Special Relativistic Gravitational Force Law

[Vrbic]

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?

 

[mfb]

I think (i) is much easier. What does the principle of relativity tell you if you compare two objects of different mass?

 

[Vrbic]

I think (i) is much easier. What does the principle of relativity tell you if you compare two objects of different mass?

Sorry for the delay, I missed an alert... In more massive one is more energy. Are you pointing there?

 

[mfb]

In more massive one is more energy.

That is right, but not the interesting point here.

They should have the same acceleration. What does that tell you about the force as function of the object's mass?

 

[Vrbic]

That is right, but not the interesting point here.

They should have the same acceleration. What does that tell you about the force as function of the object's mass?

If I want to have same acceleration there is a linear dependence of force on mass. Unfortunately I'm not sure where you are pointing so I hope I'm understanding what you are asking.
(i) Also says that natural laws are same (same form) in all reference frames.

 

[mfb]

If I want to have same acceleration there is a linear dependence of force on mass.

That is what I meant.

 

[Vrbic]

That is what I meant.

Ok and how does it help me to find such law? Does it mean that I'm looking for linear law? What item (ii) and (iii)?

 

[mfb]

It is one piece of a law. (ii) and (iii) give other pieces.

 

[Vrbic]

It is one piece of a law. (ii) and (iii) give other pieces.

Ok so what is right and helpful and what not?
(i) I'm looking for linear law
(ii) Equations of same order as for Newton case
(iii) ##\frac{dm}{dt}=0##

 

[mfb]

(ii) Equations of same order as for Newton case

The equation cannot be the same, as you are looking for a four-vector.

All three parts are reasonable conditions for such a force law. The question is then how can a force law look that satisfies all three conditions.