The postulates of the General Theory of Relativity are:

1. Principle of General Covariance: The laws of physics take the same form in all reference frames.
2. Principle of Equivalence: The effects of a uniform gravitational field and uniform acceleration are equivalent or inertial mass = gravitational mass.

The second postulate is not so surprising in light of what we have learned in Newtonian physics. In Newtonian physics, the equality of gravitational and inertial mass was an experimental result. In the General Theory, it is promoted to a postulate. It is now the first postulate which is somewhat surprising. Why should the laws of physics look the same in any reference frame. In a rotating frame, you feel "fictitious" forces (such as being thrown out) which occur because the reference frame is rotating. From outside the rotating frame, an object you release will travel on a straight line path since no force acts on it, but from inside the rotating frame the path will be curved. On the other hand, all objects released will have the same acceleration as viewed from the rotating frame – this is the signature of a gravitational field. Thus there is some chance of writing the laws of physics including gravity in a way that will be the same in all reference frames. In this manner Einstein was led to a new theory of gravity.

Before looking at some of the consequences of the postulates, it is important to stress that gravity can never be totally removed by going into a free-falling reference frame. Gravity is produced by mass and mass is absolute. Only a uniform (constant) gravitational field could be totally removed by going into an accelerating frame, but a totally uniform gravitational field does not exist. An approximately uniform gravitational field can exist, for example, if we confine ourselves to a small region of space at some distance from the Earth. Imagine you are free falling in an elevator above the Earth. You release two balls. Since they are moving towards the center of the Earth, the distance between the balls will slowly decrease – they won’t exactly float in front of you. The difference in the forces within a free falling frame at different points in the frame are referred to as tidal forces since this is the same mechanism responsible for the tides. In some sense, the effects produced by masses are absolute. We will see that they are absolute in that they lead to a curvature of space-time that is absolute.

The General Theory of Relativity is a complicated mathematical theory. However, we can get some idea of the consequences of the postulates by adopting the following procedure:

1. Look at some events in an accelerating reference frame.
2. Use the Special Theory of Relativity to view these events from an inertial frame relative to which the other frame is accelerating.
3. From within the accelerating frame, attribute the effects to a gravitational field.
4. Conclude that the same results would occur in a true gravitational field.

To see how the method works, let’s look at some examples.

Bending of light in a gravitational field. Consider first an accelerating rocket in outer space. We have already seen that if you release a ball in this rocket it accelerates to the bottom of the rocket as viewed in the rocket frame. Imagine that you throw a ball horizontally in the rocket frame. From the reference frame outside the rocket (inertial frame) the ball moves on a straight line since no forces act on it. From inside the rocket frame, the ball will move on a curved arc, just as in a gravitational field. Now instead of throwing a ball, flash a laser beam on and off in the horizontal direction. From outside the rocket frame, the light will propagate horizontally at the speed of light since the speed of light is the same in all inertial frames. However from inside the rocket frame, just as for the ball, the light will follow a curved arc (the curve is much smaller for light since its horizontal velocity is enormous compared to that of the ball). The observer inside the rocket attributes this curvature to a gravitational field. Thus we can expect that light will bend in a true gravitational field and that the speed of light is not constant in a true gravitational field.

Gravitational red shift. Now imagine a rotating reference frame (disk) in outer space. There are identical clocks at the center of the disk and on the circumference of the disk. From an inertial frame outside the disk, the clock on the circumference is moving at a speed v relative to the inertial frame, while the clock at the center is stationary relative to the inertial frame. From the Special Theory, it then follows that the clock on the circumference runs slower than the clock at the middle. In the rotating frame, the observer attributes this difference in clock rates to a gravitational field which is stronger the further you are from the center of the disk. In a stronger field, clocks run slower. Thus we can conclude that in a true gravitational field that clocks run slower closer to the mass producing the field. This known as the gravitational red shift.

Since clocks run at different rates at different points in a gravitational field, it is impossible to synchronize clocks in a gravitational field. The gravitational red shift can also be used to "explain" the twin paradox. When the space twin turns around, she must accelerate. In her frame there appears to be a strong gravitational field and she is much closer to the center of the field than the Earth twin. Therefore, when this field is on, the Earth twin’s clock is moving much faster than hers and this "explains" why the Earth twin is older. Of course this does not explain anything. It gives a consistent picture of the result, but that is all. There is no true gravitational field in the twin paradox. The fact that the space twin is younger is related to the structure of space-time.

Non-Euclidean geometry in a gravitational field. Back on the disk, let’s measure the ratio of the circumference of the disk to its diameter. To do so we lay meter sticks around the circumference and along the diameter. Viewed from outside the disk in an inertial frame, the meter sticks along the circumference are shorter than their rest length, while those along the diameter are equal to their rest length (since the velocity is in the tangential direction, only length in the tangential direction is modified). Thus it will take more meter sticks than "normal" to go around the circumference and the ratio of the circumference to the diameter will be greater than p. Inside the rotating frame, the observer attributes the fact that geometry is non-Euclidean (in Euclidean geometry the ratio of circumference to diameter equals p) to the presence of a gravitational field. Thus, we conclude that in a true gravitational field, space, or, more generally, space-time, is non-Euclidean.

It should also be apparent that, since the length of meter sticks changes in the rotating frame, it is impossible to have uniform meter sticks in a gravitational field.