Demonstration of the Brans-Dicke's Lagrangian

[fab13]

TL;DR Summary

I try to find the formula of the Brans-Dicke's Lagrangian

Helo,

The Lagrangian in general relativity is written in the following form:

\begin {aligned}
\mathcal {L} & = \frac {1} {2} g ^ {\mu \nu} \nabla \mu \phi \nabla \nu \phi-V (\phi) \\
& = R + \dfrac {16 \pi G} {c ^ {4}} \mathcal {L} _ {\mathcal {M}}
\end {aligned}

with $g ^ {\mu \nu}:$ the metric
$\phi:$ non-gravitational scalar field
$R$: Ricci scalar
$\mathcal {L} _ {\mathcal {M}}:$ Lagrangian of the density of matter
By replacing the gravitational constant $G$ by its new definition, $\dfrac{1}{\varphi (t)},$ How to prove that the Lagrangian within the framework of Brans-Dicke's model becomes:

$\mathcal {L} = \varphi R + \dfrac {16 \pi} {c ^ {4}} \mathcal {L} _ {\mathcal {M}} - \omega_ {BD} \left (\dfrac {\varphi_ {, i} \varphi ^ {, i}} {\varphi} \right)$

Regards

 

[AndyW]

,Hello,

Thank you for your post. The Brans-Dicke model is an alternative theory of gravity that extends general relativity by introducing a scalar field, $\phi$, in addition to the metric tensor $g ^ {\mu \nu}$. The field $\phi$ is responsible for the dynamics of gravity, and its behavior is determined by the Brans-Dicke parameter $\omega_{BD}$.

To prove that the Lagrangian in the Brans-Dicke model takes the form you have mentioned, we need to start with the original Lagrangian in general relativity and then make the necessary substitutions. Let's begin by writing the Lagrangian in its original form:

\begin{aligned}
\mathcal{L} &= \frac{1}{2}g^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi - V(\phi) \\
&= R + \frac{16 \pi G}{c^4} \mathcal{L}_\mathcal{M}
\end{aligned}

Now, we can replace the gravitational constant $G$ with its new definition, $\frac{1}{\varphi(t)}$. This substitution is equivalent to replacing the Ricci scalar $R$ with $\varphi R$, since $\varphi$ is a function of time only and does not depend on the spacetime coordinates. This gives us:

\begin{aligned}
\mathcal{L} &= \frac{1}{2}g^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi - V(\phi) \\
&= \varphi R + \frac{16 \pi}{c^4 \varphi} \mathcal{L}_\mathcal{M}
\end{aligned}

Next, we need to include the Brans-Dicke term, which is given by $-\omega_{BD}\left(\frac{\varphi_{,i}\varphi^{,i}}{\varphi}\right)$. This term is added to the Lagrangian to account for the dynamics of the scalar field. Including this term gives us the final form of the Lagrangian in the Brans-Dicke model:

\begin{aligned}
\mathcal{L} &= \varphi R + \frac{16 \pi}{