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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

1

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

2

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

2Zdxu

ds

dxv

ds

∂guv

δxσ

δxσ+guv

dxu

ds

d

dsδxv+guv

dxv

ds

d

dsδxuds =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

2Zdxu

ds

dxv

ds

∂guv

δxσ

δxσ+guσ

dxu

ds +gσv

dxv

ds d

dsδxσds = 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)=δdxσ

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 =dguσ

dxu

ds +gσv

dxv

ds

x=dxσ

ds dx =d

ds (δxσ)

Equation 2 is then

1

2Zdxu

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

3

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

4

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

5

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

ds2

+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

ds2

−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

6

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

c2dφ

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.

7

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

8

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

2λ01

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−λ1+1

2r(φ0−λ0)−1 = 0 (20)

R33 = sin2θe−λ1+ 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

c2dφ

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

21−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

r21−2m

r−1

Γr

θθ =Γ1

22 =−r1−2m

r

Γr

φφ =Γ1

33 =−r1−2m

rsin2θ

Γr

tt =Γ1

44 =m

r21−2m

r

Γt

rt =Γ4

14 =m

r21−2m

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

r21−2m

r−1dr

ds2

+r1−2m

rdφ

ds 2

−m

r21−2m

rdct

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

c21−2m

r−2dr

dτ 2

+r2

c21−2m

r−1dφ

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

c2dφ

dτ 2

(34)

d2φ

dτ2=−2

r

dr

dτ

dφ

dτ (35)

dt

dτ 2

=1−2µ

c2r−1

+1

c21−2µ

c2r−2dr

dτ 2

+r2

c21−2µ

c2r−1dφ

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

c2dφ

dτ 2

(37)

d2φ

dτ2=−2

r

dr

dτ

dφ

dτ (38)

dt

dτ 2

=1

c21−2µ

c2r−2dr

dτ 2

+r2

c21−2µ

c2r−1dφ

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¯r2

¯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¯r2

1+ µ

2c2¯r2dt2−1

c21+ µ

2c2¯r4d¯r2+r2d¯

φ2

and this is approximated by

d¯s2=1−2µ

c2¯rdt2−1

c21+ 2µ

c2¯rd¯r2+r2d¯

φ2

The exact isotropic Schwarzschild equations of motion for a spacecraft become

d2¯r

dτ2=−µ

¯r21+ µ

2c2¯r−4

+1−µ2

4c4¯r2−1

×(µ3

2c4¯r51+ µ

2c2¯r−4d¯r

dτ 2

+1+ µ

2c2¯r2¯r−3µ

c2dφ

dτ 2)

d2¯

φ

dτ2=−1−µ2

4c4¯r2

1+ µ

2c2¯r2

2

¯r

d¯r

dτ

d¯

φ

dτ

20

d2¯

t

dτ2=1+ µ

2c2¯r2

1−µ

2c2¯r2+1

c21+ µ

2c2¯r6

1−µ

2c2¯r2"d¯r

dτ 2

+¯r2d¯

φ

dτ 2#

and these may be approximated by

d2¯r

dτ2=−µ

¯r21−2µ

c2¯r+¯r−2µ

c2dφ

dτ 2

(42)

d2¯

φ

dτ2=−1−µ

c2¯r2

¯r

d¯r

dτ

d¯

φ

dτ (43)

d2¯

t

dτ2=1+ 2µ

c2¯r+1

c21+ 4µ

c2¯r"d¯r

dτ 2

+¯r2d¯

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¯r3d¯r

dτ 2

+1+ µ

2c2¯r2

¯r−3µ

c2dφ

dτ 2)

d2¯

φ

dτ2=−1−µ2

4c4¯r2

1+ µ

2c2¯r2

2

¯r

d¯r

dτ

d¯

φ

dτ

d2¯

t

dτ2=1

c21+ µ

2c2¯r6

1−µ

2c2¯r2"d¯r

dτ 2

+¯r2d¯

φ

dτ 2#

and these may be approximated by

d2¯r

dτ2=¯r−2µ

c2dφ

dτ 2

(45)

d2¯

φ

dτ2=−1−µ

c2¯r2

¯r

d¯r

dτ

d¯

φ

dτ (46)

d2¯

t

dτ2=1

c21+ 4µ

c2¯r"d¯r

dτ 2

+¯r2d¯

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

¯r21−2µ

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

2c2r2

1+ µ

2c2r2c2dt2−1+ µ

2c2r4dr2+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

c1+ µ

2c2r3

1−µ

2c2rdx2+dy2+dz2

1

2

Expanding in a Taylor series and retaining terms of order c−5,

dt =1

c1+ 2µ

c2r+7

4

µ2

c4r2dx2+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

c1+ 2µ

c2r+7

4

µ2

c4r2dz +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

z11+ 2µ

c2r+1

2

dy2

dz2+7

4

µ2

c4r2dz (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

z1δf−2µ

c2R(z0

√z02+R2−z1

pz2

1+R2)dz0

and

δy=δf(z−z1)−2µ

c2Rpz2+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

c2Rqz2

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

2δf(z−z1)−2µ

c2Rpz2+R2−zz1+R2

pz2

1+R2dz

∆trr =2µ

c3R2(1

pR2+z2

2δf(R3+z1z2R)+2µR

c2arctanz2

RqR2+z2

2

+2µ

c2R2sin φ1+z1z2sin φ1−z2qR2+z2

1

−1

pR2+z2

1δf(R3+z2

1R)+2µR

c2arctanz1

RqR2+z2

1

+2µ

c2R2sin φ1+z2

1sin φ1−z1qR2+z2

1)

∆trr =2µ

c3R(δfhz1z2+R2

pz2

2+R2−qz2

1+R2i

−2µ

c2arctanz1

R−arctanz2

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

z1dy

dz 2

dz

Substituting Equation 55 for the slope into the above equation gives

∆tb=1

2cZz2

z1δf+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 φ1qR2+z2

2−qR2+z2

1

+4µ2R

c4arctanR

z2−arctanR

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

+Rharctanz1

R−arctanz2

Rii)(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

Ri (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−z12µ

c2Rsin φ1(z2−z1)−2µ

c2Rqz2

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

c2arctanz1

R−arctanz2

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

+Rharctanz1

R−arctanz2

Rii)

+7

4

µ2

c5Rharctan z2

R−arctan z1

Ri

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

Ri

δf≈1

z2−z1

2µ

c2Rqz2

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

c2Rqz2

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−1Vrcos 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

30

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.

31