Similarity between the Coulomb force and Newton's gravitational force

Question

Coulomb force and gravitational force has the same governing equation. So they should be same in nature. A moving electric charge creates magnetic field, so a moving mass should create some force which will be analogous to magnetic force.

Answer 1

It's a good observation that the electric and gravitational fields both satisfy Poisson's equation $$ \nabla^2\Phi_G = 4\pi\rho_G, \qquad \nabla^2\Phi_E = -\frac{\rho_E}{\epsilon_0} $$ where $\Phi_G, \Phi_E$ are the gravitational and electric potentials and $\rho_G,\rho_E$ are the mass and charge densities. It would seem from the perspective of Newtonian gravitation that this is where the analogy stops. The magnetic field in electrodynamics arises due to moving charges because of the following Maxwell equation $$ \nabla\times\mathbf B - \frac{1}{c^2}\frac{\partial \mathbf E}{\partial t} = \mu_0 \mathbf J $$ There are no such analogous equations for moving masses in Newtonian gravitation.

However using general relativity, and specifically Einstein's field equations, one can show in certain situations (where gravity is weak and the spacetime is reasonably flat), that gravity does in fact behave like electrodynamics. In fact, there are fields called gravitoelectric and gravitomagnetic fields that obey analogous equations to the Maxwell equations. The result is what is aptly called gravitoelectromagnetism (GEM).

This https://arxiv.org/abs/gr-qc/0311030 seems like a pretty nice overview.

Answer 2

To add to the other responses, you can't generally represent gravity with Maxwell-esque equations for two very fundamental reasons: (1) gravity's field equations are non-linear, and (2) the gravitational field couples to a rank-2 tensor.

Maxwell's equations are linear, meaning the superposition principle applies and the field doesn't interact with itself. The Einstein field equation, however, are nonlinear, so the superposition principle does not apply and the field will self-interact.

The electromagnetic field couples to charge density and current density, which can be written as a Lorentz-invariant four-vector (a rank-1 tensor). The metric (gravitational) field couples to the stress-energy tensor, which is a Lorentz-invariant rank-2 tensor. This is why photons have spin-1, while theoretical gravitons have spin-2. Since the Maxwell-esque gravitomagnetic field equations involve only four components of the stress-energy tensor (mass-density and mass-current), they cannot possibly be Lorentz-invariant. Thus, a simple change in reference frame will lead to predictions that contradict experiment.

Answer 3

In an inertial reference frame at rest to a charge, one only sees a Coulomb field. When transformed into another inertial reference frame in motion relative to the charge, one gets a magnetic field. The detailed transformation formula can be seen at this wiki page.

Now the question is, can we do the same for gravity (assume we are only in the regime of special relativity and pretend general relativity does not exist)?

The answer is "no" because of an essential difference between charge and mass: the charge is an invariant of Lorentz transformation, which means different inertial reference frames see the same amount of charge, while the mass is not. We don't know (do we?) what the "Coulomb" part of the gravitational force should be like when the mass is moving. Should we use the rest mass, or the "inflated mass" ($m_0/\sqrt{1-v^2/c^2}$)? And which distance should we put into the Coulomb's law? The delayed one or the instantaneous one? Even if we try to mimic in every aspect of the Coulomb force, namely, we use $m_0$ and the delayed distance, and arrive at a set of transformation rules for the "electromagnetic-like" gravitational force, the theory is not self-consistent, simply because the mass density and mass flux together do not form a relativistic four-vector, while the charge density and charge flux do. Intuitively speaking, the difference is, energy can create rest mass, while nothing can create a net charge.

Update: I just noted that my argument that (mass density, mass flux) does not form a four-vector while the (charge density, charge flux) does is kind of equivalent to the answer of joshphysics that the Maxwell equation involving $\mathrm{J}$ does not exist for gravity.

One may get something similar to electromagnetism in the framework of general relativity (as pointed out in the comments), but to me that's another story...

I kind of guess (but not sure because I don't know what really happened to Einstein during 1905-$\sim$1915; correct me if I am wrong) what I talked about above might be one reason for Einstein to abandon such an "apparent" approach to accomodate gravity into the framework of special relativity, and to choose the much more indirect, nontrivial, and revolutionary "geometrization" approach, namely, the theory of general relativity.

Answer 4

The fundamental notion, at least classically, is that forces are represented by curvature. That of gravity by the curvature of spacetime and that of the other forces by the curvature of an internal space.

This idea is exploited in non-commutative geometry to write the entire standard model - with roughly a hundred terms - as the spectral Einstein-Riemann action of spacetime multiplied by a small non-commutative space whose classical dimension is zero but whose non-classical dimension is 6. This is notable because string theory compactifies over 6d spaces. This is Einstein's dream of a unified classical theory as well as Clifford's who speculated fifty years before Einstein that all forces of nature could be represented by the ebb and flow of the curvature of space.

Since Maxwell's equations arise from the curvature of an internal space, one might guess that we can write the gravitational field in a similar way. This is correct and was first done by Heaviside whilst he was developing vector analysis. In his 1893 paper, published in The Electrician, and titled A Gravitational and Electromagnetic Analogy, he says:

Now bearing in mind the successful manner in which Maxwell's localisation of the electrical amd magnetic energy in his ether lends to theoretical reasoning, the suggestion is very natural that we should attempt to localise gravitational energy in a similar manner, its density to depend on the square of the intensity of the force, especially since the law of the inverse squares is involved throughout.

He goes on to write down gravity in a form similar to Maxwell that Behera & Barik call Heaviside gravity. In fact, Maxwell himself made an abortive attempt in his papers which laid out his equations.

Alternatively, Hehl & Obukhov have derived Maxwell's spatial equations from the conservation of electric charge and magnetic flux. This, in a sense, is a formal exercise arising from two kinds of conservation laws. Now, we can replace electric charge and current with mass and mass current to get the inhomogeneous equations of Heaviside gravity. But what about the homogenous equations? This requires a magnetic analogue of gravity. This is given by a certain double contraction of the Riemann tensor.

Philosophically speaking, all physical order in the universe is an aspect of a unitary order in nature. Spinoza, for example, simply signified this by his notion of extension. So it's no surprise that we have found a unified law - it was to be expected.