# Gauss's and Coulomb's law forms and gravitational field

1. Jan 17, 2008

### user01

Consider int(E.dA)=q/e, guass law relating the electric field to the charge enclosed.

One can also derive (using the more mathematical version of guass' law - involving the double integral) this same formuala for a graviational field. Here the permitivitty constant would be replaced by another arbitrary constant.

Does anybody know where one can obtain the derivation to Coulombs law,
F = 1/4pi E qq/r^2, more specifically relating to the 'electrostatic constant' k = 1/4pi E?

Considering that the electric and gravitational fields are similar in terms of flux and field lines, wouldn't there be similar equations for gravity as maxwell's laws describe EM. Is it possible for the gravitational field to induce another field?

2. Jan 17, 2008

### nicksauce

From Gauss's law, whose differential form is $$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$, one can derive Coulomb's law using $$\rho = q\delta^3(\vec{r})$$ (ie a point particle).

3. Jan 17, 2008

### K.J.Healey

4. Jan 30, 2008

### user01

Thanks for the link to wiki (K.J H), that's quite useful.
I was thinking more along the lines, that if E induces B, and vice versa, then it might be possible that the gravitational field induces another field also.

In maxwells four equations (simplified), we have gauss, int(E.dA)=q/e and int(B.dA)=0 as the first two (not using the differential forms to keep it simple as one can).

Then one can derive gauss in the same way for the gravitational int(G.dA)=m/w, where G is the graviational field, m is the mass instead of charge, and w is a constant similar to the permitivitty. (derived using 'linking number' double integral)

Also the strong field can be written as int(S.dA)=0 since there is always to nucleon constituents present whenever the strong force is present (net flux=0).

Then obviously the question arises if the two fields (S and G) can induce one another?

5. Jan 30, 2008

### rbj

yes. Gauss's law applies to any field that is an inverse-square field. it arises naturally from the inverse-square relationship.

well, it's not an arbitrary constant (given our units of measurement). it's related to G just like in Coulomb's law it's related to "k".

consider the density of those flux or field lines (called "flux density") for the same charge (electric) or mass (gravitational) distributed on the surface areas of increasing concentric spheres. if you double the radius of the sphere, the surface area increases by a factor of 4, and the same number of flux lines is now distributed over an area 4 times as much. so now the flux density is 1/4 as large. that is fundamentally where the inverse-square relationship comes from in 3-dimensional space.

as someone else pointed out, it's called gravitoelectromagnetism (GEM) and has equations that look like maxwell's equations. they really come from a limiting case (reasonably flat spacetime) of Einstein's GR equation and i don't know enough about GR to derive it. but if you compare to E&M, think about it this way: in both E&M and GEM, you have this inverse-square relationship when things are static. in EM, you can derive the magnetic effect directly from electrostatics but also keeping the effects of special relativity in mind (i.e. magnetic forces are really just relativistic manifestations of electrostatic forces). then take the "gravitostatic" force action (Newton's law), apply the same song-and-dance (of how electrostatic becomes electromagnetic) to the gravitostatic field, and you get something that resembles GEM, that is an inverse-square field with gravito-magnetic effects and a speed of propagation of c. for some reason that i do not understand, there is a factor of 2 tossed in because gravity is a "spin-2" action not a spin-1 action as EM. i've asked the physicists here for a little bit of explanation long ago but never understood any reply.

this thread should maybe go into the SR/GR forum, because that's what it's about. maybe some admin would like to move it.

6. Jan 31, 2008

### CPL.Luke

the relationship between gravito magnetism and maxwell's equations is pretty simple, you could effectively postulate a lorentz invariant form of newtonian gravity by just making the analogue to the electromagnetic field and defining an additional vector field which would play the same role as the magnetic field.

the reason why such equations beak down is pretty simple. from special relativity energy is related to mass which in turn creates a gravitational field, however this gravitational field also contains energy which is related to mass and thus creates a bigger gravitational field, etc. etc.

so the full theory of gravity must be non-linear. Einstein developed this as GR, however in the case of a weak field, where space time is almost minkowskian, then that maxwell-analogue becomes accurate again and GR actually reduces back down to it.

7. Feb 1, 2008

### user01

The latter of maxwell's equations, which describe changes in flux that produce certain fields really unify E & B fields. Also, simple experiments on a macro level prove this.
I wasn't aware that gravity can induce another field, or vice versa. Where the strong field is unity, EM is on the order of a few hundreth in comparitive strength, but gravity is on the order of 10^-39. How can one prove any relationship between these other forces (as postualted in GEM) experimentally?