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There is Nordstrom theory of the particle moving in a scalar field $\varphi (x)$: $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d \lambda}\frac{dx^{\beta}}{d \lambda}}d\lambda . $$ How to get the equation of motion for massless objects? In this case I may introduce new parameter $\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d \lambda}\frac{dx^{\beta}}{d \lambda}}d\lambda = \frac{d\tau }{m}$.

But when I try to derive (like this way) the equations of motion, I get $$ \partial_{\alpha } \varphi = m^{2}\frac{d }{d \tau}\left( u_{\alpha} e^{\varphi}\right)e^{-\varphi}. $$ Unfortunately, this equation doesn't predict the absense of deflection of light. But Nordstrom theory really predicts the deflection of light. So where is the mistake in my reasoning?

Addition.

I got the mistake. By introducing new parameter $\tau$ I must rewrite my equation in a form $$ \partial_{\alpha } \varphi = \frac{1}{m^2}\frac{d }{d \tau}\left( p_{\alpha} e^{\varphi}\right)e^{-\varphi}. $$ So it really preficts no deflection of light.

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You should add your edit as an answer. –  jinawee Feb 2 at 11:49
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