Principal Difference between electrostatic potntial and gravitational

[birulami]

A scalar potential $\phi: \mathbb{R}^4\to\mathbb{R}$ has the physical unit of energy per particle property, which can be charge or mass. Take the positional derivative and multiply by the particle property to get the force on the particle. So far gravitational and electric potential are the same.

Now, when the source of the potential, an electron on the one hand or just some mass on the other, is accelerated, the effects are quite different. For the electomagnetic case, the magnetic field, seemingly, pops out of nowhere. We suddenly have two interacting force fields, the electric and the magnetic.

There seems to be nothing equivalent for the gravitational field.

Can someone explain where the crucial difference is between the two potentials?

 

[Bill_K]

For the electomagnetic case, the magnetic field, seemingly, pops out of nowhere. We suddenly have two interacting force fields, the electric and the magnetic.

There seems to be nothing equivalent for the gravitational field.

See the Wikipedia article on gravitomagnetism.

 

[The_Duck]

Now, when the source of the potential, an electron on the one hand or just some mass on the other, is accelerated, the effects are quite different. For the electomagnetic case, the magnetic field, seemingly, pops out of nowhere. We suddenly have two interacting force fields, the electric and the magnetic.

There seems to be nothing equivalent for the gravitational field.

Can someone explain where the crucial difference is between the two potentials?

There are analogous phenomena for gravity, as Bill_K's link discusses. You can think of magnetic and gravitomagnetic effects as being special relativistic effects (they are proportional to v/c, so are relatively weak except at velocities near c).

In general, theories of instantaneous action at a distance are incompatible with special relativity. Instead you need some sort of field to mediate forces. Maxwell's equations give a field theory for electromagnetism that is compatible with special relativity. The equivalent equations for gravity are the Einstein field equations of general relativity.

There is not a perfect analogy, though. In the jargon, the electromagnetic field is a "vector" field while in general relativity the gravitational field is a "tensor" field. Vector field theories and tensor field theories are two different possible ways to make field theories that are compatible with relativity. But at low velocities, both theories look the same, which is why both gravity and electromagnetism follow similar 1/r^2 laws.