Does the Universe Have a Speed Limit?

(A new way to look at relativity, space-time, and
aether)

Tom Van
Flandern

Meta
Research / <[email-masked]>

"The problem with relativity isn't that it's hard to
understand, it's that it is hard to believe." – author unknown,
conveyed by Larry Burford

(This paper is
dedicated to the memory of Richard Hazelett, 1923-2002 [[1]];
and J.P. Vigier, 1920-2004. [[2]])

Abstract. What gives physics, as contrasted with
mathematics, its intellectual appeal is its ability to explain nature with
simple models. Some of that appeal has been lost through an increasing reliance
on mathematical and geometric interpretations of relativity. However, such
interpretations are not mandatory, and indeed may violate the logical
constraints imposed by physical principles. Pursuing only physically viable
interpretations of the same equations leads us back to the kinds of simple
models that attracted students to physics in the first place.

Introduction

Newton’s universal
law of gravitation was a triumph for both physics and mathematics. Although the
law did not reveal the mechanism of gravity, it allowed practical prediction of
the motions of bodies in gravitational fields that, for most applications, was
accurate to a few parts in 100 million. Gravitation became the most precise
science. Its predictions were so dependable that, when eclipses of the moons of
Jupiter by Jupiter’s shadow in the 17th century were found to
occur early or late by as much as eight minutes, deficiencies in the law of gravity
were not high on the suspect list of possible causes. Roemer eventually proved
that the cause was the finite speed of light, and obtained the first good
estimate of light’s velocity – about 8 minutes to traverse the
radius of the Earth’s orbit. The eclipses would appear early when the
Earth was on the same side of the Sun as Jupiter, and late when Earth was on
the far side, because the travel time for light from Jupiter to Earth, which
averages over 40 minutes, would get shorter or longer as Earth’s distance
from Jupiter decreased or increased, respectively. See Figure 1.

Newton’s law
remained the paradigm for gravitational dynamics until it was superceded by
Einstein’s general theory of relativity (GR). That theory was able to
explain a small effect that Newton’s
theory did not: a slow advance of the perihelion (point closest to the Sun) of
Mercury’s orbit. In addition, it predicted certain new effects on
electromagnetic phenomena: the bending of light as it moved past the Sun by
twice the amount that the Sun’s gravity would have on a particle moving
at that speed; a redshift of light (or a slowing of clocks that use
electromagnetic phenomena as a basis for keeping time) in a gravitational potential
field; and an extra propagation delay for light or radar beams passing the Sun
(or any significant mass). One by one, these predictions were verified to
within 3% accuracy or better with modern, high-precision observations
(very-long-baseline interferometry, lunar and satellite laser ranging,
planetary radar ranging, spacecraft orbiting planets, etc.). So GR has now
become the accepted model for predicting gravitational phenomena and the
motions of bodies in space.

The
first introduction of major points made in this paper is marked by letters in
the left margin. See Table
I for a description of these points.

The existence of two physical interpretations
of GR

GR’s
outstanding success at mathematically predicting phenomena influenced by
gravity naturally led to a desire to have a deeper understanding of the physics
of gravity. Two different interpretations arose to explain the physics behind
Einstein’s equations. These have come to be called the “field
interpretation” (gravity is a classical force) and the “geometric
interpretation” (gravity is just "curved space-time").
Eddington’s 1920 book refers to the possibility of explaining at least
some GR phenomena with an optical medium instead of curved space-time [[3]], indicating an awareness of two
interpretations of the theory even that early. But many later physicists seemed
to adopt one or the other interpretation and use it without further
qualification. This was possible because the mathematics of GR was the same
either way. Indeed, many scientists came to believe that that the physical
interpretation was not of great importance as long as the mathematics of the theory
continued to predict phenomena correctly.

In
more recent times (i.e., over the last generation), the geometric
interpretation of GR has become so dominant that the field interpretation is
often no longer taught to today’s physics students and future
relativists. But it should be taught because it is an important element in
achieving a deeper physical understanding of the theory. Many of the great
physicists of the last century, including Einstein himself, were partial to the
field interpretation. More recently, Nobel laureate physicist Feynman described
the two interpretations with these words:

“It
is one of the peculiar aspects of the theory of gravitation, that it has both a
field interpretation and a geometrical interpretation. … the fact is that
a spin-two field has this geometrical interpretation: this is not something
readily explainable – it is just marvelous. The geometrical
interpretation is not really necessary or essential to physics. It might be
that the whole coincidence might be understood as representing some kind of
invariance. It might be that the relationships between these two points of view
about gravity might be transparent after we discuss a third point of view ... to
get a feeling for some directions which we might take in attempting to
understand how gravity can be both geometry and a field.” [[4]]

In
a meaningful sense, this paper discusses just such a possible “third
point of view” as Feynman envisioned. Perhaps we have a better
understanding now than Feynman did then about why it was almost inevitable that
two different interpretations of the theory would arise. To state the point
succinctly, the expression “gravitational field” is used with two
entirely different meanings that are rarely distinguished by those using the
phrase. The modern relativist normally means “gravitational potential
field”, and the modern dynamicist or celestial mechanician almost always means
“gravitational force field”, or equivalently, “gravitational
acceleration field”. As someone specializing in celestial mechanics, I
came to be keenly aware of the problem when I tried to have discussions with mathematical
relativists and found we were virtually unable to communicate because we were
speaking of different concepts while using the same words.

Indeed,
those trained in these respective disciplines seem at times to have not even
the math of GR in common. The 4-dimensional field equations and their solutions
most mathematical relativists use when discussing implications of the theory (such
as black holes) deal with the gravitational potential field. By contrast, the
equations of motion (expressions for ordinary accelerations in flat, Euclidean
space) that most dynamicists use to predict motions and compare with
observations deal primarily with gravity as a force or, equivalently, as an acceleration.
(Force is just mass times acceleration.) The former math uses 4-D tensors, the
latter uses 3-D vectors.

The
relativists point out that their solutions to the field equations are often
exact, and in any case are closer to the Einstein equations. The dynamicists point
out that their equations of motion, although approximate, can always be made at
least as accurate as the observations and experiments; are essential to the
comparison of theory and observation; and are what ultimately give the entire
theory legitimacy (i.e., take it from the realm of pure math to applied math). Unfortunately
for communication, each side describes the object of its equations as “the gravitational field”. In this
paper, we will qualify the expression “gravitational field” by
inserting the word “potential” or “force”, as
appropriate. And where the meaning of “field interpretation”
(defined above) might be ambiguous, we will associate it with forces and
ordinary (i.e., 3-space) accelerations, while associating the geometric
interpretation with the geometry of the potential field.

The geometric interpretation of GR

In
classical physics, space and time are simply dimensions, mental concepts useful
for the measurement of extent and change, respectively. As pure concepts, they
are of course not susceptible to manipulation by material, tangible bodies or
by their states of motion. In Euclidean geometry, space is flat, axes are
straight and orthogonal, and time is unique and uniform. So to determine the
distance  between any two
points in space at some fixed time, we simply use the Pythagorean theorem: . Here, the new quantities with the Greek capital delta
prefix are the three spatial components of the interval between the two points.
Hence, we describe positions and motions using just these coordinates as in
“3-space”. In geometric GR, this Euclidean distance is generalized
to a “space-time” path length : , where  is the speed of
light. Such a space-time path is called a "geodesic" path to
distinguish it from a distance in space. Because there are now four
coordinates, we refer to descriptions in this form as in “4-space”.

The
core idea behind geometric GR is that “gravity is just geometry”, in
explicit contrast to the idea that gravity is a force. Indeed, it is often said
that “gravity as a force does not exist” when discussing geometric
GR. More generally, geometric GR posits that bodies follow the geometry of
space-time on geodesic paths because the geodesic path is the nearest
equivalent of a straight line available in space-time (which here is the
equivalent of “4-space”, a purely mathematical combination of 3-dimensional-space
and imaginary time[*]) when space-time
is curved by the presence of mass. The mantra of geometric GR is, “Mass
tells space how to curve, and space tells matter how to move”. However,
even in that mantra we have the seeds for later confusion and disagreement. That
seemingly innocuous word-switch here from “space-time” to
“space” leads to problems with the physical interpretation of GR,
as we will shortly see.

Because
geometric GR is primarily about the gravitational potential field, it
implicitly assumes that the normal relation “acceleration is the gradient
of potential” means that potential causes acceleration. However, we must
not forget this is just an assumption, and one that has considerable influence
on which physical models for gravitation are considered viable. The causality
might well be the other way around, with gravitational force causing the
potential field to have a gradient (a slope through space).

Geometric
GR has no propagating gravitational force because acceleration is imagined to
be a consequence of curved space-time geometry. So a static gravitational (potential)
field is like a frozen waterfall, having no moving parts (Figure 2). As such, the field carries no momentum, and
deposits no energy into the bodies it affects.

The
importance of the potential field is that gravitational potential causes
gravitational redshifts and clock slowing, propagation delay, and bending for
electromagnetic waves, and is a factor in determining the “perihelion
advance” effect on orbits as predicted by GR. Otherwise, potential is not
a factor in determining the motions of material bodies through 3-space. Newton’s
gravitational force law, the main contributor to ordinary gravitational
acceleration, would work just as well if there were no such thing as
gravitational potential. Although we often mathematically derive accelerations
from changes in potentials, nothing compels us to do it that way. And we will
shortly see advantages to reversing the direction of causality, with
gravitational force causing the changes in potential.

The field interpretation of GR

If
half-a-dozen relativists were asked to define the field interpretation, we
might hear as many answers. Indeed, the meaning of the concept has evolved as
discussions of the theory have evolved over time. Here we will adopt the view
that, in field GR, gravity is a classical acceleration-producing force that
propagates from the source mass to a target body. The gravitational field
around a source mass is a field of force or acceleration vectors. The force
field and the potential field are separate entities. Although the direction of
causality is often not of concern, physical models work best when the
force/acceleration field acts on and shapes the potential field, rather than
vice versa. (This is analogous to acceleration changing velocity, not vice
versa.) In some of these physical models, the potential field has all the
properties of an optical medium. It also has some properties in common with the
“light-carrying medium”, “luminiferous ether”, or
“elysium”, which are variants on the same basic concept.[†][[5],[6]]
None of these concepts is required to produce ordinary gravitational
acceleration.

Meanwhile,
the gravitational-force-carrying field itself, even when “static”
as for a non-moving source mass, is always a dynamic field in the same sense as
a flowing waterfall (Figure
3). It may look unchanging from a distance. But when examined
closely, we see that every drop of water is continually being replaced by
another from behind. As such, the acceleration field of GR does carry momentum
from the source mass to the target body, and does deposit energy in any body it
affects. This follows from the definition of force in classical physics: the
time rate of change of (3-space) momentum; and from the necessity (from the principles
of physics) that all forces involve contact pushes. [[7]] Inasmuch as momentum is
proportional to velocity, the momentum-carrying entities must have some
velocity, like the water droplets in the flowing waterfall.

Experimentally,
we know that force and acceleration have no direct effect on gravitational
redshift or clock rates or electromagnetic wave propagation delay or
light-bending – effects first introduced into gravitation by GR and
related to gravitational potential. This absence of force effects remains true
even at accelerations as high as 1019g, where g is the
acceleration of gravity at Earth’s surface. [[8]]
However, gravitational acceleration is the primary determinant of the motion of
bodies in a gravitational force field. The special GR effects over and above
ordinary gravitational force must then arise only in the gravitational
potential field, which apparently has all the properties of an optical medium.

Gravitational potential vs.
gravitational acceleration

The
two interpretations of GR use two different concepts for “gravitational
field”. Let’s now examine the relationship between these two
concepts. Are they two different manifestations of the same thing, or are they
two independent things that appear to be related because they interact? We will
examine this from the perspectives of physics and mathematics.

Physically,
it is important to note that the strength of the local gravitational potential itself
(as contrasted with its gradient) is almost completely irrelevant to the motion
of a target body. That motion is determined to great precision by the local
gravitational acceleration (which is the force per unit mass) alone, regardless
of the strength or weakness of the local potential. By contrast, if we are
seeking the effect of gravity on the rate of ticking of a clock, only relative
speed and gravitational potential matter, but not acceleration or force.

For
example, Figure
4 shows a clock inside a uniform spherical shell. Such
a clock feels no gravitational force because, as is well known in dynamics,
gravitational forces from all parts of the shell cancel. However, the
gravitational potential does not cancel, although it is uniform throughout the
shell interior. So the rate of the clock is slowed by the amount of the
potential, which depends on the mass of the shell and could be substantial. A
similar example is presented by an infinite wall of uniform density. Away from
the wall, the force is everywhere constant and unrelated to the absolute strength
of the potential, which likewise tends toward some constant value near the
wall.

Figure 5 shows a pair of identical clocks suspended from a
tower. The one on the right is released and begins a free fall. Before the
acceleration has a chance to change that clock’s velocity and therefore
its rate of ticking, there is no effect on the rate purely as a consequence of
being in a state of free fall. The two clocks tick at the same rate until the
falling one picks up speed.

Now
let’s examine the math of potentials and accelerations. Letbe the gravitational constant, andbe the mass of a source producing a field around itself. If
we measure that field at some point at distancein directionfrom the source mass (where is the vector from the source mass to the field point), then
the Newtonian gravitational potentialat that point is. The Newtonian approximation is sufficient for the
qualitative distinction being made here. Of course, potential is a scalar and
has no associated direction. At the same point, the Newtonian gravitational
acceleration is , where is the acceleration vector of a target body or massless force
field point, directed toward the source mass. (Each dot represents a derivative
with respect to time. Bold symbols with arrows are 3-space vectors, and bold
symbols with carets are unit vectors.)

{Note
that celestial mechanics actually deals with accelerations (which are
observable) rather than forces (which are usually only inferable). To convert
the acceleration into an inferred force, we would have to put a target body with massat the force field point, and use Newton’s second law, . However, that is an unneeded complication because the
motion of a body in a gravitational force field is independent of its own mass.}[‡]
[7]

The
mathematical relationship between these two concepts is that gravitational
acceleration is the negative of the gradient of gravitational potential: , where , and  are the
components of in the orthogonal directions  respectively.
These three components are unit vectors centered on the source mass. Physically,
the gradient is essentially the slope of the potential field at the point of
interest, and its negative naturally points toward the source mass.[§]

At
this point, we come to a problem that is usually ignored in both the geometric
and field interpretations of GR, but which is critical for any meaningful
physics that might be associated with the concepts of potential and
acceleration. If the point of interest is fixed relative to the source mass,
then the instantaneous and retarded directions of the source mass from the
point of interest are the same, and the above relation works without ambiguity.
However, if the given point of interest represents a target body with a
transverse motion relative to the source mass (as for the Earth orbiting the
Sun), then the instantaneous and retarded directions of the source mass from
the target body are two different directions in space. That leaves us with an
open question: Should the gradient relating acceleration and potential use
instantaneous or retarded coordinates? I.e., should the negative of the gradient,
and therefore the acceleration, point toward the instantaneous or retarded
source mass location?

The
answer, as we will see, depends on which interpretation of GR is being invoked.
To see why, we next examine the effect of propagation delay, which will show
why the answer to our question makes a big difference.

The effect of propagation delay

Propagation
delay can cause a time lag in the application of the force to a target. The
main signature of propagation delay when the target is moving relative to the
source is to change the effective direction of application of the force to the
target. In astronomy, this change in the angular direction of a propagating
entity due to a relative motion of the target is called
“aberration”. See Figure
6. If the entity (e.g., our force) has a propagation
speed of and the transverse component of the target’s relative
speed is, then the aberration angle is . For cases of interest here, >>, so the aberration in radians is simply. The source sees that as the angle through which the target
moves during the projectile transit time, or as the angle by which the
projectile must lead the target when launched in order to arrive at the same
place as the target at the same time. The target sees it as the angle moved by
the source during the same time interval; i.e., the difference between the
retarded and instantaneous directions of the source.

All
motion is relative, so no experiment can tell the difference between a
fixed-source, moving-target situation and a moving-source, fixed-target
situation. I.e., both perspectives are equally valid. Moreover, the aberration
angle represents a real, physical difference of path direction for the
projectile depending on the reference frame, not just an illusion; and as a
consequence, any force applied to the target by the projectile acts from the
direction of projectile travel seen by the target, not that seen by the source.
That is obvious for sub-light-speed projectiles; for example, encounters among balls
moving on a pool table or arrows shot at a moving target or water waves acting
on a moving boat. It is equally true for projectiles moving at lightspeed, such
as the force of solar radiation pressure acting on dust particles or balloon
satellites. Logically, it should be true for projectiles of any speed.

To
see exactly why a force applied to a moving target must act at an angle to the
radial direction from a fixed source, visualize the target in Figure 6 as having a finite width and open windows on both
sides. Then the projectile will enter from a forward window on the source-ward
side, and leave by a more rearward window on the opposite side, because of the
forward motion of the target during the time it takes the projectile to cross
the width of the target. If the projectile enters via a rearward window, it may
hit the back wall of the target, showing that a component of its force must act
to oppose the forward motion of the target. (See an animation of this figure and
moving geometry at http://metaresearch.org/media%20and%20links/animations/aberration04.swf.)

So
the principal manifestation of propagation delay is aberration, which must
always exist for any type of projectile (whether particle or wave or wavicle or
other) moving linearly between a source and a target with relative motion. And
the aberration of projectiles acts to oppose the relative motion of the target,
thereby assuring that the angular momentum balance between source and target
cannot, in principle, be preserved. (I.e., some of the linear momentum
delivered to the target by the projectile changes its angular momentum also.) The
target angular momentum can be preserved in practice only by having projectile
speeds so high compared with the relative motion of source and target that
aberration is negligible.

Therefore,
the answer to the question posed in the preceding section about the direction
of the negative of the gradient must be answered in favor of the retarded
direction of the source. However, in both gravitation and electrodynamics, this
answer will change orbital angular momentum, causing all orbits to be spirals
(inward for repulsive forces and outward for attractive forces). In light of
this inconvenient fact, three propositions exist to justify the use of
instantaneous coordinates for the gradient:

(1)  Deny the existence of momentum carriers (projectiles)
between a source mass and a target body, thereby denying that gravity is a
force. Then postulate that “gravity is just geometry”, the approach
taken in geometric GR.

(2)  Postulate the existence of a velocity-dependent force
arising from the force field, as needed to conserve angular momentum. This
force must exactly cancel the main effect of propagation delay, aberration. This
approach avoids the necessity of having momentum carriers propagating faster
than light in forward time, in violation of SR.

(3)  Postulate momentum carriers (projectiles producing forces)
that propagate so fast relative to velocities of target bodies that aberration
is negligible. This requires propagation at speeds much faster than light in
forward time.[**] While
not allowed by SR, this solution is allowed by the mathematically equivalent
and observationally viable Lorentzian Relativity (LR).

Problems with geometric GR

The
word “space-time” has led to considerable misunderstanding of GR
because it begs the student to think of 4-space as simply 3-dimensional space
plus time. However, in reality, the space-time metric (e.g., the formula for
the square of the distance between two points) consists of the square of the
product of the speed of light times the elapsed time interval between the two
points minus the sum of the squares
of the x, y, and z components of the spatial distance. That minus sign makes each
of the spatial coordinates unlike the scaled time coordinate, and the metric
unlike what it would be if time were simply a fourth space-like dimension.

Consider
a practical example of this difference. Earth’s orbit around the Sun is
an ellipse with the Sun at one focus, to good approximation. Consider any two
points along that orbit, A & B, occupied by the Earth at different times. See
Figure
7. If gravity were “just geometry” and time
were like a fourth spatial dimension, then the shortest distance between A and
B would be that described by a taut rope stretched between those two points.
The actual path followed by the Earth is a considerably longer distance
requiring more time to traverse than if the Earth simply moved on a straight
line through space.

This
example shows us two things. First, any hypothetical “curvature of
space” is not a significant factor in determining the Earth’s
trajectory because taut ropes (which must follow any curvature of space) do not
curve. And second, whatever “space-time” means, it is not simply
space plus time. Yet in GR, the metric is an extremum along the path a body
actually follows through a gravitational force field (e.g., along the orbit in
the figure). What, therefore, does the metric mean, if its square is not simply
the squares of spatial and temporal distances combined?

Consider
the simple case of flat space-time (no gravitational field). Let the 4-space
coordinates be, and the space-time path length be. Then the metric is:

To see the true nature of a
space-time path, we divide everything by  and note that, where  is spatial velocity.
Then the metric becomes. The coefficient on the right side may be recognized as closely
related to the relativistic time dilation factor, whereis always greater than or equal to unity. So the metric
becomes simply.

In
this simple form, we can see that both sides of the metric equation describe
time intervals. is of course an interval in “coordinate time”,
unaffected by gravitational fields or by motion. is an interval of “proper time”. We will denote this
interval by. According to relativity, proper time describes the actual readings
on any clock whose rate may be slowed by motion relative to the adopted
coordinate frame and/or by a gravitational potential. So is the same as. This makes  the distance
traveled by a photon during a proper time interval. The important point here is
that  in particular and
space-time in general have a primarily time-like character rather than a
space-like one or a combination of the two. So when we speak in relativity of
“space-time curvature”, we are speaking primarily of a
clock-slowing effect, and not something that can be thought of as a curvature
of space. This is true even when a gravitational potential field is present.
The primary effect of such a gravitational potential field is a further
difference between coordinate time and proper time arising from gravitational
potential; specifically, , where  is always less
than or equal to unity. Note that the two relativistic factors and differ from unity by quantities of the same order. This is
because and are of the same order, and indeed are equal for circular
orbits.[††]

These
considerations have important implications for the geometric interpretation of
GR. The most basic is that the idea that “gravity is just geometry”
in geometric GR may now be seen as having dubious merit because “curved
space-time” is primarily a description of clock behavior; because time (being
one-dimensional, unidirectional and intangible) by itself has no geometry; and because
gravitational acceleration is not caused by any curvature of space. Note that
this point does not question any aspect of GR as a valid mathematical theory,
but only questions the physical meaningfulness of one particular interpretation
of that theory.

Because
the teaching of GR has evolved toward the geometric interpretation exclusively,
some of today’s relativists have come to think of the 4-space model as
reality. They are surprised to learn that relevant astronomical observations always
involve 3-dimensional spatial accelerations measured against the proper time of
the observer. It is therefore only through the vehicle of mathematical
expressions for this 3-space acceleration, called “equations of
motion”, that GR is validated through successful comparisons between
theory and observations. Solutions to the Einstein field equations are of no
practical value in that regard, and must be converted to equations of motion before
comparisons can be made.

That
is done by using solutions (such as the well-known “Schwarzschild
solution” to the field equations to form a Hamiltonian or a Lagrangian,
then taking partial derivatives to derive equations of motion. This is a
process quite similar to forming the gradient of a potential, and involves the
same decision, whether made implicitly or explicitly: Should the partial
derivatives use instantaneous or retarded coordinates? While this issue is usually
not raised explicitly, instantaneous coordinates are always assumed in
gravitation (and electrodynamics) because the alternative leads to equations of
motion that quickly depart from reality. Once again, one of the same three
choices previously mentioned must be invoked, at least implicitly, to justify
this choice.

Geometric GR is not physically viable

Geometric
GR is often taught by referring to the “rubber sheet analogy”,
found in many textbooks, wherein the gravitational potential field is likened
to a rubber sheet with a huge dent located at a source mass M. See Figure 8. A target body sits on the sloping side wall of this
dent at some point P. In the static situation, there is no ambiguity. The
rubber sheet is like the gravitational potential field of M, and the slope of
the side wall is like the gradient of the potential. In the static case, the
gradient at P (the analog of gravitational force) is unique and unambiguous,
and its negative points at M.

Consider
the case where M remains fixed on the rubber sheet, and P has a transverse
motion. Then the direction of the negative of the gradient would still always point
toward the instantaneous position of M because it is independent of the
transverse motion of P; i.e., it has no aberration. This is what the geometric
interpretation of GR proposes.

In
this example, nothing propagates from M to P. So aberration is not applicable,
and apparent agreement with experiment is secured. Note that this agreement
does not result from a propagation speed of , but from no propagation at all. However mathematically
elegant this solution might be, a gradient, a slope, or a curvature cannot
induce a body at rest in a potential field (e.g., at point P) to begin moving toward
the source mass unless a force acts on that body. This is because, without a
force to initiate motion, the body has no reason to choose any one direction
over any other direction. Moreover, momentum would not be conserved if the potential
field had no moving parts, yet a body at rest in it began moving in some
direction without a force acting. (By "force", we assume the
classical meaning: "the rate of transfer of momentum".)

Such a hypothetical motion
would be initiated without a cause, violating the causality principle; and
would create momentum from nothing, violating the “no creation ex nihilo” principle. [[9]] While such violations are allowed
in math and are considered in philosophy, they are forbidden in deep-reality
physics because an effect without a proximate, antecedent cause or creation ex nihilo is a form of magic or miracle,
with no hope of ever being understood or explained in physical terms. It is
therefore the logical equivalent of “God made it that way.”

In
the rubber sheet analogy, if the small target body is at rest on the side of a
dent caused by a source mass, it will remain at rest forever until some force
acts. The rubber sheet analogy works in our imaginations only because we
instinctively imagine gravity under the rubber sheet, with its pull providing a
meaning to the concept of “downhill”. But without pre-existing
gravity, the small body has no cause to accelerate, either on the rubber sheet
or in the geometrical interpretation of GR. So the idea behind geometric GR, as
easy as it is to implement mathematically, violates two principles of physics
and is therefore physically impossible.

But
there is another allowed interpretation of Figure
8 and of geometric GR. Consider M to have some
transverse motion relative to the rubber sheet and relative to point P while keeping
point P fixed on the rubber sheet. As M moves, the dent it causes moves too.
Let's assume the rubber sheet takes some finite time to adjust its shape to the
new location of M. If the adjustment wave propagates out at speed , and if point P is a distance r away from M, then the
propagation delay is a time interval . If M moves at speed , P will always feel a gradient (the “downward” slope
of the dent) that lags behind the true, instantaneous position of M by the
approximate angle  in radians.

In
this interpretation, the negative of the gradient cannot point toward the new
position of M until the adjustment wave has time to propagate from M to P. The
conventional way around this difficulty is to hypothesize a “velocity
field” in the rubber sheet paralleling the motion of M and maintaining
the needed gradients. One difficulty with that hypothesis is that the velocity
field has independent existence. (If it was caused by M, it too would have
propagation delay.) So what would make it ever pass out of existence? Even if M
sets off a new velocity field based on some new motion, that shouldn’t
affect the old velocity field because the old field is no longer controlled by
M. If one hypothesizes a characteristic decay time for the old velocity field,
it would go away too soon at large distances from the source and too late at
small distances, so it could not imitate the behavior predicted by GR and
supported by experiments. In other words, any such "velocity field"
is a deus ex machina, lacking any physical
justification.

As
we explained earlier, static potential fields are either continually
regenerated by a source mass, or are rigid in the sense of having no moving
parts. However, for gravitation we require not only a velocity field but also
an “acceleration field” to remain in conformity with observations. This
weighs heavily against the rigid potential field interpretation, which we have
already shown to be non-viable. But the velocity field fares no better because
even the Sun accelerates as it moves around the solar system barycenter, yet
planets respond almost instantly to such accelerations. Binary pulsars are an
even more dramatic demonstration of the effect, and most clearly demonstrate
the need for an acceleration field too. Lastly, velocity fields would clearly be
unable to continually update as needed in the case of two black holes in
mutual, elongated elliptical orbits, because no communication of mass or
location information across the event horizon is permitted. [[10]] We therefore conclude that the
“velocity field” interpretation is not viable on physical grounds
either.

Although
it is obvious to anyone familiar with the GR equations of motion of an n-body
system, we take special note that these equations contain no terms that are
first order in (v/c), and therefore have no aberration. [[11]] Some relativists do not dispute that
aberration is an essential requirement of physics, but claim that aberration
has been cancelled by a “velocity-dependent term” (same as a
velocity field). [[12]]
However, the derivation of the equations of motion shows that neither
aberration nor a canceling term was ever considered. [[13],[14]]
The needed partial derivatives simply used instantaneous coordinates with no
propagation delay, thereby implicitly adopting an infinite propagation speed
for gravitational force.

Finally,
we note the existence of one experiment designed to test the weak equivalence
principle, which is the idea that “gravity is just geometry” and is
therefore always independent of the mass of the target body. In a neutron
interferometer experiment, a target-body mass-dependence was observed, thereby
falsifying the geometric or “curved space-time” interpretation of
GR experimentally. [[15]]
The experiment is not better known today mainly because its result flies in the
face of conventional wisdom, and no one has attempted to replicate it.

In
summary, we have a conflict of conclusions about the validity of geometric GR
taught almost exclusively today as a mathematical theory describing potential
fields, and the constraints imposed by the principles of physics and logic that
exclude the geometric interpretation of GR in favor of the field interpretation.
The two interpretations began to diverge in a serious way with the popular book
Gravitation [10], which emphasized the geometric interpretation to
the exclusion of the field interpretation. This view has perhaps reached its
culmination with several books by Wald, who denies the viability of the field
interpretation altogether with these words: “The basic framework of the
theory of general relativity arises from considering ... that we cannot in
principle -- even by complicated procedures -- construct inertial observers in
the sense of special relativity and measure the gravitational force. This is
accomplished by the following bold hypothesis: The space-time metric is not
flat, as was assumed in special relativity. The world lines of freely falling
bodies in a gravitational field are simply the geodesics of the (curved)
space-time metric. In this way, the ‘background observers’
(geodesics of the space-time metric) automatically coincide with what was
previously viewed as motion in a gravitational force field. As a result we have
no meaningful way of describing gravity as a force field; rather, we are forced
to view gravity as an aspect of space-time structure. Although absolute
gravitational force has no meaning, the relative gravitational force (i.e.,
tidal force) between two nearby points still has meaning and can be measured by
observing the relative acceleration of two freely falling bodies. ...” [[16]]

But
as we have seen above, this geometric view is either in conflict with the
principles of physics or requires hypothesizing the existence of an unphysical,
undetectable “velocity-dependent force” whose sole reason for
existing is to explain the absence of the required aberration from the theory.
And in either case, this interpretation is contradicted by the neutron
interferometer experiment. This is no cause for concern because the field
interpretation is still viable, notwithstanding Wald’s opinion. Moreover,
when we follow where these considerations logically take us, we will see that
some ideas about relativity thought to conflict with ideas about quantum
mechanics may now be seen as reconciled.

Gravitational force propagates faster
than light

Physics
requires that both gravitational and electrodynamic forces, like anything
propagating, must experience both propagation delay and its principle
manifestation, aberration. Yet experiments are unambiguous and undisputed in
indicating that neither force displays aberration at a detectable level. We
described three possible ways to resolve this apparent contradiction, and have
now eliminated two of them. The physical model that remains on the table is the
field interpretation of GR and momentum carriers that propagate much, much
faster than light. This reduces aberration below presently detectible levels by making very large. This model has implied momentum carriers, and
therefore has a cause for inducing motion in the target body, and requires no
creation of motion from non-motion (momentum from nothing). It avoids the
logically inconceivable “instantaneous action at a distance” by
retaining finite force propagation speeds, however large they might be. It
makes gravitational force fields dynamic carriers of energy able to deposit
heat in affected masses. And it implies that these force fields create
gravitational potential changes, for example, by altering the density of an
underlying “light-carrying medium”, hereafter referred to as
“elysium”. This interpretation therefore has no need of a
“velocity-dependent force” in the force field separate from
gravitation or electrodynamics. And it has recently been developed into a
complete model for the nature and origin of gravity, including its relativistic
effects. [[17]]

However,
SR forbids propagation at faster-than-light speeds in forward time. So what
remains to be discussed is how this interpretation can survive SR’s constraint.
We will begin with a review of the experimental evidence that compels us to
this conclusion of faster-then-light force propagation speeds, followed by a brief
discussion of the role of gravitational waves.

First,
we take notice of a simple, basic fact: gravitational force and light from the
Sun are not parallel in the Earth’s frame of reference. The Sun’s light
and gravity may be presumed to originate at the same place at the same time,
and to propagate along the same radial path to the same target body, say, the
Earth. Yet when they arrive, they appear to come from directions that differ by
20 arc seconds on the sky – the angle of aberration for light. This
reflects the fact that light takes about 500 seconds for this journey, during
which time the Sun has moved through an angle of 20 arc seconds from
Earth’s perspective; whereas the Sun’s gravity always acts toward
the Sun’s true, instantaneous direction, a direction in which the Sun
will not be seen by the observer until 500 seconds in the future. When sunlight
applies a force to a target body (the force of solar radiation pressure), the
direction of that force is likewise offset by 20 arc seconds from the direction
of gravitational force. This difference indicates that the speed of
gravitational force in the Earth’s frame is so fast that no offset from
the Sun’s instantaneous direction can yet be detected; whereas light
propagates so comparatively slowly that the main effect of its propagation
delay, aberration, is readily seen both directly (for eclipses) and indirectly
(through radiation forces).

If
gravitational force had a propagation speed as slow as light, it too would have
an aberration of 20 arc seconds and would appear to parallel light from the
Sun. Moreover, the force of gravity would be applied in an offset direction
also because of Earth’s orbital speed. This offset would cause orbits to
spiral outward. For Earth’s orbit, the average distance from the Sun
would double in 660 revolutions. For Mercury, about 100 revolutions would
suffice. For binary pulsar PSR1913+16, the orbital radii would double in about
a month. The absence of such an acceleration in binary pulsar PSR1534+12 sets
the most stringent lower limit available for the speed of gravity: 2x1010
c. [9]

Experiments
that show the absence of an expected effect are revealing and important. But in
view of our discussion of potential fields with no moving parts, it is
reassuring that we also have an experiment with a non-null result. Total solar
eclipses reveal that the gravitational maximum of the Sun’s force on the
Moon follows the maximum visual eclipse by about 40 seconds of time on average.
This coincides with the time it takes the Moon to traverse along its orbit
through the angular distance between the apparent Sun and the true,
instantaneous Sun. Interestingly, the visual eclipse corresponding to the
propagation-delayed image of the Sun occurs first, followed by the
gravitational maximum of the Sun’s force on the Moon. [9] If the Moon had a retrograde orbit, the visual
eclipse would occur after the gravitational maximum.

All
six experiments (described in the preceding reference), whose results bear on
the speed of gravity question, yield a lower limit to that speed well in excess
of the speed of light.

The physical meaning of potential

Until
now, it has usually been assumed that force fields and potential fields,
whether gravitational or electrodynamic, were both different manifestations of
the same phenomenon. This is especially true for electrodynamics, where the
linkage between electric and magnetic forces on the one hand, and light waves
or other parts of the electromagnetic spectrum on the other, is
well-established. In the case of gravitation, the counterpart of light waves is
supposed to be “gravitational waves”. Although these hypothetical
entities would be ultra-weak and have yet to be detected from any source in our
solar system, indirect evidence for their existence in distant astrophysical
systems has been seen in the binary pulsar PSR1913+16.

It
is clear how a potential and a corresponding force are related mathematically
– as function and its spatial derivative. But what is their relationship
physically? We have seen above that the remaining physically viable
interpretation of forces is that they are the effect of a continuous stream of
tiny momentum impulses transmitted between a source mass and a target body at
speeds far greater than lightspeed. However, the spatial integral of such
forces would have no particular significance – unless space was filled
with a medium that can also be affected by the momentum carriers. In the case
of gravity, the density, or at least the pressure, in such a medium would
increase in the vicinity of source masses because the downward force of gravity
would pack more medium into that vicinity, or at least increase the downward
pressure if the medium were incompressible. Under equilibrium conditions, the
density or pressure would increase linearly with decreasing distance from the
source mass.

Consider
an atmosphere. If no forces acted on it, density everywhere would tend to
equalize within its container. But if a force such as gravity acts on it, a
density gradient will develop. We could measure atmospheric density and the
speed of sound as a function of height, yet learn nothing about the properties
of the gravitational force that created and maintained these properties of air,
except its strength.

Air density is the analog of gravitational
potential {the medium responsible for “space-time” (i.e., proper
time) in the geometric interpretation of GR}, and sound waves are the analog of
gravitational waves. The wave speed is a function of the properties of the
space-time medium, such as its permeability and permittivity. For gravitational
waves, this speed happens to be the speed of light, c. This wave speed tells us
almost nothing about the speed of gravitational force. For example, in our
analogy, if sound waves and air density changes were set off by an asteroid
impact, these properties of air would likewise be quite insensitive to the
speed of the asteroid.

Let’s assume that a gravitational potential
exists due to the presence of a single, isolated, massive star fixed in space.
The retarded and instantaneous scalar potential values (but not their
gradients) then coincide for observers in all frames. We will examine the view
from a massless particle (target body) moving on a circular orbit so that the
potential remains rigorously constant for that target body at all times. With
that setup, there is still an issue to be understood. The constant potential has
a time-variable gradient because the direction of the gradient is always
changing. So when we take the gradient, we must specify whether it will be the
instantaneous or the retarded gradient, even though the potential itself is
rigorously constant for all time. In general, it does not matter whether the scalar
potential itself is retarded or not. The whole issue of the speed of
gravitational force revolves around the gradient, which has properties not
derivable from the potential alone.[‡‡]

Elysium
is, by definition, the light-carrying medium. If we now identify the
hypothetical gravitational potential medium with elysium, we see at once that several
remarkable properties emerge. First, note that elysium, like scalar potential,
has an arbitrary background level (zero point). We are concerned mainly with
the effects of changes in that background level, but not normally with the
absolute level itself. Then if light propagates past a source mass, the
increased density of elysium there will refract the path of the light, bending
it away from the normal to the source mass. Moreover, propagation speed of the
light waves will be slowed, as will be the rate of ticking of any clock based
on electromagnetic phenomena because of the increased medium density and
consequent slowed propagation speed. Indeed, it was already known to Eddington
in 1920 that such an optical medium would give exactly the same predictions (to
first order in the potential) as geometric GR for the tests of GR that describe
the behavior of light in a gravitational potential field. [[18],2,17] And it has since been shown that the same
optical medium can yield the same results for the fourth test of GR, the
advance of the pericenter of orbits in general or of the perihelion of Mercury
in particular.[§§]
[[19],[20]]

The
upshot of these considerations is that the potential field may be visualized as
the elysium, with its density increasing as any source mass is approached. We
then have a clear visualization of why a gravitational potential field and a
force field are physically different entities, yet related to each other as a
function and its derivative. We also see why “space-time curvature”
(now visualized as optical refraction) is so important to potential fields, yet
has no discernible effect on the basic Newtonian component of gravitational
force. The clock-slowing effects of speed and potential can either cancel or
combine depending on the resultant effective density of the elysium.[***] The
radial velocity between a source and an observer does not affect the wave speed
or path, but only its wavelength and perceived energy. By contrast, aberration
does affect the direction of the path of a light wave, and depends on the transverse
velocity between source’s local potential field and the observer. “Local”
is important. For example, aberration is the same for both components of a
double star, even though each star has a different velocity relative to Earth. But
each component of the double star is immersed in a larger, combined potential
field. So the wave speed depends on the local density all along its path, and
aberration depends on propagation delay along the same path.

Gravitational waves propagate at the
speed of light

Lastly,
we readily see why the subject of the “speed of gravity” has led to
so much confusion and debate. That speed is necessarily equal to the speed of
light for disturbances of gravitational potential fields, and is necessarily
much faster than light for gravitational force fields.[†††]
Today, there is little reason to doubt the existence of gravitational waves.
But inasmuch as these entities are the result of disturbances of a
gravitational potential field, they have nothing to do with changes in gravitational
forces or force fields. Only the belief that potential and force fields are two
manifestations of a single phenomenon implies such a relationship. But as we
have seen, such a belief is ill-founded. We now have reason to suspect that
gravitational waves are simply very-long-wavelength electromagnetic waves, and
that the hypothetical spin-2 entity in quantum physics now called a
“graviton” actually describes the basic constituent of the
potential field, and as such would more appropriately be named the
“elyson”. When two bodies with a relative speed orbit one another, the
smaller will experience more drag from the elysium medium than the larger,
which would manifest itself to us as gravitational waves (disturbing the
potential field of the other mass) and losing angular momentum as a consequence,
as observed in binary pulsar PSR1913+16. Likewise, the elysium field would
produce a torque on a rotating body, analogous to the
“frame-dragging” effect predicted by GR. This is because anything
not on an equipotential surface will tend to have a horizontal torque as its
path around the center tries to precess because atomic processes are being
forced to occur at different rates for different parts of the body. [‡‡‡]

Some
textbooks suggest that gravitational waves are simply changes in gravitational
forces. However, that interpretation cannot be correct because gravitational
waves originating in the solar system have not yet been detected. Consider
these examples:

(1)  When we walk into a room, a gravimeter can track
changes in the gravitational acceleration we produce. These are easily
detected.

(2)  In a static potential field, an orbiting body is
continually changing its acceleration. We can easily measure these force/acceleration
changes.

(3)  The GR equations of motion are expressions for
accelerations of bodies as a function of location, velocity, and potential. As
such, they describe changes in acceleration through second order in potential
or fourth order in velocity. Yet there is no gravitational wave component in these
equations.

Moreover, gravitational waves
passing through a body merely cause its atoms to oscillate slightly, but
produce no net force on the body. It is therefore safe to conclude that gravitational
waves have nothing to do with ordinary gravitational acceleration or with
changes therein.

We
have seen that geometric GR is based primarily on properties attributable to
potential fields that are then assumed to apply also to force fields. One of
those is that fields are “fossilized”; i.e., have no moving parts
unless disturbed, like the frozen waterfall of Figure 2. By contrast, the force field interpretation of GR
with momentum-carrying entities has fields that continually regenerate, and as
such are a potential source of energy and can do work, much like the flowing
waterfall of Figure
3. In further support of the latter interpretation, we
note how much easier it is to understand the properties of gravitational forces
with the force field interpretation. [17]

For
example, while it might be convenient to describe the Sun’s gravitational
force field as “static” to explain why planets accelerate toward
the true, instantaneous Sun rather than the light-time-retarded Sun, reality is
that the Sun is in a constant state of acceleration relative to the solar
system barycenter, with amplitude great enough that the barycenter is often
located outside the physical body of the Sun. This is more acutely pertinent in
the case of binary pulsars, where the acceleration of each component is toward
the true, instantaneous position of the other to better accuracy that even the
linearly-extrapolated-retarded position can provide. And the fossilized force fields
explanation is never more strained than when it tries to explain how binary
black holes in mutual elliptical orbits can continually update the curvature of
space-time when the mass and location information needed to do so is hidden
behind a pair of event horizons.

@@@ [gravitational wave diagram and physical explanation to be added
here]

Is faster-than-light propagation allowed
by the laws of physics?

The
proof that faster-than-light propagation is not
allowed is simple. Special relativity forbids it because time slows and approaches
a cessation of flow for any material entity approaching the speed of light. So
no matter how much energy is brought to bear, the entity cannot be propelled
all the way to, much less beyond, the point where time ceases. The
entity’s inertia simply increases towards infinite as the speed barrier
is approached. Hypothetical mathematical entities with imaginary masses (whatever
that means) might exist, according to the equations. These “tachyons”
would always travel faster than light, but must always propagate backward in
time and could never be slowed enough to cross the light-speed barrier either. Relativists
are confident that SR is a valid theory because it has passed eleven
independent experiments testing most of its features and predictions. And the
very successful theory of general relativity is based on SR, and has likewise
passed several major experimental tests.

As
solid as this reasoning appears to be, it has a logical flaw, because another
theory exists about which the same supporting claims can be made, but which has
no universal speed limit. This is the so-called “Lorentzian
relativity” (LR). Let’s briefly review what this theory is, how it
differs from SR, and what the experiments have to say about it.

Lorentzian
relativity is a modern updating of the Lorentz Ether Theory, first published in
1904 a year before Einstein published SR. [[21]] It
is based on the relativity principle, first formulated at least a generation
earlier; and on the famous transformations named after Lorentz, thereby having
the same mathematical form as SR. In essence, LR is relativity for the aether.
Einstein’s innovation in SR was to abolish the need for aether, or more
specifically, the need for a preferred frame, by making all inertial frames
equivalent, with each having the same speed of light. In SR, the Lorentz
transformations apply to time, space, and mass; whereas in LR, they apply to
clocks, meter sticks, and momentum. That is why there is no universal speed
limit in LR – nothing ever happens to time itself, just to clocks
attempting to keep time.

Because
time is never affected, LR recognizes a “universal time” applicable
to all frames, and a universal instant of “now”. In SR, all
inertial frames are equivalent, so the Lorentz transformations apply reciprocally
(i.e., both ways between two frames); whereas in LR, the local gravitational
potential field constitutes a preferred frame, and the Lorentz transformations work
just one way from the preferred frame, but not reciprocally.

Eleven
independent experiments confirm the basic behavior of nature predicted by SR,
but also predicted by LR. [[22]]
Just as with the two interpretations of GR, the physical interpretations of the
two variants for the relativity of motion differ, but not the mathematical form
or the observable phenomena in most instances. Although claims have been made
over the years that various experiments on this list falsified either SR or LR,
subsequent discussion indicated that was not the case. It is now widely
believed that no experiment dealing with lightspeed or slower phenomena can
distinguish the two theories.

To
cite just two examples, particles in accelerator experiments do seem to
approach the speed of light indefinitely closely without ever exceeding it, no
matter how much energy is supplied, which appears to favor the SR
interpretation. However, LR notes that one cannot expect to push particles
faster than the speed of light using forces which themselves propagate only at
lightspeed. This would be like trying to use propellers alone on a plane, with
no gravity assist, to exceed the speed of sound. And in the Global Positioning
System (GPS), all atomic clocks aboard satellites with a variety of orbital
planes, and all atomic clocks all over the rotating Earth, are all synchronized
with one another, and remain synchronized, despite being in many different
inertial frames. This appears to be a practical realization of Lorentz’s
universal time. But SR points out that the clocks had to be adjusted in rate to
achieve this synchronization, and that the measured speed of light is then not
constant in frames other than the local gravitational potential field. If the
two postulates of SR are adhered to, then the clocks behave in all frames just
as predicted by SR, albeit at the cost of adding considerable complexity to the
system. That is avoided by synchronizing each clock to a momentarily co-located
imaginary clock at rest in the local gravitational potential field, and working
only in that Earth-centered inertial frame.

Conspicuously
missing from the list of experimental results is any experiment testing
reciprocity of the Lorentz transformations. Specifically, GR is built on SR
using only one-way Lorentz transformations relative to the local gravitational
potential field (center-of-mass reference frame), and is therefore just as
consistent with LR as SR. The famous Twins Paradox, an attempt to show that SR
had an inconsistency, has no counterpart in LR because LR’s
transformations work only one way. [[23]] It
now seems that only an experiment demonstrating a real phenomena propagating
faster than light in forward time could decide between SR and LR. That issue now
appears to have been decided in favor of LR by experiments implying that
gravitational and electrodynamic forces propagate faster than light in forward
time. [2] In any case, the mere existence of LR as a
viable alternative to SR means that nature need not have a speed limit.

Gravitational force has no inertia

@@@ [include material from “inertia article and add discussion of equivalence
principle]

Conclusion

General
and special relativity are elegant mathematical theories with non-unique
physical interpretations. When physical principles are brought to bear, the
interpretation ambiguities can be resolved in favor of the force field
interpretation of GR and the Lorentzian variant of the relativity of motion,
LR. Both of these interpretations allow faster-than-light propagation of
material entities in forward time. The tightest experimental constraints
indicate that the propagation speed of gravitational force must be no less than
2x1010.

This
change of perspective does not yet require any change in the math of either the
relativity of gravitational fields or the relativity of motion. Yet it
potentially provides the missing key to unification of the fundamental forces
of nature, and eliminates the locality requirement for quantum phenomena such
as those involved in the EPR paradox. It also permits developments in the field
of quantum gravity that were previously blocked. [17]

A

GR has two physical
interpretations – field and geometric.

B

“Gravitational
field” has two meanings – force field and potential field.

C

“Space-time”
is not a combination of 3-dimensional space plus time. (See also G.)

D

Potential affects
clock rates, acceleration and free fall do not.

E

Physics requires that
gradients of potentials must be retarded gradients.

F

Retardation implies
orbit spiraling that can be avoided in three known ways.

G

Orbital motion is not
caused by any curvature of space.

H

The physical
interpretation of “space-time” is usually just proper time.

I

Gravity acceleration
is not caused by geometry.

J

The geometric
interpretation of GR is not physically viable.

K

“Velocity
fields” are not a viable explanation for the lack of observable
aberration.

L

Gravitational
force propagates faster than light in forward time.

M

Gravitational
potential has the character of density of a light-carrying medium.

N

Gravitational
waves have nothing to do with ordinary gravitational acceleration.

O

Lorentzian
relativity allows faster-than-light propagation in forward time.

P

Gravitational
force has no inertia.

Table I. Major points made in this paper, and
locations in the text where they are first raised.

If
these conclusions are, in the main, valid, then our chief concern would be the progress
this represents toward the scenario forecast by Douglas Adams: "There is a
theory which states that if ever anybody discovers exactly what the universe is
for and why it is here, it will instantly disappear and be replaced by
something even more bizarrely inexplicable. There is another theory which
states that this has already happened.” [[24]]

Acknowledgments

The
author thanks the Meta Research Board and members for financial support, with a
special thanks to Tim Seward and the
late Richard Hazelett. Many of these ideas were inspired and supported by the
late J.P. Vigier.

[last revised 2005/12/30]

[*] To be a
physical combination instead of a purely mathematical one, space and time would
have to be combined in some additive way, as would be expected if time (scaled
in some suitable way) were a fourth dimension with a space-like character.
However, this mathematical combination called “space-time” subtracts the sum of the squares of the
three space coordinates from the square of the time coordinate. So time is not
at all like a fourth spatial coordinate, which of course would be additive to
the other three coordinates.

[†] Although
GR is often today considered to need no aether, the aether concept has many
variants. One of those is the sense in which Einstein used the term when he
said “According to the general theory of relativity, space without ether
is unthinkable”, and equated it to “the gravitational field”
(meaning gravitational potential field). He went on to explain: “The
aether of the general theory of relativity is a medium without mechanical and
kinematic properties, but which codetermines mechanical and electromagnetic
events.” Dirac held a similar view.

[‡]
Although we will make no use of the distinction between gravitational mass and
inertial mass here, it is important in other contexts to note that the target
body mass in Newton’s
second law is always its inertial mass. So if we convert the expression for
gravitational acceleration (which comes directly from observations) into one
for gravitational force by multiplying both sides by and dropping the directional component, we arrive at Newton’s universal
law of gravitation in the usual form. In this expression, is the gravitational mass of the source. However, contrary to
custom throughout the field, must necessarily be the inertial
mass of the target body, not its gravitational mass, because one must multiply
both sides of the equation derived from observations by the same quantity. This
means that most experiments purporting to test the equality of gravitational and
inertial masses (e.g., Eötvös experiments) do no such thing becausein the preceding equation is already the inertial mass.

[§] Note
that, despite the gradient relationship, changes in potentials and changes in
forces need not coincide. For example, velocity and acceleration bear the same kind
of derivative relationship. Yet changes in velocity do not require changes in
acceleration, and vice versa (as for angular acceleration).

[**] Note
that the propagation speed cannot be infinite. Any entity with finite mass and
infinite speed would have infinite momentum.

[††]
Consider the Earth’s orbit around the Sun – clearly a curved path
through space, called a “geodesic path”. Choose two points along
the orbit and stretch a taut rope between them. That is obviously a shorter
path through space than the geodesic path. The geodesic path between two points
at a certain initial speed is the one for which the smallest amount of proper
time elapses. If the body deviates too far from a straight line, the elapsed proper
time at that speed will be greater because the distance is greater. But if the
exact straight line path were taken, the elapsed proper time would not be a
minimum because that path goes through a stronger gravitational potential (e.g.,
closer to the Sun) than an almost-straight line that curves enough to keep a
greater average distance from the Sun. So the geodesic path is the one where
the elapsed proper time (not the “distance”), considering the
clock-slowing effects of speed and gravitational potential, is a minimum.

[‡‡]
Note that GR treats potential fields and force fields inconsistently by using
retarded potentials and instantaneous gradients.

[§§]
Actually, although the single-source-mass pericenter advance rate is exactly
the same as in geometric GR, the interpretation described here implies a small
difference in predicted pericenter rates when two or more significant masses
interact. The predicted angular rate is about 16% smaller in the case of binary
pulsar PSR1913+16, which is within the range of present observational
uncertainty, but will soon be detectible.

[***] Motion
through a medium increases the number of constituent particles encountered per
unit time, and hence the effective density of the medium. For example, the
effect of Earth’s rotation speed on clocks just cancels the effect of
Earth’s equatorial bulge, making Earth’s sea-level surface an
equichron (equal time) surface. But these two effects for an orbiting satellite
in an eccentric orbit combine and produce changes in clock rates around the
orbit.

[†††]
The potential field and the force field cannot be the same medium because the
former has wave properties and the latter has only particle properties; and
because the speeds of action are so vastly different. The speed at which waves
propagate through any one medium is always close to the speed of motion of the
principal constituents of that medium. For example, for a gas, the wave speed
is  of the average speed
of the particles in that gas. But the propagation speeds for gravitational
force and potential differ by at least ten orders of magnitude.

[‡‡‡]
There is no immediately obvious reason why the predictions of the two
interpretations for “frame-dragging” should differ.

[1] D. Turner & R. Hazelett, eds. (1979), The Einstein Myth and the Ives Papers,
Devin-Adair Co., Old Greenwich, CT.

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