various localized energy sources, we shall find that these terms can generate contact interactions amongst these sources.