Tom Van Flandern / Meta Research
Gravity makes heavy and light bodies fall at the same rate. Gravity obeys the
“equivalence principle”, and is just "curved space-time geometry" in geometric
general relativity. But space-time curvature alone cannot initiate motion, and
changes in momentum still require a force acting. Moreover, gravity can deviate
slightly from the “equivalence principle”, and “space-time” is really just
proper time and does not involve any curvature of space. The Le Sage “pushing
gravity” concept is a better way to explain the physics of gravity. For forces
other than gravity, the momentum transferred must be shared by all particles in
the target body, producing what we call “inertia” -- a simple dilution of
momentum. Gravity obeys the “transparency principle”, allowing momentum to be
transferred directly to each particle. Without need for dilution of momentum,
gravity has no inertia. «
different from the other known forces of nature. All bodies, big and small,
accelerate at equal rates in any given gravitational field. That property is
opposite to our everyday experience, in which more massive bodies require more
work to move or accelerate than less massive ones. That gravity accelerates
masses of all size with equal ease is so anti-intuitive that people universally
believed otherwise until Galileo’s demonstration at the Leaning Tower of Pisa.
He simultaneously dropped a heavy and a light mass (both heavy enough that air
resistance was not a factor), and observers below tried to time which hit first
and by how much. But to the astonishment of the observers, who were certain that
the heavier body would fall faster, the two masses reached the ground at the
David Scott, an
Apollo 15 (1971) astronaut, did a unique, modern-day version of this same basic
experiment by simultaneously dropping a hammer and an eagle feather while
standing on the Moon. Because the Moon has no atmosphere, the fall of the
feather is due to gravity only, not slowed by air resistance. And because the
Moon has only 1/6 of Earth’s surface gravity, the falls of both objects were
slowed by a factor of six. But again, the heavy and light objects hit the ground
at the same time.
When the nature
of gravitation was being considered around the beginning of the 20th
century, the uniqueness of this property played a major role in that thinking.
Einstein formulated the principle of equivalence – that a uniform acceleration
and a gravitational field were indistinguishable to an enclosed observer. No
experiment from inside a closed box could tell, he reasoned, whether the box was
resting on the surface of a massive body or was being accelerated by rockets
through outer space. Either way, a downward, gravity-like force would be felt.
Likewise, someone in a freely falling elevator would accelerate downward but
feel no force, and might conclude that he was floating in outer space without
used this equivalence principle to conclude that gravity is not a force like the
other forces of nature, but is instead a curvature of space-time near large
source masses. Because all parts of a target body simultaneously experience this
curvature as they move, the change in the target body’s motion would be the same
no matter how big or small that body is. This was a new and unique way of
thinking about gravity.
The problems with this curved
space-time view are several. But the most basic of them all is that a body at
rest in a gravitational field has no cause to commence motion because curvature
does not induce motion unless a force acts. For example, if “curved space-time”
were visualized as a rubber sheet with a dent, then a body at rest on the side
of the dent would remain stuck there unless a force (such as gravity underneath
the rubber sheet) acted to make it move. In open space with no gravity acting,
the body would have no sense of which direction was “down” and no reason to move.
second problem with curved space-time (or curved anything) causing motion by
itself is that motion is momentum, and momentum cannot be created from nothing.
So the curved space-time (or curved whatever) must still apply a force; i.e., it
must have momentum of its own in the form of moving parts. Creating momentum or
anything from nothing requires a miracle, and postulating that the curved
space-time applies a force or has moving parts defeats the value of this “pure
geometry” mechanism as an explanation for gravity. With a force applied, we
would be back to wondering why big masses and small all fell (or curved) by the
same amount, unlike the action of other forces of nature. [
problem with curved space-time is that the acceleration of a body in a
gravitational field is not completely independent of the body’s own mass. (This
point is unrelated to the back-acceleration any small body produces on the
source mass, which may affect the relative acceleration between two bodies, but
not the acceleration of the target body relative to an inertial frame such as
that provided by the distant stars.) Equations of motion in general relativity (GR)
show that a body’s mass does (at order )
very slightly affect its own acceleration, violating the equivalence principle.
Experimentally, this violation of the equivalence of acceleration and
gravitational fields has been observed with neutron interferometers. []
So in addition to the difficulty this idea poses for consistency with physical
principles, it is theoretically and experimentally incorrect too.
relativists may argue that “space-time” is not simply space plus time, but a
higher-level concept that includes the notion of “time”, so the physical
principles do not apply. However, the physical principles arise from logic alone
and should be immutable, in contrast to the laws of physics, which can change as
knowledge improves. []
Moreover, “space-time” is a mathematical concept, which amounts to a fancy way
of referring to proper time in relativity (the time kept by perfect clocks), and
does not involve any curvature of space. To show this, consider the following
mathematical and physical arguments.
Let dT be a coordinate time interval (an
idealized time in some specific reference frame) for a moving body, and let
(dX, dY, dZ) be the change in the body’s space coordinates during that time
interval. Next, let ds be a path length in “space-time” for the body during
the same interval; and let c be the speed of light. Then the standard
relation between space, time, and “space-time” (with no gravity acting) is:
Multiplying the coordinate time interval
by the speed of light has turned time into a length-like coordinate (meaning
a time interval measured in meters, not to be confused with a "space-like
interval" as used in GR). This change of units allows the time interval to
be combined with the coordinates for the three spatial dimensions. However,
the presence of a minus sign makes the combination un-length-like; i.e., not
the equivalent of space plus time treated as comparable length-like
coordinates. So to see the physical meaning of the space-time parameter s,
first note that the parentheses enclose the square of the distance traveled
by the body. But distance is just velocity v times the time interval dT: .
So  becomes: .
Moreover, if the body travels through a
gravitational field having potential f (;
note the minus sign, following the astronomical convention wherein
potentials are negative) at a distance r from a source mass M, then s is a
“curved” space-time path length along a geodesic path (a free-fall path
through a gravitational field), and our preceding formula generalizes to
what is called the “Schwarzschild metric”:
Finally, divide each term by c2, which
converts the length-like interval ds into a time-like interval that we can
readily identify as the elapsed proper time
for the body, dt, as defined in the
theory of relativity:
In this form, we can see the space-time
interval ds as a purely time-like interval dt
that was merely made to look length-like through multiplying it by c. This
is what we mean by saying that curved space-time does not involve a
curvature of space. The only effects in the relation between coordinate time
and “space-time” are the clock-slowing effects of velocity and gravitational
Equations ,  can also be used in a
different way by setting the elapsed coordinate time dT = 0 to give what GR
calls a “space-like interval”. But we must first cancel the explicit dT in
 with the implicit dT in the denominator of ,
where du is the Euclidean distance traveled in time dT. We can simplify what
is left by introducing the contraction parameter .
Then  reduces to: ,
where t is proper time. This is an expression for the velocity of a lightwave
through a potential field, and shows that lightwaves slow when passing
through a stronger gravitational potential. It still does not involve space
Because the point about spacetime not including space is of some importance, we
will illustrate it physically as well. Consider the geodesic (orbital) path of
the Earth with respect to the Sun in Figure 1. If
we choose any two points along that path (call them A and B), note that a
straight line between A and B (as could be represented by a taut rope) is a
shorter path through space than the geodesic path. Precisely the same remarks
would be true if the Earth were replaced by a photon whose path is bent with
respect to space as it passes the Sun – a taut rope takes a shorter path through
space than the photon does. The extra bending is most easily explained as a
refraction effect in the space-time or light-carrying medium. [,]
This again illustrates that “curved space-time” geodesic paths do not involve
any curvature of space.
The contrary viewpoint in many textbooks has been a source of confusion for
physics students for the last generation. For an extreme expression of this
contrary viewpoint in support of the geometric interpretation of GR, see
opinions by Robert Wald. []
Yet the fact that space does not curve just because "space-time" does would come
as a surprise to many students of the history of relativity, who tend to think
as Riemann did. However, even in Misner, Thorne & Wheeler's text that
popularized the geometric interpretation of GR, []
we find the following caution: "But if there was one reason more than any other
why [Riemann] failed to make the decisive connection between gravitation and
curvature, it was this, that he thought of space and the curvature of space, not
of spacetime and the curvature of spacetime."
This is an important concept. If the curved path
of a body through space is not caused by a curvature of space, then space
remains Euclidean (flat) and an external force is still required to produce and
explain any deviation from straight line motion. Moreover, some explanation
other than curved space is needed to understand the equivalence-principle-like
property of gravity.
Fortunately, another explanation of
the equivalence principle and of gravitation itself, consistent with general
relativity, is available. It is based on the Le Sage model, in which space is
filled with a flux of extremely tiny, extremely fast particles called
The apple falls from the tree because it is struck by more gravitons from above
than from below because Earth blocks some gravitons from getting through from
below. And any two bodies in space shadow one another from some graviton
impacts, resulting in a net push toward one another. The special GR effects
(light-bending, gravitational redshift, radar time delay, pericenter advance)
are provided by an optical, light-carrying medium called “elysium” through the
phenomenon of refraction because gravity makes the medium denser near masses.
To understand why gravity appears
to obey an equivalence principle, we first need to understand why other forces
of nature do not. Visualize what happens to a body composed of innumerable atoms
when we push it. Obviously, the push makes direct contact only with a relatively
small number of atoms. Those contacted atoms are set in motion by the push. But
before they travel very far, they collide with other atoms and pass along some
of their momentum. Those atoms in turn collide with other atoms, and so on,
until all atoms comprising the body are set into motion. This transfer of
momentum from atom to atom occurs so rapidly that it appears to be instantaneous
to our senses. But of course, the pressure wave resulting from the original push
travels through the body at the speed of sound for that body, always less than
the speed of light; and the far side of the body does not begin to move until
the pressure wave arrives there. For example, the speed of sound in iron or soft
steel is about 5000 m/s.
force is applied to the original points of contact, this force transfers
momentum that must ultimately be shared equally by all the atoms of the body.
The more atoms present, the more sharing, with correspondingly less momentum for
each atom. Because the mass of the body is the sum of the masses of all its
atoms, we can now see in an intuitive way why the resistance to new motion
(acceleration) of the body is inversely proportional to its own mass. The more
mass, the greater the division of any momentum applied to the body among its
atoms, leaving less momentum for each atom.
The essence of “inertia” is the
resistance of a body to change from a state of rest or steady linear motion. The
dilution of momentum just described is why bodies appear to have inertia in
proportion to their own masses. In contrast to Mach’s famous conjecture that
inertia originates in the distant mass of the universe, we see here that inertia
is produced entirely within the affected body and is caused by the dilution of
momentum among more constituents of a body than are directly affected by the
Gravity has no such dilution. The
obvious explanation for this characteristic is that Le Sage-type momentum
carriers of gravitational force are so small that they easily reach every part
of the interior of the affected body, yet move so fast that they still carry
appreciable momentum despite their small size. We call this the “transparency
principle”, wherein every constituent of a body is equally accessible to a
force. Although gravitons are theoretical, the concept of transparency is not.
Neutrinos are an example of entities that usually fly easily through
planet-sized masses without noticing, but occasionally are absorbed.
When the transparency principle
operates, a force is applied equally to every constituent of the body. There is
therefore no need for constituents to carry a pressure wave to their neighbors
because all constituents are affected equally. Under those circumstances, it
does not matter how many constituents are present. There is no dilution of
momentum, so gravitational acceleration is the same for bodies of any mass.
And that is a sufficient reason
for gravitation to operate as if the equivalence principle were in effect.
Bodies of all masses fall at the same rates from the Tower of Pisa because the
acceleration applied by Earth’s gravity is the same for each constituent, and
does not depend on the number of constituents or on the body’s mass. Inertia
(the amount of resistance to a change of motion for a target body) is a
characteristic of the particular force being applied, and not something
intrinsic to the body that would affect its response to all external forces.
answer our title question in the negative. In gravitation, any momentum
transferred to a body by an external force suffers no dilution and is applied
undiminished to each body constituent. Each such momentum transfer is an
impulse. A continuum of impulses produces acceleration. And gravitational
acceleration is independent of the mass of the affected body. So gravitation
operates without inertia.
T. Van Flandern & J.P. Vigier (2002), “Experimental Repeal of the Speed
Limit for Gravitational, Electrodynamic, and Quantum Field
Interactions”, Found.Phys. 32(#7): 1031-1068.
C.W. Misner, K.S. Thorne & J.A. Wheeler (1973), Gravitation, W.H.
Freeman & Co., San Francisco, 1095.
D.M. Greenberger & A.W. Overhauser (1980), “The role of gravity in
quantum theory”, Sci.Amer. 242 (May): 66.
T. Van Flandern (2001), “Physics has its principles”, in Gravitation,
Electromagnetism and Cosmology, K. Rudnicki, ed., C. Roy Keys Inc.,
Montreal, 87-101; also (2000), MetaRes.Bull. 9: 1-9.
Sir A. Eddington (1920), Space, Time and Gravitation, Cambridge
Univ. Press (reprinted 1987), Cambridge, 109.
T. Van Flandern (2002), “Gravity”, in Pushing Gravity: New
Perspectives on Le Sage’s Theory of Gravitation, Apeiron, Montreal,
R.M. Wald (1984), General Relativity, U. of Chicago Press,
C.W. Misner, K.S. Thorne & J.A. Wheeler (1973), op.cit., 32-33.
M. Edwards, ed. (2002), Pushing Gravity: New Perspectives on Le
Sage's Theory of Gravitation, Apeiron Press, Montreal, 93-122.