how to deal with lovelock gravity?

Question

For Lovelock gravity,

$$L=R_{a_{2}b_{2}}^{c_{2}d_{2}}\cdots R_{a_{m}b_{m}}^{c_{m}d_{m}}\delta_{c_{2}d_{2}\cdots c_{m}d_{m}}^{a_{2}b_{2}\cdots a_{m}b_{m}}$$

where

$$ \delta_{{jc_{1}d_{1}...c_{m}d_{m}}}^{{ia_{1}b_{1}...a_{m}b_{m}}}=\det\left[\begin{array}{c|ccc}\delta_{j}^{i} & \delta_{{c_{1}}}^{i} & \cdots & \delta_{{d_{m}}}^{i}\\hline \\delta_{j}^{{a{1}}}\\vdots & & \delta_{{c_{1}d_{1}...c_{m}d_{m}}}^{{a_{1}b_{1}...a_{m}b_{m}}}\\delta_{j}^{{b\{m}}}\end{array}\right] $$

when $m=2$, we get Einstein gravity; when $m=3$, we get Gauss-Bonnet gravity. But how to deal with it more generally? More concretely, I don't know how to do it use for loop.

Answer 1

Hi Eureka.

In the article here, we provide an algorithm to find the Lagrangian and field equations of the Lanczos--Lovelock models.

At the time we wrote the article going beyond $m=4$ would take a virtually infinite amount of time.

I invite you (or any person interested) to go through our algorithm, and we could discuss ways to improve its performance.

Cheers, Dox (aka Oscar)

Comments

Thanks for your infromation, it's useful. I'm interested in your algorithm, and discussion is welcome.

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Hi Eureka.

Please contact me through email, to continue our discussion.

Cheers.

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Ok , I have sent you a email for this.