Pressuron

The action of the scalar–tensor theory that involves the pressuron ${\displaystyle \Phi }$ can be written as

${\displaystyle S={\frac {1}{c}}\int d^{4}x{\sqrt {-g}}\left[{\sqrt {\Phi }}{\mathcal {L}}_{m}(g_{\mu \nu },\Psi )+{\frac {1}{2\kappa }}\left(\Phi R-{\frac {\omega (\Phi )}{\Phi }}(\partial _{\sigma }\Phi )^{2}-V(\Phi )\right)\right],}$

where ${\displaystyle R}$ is the Ricci scalar constructed from the metric ${\displaystyle g_{\mu \nu }}$, ${\displaystyle g}$ is the metric determinant, ${\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}}$, with ${\displaystyle G}$ the gravitational constant^[4] and ${\displaystyle c}$ the velocity of light in vacuum, ${\displaystyle V(\Phi )}$ is the pressuron potential and ${\displaystyle {\mathcal {L}}_{m}}$ is the matter Lagrangian^[5] and ${\displaystyle \Psi }$ represents the non-gravitational fields. The gravitational field equations therefore write^[2]

${\displaystyle R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R=\kappa ~{\frac {1}{\sqrt {\Phi }}}T_{\mu \nu }+{\frac {1}{\Phi }}[\nabla _{\mu }\nabla _{\nu }-g_{\mu \nu }\Box ]\Phi +{\frac {\omega (\Phi )}{\Phi ^{2}}}\left[\partial _{\mu }\Phi \partial _{\nu }\Phi -{\frac {1}{2}}g_{\mu \nu }(\partial _{\alpha }\Phi )^{2}\right]-g_{\mu \nu }{\frac {V(\Phi )}{2\Phi }},}$

and

${\displaystyle {\frac {2\omega (\Phi )+3}{\Phi }}\Box \Phi =\kappa {\frac {1}{\sqrt {\Phi }}}\left(T-{\mathcal {L}}_{m}\right)-{\frac {\omega '(\Phi )}{\Phi }}(\partial _{\sigma }\Phi )^{2}+V'(\Phi )-2{\frac {V(\Phi )}{\Phi }}}$.

where ${\displaystyle T_{\mu \nu }}$ is the stress–energy tensor of the matter field, and ${\displaystyle T=g^{\mu \nu }T_{\mu \nu }}$ is its trace.

If one considers a pressure-free perfect fluid (also known as a dust solution), the effective material Lagrangian becomes ${\displaystyle {\mathcal {L}}_{m}=-c^{2}\sum _{i}\mu _{i}\delta (x_{i}^{\alpha })}$,^[6] where ${\displaystyle \mu _{i}}$ is the mass of the ith particle, ${\displaystyle x_{i}^{\alpha }}$ its position, and ${\displaystyle \delta (x_{i}^{\alpha })}$ the Dirac delta function, while at the same time the trace of the stress-energy tensor reduces to ${\displaystyle T=-c^{2}\sum _{i}\mu _{i}\delta (x_{i}^{\alpha })}$. Thus, there is an exact cancellation of the pressuron material source term ${\displaystyle \left(T-{\mathcal {L}}_{m}\right)}$, and hence the pressuron effectively decouples from pressure-free matter fields.

In other words, the specific coupling between the scalar field and the material fields in the Lagrangian leads to a decoupling between the scalar field and the matter fields in the limit that the matter field is exerting zero pressure.

The pressuron shares some characteristics with the hypothetical string dilaton,^[3][7] and can actually be viewed as a special case of the wider family of possible dilatons.^[8] Since perturbative string theory cannot currently give the expected coupling of the string dilaton with material fields in the effective 4-dimension action, it seems conceivable that the pressuron may be the string dilaton in the 4-dimension effective action.