Field Equations & Equations of Motion
(General Relativity)

Velocity is a vector (tensor) or vector (tensor) field. In familiar notation, the velocity v is represented by

where vⁱ represent the components of the velocity, and e_i represent basis (unit) vectors in the selected coordinate system. (As usual in tensor notation, summation is assumed over all repeated indices unless otherwise specified.)

Acceleration is the first time-derivative of velocity, and involves derivatives of both the vⁱ and the e_i :

The second term may be further expanded as

vⁱ(de_i/dt) = vⁱ(_i^j_k dx^j/dt e_k )

where _i^j_k are the appropriate Christoffel symbols. Substituting, the expression for acceleration becomes

a = (dvⁱ/dt + v^s_sⁱ_j dx^j/dt )e_i

(with suitable change of indices on v^s_sⁱ_j).

If, in a Euclidean space, the components of velocity, v_i , are referred to an inertial (non-accelerated) Cartesian (geodesic) coordinate system, then the _jⁱ_k all vanish (i.e., _jⁱ_k = 0 values of i, j, & k) and the expression for acceleration has the form

If a non-Cartesian inertial coordinate system is used, say a polar or a spherical coordinate system, then the _jⁱ_k do not all necessarily vanish, and the expression for acceleration may involve non-zero values of some of the v^s_sⁱ_jdx^j/dt e_i.

[eg.: In the case of an inertial polar coordinate system, the non-zero values of v^s_sⁱ_jdx^j/dt e_i simply reflect the fact that the base unit vectors, and depend for their direction on their location in the space. Specifically, with i and j being unit vectors in the Cartesian coordinate system, the familiar transformations are: = i sin + j cos , and = -i cos + j sin ,where, for a moving object, = (t) and = (t) .]

If the coordinate system to which the vi are referred is non-inertial (i.e., accelerating: say it is rotating or linearly accelerating (or both)), then the _jⁱ_k do not all vanish, and the expression for acceleration again involves non-zero values of the terms v^s_sⁱ_jdx^j/dt e_i . In this case, these non-zero values are associated with the so-called inertial accelerations, i.e., "g's", and the Coriolis and centrifugal accelerations. These accelerations are independent of any applied forces, and are due only to the accelerated motion of the coordinate system.

In a non-inertial system, the total force, ma, is the vector sum of

Even if the applied force is zero, we still have the inertial acceleration(s):

a = v^s_sⁱ_jdx^j/dt e_i .

These accelerations have the characteristic that if several different test masses are sequentially placed at a point in the system, they will all experience the same inertial acceleration (i.e., the inertial force on the various test masses will be proportional to the masses only, with the acceleration being a constant). Gravitational acceleration exhibits identical behavior in this regard; i.e., in classical mechanics, the gravitational force on a body is proportional to its mass only, the acceleration being a constant at every point in the field. This observation leads to the identity of gravitational and inertial mass, noted by Newton, and used as a motivation toward General Relativity by Einstein. Let me now present a heuristic approach to the equations of General Relativity.

One method of setting up the equations of motion for bodies in classical circular orbits is to set the gravitational force equal to the centrifugal force in a coordinate system which is revolving with the body:

(where u is a unit vector). This expression is equivalent to setting the total force on the orbiting body equal to zero, and results in the usual equations of motion for the orbiting body:

These equations may be solved if a field law is given for the gravitational field g. In classical mechanics, this law is

(where m is the field-generating mass).

The same reasoning may be applied to the tensor equations developed above. We first set the total force equal to zero everywhere in the gravitational field so that

dvⁱ/dt = - v^s_sⁱ_jdx^j/dt .

Using the relationship vⁱ = dxⁱ/dt , substituting, and rearranging terms, we then obtain

0 = d²xⁱ/dt² + _sⁱ_j(dx^s/dt) (dx^j/dt) e_i .

This expression is the differential equation for a straight line in Euclidean space, or a geodesic in a non-Euclidean space. If the classical requirement that physical space be Euclidean is relaxed, and non-Euclidean spaces are introduced, the motion of bodies in the gravitational field may be described by this equation (equation of motion) without recourse to any gravitational 'force'; i.e., the law of motion becomes: The paths followed by bodies in a gravitational field are geodesics in a [suitable] non-Euclidean space [space-time]. The problem becomes one of properly selecting the values of the _sⁱ_j (components of the gravitational field). As before, this problem may be solved by specifying a field law. Einstein chose the expression

where R_ij (= _h(_i^h_j ) - _j(_h^h_i ) + _h^h_li^l_j - _i^l_hj^h_l , with _m = /x^m defined for notational convenience) is the contracted Riemann-Christoffel curvature tensor (R_i^s_s j , a.k.a. the Ricci tensor), R is the associated scalar g^{i j}R_{i j} , g_{i j} is the fundamental tensor, and T_{i j} is the stress-energy tensor. (The expression on the l.h.s. has a vanishing divergence, satisfying the conservation of mass-energy in the gravitational field). When these equations are used with the equations of motion¹⁸

0 = d²xⁱ/dt² + _sⁱ_j (dx^s/dt) (dx^j/dt) .

the orbits of bodies and beams of light are accurately described.

To a first order of approximation, for speeds that are small compared with the speed of light, and mass densities which are comparable to those observed in our solar system, General Relativity gives results in agreement with the equations of Newton. When the field equations T_{i j} = 0 (Schwartzchild¹⁹) are used (specifying zero mass-energy density in the space surrounding the sun or any star - a good approximation for our solar system), the "classical tests" of the General Theory result: i.e.,

Finally, for cases of very high velocities (approaching the speed of light) and/or very large mass-energy densities, the predictions of General Relativity significantly diverge from those of Newton, but are confirmable by astronomical observations.

¹⁹ T_ij=0 R_ij - (1/2)g_ij R=0. Multiplying the left hand side by g^mi and summing yields R_j^m-(1/2)_j^mR=0. Next, setting j = m and summing yields R - 2R = 0 or R = 0. But R = 0 R_j^m = 0 R_ij = 0 ; therefore, T_ij = 0 R_ij = 0. The last expression is the set solved by Schwartzchild and is known as Schwartzchild's equation.