# Special Relativistic Gravitational Force Law

1. Aug 8, 2017

### Vrbic

1. The problem statement, all variables and given/known data
In Newtonian theory the gravitational potential Φ exerts a force F = dp/dt = −mΦ 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?

2. Relevant equations

3. 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 $d^2\Phi/dt^2$.

2. Aug 8, 2017

### Staff: Mentor

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

3. Aug 10, 2017

### Vrbic

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

4. Aug 10, 2017

### Staff: Mentor

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?

5. Aug 11, 2017

### Vrbic

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.

6. Aug 11, 2017

### Staff: Mentor

That is what I meant.

7. Aug 14, 2017 at 10:35 AM

### Vrbic

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)?

8. Aug 14, 2017 at 1:00 PM

### Staff: Mentor

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

9. Aug 15, 2017 at 9:15 AM

### Vrbic

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) $\frac{dm}{dt}=0$

10. Aug 15, 2017 at 10:01 AM

### Staff: Mentor

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.