Principle of Equivalence in Mathematical Form

In general relativity, gravity is no longer thought of as just a force. It is thought of as spacetime curvature. General relativity yields the special theory of relativity as an approximation consistent with the Principle of Equivalence. If we focus our attention on a small enough region of spacetime, that region of spacetime can be considered to have no curvature and hence no gravity. Although we cannot transform away the gravitational field globally, we can get closer and closer to an ideal inertial reference frame if we make the laboratory become smaller and smaller in spacetime volume. In a freely falling (non-rotating) laboratory occupying a small region of spacetime, the laws of physics are the laws of special relativity. Hence all special relativity equations can be expected to work in this small segment of spacetime.

In special relativity the invariant expression that defines the proper time, , in terms of the invariant 4-D length s is

 (1.1)

where

 (1.2)

We are using the Einstein summation convention here with  and  running from 0 to 3 connecting  to the general spacetime coordinates . The 4-D coordinates  are related to the standard space and time coordinates cT, X, Y, Z in the following way:

 (1.3)

The explanation of the expression in (1.1) is as follows. The invariant 4-D length between two distinct events placed an infinitesimal distance apart looks like.

 (1.4)

This pattern defines what a special relativistic frame transformation has to obey to be a legitimate transformation . Any transformation that keeps this pattern satisfied is called a Lorentz Transformation. Let us assume that there is a frame where the two events occur at the same space location but at different times. This frame is said to be the proper frame for these events. The time interval between the two events in this frame is said to be the proper time . In this proper frame the expression (1.4) must take the form

 (1.5)

Hence, expression (1.1) is just a statement that the 4-D length in the proper frame and the nonproper frame (having coordinates cT, X, Y, Z) must be equal. The matrix (1.2) has been used in (1.1) to make the expression more succinct and general. This general matrix form will prove to be very useful later in the context of general relativity.

Now we analyze what occurs in more general coordinate systems. Say we change the coordinates to

 (1.6) .

We allow this function of the coordinates to be any arbitrary function. Since it's arbitrary we no longer will be in an inertial frame of reference necessarily.

The invariant 4-D length equation now takes the new form:

 (1.7)

where  is some connecting array. Therefore, from 4-D length invariance

 (1.8)

Using the derivative relation

 (1.9)

we get an expression for the quantity :

 (1.10)

In the freely falling frame we must have the motion equation:

 (1.11)

if we consider the motion of a free particle. Just as long as the free-float frame in a limited region of space lasts this equation must be in effect. In terms of the arbitrary coordinates  that we introduced above we can use the chain rule to obtain a new motion equation :

 (1.12)

where we have defined the  quantity as

 (1.13)

Note that equation (1.12) can be rewritten in a form that corresponds with the Newtonian 2nd law of motion when an external force is present:

 (1.14) .

We have introduced the mass of the moving object here as . The messy part that came in when we switched to arbitrary coordinates can be viewed as a force if we insist on thinking of ourselves as being in a simple inertial frame where (1.11) is in effect in a force free situation. Such coordinate-dependent forces are called fictitious forces.

 (1.15)

Note that the fictitious force is proportional to the mass of the object of attention. Einstein noticed that this was very much like the way that the mass enters into standard Newtonian gravitational theory.

 (1.16)

As far as we know, no other physical forces are proportional to mass like the gravity force or the fictitious forces are. Hence, Einstein thought deeply about what was similar and what was different between a fictitious force and the gravitational force. This thinking led to the more general theory of relativity that he had set out to find after his success with the theory of special relativity.

Einstein's proposals for general relativity were as follows. Proper time in general relativity (defined in terms of any coordinates  ) is given by

 (1.17)

and the equation of motion in general relativity is given by

 (1.18) .

In contrast, however, to special relativity,  no preferred coordinates in general relativity that will reduce these two equations to the form , and , globally. The array  is taken to represent the gravitational field. Although locally you can always manage to make this quantity look like the Minkowski metric (1.2), globally it can never be equivalent to the Minkowski metric since the Minkowski metric implies a spacetime that has no curvature, and therefore has no gravity.