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GENERAL RELATIVITY THEORY FOR NAVIGATION
James K. Miller†and Gerald R. Hintz‡
The General Theory of Relativity, as it relates to navigation of spacecraft, may
be separated into two parts. The first part involves derivation of a set of differential field
equations that may be solved for the metric tensor and the second part involves inserting
the metric tensor into the equation of geodesics to obtain the equations of motion that may
be solved for formulae describing the precession of Mercury’s orbit, the bending of light,
radar time delay, gravitational red shift, and the time measured by clocks. In this paper,
the solution for the metric tensor is obtained from equations that provide a statement of the
theory’s fundamental assumtions. These assumptions are simply that the speed of light is
constant, matter or energy curves space and the universe has some symmetrical properties.
These assumptions are observed and cannot be proved.
Two methods are used to solve for the metric tensor. The first is a computer solu-
tion that involves parameterizing the metric tensor and solving for these parameters using an
orbit determination filter. The second is the analytic solution developed by Einstein involving
defining a covariant derivative and differentiating to obtain the Riemann tensor, Ricci tensor
and Einstein’s field equations which may be solved for the metric tensor.
Introduction
The Einstein field equations have been solved exactly for the case of spherical symmetry by Schwarzschild
and this solution and Einstein’s solution have spawned a number of formulae describing the precession of
Mercury’s orbit, the bending of light, radar time delay, gravitational red shift, and several more that relate
to special relativity. The Schwarzschild solution has been transformed to a form such that the equations of
motion look like Newton’s equations of motion with a small relativistic perturbation. For orbit determination,
these equations have been programmed into software that is used for navigation. One might question whether
this is really necessary since the perturbations due to general relativity are so small. The justification is
that the orbit solution used for prediction of a spacecraft orbit is obtained after analysis of data residuals,
the difference between the real world and the world computed by a mathematical model. Since the data is
very high precision, a very small modeling error will show up as a signature in the data residuals. If there
is an error in the model such as an incorrect gravity harmonic, the signature will grow in magnitude with
time. If there was no relativity modeling, a navigation analyst may initially conclude that the signature is
caused by relativity or some other error source such as a clock failing to keep the right time. Eventually the
signature will grow in magnitude and the alarm bells will be rung indicating there is a problem. The earlier
the problem is detected, the more likely a solution can be found before the spacecraft crashes into something.
The problem of an errant gravity harmonic caused an exponential rise in the Doppler signature on the Near
Earth Asteroid Rendezvous (NEAR) mission which was detected early and corrected before anything bad
happened. For this reason, general and special relativity are programmed into the navigation operational
software.
In the late 1960s, general relativity was programmed into the Orbit Determination Program (ODP) at
the Jet Propulsion Laboratory (JPL). At the time, those outside of navigation thought this was not needed.
Since that time, many orbits have been determined using the ODP and little attention has been given to
general relativity. The ODP is treated as a black box. When comet and asteroid missions came along, a new
†Navigation Consultant, Asso ciate Fellow AIAA, jkmxxxx@gmail.com
‡Adjunct Professor, Department of Astronautical Engineering, University of Southern California, Los Angeles, CA, 90089, Associate
Fellow AIAA
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orbit determination program was needed and this effort required implementing general relativity. Finding
the equations became a big problem. Understanding them became a bigger problem. After consulting many
sources including relativity experts at JPL, the correct equations were programmed into the software used for
the NEAR mission. We know the equations are correct because the spacecraft did not crash into anything.
The derivation of the relativity equations of motion was initiated from the metric tensor which was
assumed to be correct. The goal of deriving the equations from Einstein’s original assumptions, the speed
of light is constant and matter curves space, has been difficult to achieve. The equations of motion were
worked out long before Einstein’s death. His theory written in books published up to that time were close
to his original 1916 paper1. After his death, cosmologists got hold of the theory, and engineers had difficulty
understanding the mathematics. The main source of confusion was the normalization of coordinates removing
c, the speed of light, and Gthe gravitational constant from the equations. Einstein did this to make his
theory look more profound and mathematical. In this paper, the part of Einstein’s theory pertaining to
navigation of spacecraft in the solar system has been extracted from Einstein’s original paper1, Eddington’s
book2written in 1923, Harry Lass’s book3on tensors written in 1950 and Sokolnikoff’s book4written in
1951. What goes on inside the sun, earth or black holes is not relevant to navigation of spacecraft in the
solar system. Einstein’s paper is difficult to understand but all the essential equations are there. Einstein’s
audience was other mathematicians and physicists. Eddington, who was a mathematician, explained some of
the theory in a clear way that is relatively easy to understand. His audience was much wider than Einstein’s.
Sokolnikoff shows how the Riemann tensor is put together and Schwarzschild’s solution is obtained. Harry
Lass was one of the greatest teachers CalTech has ever produced. CalTech is noted for teaching and Richard
Feynman is a good example.
Geodesic Equation
The equation for a line element in Einstein’s summation notation is given by
ds2=guvdxudxv(1)
and since the order of differentiation is arbitrary, guv is symetric. If the elements of guv are put in a matrix
with rows defined by uand columns defined by v,weget
ds2=[[guv ]dX]TdX
where dX is a column matrix which becomes a row matrix when transposed. For regular three dimensional
Cartesian coordinates, where guv is the Kronecker delta or identity matrix, we have for dX
dX =

dx
dy
dz 

and
ds2=dx2+dy2+dz2
In four space we have the Minkowski metric
ds2=c2dt2−dx2−dy2−dz2
The shortest distance between two points on a curved surface is called a geodesic. When an airplane
flys over the North pole on its way to Europe, it is following a geodesic or great circle arc. For curved space,
the metric tensor (guv) defines a line which is a collection of points strung together. The elements of guv are
functions that define guv at some point in space. The integral of the line element (ds) defines the distance
between two points or the length of the curve connecting them. Consider two points A and B. We may define
a coordinate system somewhere in space that may be used to locate the two points relative to one another.
The points have to be real physical bodies for this definition to mean anything. Empty space is meaningless.
Our universe consists of only two bodies and they have zero mass and this constitutes the real world. Since
the reference coordinate system is arbitrary, the coordinates of the bodies are of little use. The only useful
physical reality in this world is the distance between the two bodies. The metric tensor can be integrated to
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determine the length of this line. Since we do not know the elements of the metric tensor, we simply assume
aguv. Next, we draw a line between the two points that is the shortest distance. The variation of the path
length with respect to the coordinates must be zero since only one path is the shortest. Thus we have from
Equation 1
2dsδ(ds)=dxudxvδguv +guvdxuδ(dxv)+guv dxvδ(dxu)
and
2dsδ(ds)=dxudxv
∂guv
∂xσ
δxσ+guvdxud(δxv)+guvdxvd(δxu)
and the stationary condition is Zδ(ds)=0
1
2Zdxu
ds
dxv
ds
∂guv
δxσ
δxσ+guv
dxu
ds
d
dsδxv+guv
dxv
ds
d
dsδxuds =0
The dummy indices on the last two terms can be changed to be in the same order as the first term. This
trick is a property of summation notation and amounts to changing the order of the rows in matrices and a
vector that is to be multiplied such that the vector can be factored out and the matrices summed before the
multiplication. For more information on this property, consult Einstein’s twenty pages on tensor algebra in
his 1916 paper or Harry Lass’s book on tensors. Here, we perform the operation and rely on the references
by Einstein and Harry Lass to obtain
1
2Zdxu
ds
dxv
ds
∂guv
δxσ
δxσ+guσ
dxu
ds +gσv
dxv
ds d
dsδxσds = 0 (2)
We adopt Einstein’s summation notation matrix notation in this equation in place of matrix notation.
Integration by parts is defined by the following equation.
Zx2
x1
ydx +Zy2
y1
xdy =xy
x2y2
x1y1
=x2y2−x1y1
The geometrical interpretation of this equation is that the area between the xaxis and curve between the
limits x1and x2plus the area between the yaxis and the curve is equal to the rectangular area x2y2minus
the rectangular area x1y1. This result is obvious if you draw a picture. The following property of differentials
is provided by Einstein in his 1916 paper.
d
ds(δxσ)=δdxσ
ds 
In his derivation, Eddington omitted this equation probably because he thought it was trivial. Einstein
included this equation because it is important to the understanding even if it looks strange. Einstein liked
to make statements that are counter intuitive and reading his paper for the words is satisfying even if the
equations are not understood. If we let
y=guσ
dxu
ds +gσv
dxv
ds dy =dguσ
dxu
ds +gσv
dxv
ds 
x=dxσ
ds dx =d
ds (δxσ)
Equation 2 is then
1
2Zdxu
ds
dxv
ds
∂guv
δxσ−d
ds guσ
dxu
ds +gσv
dxv
ds δxσds = 0 (3)
This equation must hold for all arbitrary displacements of δxσ. If we make δxσextremely small, the
difference between the xy rectangles defined above go to zero and may be discarded. We then make ds
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infinitely smaller than δxσand we are left with the term in the brackets. This term does not go to zero
because we must add up the same infinity of ds intervals, a Riemann sum. The terms in the bracket must be
zero. The line integral is an increasing monotone from point A to point B. The line segment keeps getting
longer. Thus, every interval of the integrand must be zero because if any interval is not zero, there can never
be a negative interval to restore the total integration to the path length. Christoffel obtained the following
result by defining an integral and then arguing that the integration must be zero, not by actually integrating.
The secret to relativity theory is to define things that are zero and avoid doing any real mathematics. This
approach makes the theory difficult to understand but is probably the only way the problem can be solved.
Carrying out the differentiation indicated in Equation 3 and setting the integrand as required by Equation
3 gives
dxu
ds
dxv
ds
∂guv
δxσ−dguσ
ds
dxu
ds +dgσv
ds
dxv
ds +guσ
d2xu
ds2+gσv
d2xv
ds2=0
The chain rule applied to summation notation is
dguσ
ds =∂guσ
∂xv
dxv
ds
dgσv
ds =∂gσv
∂xu
dxu
ds
and since the metric tensor is symetric (gvσ =gσv)
dxu
ds
dxv
ds ∂guv
δxσ−∂guσ
δxv−∂gvσ
δxu−2geσ
d2xe
ds2=0
The next step is to multiply through by the contravariant metric tensor gασ .
gασgeσ =δα

In matrix notation, this is the same as multiplying the metric tensor by its inverse.
[geσ]−1[geσ ]=I
In Einstein’s description of the contravariant fundamental tensor (the inverse of the covariant metric tensor),
he describes the matrix inversion process which involves cofactors and determinants and the Kronecker delta
which is the identity matrix. Eddington had a similar description. One advantage of summation notation is
that the order of multiplication is arbitrary, so
dxu
ds
dxv
ds gσα ∂guv
δxσ−∂guσ
δxv−∂gvσ
δxu−2d2xα
ds2=0
The equation for a geodesic is thus
d2xα
ds2+{uv, α}dxu
ds
dxv
ds = 0 (4)
and the Christoffel symbols are defined by
{uv, α}=1
2gσα ∂guσ
δxv
+∂gvσ
δxu−∂guv
δxσ(5)
Summary of General Relativity Fundamental Asumptions
The fundamental assumptions of General Relativity are stated in equations without proof. Perhaps
they can be replaced in the future by assumptions that are more fundamental. The first assumption is that
the speed of light is constant defined by cand the observed speed of light defined by the path length ds is
also constant and equal to c.
ds2=c2dt2−dx2−dy2−dz2
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The line element is defined by the metric tensor
ds2=guvdxudxv(6)
which gives us the equation of geodesics when integrated along the path length and minimized to obtain the
shortest distance.
d2xα
ds2+{uv, α}dxu
ds
dxv
ds = 0 (7)
{uv, α}=1
2gσα ∂guσ
δxv
+∂gvσ
δxu−∂guv
δxσ
It is also necessary to define a measurement (Z) which is the projection of the observed acceleration on the
trajectory of a point mass in the gravity field defined by the metric tensor.
Z=Au
dxu
ds
This scalar measurement gives us one equation, but there are 10 elements of the metric tensor and we need
nine more equations. For an analytic solution, we can differentiate this measurement with respect to the
assumed coordinates to obtain 4 more equations that can be measured. Differentiating again gives four more
equations that define the curvature and can also be observed. We need one more equation to solve for the
metric tensor. This equation is obtained by assuming the scale is proportional to mass. The assumption
that the curvature of space is proportional to mass may be satisfied by placing a boundary condition on
the solution to the Einstein field equations or solving the Einstein tensor by equating it to the stress energy
tensor. For a spherically symmetric body, the acceleration as rapproaches infinity, referred to as the weak
field solution, is given by Newton’s law of gravity and this is the easy way to obtain the tenth equation.
a=µ
r2
The solution from Einstein’s tensor, which applies to any mass distributon, is obtained from G=8πT .
There are a few other assumptions associated with mathematics that are difficult to state in simple
equations. These include symmetry, linearity and continuity. Not only the trajectory of a particle but all
the higher order derivatives must be continuous. They trace a smooth curve when drawn on graph paper
and they have slopes and areas under the curve. Once the above fundamental equations are defined, the
work of the scientist is complete. For a solution, we turn the problem over to mathematicians. Einstein was
the essential bridge between the two camps.
Computer Solution for Metric Tensor
The metric tensor is symmetric and has 10 independent elements in four space. If we knew the location
of 10 points in the real world, we could use the definition of the metric to solve for guv. We only know one,
the vector from the sun to the spacecraft. We can get around this problem by assuming that the two bodies
in the real world have mass and the mass of the central body or sun is much greater than the mass of the
spacecraft. Now, our imaginary universe is real and we can assume that there is a force at every point along
the geodetic line that results in an acceleration of the spacecraft that can be observed. We can take the dot
product of this force vector with the line element, which is in the direction of the velocity vector, and this
gives us a measurement or quantity that can be observed. We can use this measurement at various points
along the path to solve for the metric tensor. We need at least 10 points and by assuming a coordinate
system all the mathematics associated with Einstein’s solution are bypassed.
Given the equation of geodesics, our main objective is to determine the metric tensor in an assumed
coordinate system. Once the metric tensor is known, we have the equations of motion of the spacecraft.
A direct approach is to parameterize the metric tensor as a function of the coordinates and solve for these
parameters with an orbit determination filter. This approach is only practical if we have very high precision
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measurements. However, once we formulate the equations, we have the equations that may be solved ana-
lytically for the metric tensor. This is not an easy task but Einstein together with a host of mathematicians
figured out how to solve this system of equations.
Consider a spacecraft in orbit about the sun somewhere near Mercury’s orbit but far from Mercury. The
estimated parameters would be the initial spacecraft state and the parameters that characterize the metric
tensor. The measurements would be Doppler and range data from the Deep Space Network. Thus we have
for the metric tensor
g11 =eφ
φ=A0+A1r+A2r2+A3r3+···
g22 =−r2
g33 =−r2sin θ
g44 =eλ
λ=B0+B1r+B2r2+B3r3+···
and all the other elements of the metric tensor are zero. This is essentially the same metric that Schwarzschild
assumed for his solution except that g11 and g44 are parameterized here as a function of r. Here we have
interchanged Schwarzschild’s definitions of λamd φto avoid confusion with the coordinate φ. We know
from symmetry that these terms must be a function of only r. The curvature of space is static and therefore
there is no time dependence. Since we do not have a spacecraft in the desired orbit, we can use the exact
Schwarzschild equations of motion to simulate the spacecraft trajectory. The computed equations of motion
are obtained by substituting the Christoffel symbols computed from the parameterized metric into the
equation of geodesics. We thus obtain for the computed equations of motion
d2r
ds2=Γ
1
11 dr
ds2
+Γ
1
22 dφ
ds 2
+Γ
1
44 dct
ds 2
where
Γ1
11 =−1
2
1
g11
∂g11
∂r
Γ1
22 =−1
2
1
g11
∂g22
∂r
Γ1
44 =−1
2
1
g11
∂g44
∂r
and
d2r
ds2=−1
2g11
∂g11
∂r dr
ds2
−r
g11 dφ
ds 2
+1
2g11
∂g44
∂r dct
ds 2
The line element is given by
ds2=g11dr2+g22dφ2+g44 c2dt2
and, for ds2=c2dτ 2, where τis the proper time, we obtain
dt
dτ 2
=1
g44 −g11
c2g44 dr
dτ 2
−g22
c2g44 dφ
dτ 2
The equations of motion, after substituting the metric equation, become
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d2r
dτ2=1
2
1
g44
∂g44
∂r dr
dτ 2
−r
g11 dφ
dτ 2
−1
2
∂g44
∂r c2−1
2g44
∂g44
∂r dr
dτ 2
−1
2
∂g44
∂r
g22
g11g44 dφ
dτ 2
d2φ
dτ2=−2
r
dr
dτ
dφ
dτ
where
g11 =−1
g44
Observe that the first and fourth terms which are functions of radial velocity cancel when g11 is the
negative reciprocal of g44. This is a result of the assumption that there is no gravity drag. There is no ether
to slow down the planets resulting in the planets falling into the sun. Also observe that the third term has c2
in the numerator. The partial of g44 with respect to rmust have c2in the denominator or the equation for
racceleration will blow up or at least become very large. The third term is the Newtonian acceleration in
flat space. Einstein commented on the unusual mathematical quirk that the Newtonian acceleration comes
from the g44 term of the metric tensor. Even a great scientist like Einstein is sometimes amazed by the
mathematics. The final form of the equations of motion are
d2r
dτ2=−µ
r2−r
g11 −r2
2
∂g44
∂r dφ
dτ 2
d2φ
dτ2=−2
r
dr
dτ
dφ
dτ
If we borrow the exact solution for g44 from Schwarzschild which is
g44 =1−2µ
c2r
we get for the radial acceleration (Equations 34 and 42)
d2r
dτ2=−µ
r2+r−3µ
c2dφ
dτ 2
(8)
Next, we insert the parameterized metric into the modeled equation of geodesics and integrate the equations
of motion for a few weeks. Along with the equations of motion, we could integrate the variational equations to
obtain the partial derivatives of Doppler and range measurements with respect to the estimated parameters
which are the metric polynomial coefficients and initial spacecraft state. To make this demonstration simple,
we assume that we can measure ¨rdirectly and the spacecraft trajectory is known with high precision.
The solution for the polynomial coefficients is obtained by processing several hundred data points using
a weighted least square data filter. The data points are obtained from the exact Schwarzschild solution. In
theory we do not need the covariant derivative, Riemann’s tensor, Ricci’s tensor and Einstein’s tensor to do
navigation. In practice, the accuracy of the data would limit the accuracy of the parameterized metric. The
result of processing an orbit of a spacecraft in Mercury’s orbit is shown in Figure 1.
The top curve is a plot of the base 10 logarithm of the natural logarithm of g44 as a function of distance
from the sun.
λ=−0.2876563897976650D−06 + 0.1126202037576127D−13 r−0.2203417321282424D−21 r2
+0.2154331059460343D−29 r3−0.8420743154620013D−38 r4
g44 =eλ
The bottom curve is the base 10 logarithm of the difference between ln(g44) and the orbit determination
solution for the polynomial coefficients or metric tensor. Over the range from 47 ×106km to 57 ×106km
the fit is at the limit of computer precision. We see numerical noise at around 15 decimal places of accuracy.
Over the range from 30 ×106km to 70 ×106km which covers the entire orbit of the spacecraft, the error
is less than 1%. The precession of the orbit was 496.62 nanoradians per revolution about the sun which
compares favorably with the Einstein formula of 479.98 nanoradians. The precession is about 40 arcseconds
per century.
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Figure 1 Metric Tensor Estimation Error
Covariant Derivative of a Vector
Since the metric tensor is only a function of the distribution of matter in space and is independent
of the method used to determine the orbit, it should be possible to eliminate the measurement from the
differential equations for the metric tensor. Einstein and a host of mathematicians came up with a solution for
eliminating the observations from the orbit determination solution and thus obtained differential equations
that can be solved directly for the metric tensor. The metric tensor is obtained by placing the appropriate
boundary conditions on these differential equations. This is the same idea as is used to solve Laplace’s
equation for gravity harmonics.
Consider the following product of a covariant vector with a contravariant vector.
Au
dxu
ds
In matrix notation this would be a row vector times a column vector or the dot product. This dot product
is a scalar function of the coordinates and represents a measurement of the spacecraft motion. The vector
Ais arbitrary in that there are many different measurements that may be used to determine the spacecraft
motion. However, to make the problem simple, the vector Ais projected onto the observed motion. This is
equivalent to determining an orbit by observing the one-dimensional range or range rate between a spacecraft
and a tracking station. For the above orbit determination solution the orbit is sampled at points along
the trajectory. This would make the analytic solution difficult because it would be necessary to map the
measurement in space and time. Another approach is to define alternate measurements at a point in space-
time. We could measure the first and higher order derivatives of Aand this is the approach Einstein came
up with. Whether it was his original idea or not, we will probably never know. Whatever Ais, it must be
eliminated from the equations to obtain a solution for the metric tensor. Therefore, we make the observation
mathematically simple and thus make eliminating it simple.
The vector Auand the velocity or direction of motion is dependent on the assumed cordinate system.
The vector Auis not acceleration but a measure of acceleration and is thus non dynamic. The motion of
a body is not dependent on the measurement of the motion. The dot product of the vector Auwith the
velocity vector is independent of the assumed coordinate system. The projection of acceleration on velocity
is the same if viewed from any vantage point. Therefore, the derivative of the projection, which is a scalar,
is zero or in mathematical terminology is invariant and
d
ds Au
dxu
ds =0
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The problem is to find a mathematical solution for the metric tensor either by processing observed motion
in an orbit determination program or by solving the above equations. Which method of solution is best
depends on the problem. The above computer approach requires immersion of a spacecraft in the gravity
field which can be accomplished by passing an electromagnetic wave between two spacecraft safely away
from the distributed mass. The analytic approach requires observation of the mass distribution and solution
of differential equations that are really difficult to solve. It may be difficult, if not impossible, to observe the
distribution of mass in a black hole. Performing the differentiation,
∂Au
∂xv
dxv
ds
dxu
ds +Au
d2xu
ds2= 0 (9)
This result is of practical use if it is applied to a geodesic which is also invariant with respect to the assumed
coordinate system. From Equation 4, we have
Aα
d2xα
ds2=−Aα{uv, α}dxu
ds
dxv
ds
Applying this result to Equation 9 gives
dxu
ds
dxv
ds ∂Au
xv−Aα{uv, α}=0
The expression in the brackets is the covariant derivative of a vector and is given by
Ai,j =∂Ai
∂xj−{ij, α}Aα= 0 (10)
The covariant derivative is the xjth derivative of Aiwith respect to the metric tensor gij . The covariant
derivative cannot be solved for the metric tensor because of the presence of the vector A. We need to get
rid of A. One more differentiation is needed. We differentiate position to get velocity and then differentiate
velocity to get acceleration in the classical world. We then write equations for acceleration and then integrate
twice to get position. We do the same thing in the curved space world. The location of mass in the real
world defines the curvature of space or the metric tensor. The metric tensor gives us the equations of motion
through the equation of geodesics.
Covariant Derivative of a Tensor
Consider the following product of a covariant tensor with a contravariant vector.
Auv
dxu
ds
dxv
ds
Differentiating with respect to the coordinates as was done for the covariant vector, we obtain
∂Auv
∂xσ
dxσ
ds
dxu
ds
dxv
ds +Auv
dxv
ds
d2xu
ds2+Auv
dxu
ds
d2xv
ds2=0
Substituting Equation 4, we obtain the covariant derivative of a tensor.
d2xu
ds2=−{uσ, α}dxu
ds
dσ
ds
d2xv
ds2=−{vσ, α}dxv
ds
dσ
ds
9

∂Auv
∂xσ−Aσv {uσ, α}−Auσ {vσ, α}dxσ
ds
dxu
ds
dxv
ds =0
Auvσ =∂Auv
∂xσ−Aσv {uσ, α}−Auσ {vσ, α}
or
Ai,jk =∂Ai,j
∂xk−{ik, α}Aα,j −{jk,α}Ai,α (11)
Riemann-Christoffel Tensor
Substituting the covariant derivative of a vector into the covariant derivative of a tensor, we obtain
Ai,jk =∂
xk∂Ai
∂xj−{ij, α}Aα−{ik, α}∂Aα
∂xj−{αj, β }Aβ−{jk,α}∂Ai
∂xα−{iα, γ}Aγ
If we reverse the order of differentiation, we get
Ai,kj =∂
xj∂Ai
∂xk−{ik, α}Aα−{ij, α}∂Aα
∂xk−{αk , β}Aβ−{kj, α}∂Ai
∂xα−{iα, γ }Aγ
Carrying out the differentiation
Ai,jk =∂2Ai
∂xkxj−∂{ij, α}
∂xk
Aα−{ij, α}∂Aα
∂xk−{ik , α}∂Aα
xj
+{ik, α}{αj, β}Aβ−{jk, α}∂Ai
∂xα−{jk, α}{iα, γ}Aγ
Ai,kj =∂2Ai
∂xjxk−∂{ik, α}
∂xj
Aα−{ik, α}∂Aα
∂xj−{ij, α}∂Aα
xk
+{ij, α}{αk, β}Aβ−{kj, α}∂Ai
∂xα−{kj, α}{iα, γ}Aγ
The order of differentiation should not make any difference so, if we subtract, the result should be zero. We
are looking for a metric tensor that gives this result.
Ai,jk −Ai,kj ={ik, α}{αj, β}Aβ−∂{ij, α}
∂xk
Aα−{ij, α}{αk, β}Aβ+∂{ik , α}
∂xj
Aα=0
Interchanging the αand βdummy indexes associated with the Aβterms and factoring out Aαwe obtain
the Riemann-Christoffel tensor,
Rα
ijk ={ik, β }{βj,α}−∂{ij, α}
∂xk−{ij, β}{βk,α}+∂{ik, α}
∂xj
=0
This tensor has the property we need to solve for the metric tensor. The arbitrary measurement vector A
has been eliminated and the Christoffel symbols are a function of only the metric tensor.
Ricci Tensor
The Riemann-Christoffel tensor has 256 elements and each element is a function of the coordinates. We
would have to solve 256 simultaneous differential equations to obtain a solution. Due to symmetry, most of
the elements of the Riemann-Christoffel tensor are equal to or multiples of a subset of independent elements.
All the elements of the Riemann-Christoffel tensor are zero if the independent elements are zero. In tensor
10

algebra, the independent elements may be isolated by what is called contraction. In Einstein summation
notation, contraction is performed by making the kand αindices equal. The result is what is called the
Ricci tensor.
Rij ={iα, β}{βj,α}−∂{ij, α}
∂xα−{ij, β}{βα,α}+∂{iα, α}
∂xj
(12)
The Rsymbol stands for Riemann, not Ricci. Einstein does not mention Ricci when he contracts the
Riemann tensor. He just makes the indices equal and calls it contraction. Einstein does mention Ricci in
his tutorial on tensors. The Ricci tensor is solved for the metric tensor and when the boundary condition
at infinity is applied we obtain the equations that may be verified by experimentation and are used for
navigation of spacecraft.
Einstein Tensor
The Ricci tensor only applies where there is no mass. Outside the sun, we can force the Ricci tensor to
satisfy Newton’s law of gravity and we are done. Navigation of space ships does not need to know what goes
on inside the sun because we have no plans to go there in the foreseeable future. For the same reason, we can
forget about black holes. A problem with the Ricci tensor is that it models the curvature of space but does
not account for the scale. This problem with formulating measurements is common when determining orbits.
For example, when tracking landmarks to determine an orbit about an asteroid, the angle measurements
are not sufficient to determine the scale. Angles cannot determine length. Doppler or range data must be
introduced to determine length. Another example is determining the orbit of planets by measuring angles
obtained from a photographic plate on a star background. The astronomical unit or distance of the Earth
from the sun must be obtained from other sources of data. A more relevant example is determining the
inertia tensor of a rotating body by observation of its rotation. The trace of the inertia tensor can only be
determined if some known external torque is applied to the body. The Einstein tensor acknowledges this
problem and adds a term to the Ricci tensor. This term involves the trace of the Ricci tensor mapped from
space-time coordinates to observed space-time coordinates by the metric tensor.
The Einstein tensor applies both inside and outside the sun. In the limit of small mass, the Ricci tensor
also applies inside the sun. Therefore the challenge for Einstein was to find a term that when added to
the Ricci tensor satisfies the boundary condition at the surface of the sun and scales the Ricci tensor. The
Einstein tensor is
G=Guv =Ruv −1
2guv R(13)
R=guv Ruv (14)
G=8πT(15)
The Gsymbol stands for geometry. This is appropriate because Einstein’s main contribution was in linking
the Gtensor to T, the stress energy tensor. Outside the sun, Tis equal to zero. Inside the sun
T=


p000
0p00
00p0
000ρ


(16)
The variable ρis the scalar invariant density of matter and the variable pis the pressure that is obtained
in hydrostatic equilibrium. The pressure term is necessary because some of the energy is stored in the
compression of gas or the solid body. Sometimes, in the literature, the Ricci tensor stands alone and the
scalar curvature term is on the right side with T. Modern convention has all the geometry terms on the left
side. I would put the 8πon the left side. When πis in an equation used for engineering or physics, a circle is
generally involved and a circle is geometry. One may argue that there is no circle in the normal probability
distribution, but there is a pi. The circle was provided by Poisson when integrating the exponential function
11

at infinity. He could have selected any closed curve that encompassed all of infinity but chose the circle since
it is easier to integrate to get 2π.
Summary of Einstein’s Theory
Before going on to solve the field equations for a specific metric tensor and applying this result to the
equation of geodesics to obtain the equations of motion, a review of Einstein’s and classical potential theory
may be useful. Comparison of the general relativity approach with Newton’s classical theory reveals some
striking similarities. Newton starts with his inverse square law and the potential and divergence follow
in a straight forward mathematical derivation. The divergence may be solved for the potential and the
equations of motion follow from the gradient of the potential function. The inverse square law is given and
is not proved. Einstein starts from a much simpler equation and after a much more difficult mathematical
derivation obtains the Einstein tensor which may be solved for the metric tensor. The simple equation he
starts from is difficult to interpret in the physical world. Eddington has no problem in not understanding in
that he argues that if the resulting equations of motion can be verified by experiment, the theory must be
correct. Einstein offered a physical explanation of the simple starting point in terms of tensors which is hard
to understand. The difference between Einstein and Eddington is minor and in the end they are together.
Eddington probably preferred the experimental approach since he conducted the light bending experiment
that proved the theory. It appears Einstein was trying to convince mathematicians that he was one of them
and Eddington, who was a mathematician, was trying to convince physicists that he was one of them. This
was an exercise in futility.
Table 1 contains the key mile stones in the development of Newton’s classical theory of gravity and
Einstein’s General Theory of Relativity. For the classical theory, the first row contains a scalar potential,
the second row contains the first derivative or acceleration of the scalar potential and the third row contains
the second derivative or divergence of the scalar potential. For general relativity. the first row contains a
scalar which may be thought of as a measurement of motion, the second row contains the first derivative or
covariant derivative of the measurement and the third row contains the second derivative or curvature of the
measurement which is the Einstein tensor.
Table 1
Comparison of Newton’s Theory with Einstein’s Theory
Newton0sTheory Einstein0sTheory
P∞
i=0 −µi
|r0−ri|U Audxu
ds
P∞
i=0 −µi(r0−ri)
|r0−ri|3∇U∂Ai
∂xj−{ij, α}Aα
P∞
i=0
3µi
|r0−ri|3−3µi
|r0−ri|5|r0−ri|2=0 ∇·∇U=∇2U Ruv −1
2guv R
For Newton’s theory, the inverse square law is given and the potential function and divergence may be
obtained by mathematical operations on the inverse square law. We simply integrate to obtain the potential
and differentiate to obtain the divergence. It is not necessary to assign any physical meaning to potential or
divergence. For general relativity, the term in the first row is an artificial measure of motion when applied
to the equation of geodesics. The covariant derivative and Einstein tensor are obtained by mathematical
operations and require no physical explanation. One of the short comings of general relativity theory is
the lack of an equation for the gravitational constant. Maxwell was able to obtain an equation for the
speed of light as a function of the electric and magnetic field constants. These constants can be determined
independently by experiment.
The key to physically understanding theories involving the divergence theorem is to physically describe
one of the terms given in Table 1. The other two terms become locked in once one is understood since they
12

are mathematically related. For classical gravitational theory, the potential and divergence are locked to
the inverse square law and we can solve the equation for divergence and take the gradient to obtain the
equations of motion. For heat, the divergence can be obtained directly from heat flow and solved for the
scalar temperature distribution. The inverse square relationship is inside the body and is generally ignored.
For general relativity, a measurement is defined that consists of the product of two vectors.
Z=Au
dxu
ds
Aucan be thought of as acceceration and dxu
ds is in the direction of the velocity vector or tangent to the line
element at some point. Both of these vectors are observable but there components are dependent on the
assumed coordinate system. The dot product (Z) is not dependent on the assumed coordinate system. The
projection of Auon dxu
ds is the same in any coordinate system. Zis said to be invariant with respect to the
system of coordinates (ds). The derivative of Zwith respect to the coordinates (the covariant derivative) is
also invariant as are higher order derivatives. The solution of the resulting differential equations is obtained
by applying insight into the constraints associated with symmetry. One can go through all the mathematics
associated with this solution and have no understanding of general relativity. The understanding of General
Relativity theory is associated with the above fundamental assumptions.
Schwarzschild Solution
An exact solution of the Einstein field equations for a spherically symmetrical sun was obtained by
Schwarzschild about a month after Einstein published his theory. Apparently Schwarzschild was aware
of Einstein’s work long before he published. It is unreasonable to believe that Schwarzschild was able
to obtain his solution in a month. We know Hilbert was aware of Einstein’s work because they were in
communication with each other a few months before Einstein published his theory. According to Walter
Isaacson in his book Einstein, Hilbert published his own theory while Einstein was refining his theory.
Einstein objected and Hilbert withdrew his paper. According to Kip Thorne, the differences were minor.
However, Hilbert recognized the theory was Einstein’s and deferred. This little episode probably accelerated
Einstein’s publication. He probably would have preferred to wait a few more years, like Newton, because he
knew that once the theory was published he would have a lot of competition and distraction. He was right.
A valid method for solving differential equations is to guess the solution and insert it into the equations
and hope it works. Schwarzschild started with a metric tensor that was obviously close but had a couple of
undetermined functions.
gij =









−eλ000
0−r200
00−r2sin2θ0
00 0 eφ









(17)
The zeros are due to symmetry. Since all the stars in the sky and the cosmic background are evenly
distributed, it is reasonable to assume there is no preferred direction for space. The only curvature is with
respect to the rand time spherical coordinates. The functions φand λwere made exponents of esince he
knew the g11 and g44 terms of the metric tensor would approach minus one and one respectively (flat space)
as rapproached infinity. Once the metric tensor is defined the Christoffel symbols may be computed from
{uv, α}=1
2gσα ∂guσ
δxv
+∂gvσ
δxu−∂guv
δxσ
and
{11,1}={rr, r}=1
2
dλ
dr
{22,1}={θθ, r}=−re−λ
13

{33,1}={φφ, r}=−re−λsin2θ
{44,1}={tt, r}=1
2eφ−λdφ
dr
{14,4}={rt,t}=1
2
dφ
dr
{13,3}={rφ,φ}=1
r
{23,3}={θφ,φ}= cot θ
{12,2}={rθ, θ}=1
r
{33,2}={φφ, θ}=−sin θcos θ(18)
The Christoffel symbols are inserted into the Ricci tensor. This is a tedious process and will be done
for the R11 term. The other terms are easy to obtain once we have the R11 term.
R11 ={1α, β}{β1,α}−∂{11,α}
∂xα−{11,β}{βα,α}+∂{1α, α}
∂x1
Term 1
{1α, β}{β1,α}={11,1}{11,1}+{12,2}{21,2}+{13,3}{31,3}+{14,4}{41,4}
{11,1}{11,1}=1
4(λ0)2
{12,2}{21,2}=1
r2
{13,3}{31,3}=1
r2
{14,4}{41,4}=1
4(φ0)2
Term 2
∂{11,α}
∂xα
=∂{11,1}
∂x1
=1
2λ00
Term 3
{11,β}{βα,α}={11,1}({11,1}+{12,2}+{13,3}+{14,4})
−{11,β}{βα,α}=−1
2λ01
2λ0+1
r+1
r+1
2φ0
−{11,β}{βα,α}=−1
4(λ0)2+λ0
r+1
4λ0φ0
Term 4
∂{1α, α}
∂x1
=∂
∂x1
({11,1}+{12,2}+{13,3}+{14,4})= ∂
∂r 1
2λ0+1
r+1
r+1
2φ0
+∂{1α, α}
∂x1
=1
2λ00 −2
r2+1
2φ00
14

The primes indicate differentiation with respect to r. The complete Schwarzschild-Ricci tensor is then given
by
R11 =1
2φ00 −1
4λ0φ0+1
4(φ0)2−λ0
r= 0 (19)
R22 =e−λ1+1
2r(φ0−λ0)−1 = 0 (20)
R33 = sin2θe−λ1+ 1
2r(φ0−λ0)−1= 0 (21)
R44 =eφ−λ"−1
2φ00 +1
4λ0φ0−1
4(φ0)2−φ0
r#= 0 (22)
The above equations are given on page 303 of Sokolnikoff’s book4and on pages 330-331 of Lass’s book3.We
may conclude from Equations (19) and (22) that
λ0=−φ0
and
λ=−φ+constant
However, as r approaches infinity, λand φapproach zero and the constant is also zero. Therefore,
λ=−φ
Equation (20) becomes
eφ(1 + rφ0)=1
A change of variable from eφto γ, as described in Sokolnikoff’s book4, yields
γ+rγ0=1
d(rγ)
dr =1
rγ =r+constant
and
γ=1−2m
r=eφ
where 2mis a constant of integration. The metric tensor is thus
gij =










−1−2m
r−100 0
0−r200
00−r2sin2θ0
000
1−2m
r










The solution is complete once the constant mis determined. There are two ways of determining m. The
first involves substituting the metric into the equation of geodesics and obtaining the equations of motion.
The term for the weak gravity field containing mis equated with Newton’s gravitational acceleration. This
is a little tricky because the fourth coordinate of the geometry is ct and we need to factor out the cto get
the equations of motion in terms of t. The result for the acceleration of ris given in Equation 34 below and
is
d2r
dτ2=−mc2
r2+(r−3m)dφ
dτ 2
(23)
15

For a spacecraft being radially accelerated, the φcoordinate is constant and dφ is zero. Therefore,
d2r
dτ2=−mc2
r2(24)
The weak field acceleration according to Newton is
d2r
dt2=−GM
r2=−µ
r2
In the weak field, dt ≈dτ. Therefore, mc2=µand
m=µ
c2
d2r
dτ2=−µ
r2+r−3µ
c2dφ
dτ 2
(25)
The same result may be obtained by solution of the Einstein field equations inside the sun. This will
be made a lot easier by making the following substitutions in the Ricci tensor to temporarily get rid of the
exponentials.
Φ=eφand Λ=eλ
The Ricci tensor defined by Equations 19-22 becomes
R11 =−Φ00
2Φ −Φ0Λ0
4ΦΛ −(Φ0)2
4Φ2−1
r
Λ0
Λ
R22 =rΦ0
2ΦΛ +1
Λ−rΛ0
2Λ2−1
R33 =( rΦ0
2ΦΛ +1
Λ−rΛ0
2Λ2−1) sin2θ
R44 =−Φ00
2Λ +Φ0Λ0
4Λ2+(Φ0)2
4ΦΛ −1
r
Φ0
Λ
The scalar curvature of space is given by
R=Ru
u=guvRuv =g11 R11 +g22R22 +g33R33 +g44R44
R=−1
ΛR11 −1
r2R22 −1
r2sin2θR33 +1
ΦR44
Since in the Schwarzschild geometry the trajectory is planar, we may set θ=π
2and R33 =R22.
R=−1
ΛR11 −2
r2R22 +1
ΦR44
R=−Φ00
ΦΛ +Φ0Λ0
2ΦΛ2+(Φ0)2
2Φ2Λ−2
r
Φ0
ΦΛ −2
r
Λ0
Λ2−2
r2(1 −1
Λ)
G44 =R44 −1
2g44R(26)
G44 =−Φ00
2Λ +Φ0Λ0
4Λ2+(Φ0)2
4ΦΛ −1
r
Φ0
Λ+Φ00
2Λ −Φ0Λ0
4Λ2−(Φ0)2
4ΦΛ +1
r
Φ0
Λ+1
r
ΦΛ0
Λ2+Φ
r2(1 −1
Λ) (27)
16

G44 =1
r
ΦΛ0
Λ2+Φ
r2(1 −1
Λ)
At the boundary which is the surface of the sun and outside the surface of the sun where mis constant, Φ
factors out when G44 is equal to zero and G44 becomes
G44 =1
r
Λ0
Λ2+1
r2(1 −1
Λ)=0
or
G44 =1
r2−e−λ
r2−1
r
de−λ
dr =0
and this equation is satisfied if
e−λ=1−2m
r
Inside the sun, where mis not constant, Φ does not factor out of the G44 equation and this implies a different
metric tensor inside the sun. The solution inside the sun for the line element is
ds2=−(1 −2mr2
r3
s
)−1dr2+r2dθ2+r2sin2θdφ
2+1
4 3r1−2m
rs−s1−2mr2
r3
s!2
c2dt2
At the surface of the sun, the curvature just below the surface must equal the curvature above the
surface in empty space. When r=rs, the metric tensor inside the sun matches the metric tensor outside
the sun thus satisfying the boundary condition. From classical theory, it is well known that the sun may be
replaced by a point mass of infinite density with the same mass as the sun’s. The sun may also be replaced
by an infinitely thin shell at the surface also of infinite density with mass equal to the sun’s mass. This
removes the rdependence and the curvature inside the shell is the same as the curvature in free space. The
variable mis no longer constant but varies with racross the thin shell. We may thus use the G44 for free
space over the thin shell since rmay be assumed to be constant. In mathematics, this is called a squeeze
and we should have a mathematician verify that the above makes sense. Since we get the same result as is
in the literature, we chose to skip the verification. Thus we get
G=8πT
and
G44 =1
r2−e−λ
r2−1
r
de−λ
dr =8πT
44 =8πρ (28)
Since
e−λ=(1−2m
r)
we get
2m=r−re−λ
dm(r)
dr =1
21−e−λ−rde−λ
dr 
After substituting into Equation 28,
2
r2
dm(r)
dr =8πρ
17

Integrating outward to the surface of the sun (rs) we obtain
m(rs)=Zrs
0
4πr2ρdr =ρ V olume
Here we have replaced the volume integral over the thin shell by the volume integral over the entire sun. We
do this to avoid infinite density and zero thickness.
ρ V olume =Gm =µ
When we transform ct to τ,weget
m(rs)= µ
c2(29)
Schwarzschild Equations of Motion
With the metric defined, we are half way there. The other half is developing equations of motion from
the metric that may be programmed on a computer for navigation. The geodesic equation describes the
acceleration of a particle in space-time coordinates and takes the place of the gradient in classical theory.
Thus,
d2xα
ds2+Γ
α
uv
dxu
ds
dxv
ds = 0 (30)
Γu
αβ =guvΓvαβ
Γuαβ =1
2∂guα
∂xβ+∂guβ
∂xα−∂gαβ
∂xu
The Christoffel symbols for the Schwarzschild solution are given by
Γr
rr =Γ1
11 =−m
r21−2m
r−1
Γr
θθ =Γ1
22 =−r1−2m
r
Γr
φφ =Γ1
33 =−r1−2m
rsin2θ
Γr
tt =Γ1
44 =m
r21−2m
r
Γt
rt =Γ4
14 =m
r21−2m
r−1
Γφ
rφ =Γ3
13 =1
r
Γφ
θφ =Γ3
23 = cot θ
Γθ
rθ =Γ2
12 =1
r
Γθ
φφ =Γ2
33 =−sin θcos θ
To be consistent with the literature, the symbol for the Christoffel symbols {uv, α}has been changed to Γα
uv.
It is no longer necessary to recognize that the Christoffel symbols are not a tensor. The equations of motion
are obtained by substituting the Christoffel symbols into the geodesic equation. Since the motion is planar,
we may rotate to a coordinate system such that the motion is in the x-y plane. The θdependency is thus
removed and for θ=π
2, we obtain from the geodesic equation
18

d2r
ds2=m
r21−2m
r−1dr
ds2
+r1−2m
rdφ
ds 2
−m
r21−2m
rdct
ds 2
(31)
d2φ
ds2=2
r
dr
ds
dφ
ds
A clock carried on the spacecraft will provide a measure of proper time defined by the line element obtained
from the Schwarzschild solution.
ds2=−(1 −2m
r)−1dr2+r2dθ2+r2sin2θdφ
2+(1−2m
r)c2dt2
For ds2=c2dτ 2, this equation yields
dt
dτ 2
=1−2m
r−1
+1
c21−2m
r−2dr
dτ 2
+r2
c21−2m
r−1dφ
dτ 2
(32)
Substituting Equation 32 into Equation 31 gives the following equation of motion for r.
d2r
dτ2=−mc2
r2+(r−3m)dφ
dτ 2
(33)
Replacing mby µ
c2gives the following equations of motion.
d2r
dτ2=−µ
r2+r−3µ
c2dφ
dτ 2
(34)
d2φ
dτ2=−2
r
dr
dτ
dφ
dτ (35)
dt
dτ 2
=1−2µ
c2r−1
+1
c21−2µ
c2r−2dr
dτ 2
+r2
c21−2µ
c2r−1dφ
dτ 2
(36)
The trajectory of a photon differs from that of a particle or spacecraft moving at the speed of light
even in the limit of very small mass for the spacecraft. The difference arises because a photon has zero rest
mass and thus there is no force of gravity acting on the photon that gives rise to Newtonian acceleration.
This is fortunate because otherwise the Earth would increase in mass as photons from the sun collide. The
photon follows the contour of curved space and the resulting path is the called the null geodesic. Consider
the metric associated with a particle traveling at the speed of light,
ds2=0=−1
c2(1 −2µ
c2r)−1dr2+r2dθ2+r2sin2θdφ
2+(1−2µ
c2r)dt2
Since ds is zero, the geodesic equation degenerates to indeterminate forms that must be evaluated in
the limit as ds goes to zero. The indeterminate form ds/ds, which has the value of 1 for a spacecraft, has
the value 0 for a photon in the limit as ds approaches zero. We resolve the problem of ds approaching zero
in the geodesic equation by introducing the affine parameter τthat acts like a clock at rest. We know from
special relativity that an observer’s clock on the photon will not register any passage of time. The proper
time associated with a photon is simply the time that a stationary observer would measure. The difference of
the affine parameter (τ) between two points times the speed of light is the distance that one would measure
with a meter stick along the path of the photon. The equations of motion for a photon are given by
19

d2r
dτ2=r−3µ
c2dφ
dτ 2
(37)
d2φ
dτ2=−2
r
dr
dτ
dφ
dτ (38)
dt
dτ 2
=1
c21−2µ
c2r−2dr
dτ 2
+r2
c21−2µ
c2r−1dφ
dτ 2
(39)
Isotropic Schwarzschild Coordinates
In the Newtonian world, before general relativity, the trajectories of the planets were observed through
telescopes and the data fit to a model of the solar system based on Newton’s equations of motion. From this
model, the gravitational constant of the sun and the planetary ephemerides were estimated to an accuracy
consistent with the measurement and model errors. With the introduction of general relativity to the model,
the data was refit and a new set of constants and planetary ephemerides determined. However, since the
relativistic effects are small, the differences between the numerical values associated with the curved space
coordinates and the classical coordinates are also small. This small difference often results in confusion of
the two coordinate systems.
In order to make the classical system more nearly coincide with the relativistic system, a coordinate
transformation or change of variable was devised to make the local curved space coordinates come into
alignment with Euclidean coordinates. The volume element, which is a parallelepiped, in curved space
coordinates is stretched and compressed to make it a cube. This transformation makes the relativistic
coordinates look more classical, but does not really change anything. The transformed coordinate system is
called isotropic Schwarzschild coordinates and the transformation is given by
r=1+ µ
2c2¯r2
¯r(40)
φ=¯
φ(41)
where ¯rand ¯
φare the isotropic coordinates. In order to obtain the isotropic form of the equations of motion,
we simply substitute the above equation for rinto the exact Schwarzschild equations. The exact isotropic
Schwarzschild line element is given by
d¯s2=1−µ
2c2¯r2
1+ µ
2c2¯r2dt2−1
c21+ µ
2c2¯r4d¯r2+r2d¯
φ2
and this is approximated by
d¯s2=1−2µ
c2¯rdt2−1
c21+ 2µ
c2¯rd¯r2+r2d¯
φ2
The exact isotropic Schwarzschild equations of motion for a spacecraft become
d2¯r
dτ2=−µ
¯r21+ µ
2c2¯r−4
+1−µ2
4c4¯r2−1
×(µ3
2c4¯r51+ µ
2c2¯r−4d¯r
dτ 2
+1+ µ
2c2¯r2¯r−3µ
c2dφ
dτ 2)
d2¯
φ
dτ2=−1−µ2
4c4¯r2
1+ µ
2c2¯r2
2
¯r
d¯r
dτ
d¯
φ
dτ
20

d2¯
t
dτ2=1+ µ
2c2¯r2
1−µ
2c2¯r2+1
c21+ µ
2c2¯r6
1−µ
2c2¯r2"d¯r
dτ 2
+¯r2d¯
φ
dτ 2#
and these may be approximated by
d2¯r
dτ2=−µ
¯r21−2µ
c2¯r+¯r−2µ
c2dφ
dτ 2
(42)
d2¯
φ
dτ2=−1−µ
c2¯r2
¯r
d¯r
dτ
d¯
φ
dτ (43)
d2¯
t
dτ2=1+ 2µ
c2¯r+1
c21+ 4µ
c2¯r"d¯r
dτ 2
+¯r2d¯
φ
dτ 2#(44)
The exact isotropic Schwarzschild equations of motion for a photon become
d2¯r
dτ2=1−µ2
4c4¯r2−1(−µ2
2c4¯r3d¯r
dτ 2
+1+ µ
2c2¯r2
¯r−3µ
c2dφ
dτ 2)
d2¯
φ
dτ2=−1−µ2
4c4¯r2
1+ µ
2c2¯r2
2
¯r
d¯r
dτ
d¯
φ
dτ
d2¯
t
dτ2=1
c21+ µ
2c2¯r6
1−µ
2c2¯r2"d¯r
dτ 2
+¯r2d¯
φ
dτ 2#
and these may be approximated by
d2¯r
dτ2=¯r−2µ
c2dφ
dτ 2
(45)
d2¯
φ
dτ2=−1−µ
c2¯r2
¯r
d¯r
dτ
d¯
φ
dτ (46)
d2¯
t
dτ2=1
c21+ 4µ
c2¯r"d¯r
dτ 2
+¯r2d¯
φ
dτ 2#(47)
Mercury Perihelion Shift
Integration of the classical equations of motion for the orbit of Mercury reveals a shift in perihelion that
cannot be accounted for with Newtonian theory. For navigation, it is necessary to modify the equations of
motion to account for perihelion precession which is caused by the relativistic curvature of space near the
sun. This may be accomplished by use of a well-known formula or numerical integration of the relativistic
equations of motion. The results obtained by numerical integration of the relativistic equations of motion
may be compared with this formula. The well-known formula is found on the last page of Einstein’s paper1.
δφ0=24π3a2
T2c2(1 −e2)
and since the orbital period is
T=2πsa3
µs
21

we get the modern form of this equation.
δφ0=6πµs
c2a(1 −e2)
where µsis the gravitational constant of the sun, ais the semi-major axis of Mercury’s orbit, eis the orbital
eccentricity, and cis the speed of light.
A simple derivation of the precession of Mercury’s periapsis may be obtained by assuming that all the
additional potential energy from general relativity goes into increasing the period of the orbit. The addition
of the general relativity acceleration does not change the mean motion. After one revolution of the classical
orbit, the perturbed orbit and the classical orbit have the same angular orientation because the orbits have
the same angular momentum. At periapsis on the classical orbit, the perturbed orbit is descending for an
additional δP to its periapsis. The precession is thus given by
δφ0=2πδP
P
δP =3P
2aδa
δa =a2
µδC3
and
δφ0=2π
P
3P
2a
a2
µδC3=3πa
µδC3
From the Schwarzschild isotropic equations of motion (Equation 40), the radial acceleration is given by
d2¯r
dτ2=−µ
¯r21−2µ
c2¯r
Integrating the acceleration from ¯rto infinity yields the potential energy and the general relativity contri-
bution is
δEr=µ2
c2¯r2
If the average radius (¯r2) is approximated by b2=a2(1 −e2) the energy addition is
δC3=2µ2
c2a2(1 −e2)=2δEr
The factor of two is necessary because the energy orbit element (C3) is twice the actual energy. Collecting
terms, the Mercury precession is approximated by
δφ =2π
P
3P
2a
a2
µδC3=3πa
µ
2µ2
c2a2(1 −e2)=6πµ
c2a(1 −e2)
The equations of motion are integrated with the initial conditions computed from the state vector of
Mercury at perihelion. After one complete revolution of Mercury about the sun, the integrated results are
transformed to osculating orbit elements and the argument of perihelion is computed. In order to remove
the integration error, the Newtonian equations of motion are integrated by the same numerical integrator in
parallel with the relativistic equations of motion. The arguments of perihelion are differenced and compared
with the formula. The same integration is repeated, only this time the isotropic form of the Schwarzschild
equations of motion may be compared with the approximate formula. The results are displayed below.
MERCURY PERIHELION SHIFT
Perihelion Shift Formula 502.527 ×10−9radians
Exact Schwarzschild Integration 502.559 ×10−9radians
Isotropic Schwarzschild Integration 502.267 ×10−9radians
The above results indicate that the formula for perihelion shift is quite accurate. The difference of
3×10−11 radians between the formula and the exact Schwarzschild integration may be attributed to the
formula or perhaps integration error. The difference between the formula and the isotropic Schwarzschild
integration is also small (26 ×10−11 radians). This difference may also be attributed to integration error but
may be the truncation error associated with the isotropic metric.
22

Radar Delay
The transit time of a photon or electromagnetic wave between two points in space is a measurement
that is used to determine the orbits of the planets and spacecraft for the purposes of navigation and science.
Both the navigation of a spacecraft and science experiments, particularly associated with General Relativity,
require precise measurements of the transit time. Since the Deep Space tracking stations can measure times
to within 0.1 ns or about 3 cm, it is necessary to model the transit time to this accuracy.
The transit time of a photon or electromagnetic wave between two points in space is often referred to as
the radar delay. This terminology originated with radar where a radio wave is transmitted and the delay in the
reception of the reflected return is measured to determine the range. The time delay included that associated
with transmission media and the path length. Individual delay terms from the troposphere, ionosphere and
solar plasma are identified and used to calibrate the measured delay. For planetary spacecraft, the path
length is computed from the theory of General Relativity. For a round trip travel time, the additional delay
attributable to the curved space of General Relativity, over what would be computed assuming flat space,
can amount to approximately 250 µs.
ds2=1−µ
2c2r2
1+ µ
2c2r2c2dt2−1+ µ
2c2r4dr2+r2dφ2+r2sin2θdθ2
For a photon, ds2= 0 and the equation to be integrated for the elapsed coordinate time (t) is obtained by
transforming to Cartesian coordinates and solving for dt.
dt =1
c1+ µ
2c2r3
1−µ
2c2rdx2+dy2+dz2
1
2
Expanding in a Taylor series and retaining terms of order c−5,
dt =1
c1+ 2µ
c2r+7
4
µ2
c4r2dx2+dy2+dz2
1
2(48)
Figure 2 Photon Trajectory Geometry
23

The photon trajectory geometry is shown on Figure 2. The motion is constrained to the y−zplane
and targeted from y1,z
1to y2,z
2such that the photon arrives at the same ycoordinate which is taken to
be R. For this geometry, the xcoordinate is zero and the ycoordinate variation is much smaller than the z
coordinate variation. Since for this problem dy
dz ∼10−4, the line element differentials may be expanded as a
Taylor series,
dx2+dy2+dz2
1
2≈dz +1
2
dy2
dz +O(dy4
dz3) (49)
Changing the yvariable of integration to zand inserting Equation 49 into Equation 48,
dt =1
c1+ 2µ
c2r+7
4
µ2
c4r2dz +1
2
dy2
dz2dz +O(dy4
dz3)(50)
Fully expanded, there are nine terms in Equation 50 and four of them are of order 1/c5or greater. Consider
a photon grazing the surface of the Sun. A maximum error of about 10 cm or 0.3 nanoseconds is desired. To
achieve this accuracy, numerical integration of the equation of geodesics reveals that only four of the terms
in Equation 48 need be retained and these are,
t2−t1=1
cZz2
z11+ 2µ
c2r+1
2
dy2
dz2+7
4
µ2
c4r2dz (51)
In carrying out the integration, care should be taken in geometrically interpreting the results. A “straight
line” in curved space geometry, the shortest measured distance between two points, is the photon trajectory
and not the dashed line shown on Figure 2.
Consider the first term of Equation 51,
∆tf=1
cZz2
z1
dz =1
c(z2−z1) (52)
This is called the flat space term. If the end points were in flat space, ∆tfwould be the time a photon
travels from point 1 on Figure 2 to point 2. In curved space, there is no such thing as a straight line that
connects these two points. The real interpretation of the term given by Equation 52 is the mathematical
result of performing the integration on the first term of Equation 51.
The second term of Equation 51 is called the logarithmic term for reasons that will become obvious.
∆t0
log =2µ
c3Zz2
z1
dz
r
Integration requires an equation for ras a function of z. An iterative solution may be obtained by assuming
a solution for rand integrating to obtain a first approximation for tand yas a function of z. This solution
is inserted into the remainder term, the difference between the assumed and actual function, and a second
iterated solution may be obtained for tand y. This method of successive approximations is continued until
the required accuracy is achieved. As a starting function, “straight line” motion is assumed. Making use of
the approximation that
r≈pz2+R2
∆t0
log =2µ
c3Zz2
z1"1
pz2+R2+1
r−1
√z2+R2#dz =∆tlog +∆trr (53)
The first term of Equation 53 integrates to the well-known equation for the time delay.
∆tlog =2µ
c3ln "z2+pz2
2+R2
z1+pz2
1+R2#(54)
24

The second term of Equation 53, which will be referred to as the radial remainder term (∆trr ), requires a
more accurate equation for rto be evaluated. In order to evaluate the terms associated with bending of the
trajectory, an equation for yas a function of zis needed. The ycoordinate is associated with the bending of
the photon trajectory. Consider two photons in the plane of motion separated by ∆R. The plane containing
these two photons and perpendicular to the velocity vector is the plane of the wave front. The bending is
simply the distance one photon leads the other divided by their separation.
δ=c∆td
∆R
In the limit as ∆Rapproaches zero, the equation for bending is
δ=cdtd
dR
The equation for the delay is taken to be the logarithmic term given by Equation 52 and for simplicity the
bending is computed starting at closest approach (z2= 0) to the origin.
td=2µ
c3ln "z+√z2+R2
R#
Taking the derivative with respect to R,
dtd
dR ≈−2µ
c2−1
R+R
√z2+R2(z+√z2+R2)
Making use of the trigonometric approximations,
cos φ≈R
√z2+R2,sin φ≈z
√z2+R2
the equation for the bending reduces to
δ=cdtd
dR =−2µ
c2R
(sin φ+1)−cos2φ
(sinφ +1) =−2µ
c2Rsin φ
δ=2µ
c2R
z
√z2+R2
Therefore, the accumulated bending from z1to z, expressed as differentials, is given by
dy
dz =δf−2µ
c2R(z
√z2+R2−z1
pz2
1+R2) (55)
where δfis the initial angle between the photon velocity vector and the horizontal line shown on Figure 2.
Referring to Figure 2, the ycomponent of the photon is
y=R+δy
δy(z)=Zz
z1δf−2µ
c2R(z0
√z02+R2−z1
pz2
1+R2)dz0
and
δy=δf(z−z1)−2µ
c2Rpz2+R2−zz1+R2
pz2
1+R2(56)
25

The angle δfmay be determined by evaluating the bending over the interval from z1to z2. The coordinates
are rotated to target the photon to the point z=z2, where δy= 0 and the constant gravitational aberration
angle δfwas determined as
δf=1
z2−z1
2µ
c2Rqz2
2+R2−z2z1+R2
pz2
1+R2(57)
The angle δfsimply rotates the coordinates of Figure 2 such that y1and y2have the same value R.
The geometrical part of the radial remainder term, given in Equation 51, may be approximated by
making use of
1
r−1
√R2+z2=1
p(R+δy)2+z2−1
√R2+z2≈−Rδy
(R2+z2)3
2
The complete radial remainder term (∆trr) is then given by
∆trr =−2µ
c3Zz2
z1
R
(R2+z2)3
2δf(z−z1)−2µ
c2Rpz2+R2−zz1+R2
pz2
1+R2dz
∆trr =2µ
c3R2(1
pR2+z2
2δf(R3+z1z2R)+2µR
c2arctanz2
RqR2+z2
2
+2µ
c2R2sin φ1+z1z2sin φ1−z2qR2+z2
1
−1
pR2+z2
1δf(R3+z2
1R)+2µR
c2arctanz1
RqR2+z2
1
+2µ
c2R2sin φ1+z2
1sin φ1−z1qR2+z2
1)
∆trr =2µ
c3R(δfhz1z2+R2
pz2
2+R2−qz2
1+R2i
−2µ
c2arctanz1
R−arctanz2
R+R(z2−z1)
pz2
1+R2pz2
2+R2)(58)
The third term of Equation 51 is the direct contribution of the trajectory bending to the time delay.
This term is referred to as the bending term and is given by
∆tb=1
2cZz2
z1dy
dz 2
dz
Substituting Equation 55 for the slope into the above equation gives
∆tb=1
2cZz2
z1δf+2µ
c2R(sin φ−sin φ1)2
dz
∆tb=1
2cZz2
z1 δf−2µ
c2R(z
√z2+R2−z1
pz2
1+R2)!2
dz
26

Carrying out the integration
∆tb=1
2c2R(4µ2
c4+δ2
fR2+4µδfR
c2sin φ1+4µ
c4sin2φ1[z2−z1]
−4µδfR
c2+8µ2
c4sin φ1qR2+z2
2−qR2+z2
1
+4µ2R
c4arctanR
z2−arctanR
z1)
∆tb=1
2c(δ2
f(z2−z1)−4µ
c2Rδfhqz2
2+R2−z1z2+R2
pz2
1+R2i
+4µ2
c4R2hR2(z1+z2)+2z2
1z2
R2+z2
1−2z1sz2
2+R2
z2
1+R2
+Rharctanz1
R−arctanz2
Rii)(59)
The fourth and final term of Equation 51 is the c5approximation to the error in the metric. This is a
small term and contributes less than a nanosecond to the delay. The equation is given by
∆tm=7
4
µ2
c5Zz2
z1
1
r2dz ≈7
4
µ2
c5Zz2
z1
1
R2+z2dz
Carrying out the integration
∆tm≈7
4
µ2
c5Rharctan z2
R−arctan z1
Ri (60)
The complete equation for the coordinate time delay of a photon moving from (y1,z
1)to(y2,z
2)is
obtained by summing all the individual terms and
t2−t1=∆tf+∆tlog +∆trr +∆tb+∆tm(61)
Before evaluating the individual terms of Equation 61, the parameters used in the individual terms must be
determined unambiguously from the end points of the photon trajectory. If two arbitrary end points in the
y−zplane are defined by (y0
1,z0
1) and (y0
2,z0
2), the vectors from the origin to these points are given by
r1=(0,y
0
1,z0
1) and r2=(0,y
0
2,z0
2)
and the vector from point 1 to point 2 is
r12 =(0,y
0
2−y0
1,z0
2−z0
1)
The angles between the vectors r1and r2and the vector r12 are computed from the dot products.
φ1= arccos r1·r12
r2r12 ,φ
2= arccos r2·r12
r2r12 
The parameters needed in Equation 61 with the coordinates rotated as shown on Figure 2 are then
given by
27

R=r1sin φ1=r2sin φ2
z1=r1cos φ1,z
2=r2cos φ2
φ2= arctan z1
R,φ
1= arctan z2
R
δf=1
z2−z12µ
c2Rsin φ1(z2−z1)−2µ
c2Rqz2
2+R2−qz2
1+R2
δf=1
z2−z1
2µ
c2RhR2+z1z2
pR2+z2
1−qR2+z2
2i
and the angle δfis given by Equation 57. The fully expanded equation for the transit time is given by,
t2−t1≈1
c(z2−z1)+ 2µ
c3ln "z2+pz2
2+R2
z1+pz2
1+R2#
+2µ
c3R(δfhz1z2+R2
pz2
2+R2−qz2
1+R2i
−2µ
c2arctanz1
R−arctanz2
R+R(z2−z1)
pz2
1+R2pz2
2+R2)
+1
2c(δ2
f(z2−z1)−4µ
c2Rδfhqz2
2+R2−z1z2+R2
pz2
1+R2i
+4µ2
c4R2hR2(z1+z2)+2z2
1z2
R2+z2
1−2z1sz2
2+R2
z2
1+R2
+Rharctanz1
R−arctanz2
Rii)
+7
4
µ2
c5Rharctan z2
R−arctan z1
Ri
After simplification this equation takes the following form
t2−t1≈1
c(z2−z1)(1 + 1
2δ2
f)+ 2µ
c3ln "z2+pz2
2+R2
z1+pz2
1+R2#
+2µ
c3Rδfh(z1z2−z2
2)pz2
1+R2+(z1z2−z2
1)pz2
2+R2
pz2
1+R2pz2
2+R2i
+2µ2
c5R2hR2(z1+z2)+2z2
1z2
z2
1+R2−2z2(z1z2+R2)
pz2
1+R2pz2
2+R2i
+15
4
µ2
c5Rharctan z2
R−arctan z1
Ri
δf≈1
z2−z1
2µ
c2Rqz2
2+R2−z2z1+R2
pz2
1+R2(62)
Equation 62 is the time delay associated with a photon or electromagnetic wave that passes through the
gravitational field of a massive spherical body. The time delay is a function of only the gravitational constant
28

of the massive body and the parameters z1,z2and R, which may be computed directly from the isotropic
Schwarzschild coordinates of the end points.
Figure 3 Time Delay for Solar Graze
In order to determine the veracity of Equation 60, a comparison with the time delay computed from
numerical integration of the geodesic equations of motion was made and the result plotted on Figure 3.
In carrying out the numerical integration, a photon was initialized with a zcoordinate of -149,000,000 km
and y coordinate of 696,000 km. The y component of velocity was set to zero and the zcomponent to c.
The xcoordinates of position and velocity were set to zero. Thus the photon is initialized with a velocity
magnitude equal to the speed of light and parallel to the zaxis about 1 A.U. from the sun and on a flight
path that would graze the surface of the sun if there were no bending due to General Relativity. The polar
coordinates of the initial conditions were used to initialize the equations of motion and these were integrated
by a fourth order Runge Kutta integrator with fifth order error control. The integration was stopped at
various times along the flight path and Equation 60 was evaluated. The required parameters were computed
from the initial coordinates and the integrated coordinates at the time of the evaluation.
Also shown on Figure 3 are some of the individual groupings of terms from Equation 49. The linear term
has been omitted since this term would require an additional 6 cycles of logarithmic scale. The dashed curve
is the difference between the time delay computed by Equation 60 and the results of numerical integration.
This difference is attributed to error in the numerical integration algorithm. This was verified by setting the
mass of the sun to zero and integrating straight line motion in the same coordinate system. Unfortunately,
the integration error masked the error in the metric. Therefore, Equation 60 could only be verified to about
0.05 ns which is about the same level of error as the error in the metric. The integration error of about
10−14 tis consistent with the error obtained integrating spacecraft orbits for navigation. Observe that the
radial remainder term and bending term cause errors on the order of 10 ns or 37 cm.
Light Deflection
Light deflection is the bending of a photon or radio wave trajectory as it passes by a massive object. An
experiment performed during a solar eclipse in 1919 measured the deflection of star light and was the first
confirmation of General Relativity theory. For this comparison, we integrate the equations of motion for a
photon and compare it with an analytic formula. The analytic formula is for a photon arriving at the Earth
from infinity. This formula has been adapted to provide a continuous measure of the bending between any
two points and is given by
δφ =2µ
c2R{(cos(90 + φ1)−cos(90 + φ2)}
where Ris the closest approach to the sun, φ1is the angle from the yaxis to the source and φ2is the angle
29

from the yaxis to the receiver. The yaxis is in the direction of closest approach as illustrated on Figure 2.
Einstein’s formula for the total bending is simply
δφ12 =4µ
c2R
where φ1=-90 degrees and φ2=90 degrees.
Another formula for the bending is given by Equation 55.
δf=1
z2−z1
2µ
c2Rqz2
2+R2−z2z1+R2
pz2
1+R2
If we take the limit as z1approaches minus infinity and z2approaches plus infinity, δfis one half of the
Einstein bending formula. Since the total bending is the sum of the approach and departure bending, which
are equal, the δfformula, when multiplied by two, is the Einstein formula.
Comparison of the Einstein formula with numerical integration of the isotropic Schwarzschild equations
of motion is a little tricky because we must define what is meant by bending in curved space. The generally
accepted definition is the angle between the local tangent of the photon trajectory and the straight line path
that the photon would follow if the sun was removed. Thus, in isotropic Schwarzschild coordinates, the
deflection is given by
δφ = tan−1Vrcos φ−Vnsin φ
Vrcos φ+Vnsin φ
where
Vr=dr
dτ
Vn=rdφ
dτ
and the undeflected photon is assumed to move parallel to the z axis.
The equations of motion are initialized with the position and velocity of the photon. We place the
photon far from the sun on a trajectory that will graze the surface of the sun. The initial state vector is
given by
r1=149,001,625.km
φ1=−89.73236 deg
dr1
dτ =−299,789.729 km/sec
dφ1
dτ =9.3982872536 ×10−6rad/sec
and the constants are
µ=1.327124399 ×1011km3/sec2
c= 299792.458 km/sec
The equations of motion are integrated along a trajectory that grazes the sun and terminates at
τ2=954.901039554 sec (affine parameter time)
t2=954.901158130 sec (coordinate time)
r2=137,274,407.km
φ2=89.70998749 deg
dr2
dτ =299,789.146 km/sec
dφ2
dτ =11.072650234 ×10−6rad/sec
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A comparison of the total bending obtained by numerical integration with the theoretical formula derived
by Einstein gives
TOTAL LIGHT DEFLECTION ANGLE
Einstein’s Formula 8.48622 ×10−6radians
Exact Schwarzschild Integration 8.48642 ×10−6radians
References
1. Einstein A., The Foundation of the General Theory of Relativity, Annalen der Physik, 1916.
2. Eddington, A. S., The Mathematical Theory of Relativity, Cambridge University Press, 1923.
3. Lass, H., Vector and Tensor Analysis, McGraw-Hill, New York, 1950.
4. Sokolnikoff, I. S., Tensor Analysis Theory and Applications to Geometry and Mechanics of Continua,
John Wiley and Sons, New York, 1964.
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