The Gravitational Field

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Einstein made two statements in the 1916 paper which are associated with the term gravitational field [Note: hab º diag(1,-1,-1,-1)]

The case of the ordinary theory of relativity arises out of the case here considered, if it is possible, by reason of the particular relations of the gab in a finite region, to choose the system of reference in the finite region in such a way that the g_ab assume the constant values [h_ab].  We shall find hereafter that the choice of such co-ordinates is, in general, not possible for a finite region. ... the quantities gab are to be regarded from the physical standpoint as the quantities which describe the gravitational field in relation to the chosen system of reference. For, if we now assume the special theory of relativity to apply to a certain four-dimensional region with co-ordinates properly chosen, then the g_ab have the values [h_ab].  A free material point then moves, relatively to this system, with uniform motion in a straight line.  Then if we introduce new space-time coordinates x⁰, x¹, x², x³, by means of any substitution we choose, the gab in this new system will no longer be constants but functions of space and time.  At the same time the motion will present itself in the new coordinates as a curvilinear non-uniform motion, and the law of this motion will be independent of the nature of the moving particle.  We shall interpret this motion as a motion under the influence of a gravitational field.  We thus find the occurrence of a gravitational field connected with a space-time variability of the g_ab. So, too, in the general case, when we are no longer able by a suitable choice of co-ordinates to apply the special theory of relativity to a finite region, we shall hold fast to the view that the gab describe the gravitational field.

Later on in this article Einstein writes

If the G^m_ab vanish, then the point moves uniformly in a straight line. These quantities therefore condition the deviation of the motion from uniformity. They are the components of the gravitational field.

These statements, taken together, often confuse people. Consider the expression for the gravitational force as it is calculated in general relativity

where v^a º (c, v_x, v_y, v_z). Compare this to the Newtonian expression

where F is the Newtonian potential. It for these reasons that the components of the metric tensor, gab, are considered to be a set of 10 gravitational potentials in general relativity. Consider also the relationships from general relativity

Compare this from the relationships from Newtonian mechanics

For this reason Einstein referrer to the Gmab as the components of the gravitational field. It is also the reason why MTW state that No G’s means no “gravitational field”... [1] Therefore the following comparison is to be made

There is another comment that has also confused people, i.e. “We thus find the occurrence of a gravitational field connected with a space-time variability of the gab.” The first thing to notice is what the acceleration, a_k, means in Eq. (4). Consider a particle moving with constant velocity in an inertial frame of reference in flat spacetime. The components of acceleration, a_k, will vanish if and only if the spatial axes are Cartesian. In what follows the spatial coordinates will be Cartesian. The second thing to notice is that the variability of the potential is a necessary condition for the non-vanishing of the gravitational field at a specific point. It is not a sufficient condition. I.e. the potential may vary in a region and yet the field may vanish at a point in that region. To illustrate this in a Newtonian context consider two point objects of equal mass sitting on the x-axis, one at x = +a and one at x = -a. The Newtonian potential is shown in Fig. 1 below

The gravitational potential, F, is given by

Notice how F is constant nowhere on the x = axis. The gravitational acceleration is given by Eq. (4b)

Notice how a_x vanishes at x = 0. It is important to take note that Einstein's definition of the gravitational field is not the one commonly used today. The presence of a gravitational field is now synonymous with a non-vanishing Riemann tensor.

References: 

[1] Gravitation, Misner, Thorne and Wheeler, W.H. Freedman and Company, (1970), p. 467

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