Derivation of equations of motion in Nordstrom's theory of scalar gravity?
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Nordstrom's theory of a particle moving in the presence of a scalar field φ(x) is given by
S=−m∫eφ(x)ηαβdxαdλdxβdλ−−−−−−−−−−√dλ,
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=ηαβdxαdλdxβdλ−−−−−−−−−−√dλ.
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.
S=−m∫eφ(x)ηαβdxαdλdxβdλ−−−−−−−−−−√dλ,
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=ηαβdxαdλdxβdλ−−−−−−−−−−√dλ.
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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Hey dear here is the answer
Gunnar Nordström was a contemporary of Albert Einstein. Both men took a circuitous path to university careers. During Einstein's early years at the Berne patent office, Nordström became an engineer. Never a very practically oriented man, he turned into studying physical chemistry. He arrived in Göttingen in April 1906 for about one year to study chemistry under Walther Nernst. In Göttingen, the young Nordström became a wholehearted believer in relativity in its Minkowskiian formulation. After having published only one paper in chemistry, Nordström's whole remaining published work was focused almost exclusively in issues of relativity, electrodynamics and gravitation.
Hope its help you
Gunnar Nordström was a contemporary of Albert Einstein. Both men took a circuitous path to university careers. During Einstein's early years at the Berne patent office, Nordström became an engineer. Never a very practically oriented man, he turned into studying physical chemistry. He arrived in Göttingen in April 1906 for about one year to study chemistry under Walther Nernst. In Göttingen, the young Nordström became a wholehearted believer in relativity in its Minkowskiian formulation. After having published only one paper in chemistry, Nordström's whole remaining published work was focused almost exclusively in issues of relativity, electrodynamics and gravitation.
Hope its help you
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