Covariant form of Newton's law of gravity

[shubham agn]

Hii!

Newton's law of gravity is ∇.(∇Φ) = 4πGρ.

A book on GR gives a suggestion to make it Lorentz covariant by using de' Alembertian operator on 'Φ' in the LHS of above equation instead of Laplacian. Then it explains that this won't work because we have to include in 'ρ' all the energy density also in addition to mass density since mass and energy are equivalent according to Special Relativity. I am not able to understand the logic behind this. Mass may be equivalent to energy but the gravitational potential should result only from the mass. I mean if I have some energy in the form of electric field, why should I include this electric energy density in 'ρ' to obtain gravitational potential?

Thank you!

 

[DEvens]

It's a little hard to know how to answer because the context is a little unclear. Hopefully your textbook gives some more of this beyond what you give here.

But as a place to start thinking about it, consider how a scalar (as $\rho$ starts in Newton's theory) is supposed to transform under a change of coordinates. And then consider how a mass density will transform under the changes of coordinates available in special relativity. Remember it is, among other things, changes of velocity you must account for.

It may be possible to get something if you rewrite this equation in terms of something with appropriate transformation properties in place of $\rho$. But it begins to get very complicated when the mass density has a component that involves electromagnetic energy. Because the electromagnetic field has interesting properties under relativistic changes also. So if you want to construct a scalar out of E&M fields then there are quite a few constraints.

 

[PeterDonis]

Then it explains that this won't work because we have to include in 'ρ' all the energy density also in addition to mass density since mass and energy are equivalent according to Special Relativity.

This is a highly heuristic argument, and it's also incomplete; there's another important issue that it leaves out. Replacing the Laplacian with the D'Alembertian gives an equation that predicts gravitational aberration: that is, the direction that the gravitational force appears to come from is not the "instantaneous" direction of the source, but the "retarded" direction of the source--where it was one light travel time ago. So, for example, the gravitational force exerted by the Sun on the Earth should point, not to where the Sun is now, but to where it was 8 minutes ago.

The problem with this is that, if gravitational aberration is present, there are no stable orbits: the Earth would either spiral into the Sun, or fly off into interstellar space, on a time scale that's very short compared to the known lifetime of the solar system. (And satellites orbiting the Earth would either spiral into the Earth or fly off into interplanetary space on a time scale of days, whereas we know satellites can stay in stable orbits around the Earth for years.)

The solution to this dilemma is somewhat complex, but the fact that the dilemma requires a solution is part of why gravity cannot be modeled as a relativistic scalar theory such as you are describing. For a good discussion of the gravitational aberration issue, see Steve Carlip's classic paper:

http://arxiv.org/abs/gr-qc/9909087

Mass may be equivalent to energy but the gravitational potential should result only from the mass.

The problem with this is that "mass" is an ambiguous term. Do you mean rest mass (i.e., invariant mass)? If so, you have a problem: invariant mass is not additive. If I have two objects in relative motion (such as a binary star system where the two stars are in orbit about each other), the invariant mass of the system composed of both objects is not the sum of the invariant masses of the two objects individually; the system's invariant mass also includes the kinetic energies of the objects in the center of mass frame. And if I am very far away from the system, but at rest relative to its center of mass, the gravitational mass I will measure (by, for example, putting a test object into orbit around the system and measuring its orbital parameters) will be the invariant mass of the system, not the sum of the invariant masses of the individual objects.

If, OTOH, by "mass" you mean "relativistic mass", then that is just total energy, so you are indeed counting energy as a source of gravity.

 

[Chhhiral]

Hii!

Newton's law of gravity is ∇.(∇Φ) = 4πGρ.

A book on GR gives a suggestion to make it Lorentz covariant by using de' Alembertian operator on 'Φ' in the LHS of above equation instead of Laplacian. Then it explains that this won't work because we have to include in 'ρ' all the energy density also in addition to mass density since mass and energy are equivalent according to Special Relativity. I am not able to understand the logic behind this. Mass may be equivalent to energy but the gravitational potential should result only from the mass. I mean if I have some energy in the form of electric field, why should I include this electric energy density in 'ρ' to obtain gravitational potential?

Thank you!

1)What is for you the rest mass?
The rest mass of an atom (for example) is not the sum of the masses of its components but there are additional contributions that depend from the binding energy of the particles that compose it (for example, electromagnetic energy).
2) The matter density is not a Lorentz scalar, its value changes under Lorentz transformations so that the expression is not Lorentz coovariant. By Special Relativity you can see that it does transform as the 00 component of a rank-2 tensor. So that you can conclude
that the source term in the gravitational field equations should be a rank-2 tensor.
I think you reading the cap.7 of the Hobson book, some of your doubts will disappear when you read chapter 8 (I think :) )

 

[shubham agn]

Thank you all for your replies! They were indeed helpful!