General covariance has something of the character of a strategy, rather than as a fundamental principle. It probably would be possible to construct a theory of GR without resorting to tensors or other forms of general covariance, although it certainly would not be easy to do so. Equivalence, on the other hand, is absolutely fundamental to GR, constituting the connection between curvature and gravitation. As a principle underlying GR it has its basis in the observed equality of gravitational and inertial mass, so that all objects experience the same acceleration in a gravitational field, quite unlike the case with, e.g., electrical or magnetic fields. By itself, this sets gravitation apart from other forces; but Einstein extended the principle further by applying it – in some sense – to all of physics, not just gravitational dynamics. It is thus useful to distinguish between the Weak Equivalence Principle and the Strong Equivalence Principle, both of which assert that acceleration and gravitation are equivalent, but in somewhat different senses and with different consequences.
Weak Equivalence Principle
The Weak Equivalence Principle (WEP) states simply that all objects in a gravitational field experience the same acceleration. As a consequence, the accelerating effects of gravitation can be transformed away by going over to a coordinate system falling freely with the gravitational field, much like astronauts in the space station or Einstein's workmen falling from a roof. Thus, an operational definition of the WEP is:
The dynamical effects of a gravitational field can be transformed away by moving to a reference frame that is freely falling in the gravitational field.
The utility of this principle is limited by its strict applicability only to static, uniform gravitational fields, which are rare. Objects falling to the ground, for instance, experience a gravitational force that increases as they get closer to the Earth; and objects in orbit experience a gravitational field that changes direction with orbital location. Applications of the WEP are thus limited to the near locality (in space-time) of a falling object: such Locally Inertial Reference Frames are formally defined in the following section.
Even with this limitation the WEP can have interesting consequences for gravitational physics, the most important of them being the equations of motion of a freely falling object – i.e., one experiencing only gravitational forces.