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
v
= v^{i}e_{i}
where
v^{i} 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 timederivative of velocity, and involves derivatives of
both the v^{i} and the e_{i} :
a
= dv/dt = (dv^{i}/dt)e_{i} + v^{i}(de_{i}/dt)
.
The second
term may be further expanded as
v^{i}(de_{i}/dt)
= v^{i}(_{i}^{j}_{k}
dx^{j}/dt e_{k} )
where
_{i}^{j}_{k}
are the appropriate Christoffel symbols. Substituting, the expression
for acceleration becomes
a
= (dv^{i}/dt + v^{s}_{s}^{i}_{j}
dx^{j}/dt )e_{i}
(with
suitable change of indices on v^{s} _{s}^{i}_{j}).
If, in
a Euclidean space, the components of velocity, v_{i} , are referred
to an inertial (nonaccelerated) Cartesian (geodesic) coordinate system,
then the _{j}^{i}_{k}
all vanish (i.e., _{j}^{i}_{k}
= 0
values of i, j, & k) and the expression for acceleration has the form
a
= (dv^{i}/dt)e_{i} .
If a
nonCartesian inertial coordinate system is used, say a polar or a spherical
coordinate system, then the _{j}^{i}_{k}
do not all necessarily vanish, and the expression for acceleration may
involve nonzero values of some of the v^{s}_{s}^{i}_{j}dx^{j}/dt
e_{i}.
[eg.:
In the case of an inertial polar coordinate system, the nonzero values
of v^{s}_{s}^{i}_{j}dx^{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 noninertial (i.e.,
accelerating: say it is rotating or linearly accelerating (or both)),
then the _{j}^{i}_{k}
do not all vanish, and the expression for acceleration again involves
nonzero values of the terms v^{s}_{s}^{i}_{j}dx^{j}/dt
e_{i} . In this case, these nonzero values are associated
with the socalled 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
noninertial system, the total force, ma, is the vector sum of
1. The
applied force(s), m dv^{i}/dt e_{i} , and
2.
The inertial force(s) m v^{s}_{s}^{i}_{j}dx^{j}/dt
e_{i}.
Even
if the applied force is zero, we still have the inertial acceleration(s):
a
= v^{s}_{s}^{i}_{j}dx^{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:
mg
=  (mv^{2}/r)u.
(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:
f
= mg + (mv^{2}/r)u = 0 .
These
equations may be solved if a field law is given for the gravitational
field g. In classical mechanics, this law is
g
=  (Gm/r^{2}) u .
(where
m is the fieldgenerating 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
dv^{i}/dt
=  v^{s}_{s}^{i}_{j}dx^{j}/dt
.
Using
the relationship v^{i} = dx^{i}/dt , substituting, and
rearranging terms, we then obtain
0 =
d^{2}x^{i}/dt^{2} + _{s}^{i}_{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 nonEuclidean space. If the classical requirement
that physical space be Euclidean is relaxed, and nonEuclidean 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]
nonEuclidean space [spacetime]. The problem becomes one of properly
selecting the values of the _{s}^{i}_{j}
(components of the gravitational field). As before, this problem may be
solved by specifying a field law. Einstein chose the expression
R_{ij}
 (1/2)g_{ij}R = T_{ij}
where
R_{ij} (= _{h}(_{i}^{h}_{j}
)  _{j}(_{h}^{h}_{i}
) + _{h}^{h}_{l}
_{i}^{l}_{j}
 _{i}^{l}_{h}
_{j}^{h}_{l}
, with _{m}
= /x^{m}
defined for notational convenience) is the contracted RiemannChristoffel
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
stressenergy tensor. (The expression on the l.h.s. has a vanishing divergence,
satisfying the conservation of massenergy in the gravitational field).
When these equations are used with the equations of motion^{18}
0 =
d^{2}x^{i}/dt^{2} + _{s}^{i}_{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
(Schwartzchild^{19}) are used (specifying zero
massenergy 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.,
 The rotation
of Mercury's perihelion at a rate of 42.9 arcsec per century (not
predictable by classical mechanics unless an intraMercurial planet
is assumed [Vulcan]);
 The 1.75 arcsec
deflection of starlight grazing the sun's surface (measured by Eddington;
originally predicted to be only half as big by classical mechanics
in conjunction with Special Relativity); and
 The red shift
of starlight traveling outward in the gravitational field of a star
(in agreement with classical mechanics in conjunction with Special
Relativity).
Finally,
for cases of very high velocities (approaching the speed of light) and/or
very large massenergy densities, the predictions of General Relativity
significantly diverge from those of Newton, but are confirmable by astronomical
observations.
^{18}
In General Relativity, the differential time dt is no longer used because
it is not an invariant (i.e., a scalar; it is the component of a
fourvector and subject to the Lorentz transformation); rather, the invariant
interval ds^{2}= g_{ij} dx^{i }dx^{j}
is used instead.
^{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}^{m}R=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.
