If an electron orbital the earth only influence by gravity,does it emit electromagnetic waves? if so, the radius will decay,eventually falling on earth?
Are you postulating an electron orbiting the earth? Why bother with such a strange example? Using the "planetary model", an electron orbiting the nucleus of an atom is accelerating and so must be emitting waves- which would result in its losing energy until it cannot orbit the nucleus. That is one of the things that led to the development of quantum physics. The "planetary model" is invalid and an electron emits electromagnetic waves only when goes from one "energy level" to another.
Let's clear some stuff up first. The principle of equivalence says that free fall is locally equivalent to inertial movement. It also says that being at rest in a static gravitational field (meaning following orbits of the time translation symmetry) is locally equivalent to acceleration. All these things presuppose that the system one is working with is local in space-time. An electron carries an electromagnetic field which fails to satisfy this requirement so the equivalence principle does not apply here.Principle of equivalence says that free fall has to be indistinguishable from inertial movement.