Modified gravitation ; how to make it relativistic ?

[notknowing]

Suppose one is not happy with Newton's law of gravitation and finds that it should be modified and that one has an equation describing the gravitational potential
$$\phi$$ in function of radius. If one then want to obtain a relativistic description, would it be sufficient to replace in the Schwarzschild metric the term(s)

$$\left(\ 1 - \frac {2 G M} {c^2 r} \right)$$

by

$$\left(\ 1 + \frac {2 \phi} {c^2 } \right)$$

or would that be to simplistic?

Rudi Van Nieuwenhove

 

[jimgraber]

"would that be too simplistic?"
Probably (almost certainly) yes, but it would likely be a good first approximation.

Jim Graber

 

[Chris Hillman]

Too simplistic, in all likelihood. If you want more detail I think you'll have to be more specific concerning the modification you have in mind. Providing some rationale for your proposed nonrelativistic correction would probably also be appropriate!

 

[notknowing]

Too simplistic, in all likelihood. If you want more detail I think you'll have to be more specific concerning the modification you have in mind. Providing some rationale for your proposed nonrelativistic correction would probably also be appropriate!

The modification I had in mind is described in : http://arxiv.org/abs/0712.1110
I expected that it was too simplistic, but I don't exactly understand why. If I look for instance in the book Gravity, An Introduction to Einstein's general relativity by James B. Hartle, I find that the derivation of the metric for the static weak field metric (eq. 6.20 on page 126) would have exactly the same form if $$\phi$$ would have a different radial dependence, or am I wrong? And if it is too simplistic, what would be the correct way to make it relativistic? Or what objection could be made to the simple replacement of $$\phi$$ with a different radial dependence in the Schwarzschild metric? What physical principles would it violate?

 

[Garth]

Rudi,

What action do you have that generates your metric?

Garth

 

[notknowing]

Rudi,

What action do you have that generates your metric?

Garth

For the moment I have neither a metric nor an action. The only thing I have is a simple potential. What would prevent me from using the replacement described in my earlier post?

Rudi Van Nieuwenhove

 

[A/4]

Certainly, the weak field limit should produce $$g_{00} = 1+2\phi(r)$$, so in that sense it seems reasonable as a first approximation (as others have echoed already). I think it depends on the form of the potential, and how it depends on the radial coordinate. If you consider the extra-dimensional Schwarzschild metric, it follows the pattern you suggest, e.g.:

$$g_{00} = 1-\left(\frac{r_H}{r}\right)^{n+1}$$

which we know is an exact solution (derived from the action) and basically boils down to substituting the $$n+1$$-dimensional potential. I would think that as long as the potential follows a power-law behavior, you're good to go (but in that case you're not proposing something radically different).

That being said, I think you've raised an interesting question, because it's not one that I've seen explained with any high degree of satisfaction.

 

[Mentz114]

Rudi,
you start by proposing that if Newtonian gravity is not satisfactory and should be modified, that we should modify GR. Metrics are not part of Newtonian gravity so you are proposing modifying GR ( ?) because N. gravity is no good. Your question does not make sense to me.

As has been pointed out, Newtonian gravity emerges as a weak field solution in GR. But GR and Newtonian gravity are separate and different theories.