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Yaakov Friedman and Tzvi Scarr
A Novel Approach to Relativistic
Dynamics
Integrating Gravity, Electromagnetism and
Optics
Fundamental Theories of Physics 210
January 8, 2023
Springer Nature
Contents
Notation ...................................................... 5
Introduction .................................................. 7
1.1 Physics via Geometry - A Historical Perspective . . . . . . . . . . . . 7
1.2 Unification and Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Overview of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Classical Dynamics ........................................... 21
2.1 ClassicalFields......................................... 21
2.2 Motion in the Classical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 The Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
The Lorentz Transformations and Minkowski Space........... 33
3.1 InertialFrames......................................... 34
3.2 Spacetime Transformations that Satisfy the Principle of
Relativity ............................................. 38
3.2.1 The Galilean Transformations . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 The Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Einstein Velocity Addition and Applications . . . . . . . . . . . . . . . 47
3.3.1 Velocity Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Fiber Optic Gyroscopes and the Sagnac Effect . . . . . . . 50
3.4 Minkowski Space ....................................... 52
3.5 Four-vectors, Four-covectors, and Contraction . . . . . . . . . . . . . . 55
3.6 Relativistic Energy-Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7 Relativistic Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8 Lorentz-covariant Functions for a Single-source Field . . . . . . . . 67
The Geometric Model of Relativistic Dynamics ............... 71
4.1 The Relativity of Spacetime and the Extended Principle of
Inertia ................................................ 71
v
vi Contents
4.2 Geodesics on the Globe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 The Geometric Action Function and its Properties . . . . . . . . . . 84
4.4 Simple Action Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Universal Relativistic Equation of Motion . . . . . . . . . . . . . . . . . 88
4.5.1 The Equation of Motion using Proper Time . . . . . . . . . 89
4.5.2 The Equation of Motion using ˜τ. ................... 95
The Electromagnetic Field in Vacuum ........................ 99
5.1 The Electromagnetic Field Tensor . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 The Four-Potential of a Single-source Electric Field . . . . . . . . . 101
5.3 The Electromagnetic Field of a Moving Source . . . . . . . . . . . . . 104
5.4 The Electric and Magnetic Components of the Field of a
Uniformly Moving Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5 The Energy-Momentum of an Electromagnetic Field . . . . . . . . 111
5.6 The Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.7 The Four-Potential of a General Electromagnetic Field . . . . . . 116
5.8 The Field of a Current in a Long Wire . . . . . . . . . . . . . . . . . . . . 117
5.9 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.10 Orbits of Charged Particles in a Static, Single-source Field . . 122
5.11 Circular Orbits......................................... 125
The Gravitational Field....................................... 127
6.1 The Gravitational Field of a Stationary, Static, Spherically
Symmetric Body and its Geometry . . . . . . . . . . . . . . . . . . . . . . . 128
6.2 Precession of Orbits in a Stationary, Static, Spherically
Symmetric Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3 Periastron Advance of Binary Stars . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Orbits in the Strong Field Regime . . . . . . . . . . . . . . . . . . . . . . . . 142
6.4.1 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.4.2 Elliptical Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.4.3 Hyperbolic-like Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.5 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.6 Shapiro Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.7 The Gravitational Field of Multiple Sources . . . . . . . . . . . . . . . . 154
6.8 The Gravitational Field of a Moving Source . . . . . . . . . . . . . . . . 156
6.9 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Motion of Light and Charges in Isotropic Media .............. 161
7.1 The Photon Action Function of Rest Media . . . . . . . . . . . . . . . . 161
7.2 The Photon Action Function in Moving Media . . . . . . . . . . . . . 162
7.3 Refraction of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.4 Motion of a Charge in an Isotropic Medium at Rest . . . . . . . . . 166
Contents vii
Spin and Complexified Minkowski Spacetime ................. 169
8.1 History of the Spin of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.2 The State Space of an Extended Object Moving Uniformly . . 171
8.3 Complexified Minkowski Space as the State Space of an
Extended Object ....................................... 173
8.4 The Representation of the Spin of an Electron . . . . . . . . . . . . . 175
8.5 Transition Probabilities of Spin States and Bell’s Inequality . . 177
8.6 Motion of Particles with Spin in a Slow-varying
Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
The Prepotential ............................................. 183
9.1 The Prepotential and the Four-Potential of a Field
Generated by a Single Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.2 Representations of the Lorentz group on Mc............... 187
9.3 Lorentz Invariance of the Prepotential and the Conjugation . . 190
9.4 The Four-Potential of a Moving Source . . . . . . . . . . . . . . . . . . . . 192
9.5 The Symmetry of the Complex Four-Potential . . . . . . . . . . . . . 194
9.6 The Prepotential and the Wave Equation . . . . . . . . . . . . . . . . . . 195
9.7 The Electromagnetic Field Tensor of a Moving Source and
its Self-Duality......................................... 196
9.8 The Prepotential of a General Electromagnetic Field . . . . . . . . 197
References ................................................. 201
Index ......................................................... 205
List of Figures
1.1 Galileo Galilei .......................................... 7
1.2 Bernhard Riemann ...................................... 8
1.3 AlbertEinstein ......................................... 9
1.4 Isaac Newton ........................................... 12
1.5 Leonhard Euler ......................................... 15
1.6 Joseph-Louis Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 The potential energy on a bounded orbit . . . . . . . . . . . . . . . . . . . 26
2.2 The parameters of an ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 The potential energy on an unbounded orbit . . . . . . . . . . . . . . . . 29
2.4 The parameters of a hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 The coordinates of an event in a frame of reference . . . . . . . . . . 35
3.2 An accelerometer........................................ 36
3.3 Representing an event in two inertial frames . . . . . . . . . . . . . . . . 37
3.4 Standard configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Symmetric configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 The Galilean transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7 The Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8 Action of the velocity addition on Du...................... 49
3.9 A fiber optic gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.10 The light cone and four-velocoities (2D) . . . . . . . . . . . . . . . . . . . . 55
3.11 The light cone and four-velocoities (3D) . . . . . . . . . . . . . . . . . . . . 56
3.12 Energy-momentum conservation in a 1D collision . . . . . . . . . . . . 62
3.13 Photon emission and the Doppler shift . . . . . . . . . . . . . . . . . . . . . 66
3.14 Four-vectors in a single-source field . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1 The relativity of spacetime for charges . . . . . . . . . . . . . . . . . . . . . 72
4.2 The relativity of light propagation . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Geographic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Distances as Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
ix
x List of Figures
4.5 Short distances on the flat map and the globe . . . . . . . . . . . . . . . 78
4.6 Parallels and meridians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7 Adjacent circles on the flat map and the globe . . . . . . . . . . . . . . 80
4.8 Concentric circles on the flat map and the globe . . . . . . . . . . . . . 80
4.9 The action function for the globe . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.10 Great circles are geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.11 Great circles on the flat map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.12 Geodesics on the globe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1 The action function of a source charge at rest . . . . . . . . . . . . . . . 104
5.2 The influence on spacetime of a charge at rest (3D) . . . . . . . . . . 105
5.3 The influence on spacetime of a charge at rest (2D) . . . . . . . . . . 105
5.4 The direction of the electric field of a moving charge . . . . . . . . . 109
5.5 The action function for a moving source . . . . . . . . . . . . . . . . . . . . 110
5.6 The influence on spacetime of a moving source (2D) . . . . . . . . . 111
5.7 The field of a current in a long wire . . . . . . . . . . . . . . . . . . . . . . . . 118
6.1 The effect of a black hole on the light cone . . . . . . . . . . . . . . . . . 132
6.2 The action function in the strong regime for a source at rest . . 133
6.3 Planetary precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.4 Reducing a binary to a one-body problem . . . . . . . . . . . . . . . . . . 140
6.5 Energy versus eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.6 Angular momentum of circular orbits in the strong regime . . . . 144
6.7 The stability of circular orbits in the strong regime . . . . . . . . . . 145
6.8 Unbounded orbits and gravitational lensing . . . . . . . . . . . . . . . . . 148
6.9 The Shapiro Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.10 Gravitational waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.1 The Fizeau experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2 Snell’s law.............................................. 165
8.1 The Stern-Gerlach experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.2 Two Stern-Gerlach apparatuses in succession . . . . . . . . . . . . . . . 176
8.3 Transition probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.4 Decomposition of normalized spin states . . . . . . . . . . . . . . . . . . . . 179
List of Tables
3.1 Newtonian and relativistic collisions . . . . . . . . . . . . . . . . . . . . . . . 64
4.1 The factors determining an object’s spacetime . . . . . . . . . . . . . . 74
1
Preface
This self-contained monograph provides a mathematically simple and phys-
ically meaningful model which unifies gravity, electromagnetism, optics and
even some quantum behavior. The simplicity of the model is achieved by
working in the frame of an inertial observer instead of in curved spacetime.
Our approach to dynamics is geometric, and by plotting the action func-
tion of a spacetime, one is treated to a visualization of the geometry. Us-
ing these visualizations, one may readily compare the geometries of different
types of fields. Moreover, a new understanding of the energy-momentum of
a field emerges.
The reader will learn how to compute the precession of planets, the de-
flection of light, and the Shapiro time delay. Also covered is the relativistic
motion of binary stars, including the generation of gravitational waves, and
a relativistic description of spin.
The mathematics is accessible to students after standard courses in mul-
tivariable calculus and linear algebra. For those unfamiliar with tensors and
the calculus of variations, these topics are developed rigorously in the open-
ing chapters. The unifying model presented here should prove useful to upper
undergraduate and graduate students, as well as to seasoned researchers.
Acknowledgments
We would like to thank Larry Horwitz, Rainer Weiss, Bahram Mashhoon,
Dan Scarr, Tepper Gill, Tzvi Weinberger, Yakov Itin, David Hai Gootvilig,
Elazar Levzion, and Chanoch Cohen for their comments.
3
Notation
Rthe set of real numbers
Re(z) the real part of a complex number of complex-
valued function
Im(z) the imaginary part of a complex number of
complex-valued function
cthe speed of light in vacuum, c= 299,792,458
meters per second
xµ(σ), µ = 0,1,2,3 the worldline of an object, parameterized by σ
x·ythe Minkowski inner product of two four-
vectors
x◦ythe Euclidean inner product of two 3D vectors
τproper time (with dimensions of length
˜τan alternative and sometimes convenient pa-
rameter, used to simplify the Euler-Lagrange
equations
δij the Kronecker delta
uµ(σ) the four-velocity of an object
wµ(σ) the four-velocity of a source
ηµν the Minkowski metric diag(1,−1,−1,−1)
gµν an arbitrary metric
lµ(x) a four-covector appearing in the action func-
tion and representing the direction of propa-
gation of a field or the properties of a medium
Aµ(x) a four-covector appearing in the action func-
tion and representing the four-potential of an
electromagnetic field
qa charged particle or the charge of this particle
mthe mass of an object
ϵ0the permittivity of free space
µ0the permeability of free space
GNewton’s gravitational constant
h, ℏPlanck’s constant and the reduced Planck’s
constant
rsthe Schwarzschild radius
fµ,ν ∂fµ
∂xν
K, K′inertial frames
vthree-dimensional velocity
athree-dimensional acceleration
5
6 List of Tables
Λ:K→K′the Lorentz transformation from Kto K′
γ=γ(v)γ=1−v2
c2−1
v⊕uEinstein velocity addition
L(x, u) an action function
D(x, y) the distance between the spacetime points x
and y
Dxthe set of admissible four-velocities at the
spacetime position x
na unit-length 3D spatial vector, usually the di-
rection of propagation of a field
Chapter 1
Introduction
Fig. 1.1: Galileo Galilei (1636 portrait by Justus Sustermans)
This monograph on relativistic dynamics is guided by three major themes
-geometry,unification and simplicity. Our approach is geometric so that
our mathematics has physical meaning and content. We use the full strength
of Galileo Galilei’s Principle of Relativity with respect to the Lorentz trans-
formations. Our dynamics is also unifying - gravity, electromagnetism, and
optics are integrated into the same model. Our dynamics is as simple as
possible and yet can explain all known relativistic phenomena.
1.1 Physics via Geometry - A Historical Perspective
The first person to think that the laws of physics should define the geometry
of space was Bernhard Riemann (1826 - 1866) [70]. Although best known as a
7
8 1 Introduction
mathematician, Riemann became interested in physics in his early twenties.
His lifelong dream was to develop the mathematics to unify the laws of elec-
tricity, magnetism, light and gravitation. At an 1894 conference in Vienna,
the mathematician Felix Klein said:
“I must mention, first of all, that Riemann devoted much time and thought to
physical considerations. Grown up under the tradition which is represented by the
combinations of the names of Gauss and Wilhelm Weber, influenced on the other
hand by Herbart’s philosophy, he endeavored again and again to find a general
mathematical formulation for the laws underlying all natural phenomena .... The
point to which I wish to call your attention is that these physical views are the
mainspring of Riemann’s purely mathematical investigations [65].”
Fig. 1.2: Bernhard Riemann
Riemann’s approach to physics was geometric. In fact, he replaced straight
lines with geodesics, an idea later used in General Relativity (GR). As pointed
out in [82], “one of the main features of the local geometry conceived by
Riemann is that it is well suited to the study of gravity and more general
fields in physics.” Moreover, Riemann believed that the forces at play in a
system determine the geometry of the system. For Riemann, force equals
geometry.
The application of Riemann’s mathematics to gravity would have to wait
for two new ideas. While Riemann considered how forces affect space, physics
must be carried out in spacetime. One must consider trajectories in spacetime,
not in space. For example, in flat spacetime, an object moves with constant
velocity if and only if his trajectory in spacetime is a straight line. On the
other hand, knowing that an object moves along a straight line in space
tells one nothing about whether the object is accelerating. As Minkowski
said, “Henceforth, space by itself, and time by itself, are doomed to fade
away into mere shadows, and only a kind of union of the two will preserve
an independent reality [77].” This led to the second idea. Riemann worked
only with positive definite metrics, whereas the Minkowski metric of flat
1.1 Physics via Geometry - A Historical Perspective 9
spacetime is not positive definite. The relaxing of the requirement of positive-
definiteness to non-degeneracy led to the development of pseudo-Riemannian
geometry.
In 1915, fifty years after Riemann’s death, Einstein used pseudo-Riemannian
geometry as the cornerstone of GR. Acknowledging his reliance on Riemann,
Einstein said:
“But the physicists were still far removed from such a way of thinking; space was still,
for them, a rigid, homogeneous something, incapable of changing or assuming various
states. Only the genius of Riemann, solitary and uncomprehended, had already won
its way to a new conception of space, in which space was deprived of its rigidity, and
the possibility of its partaking in physical events was recognized. This intellectual
achievement commands our admiration all the more for having preceded Faraday’s
and Maxwell’s field theory of electricity [28].”
Fig. 1.3: Albert Einstein (1921 photograph by F Schmutzer)
GR is a direct application of “force equals geometry.” In GR, the grav-
itational force curves spacetime. Since, by the Equivalence Principle, the
acceleration of an object in a gravitational field is independent of its mass,
curved spacetime can be considered a stage on which objects move. In other
words, the geometry is the same for all objects.
However, GR represents only a partial fulfillment of Riemann’s program.
Since the Equivalence Principle holds only for gravitation, GR singles out
the gravitational force from other forces which are object dependent. Take
the electromagnetic force, for example. The acceleration of a charged particle
in an electromagnetic field depends on its charge-to-mass ratio, implying
that the electromagnetic field does not create a common stage on which
all particles move. Electromagnetism is an object-dependent force. Indeed,
a neutral particle does not feel any electromagnetic force at all. Thus, the
way in which spacetime is influenced by an electromagnetic field depends on
both the sources of the field and a single, intrinsic property of the object –
10 1 Introduction
its charge-to-mass ratio. This was also recognized in the geometric approach
of [22]. This raises the question: Can Riemann’s principle of “force equals
geometry” be applied to other forces? Can Riemann’s program be extended
to object-dependent forces?
In this book, we introduce several new ideas which enable us to geometrize
not only gravity, but also electromagnetism, motion in media, and even some
quantum behavior. The model presented here is thus a continuation of Rie-
mann’s program.
For us, geometry means the measurement of distances between two in-
finitesimally close points in spacetime. This lies at the core of the theory
because we use a variational principle to find stationary or “shortest” paths
of objects through spacetime.
“Many results in both classical and quantum physics can be expressed as variational
principles, and it is often when expressed in this form that their physical meaning is
most clearly understood. Moreover, once a physical phenomenon has been written
as a variational principle, ... it is usually possible to identify conserved quantities,
or symmetries of the system of interest, that otherwise might be found only with
considerable effort [86].”
In pseudo-Riemannian geometry, the distance between two infinitesimally
close spacetime points is quadratic in the temporal and spatial separations.
This is a seemingly reasonable assumption and strongly supported by the
Pythagorean Theorem and its generalizations. However, for the electromag-
netic field, one requires a linear dependence on displacements. Moreover, the
exclusively quadratic dependence was needed only to ensure positive defi-
niteness of the metric. Once we no longer require this, one should allow the
simpler, linear dependence. Thus, our first new idea is to include both linear
and quadratic dependence in our action function – the function for computing
distances in spacetime.
Second, by using geometries which are allowed to depend on the charge-
to-mass ratio of the moving object, we can incorporate electromagnetism
into our model. We can also model the motion of light in isotropic media
and between different media. In these cases, the geometry depends on the
photon’s energy or frequency. Thus, the second new idea is to broaden the
notion of geometry to include object dependence.
1.2 Unification and Simplicity
The second theme of this book is unification. A good theory unifies
concepts that had previously been treated as distinct. Einstein attempted to
unify gravity with electromagnetism, and his dream was to unify relativity
with quantum mechanics. To be sure, we have not achieved full unification
here. Nevertheless, we do have a unified approach to gravity, electromag-
1.2 Unification and Simplicity 11
netism, optics and even some quantum effects. Thus, this book may provide
a solid foundation for a complete, unified theory.
Our third theme is simplicity:
“Explanations that posit fewer entities, or fewer kinds of entities, are to be preferred
to explanations that posit more.” - William of Occam
“Scientists must use the simplest means of arriving at their results and exclude
everything not perceived by the senses.” - Ernst Mach
“A physical theory should be as simple as possible, but not simpler.” - Albert Ein-
stein
“If you can’t explain it simply, you don’t understand it well enough.” - Albert
Einstein
In physics, if two theories predict and account for the same phenomena,
the simpler theory is to be preferred. On the other hand, a theory which
accounts for more phenomena must be favored, even if it is more complicated.
For example, Special Relativity (SR) replaced Newtonian mechanics because
it explained Maxwell’s equations, the dragging of light in moving water, and
the invariance of the speed of light, while containing Newtonian mechanics as
a limit. Ten years later, GR explained what SR couldn’t - the gravitational
redshift and the anomalous precession of Mercury’s orbit. GR went on to
explain the periastron advance of a binary star, the deflection of light, the
Shapiro time delay and gravitational waves. For more than a century, GR
has been the simplest, in fact, the only theory that explains these relativistic
phenomena. Hence, any new theory must also explain these phenomena. It
must, as we say, pass the tests of GR.
In light of the above statements of William of Occam, Ernst Mach and Ein-
stein himself, it is natural to ask whether there exists a simpler theory which
also passes the tests of GR. After all, GR is a complicated theory. Working on
a curved spacetime manifold, one is immediately greeted by covariant deriva-
tives, Christoffel symbols, curvature tensors, and, last but certainly not least,
the field equations - a system of non-linear, partial differential equations with
ten degrees of freedom. This is not an easy system to solve.
This monograph demonstrates that there is a simpler theory of relativity.
Step by step, we construct relativistic dynamics in as simple a way as possible
and still obtain a theory broad enough to explain all known relativistic phe-
nomena. Our theory is objectively simpler than GR. In GR, the metric for a
gravitational field has ten degrees of freedom. In our model, it has only three
degrees of freedom. Our mathematics is also simpler than that of GR. With
a background of only multivariable calculus and linear algebra, the reader
will be able to follow the derivation of Mercury’s precession, the deflection
of light, and all of the other tests of GR. Our approach can also be used to
derive the Biot-Savart Law and Snell’s Law.
12 1 Introduction
1.3 Overview of the Model
Fig. 1.4: Isaac Newton (1689 portrait by Godfrey Kneller)
Before previewing our model, we should first define the term relativistic
dynamics. For us, relativistic dynamics is a theory of the motion of objects,
influenced by force fields and isotropic media, which is Lorentz invariant,
reduces to Newtonian dynamics when velocities are small, and explains all
known relativistic effects.
Newtonian dynamics is often a good enough approximation to relativis-
tic dynamics. If the magnitude of the measurement errors are larger than
the relativistic corrections, then relativity is not needed. When the measure-
ments are very accurate, however, one needs relativistic dynamics. It is also
possible to observe the relativistic correction for motion with extremely high
frequency, if these corrections are combined.
We note that some define relativistic dynamics as a theory of the motion
of objects whose speed is close to the speed of light c. We reject this defi-
nition because, for example, one needs relativistic dynamics to account for
the anomalous precession of Mercury, even though Mercury travels along its
orbit at less than fifty kilometers per second.
We turn now to an overview of our model. In the earlier chapters, we study
the motion of objects, ignoring any internal rotation and consider objects as
points. Rotations are handled in the later chapters. Throughout the entire
text, however, we treat, primarily, action at a distance.
We use simplicity and choose to describe the motion of objects in an iner-
tial reference frame attached to an inertial observer. Our observer describes
events in flat spacetime. This is similar to the problem of finding the shortest
route on the Earth’s surface, where it is easier to do the calculations on a flat
map, rather than comparing the length of various routes on the globe itself.
1.3 Overview of the Model 13
One way to describe the motion of an object is by a function assigning the
position of the object at different times. We work, however, with a geometric
description – the graph of this function, which is a line, called the worldline,
in spacetime. A worldline has the general form x(σ) = xµ(σ), where µ=
0,1,2,3. Here, x0is the time of the event (or a multiple of the time in order
to make x0have dimensions of length) in our inertial frame; x1, x2, x3are
the spatial coordinates in this frame, and σis an arbitrary parameter. Our
method of measuring spacetime distances will be shown to be independent of
the choice of parameter. This allows us the freedom to choose the parameter
of our liking.
The next step is to introduce the relativity of spacetime, or the notion
of an object’s spacetime. This means that spacetime is an object-dependent
notion. Each object has its own spacetime, which is defined by the forces
that affect its motion and at most one parameter intrinsic to the object. A
massive, non-charged object’s spacetime is affected by gravity but not by elec-
tromagnetic forces. Its spacetime is that of the curved spacetime induced by
nearby masses. An electron is affected by both gravity and electromagnetism.
Its spacetime reflects the combined effect of both forces and depends on its
charge-to-mass ratio. Similarly, the spacetimes of photons and charges trav-
eling through a medium are determined by the properties of the medium. A
photon’s spacetime may depend on its frequency, as we observe when a prism
splits white light into a rainbow.
Next, we introduce the Extended Principle of Inertia, which states:
Since an inanimate object is unable to change its velocity, it moves via
the shortest, or stationary, worldline in its spacetime when not disturbed by
other objects.
This principle extends Newton’s First Law from free motion in flat space-
time to the motion of an arbitrary object in its spacetime. It means that
objects move along geodesics, as suggested by Riemann and as in GR. Only
now the geodesics are with respect to a geometry (not necessarily defined by
a metric) that not only incorporates gravity, but also electromagnetic forces
and the influence of media. In GR, an object freely falling in a gravitational
field is in free motion. In our dynamics, every object is in free motion in its
spacetime.
To find stationary paths, we use the Principle of Least Action, but with
a physically meaningful action. Historically, actions have been defined differ-
ently in different areas of physics. Often, there is no physical understanding
behind an action’s definition. It simply works. Here, on the other hand, we
propose a simple, physically meaningful action function that generalizes the
Lagrangian but has no connection to the usual “kinetic minus potential en-
ergy.”
In order to define the “shortest” worldline, we need to define the “length”
of a worldline in an object’s spacetime. To do this, it is enough to define the
distance between two infinitesimally close spacetime points Pand Q. This is
14 1 Introduction
the analog of the line element of the spacetime metric in GR [78],[59]. We
propose the following definition:
Definition 1.1 The action function L(x, u) is a scalar-valued function of the
spacetime position xand a four-vector u, with the meaning that the distance
between two spacetime positions P=xand Q=x+uϵ in the object’s
spacetime is L(x, u)ϵif ϵis small.
We think of uas the direction from Pto Q. On the worldline of an object,
we will substitute its four-velocity for u.
Calculus shows us that to measure the length of a curve, it is enough to
know the approximate length of infinitesimal segments of the curve. Some-
times a linear approximation is sufficient; sometimes one needs higher-order
approximations. For us, the length is infinitesimal in ualone, and we will
see that for the electromagnetic field, linear approximations of uare enough,
while quadratic approximations are needed for gravity.
We will show that the simplest such action function L(x, u) that is Lorentz
invariant and independent of the parametrization is
L(x, u) = pu2−(l(x)·u)2+kA(x)·u, (1.1)
where l(x) and A(x) are four-vector-valued functions of the spacetime posi-
tion x,a·b=ηµν aµbνis the Minkowski inner product, and kis a parameter
which can be positive or negative. The expression under the square root must
be non-negative. This entails a restriction on the admissible four-velocities in
the spacetime under investigation.
We call A(x) the linear four-potential because it acts linearly on the four-
velocity. We call l(x) the quadratic four-potential because it act quadratically
on the four-velocity. We will see that the function kA(x) is connected to the
electromagnetic field and that k∼q/m, while the function l(x) describes the
gravitational field and the effect of an isotropic medium. For gravitational
fields, l(x) is a null vector proportional to the direction of propagation of
the field. For the motion of charges and light in isotropic media, l=l(x) is
timelike and does not depend on the spacetime position x. The exact form of
A(x) and l(x) will be obtained using Lorentz covariance and the Newtonian
limit.
Stationary worldlines will be obtained by applying the Euler-Lagrange
equations to the above action function. This entails choosing a parameter on
the worldline of the motion. We usually use proper time as the parameter,
since it is the same for all inertial observers. In some applications, however,
we do take advantage of the freedom of choice of the parameter and use a
parameter which simplify the computations.
To obtain the general equation of motion for electromagnetism and grav-
ity and motion in isotropic media, we introduce, for any four-vector-valued
function f, a first-order derivative
Fα
ν(f) = ηαλ(fν,λ −fλ,ν ),(1.2)
1.3 Overview of the Model 15
which is the antisymmetric part of the Jacobian matrix of f. If f=A, then
Fα
ν(A) is the usual electromagnetic field strength tensor.
For arbitrary A(x) and l(x), we define
bα(A, l) = p1−(l·˙x)2kF α
ν(A) ˙xν−(l·˙x)Fα
ν(l) ˙xν,(1.3)
where the dot denotes differentiation by the proper time. For l= 0, breduces
to the acceleration under the Lorentz force. In general, however, this accel-
eration must be multiplied by the gravitational time dilation, expressed by
the square root term. The second term accounts for the gravitational force,
which depends quadratically on the four-velocity.
Fig. 1.5: Leonhard Euler (1753 por-
trait by Jakob Emanuel Handmann) Fig. 1.6: Joseph-Louis Lagrange
Applying the Euler-Lagrange equations to the action function (1.1) leads
to a universal relativistic dynamics equation
¨x=b+ (b·l)l⊥+ (˙
l·˙x)l⊥,(1.4)
where l⊥=l−(l·˙x) ˙x. If there is only an electromagnetic field, then only
the first term is nonzero, and Ais a four-potential of the field. If there is a
gravitational field, the first term gives the Newtonian limit (v≪c) for a weak
gravitational field. The first two terms, taken together, give the Newtonian
limit even in the strong regime. The l⊥terms ensure that an object’s four-
velocity remains within the admissible region. Just like in the case of an
electromagnetic field, we obtain that a gravitational field splits into a near
field, falling off like 1/r2, and a far field, falling off like 1/r. The far field is
part of the third term. For a pure gravitational field, the dynamics resulting
from equation (1.4) and conservations following from the action function (1.1)
passes all of the tests of GR.
For the motion of light in isotropic media, we use the Lorentz invariance
of the action function to obtain the effect of the motion of a medium on
the velocity of light in the medium. We also derive Snell’s Law for light
propagation between two media. Then we derive the equation of motion for
16 1 Introduction
charges in an isotropic medium at rest. This equation is essentially (1.4), but
with a slight modification. Thus, equation (1.4), together with conservation
laws, describes the motion of both charged and uncharged, massive objects
and massless particles, in any electromagnetic and some gravitational fields
in vacuum, as well as in isotropic media.
We point out the advantages of the action function (1.1) over the more
standard Lagrangian Lstandard =1
2mv2−U, kinetic energy minus potential
energy. This Lagrangian depends on the mass mof the object in motion,
and yet the motion in a gravitational field is independent of the mass. In
other words, the standard Lagrangian contains a parameter which has no
bearing on the physics! On the other hand, the gravitational term of our
action function (1.1) does not depend on the mass. Moreover, the linear term
depends on the charge-to-mass ratio, a parameter on which the acceleration
due to an electromagnetic field is known to depend.
In the later chapters, we extend the description of an object from a point
to an object with internal rotation. This forces us to complexify Minkowski
space. Using Lorentz covariance, we obtain a relativistic description of spin
that predicts the correct transition probabilities between states and leads to
a relativistic dynamics equation for particles with spin. In real Minkowski
space, as we will show, there are only two types of Lorentz-covariant vec-
tors and one type of Lorentz-invariant scalar. These are needed to describe
electromagnetic and gravitational fields. In complex Minkowski space, we dis-
covered that there is an additional complex-valued, Lorentz-invariant scalar
associated to any null vector. This allows us to obtain a new relativistic de-
scription of any single-source field which propagates with the speed of light.
Our description is similar to the wave function in quantum mechanics.
1.4 Outline of the Book
The classical treatment of motion in electromagnetic and gravitational
fields is reviewed in chapter 2. Here, the reader will find explicit solutions
for both bounded and unbounded trajectories. We prove Kepler’s laws of
planetary motion and derive the Euler-Lagrange equations, which play a
central role in the book.
Since we rely heavily on the Principle of Relativity and Lorentz covari-
ance, we review these concepts in chapter 3. We use the symmetry following
from the Principle of Relativity to derive the Lorentz transformations. We
show that there are a universal speed and a metric which are invariant for
all inertial frames. We also derive velocity transformations between inertial
frames and demonstrate some of their physical applications, including the
Sagnac effect. We discuss energy-momentum of massive and massless parti-
1.4 Outline of the Book 17
cles and use it to derive the relativistic Doppler shift. We also establish here
the allowable forms of Lorentz-covariant vector-valued functions.
Although there are no new results in this chapter, one will nevertheless
find there:
•a precise and local definition of inertial frame,
•a proof that the Galilean transformations and the Lorentz transforma-
tions are the only transformations between inertial frames that satisfy the
Principle of Relativity,
•a derivation of the Lorentz transformations without assuming that the
speed of light is the same in all inertial frames,
•visualizations and geometric explanations of both the Galilean and the
Lorentz transformations,
•arguments why the Lorentz, and not the Galilean, transformations are the
true transformations between inertial frames,
•a clear explanation of the difference between vectors and covectors, and
the need for both,
•the definition of the energy-momentum of massive and massless free par-
ticles,
•a derivation of the relativistic Doppler shift based on energy-momentum,
•a characterization of the Lorentz-covariant covectors associated to the field
of a single source.
In chapter 4, we present the foundations of our approach - the relativity
of spacetime and the Extended Principle of Inertia. We introduce the action
function (1.1) and derive the relativistic dynamics equation (1.4). To help the
reader familiarize himself with action functions and stationary worldlines, we
tackle the ancient problem of finding the shortest route between two points
on the Earth’s surface. Our approach includes a method of visualizing the
geometry by plotting sections of the action function. We distinguish between
the momentum of a free particle and the momentum imparted by a field to
a particle. We derive the equation of motion with respect to two different
parametrizations of the worldline of a moving object.
Chapter 5 is devoted to the electromagnetic field. Using Lorentz covariance
and the fact that, in general, electromagnetic fields radiate, we obtain the
Li´enard-Wiechert four-potential of a single source. We describe all of the
properties of the field of a moving source and compute explicitly the near
and far fields. Our presentation includes visualizations of the geometry of the
fields and a new interpretation of the magnetic field as the angular momentum
of the source. We introduce a new definition of the energy-momentum four-
vector of a field. The connection to Maxwell’s equations are explored here as
well. We compute relativistic orbits in a static, single-source field.
Gravity is treated in chapter 6. From Lorentz covariance and the New-
tonian limit, we obtain the field of a spherically symmetric source at rest.
We present visualizations of the geometry and compare them to those of
the electromagnetic field. We note that a charge at rest does not impart
18 1 Introduction
3D momentum to test charges, while a gravitational source does imparts 3D
momentum to test objects, even when the source is at rest.
We derive the correct precession of Mercury and show that our model
passes all of the other tests of GR, including gravitational waves. The results
are extended to the field of a collection of moving, spherically symmetric
sources. Circular and elliptical orbits are computed in the strong field regime.
The motion of light and charged particles in isotropic media is handled in
chapter 7. We analyze Fizeau’s experiment and derive Snell’s Law. In chapter
8, we complexify Minkowski space and obtain a relativistic description of spin
using complex four-vectors, but without using spinors. We derive the tran-
sition probabilities between states and a relativistic dynamics equation for
particles with spin. Our model agrees with quantum mechanics with regard
to Bell’s inequality. In chapter 9, still working in complexified Minkowski
space, we find a Lorentz-invariant scalar-valued function which can be used
to describe the field of a moving source. This description is Lorentz invariant
with respect to a spin-1/2 representation of the Lorentz group and produces
a prepotential similar to the wave function of quantum mechanics. This pre-
potential leads to the Li´enard-Wiechert four-potential.
Alternative Theories
In the literature, there are other alternative approaches to reproduc-
ing the relativistic gravitational features of GR. One approach uses modified
Newtonian-like potentials. This so-called “pseudo-Newtonian” approach, in-
troduced in [81], is much simpler mathematically than GR, with no need for
covariant differentiation and complicated tensorial equations. Numerous au-
thors [1, 16, 58, 69, 72, 75, 88] have proposed various modified Newtonian-like
potentials. However, none of these potentials are able to reproduce the tests
of GR, even in the weak field regime. Moreover, as stated in [51], most of
these modified potentials “are arbitrarily proposed in an ad hoc way” and,
more fundamentally, are “not a physical analogue of local gravity and are not
based on any robust physical theory and do not satisfy Poisson’s equation.”
More recently, the above shortcomings were addressed in [52]. Using a
metric approach and hypothesizing a generic relativistic gravitational action
and a corresponding Lagrangian, the authors derive a velocity-dependent
relativistic potential which generalizes the classical Newtonian potential.
For a static, spherically symmetric geometry, this potential exactly repro-
duces relativistic gravitational features, including the tests of GR. Even
more recently, one finds a fundamental grounding to these velocity-dependent
pseudo-Newtonian potentials in [100]. The authors generalize the pseudo-
Newtonian approach to any stationary spacetime. They also include addi-
tional forces, such as the electromagnetic force. Dirac [20] combined gravity
1.4 Outline of the Book 19
and electromagnetism into one action function. Mashhoon’s [74] gravitoelec-
tromagnetism borrows ideas from electromagnetism to model gravity.
