Gravitational Force

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Define the following quantities

Note: v^a and v_a are not tensors, Upper case bold and em are 4-vectors. Lowercase bold and G are 3-vectors.  

In what follows it must be kept in mind that “no G’s mean no gravitational force.” This should not be confused with ”G means gravitational force”. The former is always true while the later may or may not be true. For example; if one uses polar spatial coordinates in an inertial frame of reference then there would be non-vanishing G’s but it would not indicate the presence of gravitational forces. An acceleration must represent an actual acceleration rather than a quirk in the coordinate system. For example; if polar coordinates are used and dq/dt is not zero for constant r then this indicates the presence of a force. However if the particle is traveling in a straight line then dq/dt and dr/dt will both be non-zero but no force will be present. Therefore one must be cautious in their interpretation. It is assumed the reader is aware of this fact in what follows.

Rearranging terms we get

Substituting

into Eq. (2) gives

The first term on the right side of Eq. (4) is the external applied force. This force moves the object off of its geodesic and allows it to remain at rest in the gravitational field (i.e. sitting on the surface of the Earth) rather than remaining in free-fall. Consider the spatial portion of Eq. (4).

fk are the components of the (external) 3-force on the particle. This force will be denoted from here on as Fex.. The quantity on the left side of Eq. (4) is the total force acting on the object and denoted as ftotal. The second quantity on the right side will be denoted as G.

Note: G is a 3-vector and not the spatial component of any 4-vector.

Therefore we have 

In general relativity, gravitational effects manifest themselves through the metric tensor gmn. According to Einstein's Equivalence Principle an accelerating frame of reference is locally equivalent to a gravitational field. This means that in the absence of a 4-force on particle, the only force acting on the particle will be equivalent to an inertial force, the force resulting entirely from observing the particles motion from a frame of reference accelerating with respect to an inertial frame. Since the first term on the right side of Eq. (4) is proportional to the 4-force it follows that the second term on the right side of Eq. (4) represents the gravitational force. Therefore

This is reminiscent of the covariant form of the Lorentz force, which in Minkowski coordinates, is given by

Minkowski coordinates are employed since we want to compare this with the gravitational force, which is a force resulting from the use of non-curvilinear spacetime coordinates. Thus we use the 3-force dp/dt in each. The spatial components can be written as

The quantities Fmn are referred to as the components of the EM field. Similarly the quantities Gakb are referred to as the components of the gravitational field [Ref 1]. Just as the EM force is velocity dependent, so too is the gravitational force.  The quantity m therefore plays the role of a gravitational charge.
    The expression for Gk can be placed in a more familiar form as follows; Rewrite the portion of gravitational force to the right of the mass as

The kth component of the gravitational force in Eq. (6) now becomes

It is shown in the appendix below that

Substituting Eq. (11) into Eq. (8) yields

This relation is reminiscent of the expression for the Lorentz 4-force in terms of the 4-potential Am defined as

where F is the Coulomb potential and A  is the magnetic potential 3-vector,. Eq. (10) now becomes

From this it can be seen that the gravitational force is determined in terms of the derivatives of the components of the metric tensor gab. Comparing Eqs. (14) And (16) it follows the metric should be interpreted as a set of ten gravitational potentials. [Ref. 1] Therefore since the metric is a tensor it is often referred to as a tensor potential.
    If the metric has constant spatial components (i.e. gij,k = 0) then we can place Eq. (14) in a more recognizable form. I.e.

If the gravitational potential F(r) is defined as

then g00 can be written as

or in vector notation we have the final form of the gravitational force (a 3-vector)

Which is exactly the form of the Newtonian equation with mass replaced by relativistic mass.

Appendix:

Recall the definition of the Christoffel symbols (of the second kind)

Multiplying both sides of Eq. (A1) and contracting over m gives

Since we want the sum of two of these terms we change indices to fit the left hand side of Eq. (13). The first term on the left side is

The second term of he left side of Eq. (13) is

Adding Eqs. (A3) and (A4) gives, upon canceling a factor of 2

References:

[1] The Foundation of the General Theory of Relativity, Albert Einstein, Annalen der Physik, 49, (1916). An English translation may be found in The Principle of Relativity, Dover Pub, see page 143.
[2] On the discovery of the gravitational field equations by Einstein and Hilbert: new materials, V.P. Vizgin, Physics-Uspekhi, 44(12) 1283-1298(2001)

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