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Consider the action, $$ S = \int d\lambda\ \phi(x) \left( -g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda} \right)^{1/2}. $$ Using the variation principle we obtain, $$ \delta S = \int d\lambda\ \left\{ \delta\phi(x) \left( -g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda} \right)^{1/2} + \phi(x) \delta \left( -g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{d x^\nu}{d \lambda} \right)^{1/2} \right\}. $$ The second term gives the usual result. See here for example. The variation $x^\mu\to x^\mu + \delta x^\mu$ would give $$ \delta\phi(x) = \frac{\partial\phi(x)}{\partial x^\sigma} \delta x^\sigma. $$ However, plugging this in and performing similar simplifications as in the usual case, I don't arrive at the expected article given in this article (eq. 20), i.e. $$ \frac{d^2x^\mu}{d\tau^2} + \Gamma_{\rho\sigma}^\mu \frac{d x^\rho}{d\tau} \frac{d x^\sigma}{d\tau} = - \left( g^{\mu\nu} + \frac{d x^\mu}{d\tau} \frac{d x^\nu}{d\tau} \right) \partial_\nu(\ln\phi). \tag{20} $$ In particular, I am having trouble getting the second term on the right-hand side. Any help will be appreciated!

Qmechanic
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MarcosMFlores
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1 Answers1

2

Hints:

  • The variable mass $$m~\propto~\phi\tag{A}$$ gives rise to an effective metric $$\bar{g}_{\mu\nu}~=~\phi^2g_{\mu\nu}.\tag{B}$$

  • The relationship between the corresponding proper times is $$-\bar{g}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}~=~ (c\mathrm{d}\bar{\tau})^2~=~(\phi c\mathrm{d}\tau)^2~=~ -\phi^2g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}.\tag{C} $$

  • Eq. (20) is a matter of transforming the geodesic equation $$ \frac{d^2 x^{\mu}}{d\bar{\tau}^2} + \bar{\Gamma}^{\mu}_{\alpha\beta} \frac{dx^{\alpha}}{d\bar{\tau}} \frac{dx^{\beta}}{d\bar{\tau}} ~=~ 0\tag{D} $$ into the unbarred variables. There are 2 effects:

    1. Non-affine parametrization $$ \frac{d^2 x^{\mu}}{d\tau^2} + \bar{\Gamma}^{\mu}_{\alpha\beta} \frac{dx^{\alpha}}{d\tau} \frac{dx^{\beta}}{d\tau} ~=~ \frac{d\ln\phi}{d\tau}\frac{dx^{\mu}}{d\tau}, \tag{E} $$ cf. e.g. my Phys.SE answer here.

    2. Transformation of the Christoffel symbols $$\bar{\Gamma}^{\mu}_{\alpha\beta}-\Gamma^{\mu}_{\alpha\beta}~\stackrel{(B)}{=}~\left(\delta^{\mu}_{\alpha}\partial_{\beta}+\delta^{\mu}_{\beta}\partial_{\alpha}-g_{\alpha\beta}g^{\mu\nu}\partial_{\nu}\right)\ln\phi. \tag{F} $$

Qmechanic
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