Nordstrom's theory of a particle moving in the presence of a scalar field φ(x) is given by S=−m∫eφ(x)√ηαβdxαdλdxβdλdλ,
where λ is the parametrization of the worldline of the particle, ignoring the free field part ∫ηαβ∂αφ∂βφd4x.
How does one derive the equations of motion in terms of the parameter dτ=√ηαβdxαdλdxβdλdλ.
uα=dxαdτ⇒uαuβηαβ=1?
My attempt:
δS=0⇒∫(∂(eφ√...)∂xαδxα+∂(eφ√...)∂(dxαdλ)ddλδxα)dλ=|dτ=√...dλ|=
=∫(√...eφ∂αφ−ddλ(dxαdτeφ))δxαdλ=
=∫(∂αφ−d2xαdτ2−dxαdτdφdτ)δxαeφ√...dλ=
=∫(∂αφ−duαdτ−uαuβ∂βφ)δxαeφ√...dλ⇒
∂αφ−duαdτ−uαuβ∂βφ=0⇒∂αφ=e−φddτ(eφuα).
Unfortunately, this equation doesn't look like the equation from Wikipedia, d(φuα)dτ=−∂αφ.
I can explain the part of differences by renaming the function, eφ→φ, in the expression for action (then my equation reduces to the form ∂αφ=ddτ(φuα)), but I can't explain why my equation has the wrong sign.
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