Canonical Formulation of Embedding Gravity in a Form of General Relativity with Dark Matter

Abstract

We study embedding gravity, a modified theory of gravity in which our space-time is assumed to be a four-dimensional surface in flat ten-dimensional space. Based on a simple geometric idea, this theory can be reformulated as general relativity with additional degrees of freedom and a contribution to action which can be interpreted as describing dark matter. We study the canonical formalism for such a formulation of embedding gravity. After solving simple constraints, the Hamiltonian is reduced to a linear combination of four first-class constraints with Lagrange multipliers. There still remain six pairs of second-class constraints. Possible ways of taking these constraints into account are discussed. We show that one way of solving the constraints leads to a canonical system going into the previously known canonical formulation of the complete embedding theory with an implicitly defined constraint.

Appendix

THE EXPLICIT FORM OF THE $\Upsilon_{ij}$ CONSTRAINT

The explicit form of the $\Upsilon_{km}$ constraint is as follows:

$$\Upsilon_{km}=\Upsilon_{km}^{(1)}-\phi^{ij}\Upsilon_{km,ij}^{(2)},$$

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where

$$\Upsilon_{km}^{(1)}=\bigg{(}L_{km,ij}+4\zeta\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}^{a}_{km}\frac{\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{\Pi}}^{c}_{a\perp}+n_{a}n^{c}}{p_{\perp}}\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}_{cij}\bigg{)}$$

$${}\times\beta^{ir}\widetilde{\beta}^{jq}e^{b}_{r}p_{b}N_{q}$$

$${}-2\zeta\frac{\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}^{a}_{km}}{p_{\perp}}\partial_{q}(Nn_{a}p_{b}e^{bq})$$

$${}+\bigg{(}2n_{c}e^{q}_{b}\partial_{k}\partial_{m}y^{b}+2\zeta\frac{p_{a}\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}_{kmc}}{p_{\perp}}e^{aq}\bigg{)}\partial_{q}(Nn^{c})$$

$${}+2n_{a}\partial_{k}\partial_{m}(Nn^{a})+2\zeta\frac{n_{a}\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}^{a}_{km}}{p_{\perp}}p_{b}e^{bq}\partial_{q}N$$

$${}+2\zeta\frac{n_{a}\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}^{a}_{km}}{p_{\perp}}\partial_{q}(Np_{b}e^{bq})$$

$${}+\frac{4\varkappa}{\sqrt{\beta}}N\pi^{lq}\pi^{ij}\bigg{(}\beta_{mj}\overline{L}_{ki,lq}+\beta_{ki}\overline{L}_{mj,lq}$$

$${}-\frac{3}{8}\beta_{km}\overline{L}_{ij,lq}+\frac{1}{2}\beta_{lq}\overline{L}_{km,ij}-\frac{1}{2}\beta_{qj}\overline{L}_{km,li}\bigg{)}$$

$${}-\frac{N\sqrt{\beta}}{2\varkappa}\overline{L}_{km,rs}\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{G}}^{rs}+4\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{D}}_{k}\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{D}}_{m}N$$

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and

$$\Upsilon_{km,ij}^{(2)}=L_{km,ij}-4\zeta\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}^{a}_{km}\frac{\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{\Pi}}^{c}_{a\perp}+n_{a}n^{c}}{p_{\perp}}\stackrel{{\scriptstyle\scriptscriptstyle 3}}{{b}}_{cij}.$$

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