Note on simple(r) equations for Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons theories

Special thanks to Helvi Witek, who originally showed me one simplified form of the “C-tensor” in Einstein-dilaton-Gauss-Bonnet (EDGB).

If you’ve ever looked into theories beyond general relativity, you’re already aware that their field equations can be very complicated. For example, here’s a clip from one paper on EDGB:

Scary equations for EDGB

Ack! That’s pretty unwieldy. But don’t despair, it turns out that the above mess can be written much more compactly.

Just to set conventions, let’s work with the action

\begin{align} \label{eq:action1} S = \int d^4x \sqrt{-g} \left[ \frac{1}{2}m_{pl}^2 R - \frac{1}{2} (\cd^a \vartheta) (\cd_a \vartheta) \right] + S_{int} \end{align}

where is a scalar (dilaton or axion) and is a non-minimal interaction term between the scalar and curvature.


For EDGB, let’s take

\begin{align} \label{eq:SEDGB} S_{int}^{EDGB} = \frac{1}{8} \varepsilon \int d^4x \sqrt{-g} F(\vartheta) \left[ R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd} \right] \end{align}

with some arbitrary coupling function F, and some dimensionless perturbation parameter . Now this above curvature combination might seem arbitrary, but it’s actually the 4-dimensional Euler density (see e.g. Bob McNees’s notes). It’s more natural to write that as

\begin{align} \label{eq:euler4} \ddR_{abcd}R^{abdc} = R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd}. \end{align}

Here we’ve defined the double-dual of the Riemann tensor. First, we dualize on the left two antisymmetric indices to define the left-dual,

\begin{align} \label{eq:leftdual} \dR^{abcd} \equiv \frac{1}{2} \epsilon^{abef} R_{ef}{}^{cd}, \end{align}

and then we further dualize on the right two antisymmetric indices to get the double-dual,

\begin{align} \label{eq:doubledual} \ddR^{abcd} \equiv \dR^{ab}{}_{gh} \frac{1}{2} \epsilon^{ghcd} = \frac{1}{2} \epsilon^{abef} R_{efgh} \frac{1}{2} \epsilon^{ghcd}. \end{align}

Now, it’s an exercise in algebraic manipulation to show that the equation for the metric from the action defined in Eqs. \eqref{eq:action1}, \eqref{eq:SEDGB} is given simply by

\begin{align} \label{eq:eom-EDGB} \boxed{ m_{pl}^2 G_{ab} + \varepsilon \cd^c \cd^d \left[ \ddR_{cabd} F(\vartheta) \right] = T_{ab} } \end{align}

where is the stress-energy tensor for matter plus the stress-energy tensor for the scalar field. That’s quite a bit simpler than the image above, isn’t it! If you want to get xTensor to verify this for you, grab EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb from the xAct examples collection.

I wrote the above in a certain way to make it look very similar to the case of dynamical Chern-Simons (DCS, below), but before moving on—recall that one reason people like EDGB is that the equations of motion are only second order in the metric. That’s not obvious from the way I wrote it, because it looks like you might get third and fourth derivatives of the metric. However, one nice property of the double-dual of Riemann is that it’s divergence free (see MTW Eq. (13.51) and exercise 13.11). This means we can rewrite

\begin{align} \label{eq:C-tensor-EDGB-idents} \cd^c \cd^d \left[ \ddR_{cabd} F(\vartheta) \right] = \cd^c \left[ \ddR_{cabd} \cd^d F(\vartheta) \right] = \ddR_{cabd} \cd^c \cd^d F(\vartheta). \end{align}

Now it’s obvious that there are only second derivatives of the metric. However, the first or second forms might give more insight, because from them you can see that this so-called “C-tensor” is itself the divergence of some tensor. That’s the kind of thing you might want to integrate over a region…

Dynamical Chern-Simons

Anyway, on to DCS. Now we use the interaction term

\begin{align} \label{eq:SDCS} S_{int}^{DCS} = \frac{1}{8} \varepsilon \int d^4x \sqrt{-g} F(\vartheta) \dR^{abcd} R_{abdc} \end{align}

with just a single dual. Again it looks kind of arbitrary, but when is integrated over the whole manifold, you get a topological invariant.

The equation for the metric in DCS is also kind of scary looking, but again some algebra shows that you can write it as

\begin{align} \label{eq:eom-DCS} \boxed{ m_{pl}^2 G_{ab} + \varepsilon \cd^c \cd^d \left[ \dR_{c(ab)d} F(\vartheta) \right] = T_{ab} } \end{align}

where means that we are symmetrizing (with a factor of 1/2) on those two indices. This looks very similar to the expression for EDGB! However, the single-dual of Riemann is only manifestly divergence-free on the left two indices, so this equation does have third derivatives of the metric. The double-divergence does vanish, so there are no fourth derivatives.

All of the above calculations are in my notebook EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb in the xAct examples collection. Hope you learned something!