Monday, 30 March 2020

homework and exercises - Derivation of equations of motion in Nordstrom's theory of scalar gravity?


Nordstrom's theory of a particle moving in the presence of a scalar field φ(x) is given by S=meφ(x)ηαβdxαdλdxβdλdλ,

where λ is the parametrization of the worldline of the particle, ignoring the free field part ηαβαφβφd4x.


How does one derive the equations of motion in terms of the parameter dτ=ηαβdxαdλdxβdλdλ.

uα=dxαdτuαuβηαβ=1?


My attempt:


δS=0((eφ...)xαδxα+(eφ...)(dxαdλ)ddλδxα)dλ=|dτ=...dλ|=

=(...eφαφddλ(dxαdτeφ))δxαdλ=
=(αφd2xαdτ2dxαdτdφdτ)δxαeφ...dλ=
=(αφduαdτuαuββφ)δxαeφ...dλ
αφduαdτuαuββφ=0αφ=eφddτ(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, eφφ, 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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