- #1
Vrbic
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Homework Statement
In Newtonian theory the gravitational potential Φ exerts a force F = dp/dt = −m∇Φ on a particle with mass m and momentum p. Before Einstein formulated general relativity, some physicists constructed relativistic theories of gravity in which a Newtonian-like scalar gravitational field Φ exerted a 4-force ##\vec{F}## = d##\vec{p}##/dτ on any particle with rest mass m, 4-velocity ##\vec{u}## and 4-momentum ##\vec{p}## = m##\vec{u}##. What must that force law have been, in order to (i) obey the Principle of Relativity, (ii) reduce to Newton’s law in the non-relativistic limit, and (iii) preserve the particle’s rest mass as time passes?
Homework Equations
The Attempt at a Solution
to (i) I have to use tensors
to (ii) I expect equation of same order
to (iii) I'm not sure how to preserve it
My first guess is something like that: ##\square \Phi=4\pi G T^i_i##. But I see that the limit i.e. I take only ##d^2/dt^2## part and say the other are negligible, the left is ##d^2\Phi/dt^2##.
Can anybody advise?