Gravitation is described as a uniquely geometric phenomenon, incompatible with the concept of force, and only analogically comparable with force by means of mathematical formalisms. Two thought experiments are employed to demonstrate that the association of gravitation with force is irreconcilable with the geometric interpretation, and without theoretical foundation or empirical support. Motion in time is identified as the dynamic source of what has been attributed as the energetic component of gravitational phenomena.
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GRAVITATION, FORCE, AND TIME
JAMES ARNOLD
University of California, Santa Cruz, USA
Abstract:
Gravitation is described as a uniquely geometric phenomenon, incompatible with the concept of force, and only analogically
comparable with force by means of mathematical formalisms. Two thought experiments are employed to demonstrate that
the association of gravitation with force is irreconcilable with the geometric interpretation, and without theoretical
foundation or empirical support. Motion in time is identified as the dynamic source of what has been attributed as the
energetic component of gravitational phenomena.
Key Words: Gravitation, Force, Time, General Relativity
Introduction:
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Conceptualization of the general theory:
Two principal mathematical analogies can be identified in the early development of
relativistic gravitation theory and implicated in its diversion. One derives from Einstein's
heuristic insight associating gravitation with geometry, apparently due to an idea suggested
by his friend Paul Ehrenfest (1909), who was himself inspired by Max Born's investigation
of relativistic rigidity (1909). Ehrenfest noted that the ratio of circumference to diameter of a
rotating disk would have to deviate from pi with relativistic accelerations at the radius. In
Einstein's subsequent pursuit of a generalization of relativity the similarity between the
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inertial effect produced at the radius of the rotating disk and the gravitational pressure we
experience at the earth's surface suggested that gravitation might be explicable as a
fundamentally geometric principle. Experimentation has confirmed the validity of that
seminal geometric insight and the service of the mathematical analogy. But in the kin ematical
similarity between objects on a rotating disk and in gravitational orbit there is also a distinct
physical difference: A test body in a box that is fixed at the edge of a rotating disk presses
against the radial wall of the box, manifesting a centrifugal "force", derivative of the actual
force that is rotating the disk; in contrast, a test body in a box orbiting an astronomical body
floats freely, following its geodesic in spacetime in parallel with the box, and gives no
indication of the presence of a force or acceleration. There is thus a mathematical analogy
due to the similar kinetics of the rotating disk and the orbiting body, but not a physical
equivalence.
The development of the field equations of General Relativity was based on another
mathematical analogy, formalizing the behavior of bodies being accelerated or pressured
toward an attractive or determinant vortex as in a field of force, and a collapsing,
concentrating sphere. The analogy holds in this case because gravity, like a field of force,
produces a typically concentric form to the motion of affected bodies. But again, the
mathematical analogy is not a physical equivalence. A neutral test body inside a charged box
that is accelerating toward the vortex of a field of force presses against the wall of the box
opposite the direction of force, and a charged body of different mass than the box accelerates
at a different rate than the box, moving consequently toward one wall or its opposite. In
contrast, a test body in a box falling or spiraling in a gravitational field floats freely,
following its geodesic in spacetime in parallel with the box, and gives no indication of the
presence of a force or acceleration.1
In both cases -- in the similarities between the rotating disk or orbiting body and between
the attractive or determinant field -- there is a discernible difference in the physical behavior
of test bodies being acted upon by a force and those moving in a gravitational field. In these
pivotal models grounding relativistic gravitation theory, the mathematical analogies between
gravitation and force are limited to descriptions of idealized curvilinear trajectories of
idealized, dimensionless particles.
Physics and Mathematics:
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The special and general theories of relativity were conceptual in origin and mathematical
only in their corroboration and utilization. The general theory has represented gravitation as a
product of the "curvature" or deformation of spacetime in the vicinity of mass, and both the
evidence and the supportive mathematics have been entirely adequate to justify its
acceptance. But the field equations of general relativity are indifferent to the dynamic basis of
gravitation, and geometry is distinctly non-dynamic. Theorists who have sought to associate
gravitation with force have consequently been compelled to develop non-geometric
extensions of the field equations, usually based on electromagnetic analogy. Gravitation has
been described in terms of the mathematics of quantum theory as a force and associated with
a hypothetical particle, without either an explanation of the relationship between geometry
and force or an explicit dissension from the geometric interpretation, and without empirical
evidence of a particle. In terms of the stated and accepted principles of science, this
represents a radical theoretical discontinuity.
Conceptual physics -- which can be considered roughly coextensive with pre-quantum
physics -- involved the initial development of coherent hypotheses, then secondarily the
employment of mathematics (and/or experiments) to support their plausibility. A
mathematical formalism without conceptual coherence would have been regarded as
irremediably provisional, if not unsatisfactory, in the former methodology. With respect to
the former physics, two thought-experiments will be employed below, without resort to
mathematics, to demonstrate that the association of gravitation with force is conceptually
flawed and without empirical support.
Two Thought Experiments:
The first experiment would be unnecessary except that the pre-relativistic association of
gravitation with inertia, and of inertia with universal mass, is still maintained on occasion, if
only tacitly, and may be the ultimate basis of the continued misidentification of gravitation
with force. The misidentification may also be a residue of one of our most familiar and
persistent experiences on the earth's surface: The pressure we feel between ourselves and the
surface (weight) is fundamental to our original concept of gravitation; we tend to regard the
pressure as a force ("the force of gravity") and our relatively static surface frame of reference
as being at rest. The following experiment may therefore be helpful in more clearly dispelling
the identification of gravitation with force and inertia, and also in prefacing the second
experiment, which will illustrate the continuity between force-free astronomical gravitation
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and gravitation at the surface of a massive body.
Imagine a spacecraft coasting on a uniform path relative to the "fixed stars"
which comes under the influence of a stellar object nearby and begins to deviate
toward it, while continuing in uniform motion by the evidence of free-floating
objects inside. In order to maintain the original course a thruster is fired, and inertial
effects are experienced onboard as the craft accelerates just enough to counter the
influence of the local gravitational field in order to maintain the intended course.
In this experiment inertial effects are associated not with gravitation, but with the
counteraction of a gravitational acceleration, and with supposedly uniform motion relative to
the distant stars, contrary to the pre-relativistic expectation. Aside from the discrimination of
inertia from any influence of the overall mass of the universe (an association that is seldom
explicitly defended now anyway), the experiment demonstrates what I hold to be most
significant, that at least in the situation just described, force becomes evident in conjunction
with gravitation only when gravitation is being resisted.
Now consider an experiment that comprehends the transition from astronomical
gravitation to an involvement with force and inertia at the surface of a massive body:
Imagine two test bodies gravitating toward the earth from some considerable
distance. For the sake of simplicity, consider the earth to be at rest with the test
bodies gravitating toward its center of mass. (They appear to be simply "falling"
from a perspective on the earth's surface.) One body is an immense hollow sphere of
negligible mass, the other is relatively small in size -- an extra-vehicular scientist,
let's say -- and also of negligible mass. Notice that while the test bodies are falling
toward the earth (or more accurately, while the three bodies are converging) there is
among them a purely relative transformation of potential energy to kinetic energy as
each moves uniformly in its own frame of reference -- there would be, at least as yet,
no occasion for an exchange of mass-energy in the form of the supposed
gravitational energy.
Let the sphere and the scientist be placed initially close together so that as they
approach the earth their geodesics converge enough to bring their surfaces in contact
some time before the larger impact. (It is the fantastic size of the hollow sphere that
allows the surfaces of the two bodies to meet somewhere above the earth's surface).
From the moment the sphere and the scientist come in contact until they reach the
surface of the earth, a static inertial acceleration between them will intensify as each
tries to conform to its own geodesic at an ever greater angle from the normal. The
situation will, if viewed in isolation, come to resemble the gravitation of a small
body pressing against a planetary surface (although the gravitation between them is
actually insignificant due to their negligible masses) and the scientist will even be
able to stand upon the sphere. This development of an increasing inertial
acceleration between the test bodies is the only aspect of the situation that changes
from the moment they meet; the earthward component of their motion continues as
before, a relative gravitation.
In a manner that is similar to the first experiment, force has developed in the
resistance to what is in this case a convergent gravitation of two bodies toward a
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third. And once the two reach the earth the situation remains essentially the same:
Each of them, now in conjunction with the entire conglomerate of the earth, presses
toward the center of mass with the same sort of conflict of geodesics as was
observed between the two when they were gravitating from a distance. Along with
the other components of the earth at and below the surface, they are resisted, and
thereby induced with a static acceleration by those further below, due to the
coincidence of the common inclination toward the center of mass and all the
subterranean obstructions.
This second experiment demonstrates that it is only in the inertial conflict of geodesics (or
as in the first experiment, in a singular inertial acceleration) that force can be observed in
association with gravitational phenomena. The intersection of geodesics and the consequent
inertial effects constitute the interruption of gravitation, and what is commonly conceived as
"the force of gravity" at a surface can be more accurately described as anti-gravitation.
The Principle of Equivalence:
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Relativity, Absolutes, and Energy:
When gravitation is isolated from circumstances where it is being resisted there is only
geodesic motion, curvilinear or straight depending on the coordinate system. In the relative
accelerations and decelerations of orbital dynamics, and in the perturbations of orbits due to
external gravitational influences, there is no indication of force or gravitational energy, there
is only the appearance of acceleration from other reference frames.2
The original goal of the generalization of relativity was to establish that inertial and
gravitational accelerations, like the special case of uniform motion, are relative. It may be
that there is now a more-or-less unconscious aversion to abandoning that aspiration to grand
simplicity. But from the perspective of a purely empirical and conceptual physics, given a
clear experimental discrimination between gravitation and inertia, a generalization of
relativity to include force and inertial accelerations is manifestly untenable. It bears
repeating: A simple experiment with a test body in a container can confirm that an inertial
acceleration is absolute, whereas an unobstructed gravitation is not.
Gravity has to be considered absolute in the aspect that a geometric vortex exists at a
center of a sufficiently large mass that cannot be transformed -- either conceptually or
mathematically -- but unless the geodesic of a body becomes obstructed, as at the surface of a
planetary body, gravitation involves uniform motion with only relative accelerations. No
force or energy can be attributed.
The problematic reliance on mathematics for conceptualization and inference discussed
earlier is nowhere more striking than in the conventional treatment of the problem where the
Field Equations presume gravitational energy but don't allow it to be identified or
mathematically expressed in local circumstances. It isn't questioned, in consequence of the
meta-mathematical approach, whether such an elusive sort of energy actually exists, it is
simply said that it cannot be "localized" (Misner, Thorne & Wheeler 1973). Thus a problem
of non-conformity between the theoretical and physical is considered nothing more than a
mathematical oddity, and thereby rendered satisfactorily unproblematic. Mathematics trumps
physics, and formulas trump observation.
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The Dynamic of Time:
There remains a most significant aspect of the distinction between gravitation and force to
be comprehended, although its full implications must be left outside the scope of this
discussion. The energy expressed in the continuous static acceleration of bodies at and below
a massive surface is rendered inexplicable in purely geometric terms when gravitation is
finally distinguished from force. If gravitation is a deformation of spacetime due to the
influence of mass, if there is no "force of gravity", what accounts for the persistent energy
pressing against a massive surface after a body has come to a relative state of rest? Recall that
in the initial appearance of force in the second experiment described above, only a conflict of
geodesics is present and resistant against the otherwise uniform motion of the test bodies. No
extrinsic source of energy can be identified, yet there is a static acceleration between the two,
even while their gravitation with the earth remains force-free.
I believe the answer lies in a curiously under-explored, if not unexplored implication of
Minkowski's (1908) interpretation of special relativity, which described space and time as a
four-dimensional continuum. His graphic representation of relativistic effects (the Minkowski
diagram) as expressed by the Lorentz transformations shows uniform motion to be motion in
time, perpendicular to space (while of course remaining in space), and relative motion to be
less in time the more rapid it is in space. It follows from this evident covariance of the spatial
and the temporal that if time is a form of motion which is normally unapparent as-such in our
world of experience, where bodies move in time with infinitesimal deviations from the
parallel, then time must be dynamic, and possessing an incessant energy, imponderable
except when a body is persistently resisted, as at a gravitational surface.4
Motion in time, the motion of matter in general, must be regarded in this view as
absolute, although relative in the incidental spacetime orientations and velocities between
individual bodies. The source of the energy usually identified as gravitational energy can thus
be attributed to an intrinsic and ceaseless dynamic of mass-energy moving in time,
independent of gravitation, and obscured by the conflation of gravitation and inertial
acceleration in circumstances when they happen to coincide (as at a gravitational surface) but
revealed by a clear recognition of their fundamental distinction.
Conclusion:
Having briefly acknowledged the implications of a consistent geometric theory of
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gravitation, that gravitation and motion in general are each in their own way both relative and
absolute, and that time is intrinsically dynamic and the source of the energy disclosed by the
opposition to gravitation in its occasional resistance, I will consolidate the findings with
regard to quantum theory and other force-based theories in the following summation:
By all evidence, gravitation is a deformation of spacetime due to presence of mass, its
effect being a geometric concentration of spacetime toward centers of mass. Bodies moving
under the influence of gravitation move uniformly in their own reference frame unless
obstructed by a body massive enough to form a spacetime vortex, when their incessant
motion in time causes them to continue to press toward the surface. Being a strictly geometric
phenomenon, gravitation cannot be a force, it cannot therefore be mediated by a particle, and
cannot radiate as mass-energy. The assimilation of gravitation by quantum theory and its
derivatives as a field of force, and the positing of a gravitational quantum of action where
none is apparent, theoretically necessary, or conceptually coherent, is entirely without
justification. 3
This is an admittedly unsettling proposition, but in consolation, its acceptance makes one
of the principle objectives of quantum theory much less complicated, as gravitation with all
its peculiarities can be disregarded in the pursuit of a unified field theory. And the concept of
time as being spatially dynamic, and a primary determinant in gravitation theory, suggests an
intriguing new area for investigation. I hope it might also signal the need to rely more upon
conceptualization, and not so heavily on mathematical formalisms, in the development of
physical hypotheses.
End Notes:
1 There may be an appearance of force if the gradient of a gravitational field is extreme
enough relative to a body's extension in the direction of the field to produce tidal stresses on
the body's molecular binding energies. (The earth's ocean tides are a dramatic instance.) But
this too is entirely geometric in its origin, and only manifests local variations in the intensity
of the distortion of spacetime. A tidal effect can be identified when a free-floating liquid test
body manifests a distinctive elongation along the axis of gravitational influence.
2 The most prominent case of hypothetical gravitational energy and its radiation is the
inspiraling binary star system, where there is evidently a loss of net relative (kinetic/potential)
energy between the companions due to their deteriorating orbital dynamics. In terms of
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gravitation as a geometric principle, the idea of a transformation of relative accelerations to
force-like radiation is incongruous; the extrinsic energy corresponding to the decrease within
the binary system should be interpreted instead as a purely relative increase of
(kinetic/potential) energy between a binary system and the rest of the universe.
3 Like energy-bearing gravitational waves, other hypotheticals -- gravitomagnetism, dark
matter, and dark energy -- can be expected to continue eluding detection, as all are based on
the presumed association of gravitation with force.
4 For a fuller description of this interpretation of time as having a kinetic aspect, see Arnold
(2013).
References:
Arnold J, "An Advancing Time Hypothesis", European Scientific Journal 9 , 12, 2013.
Born M, Die Theorie des starren Elektrons in der Kinematik des Relativitäts-Prinzipes, Ann.
Phys. (Leipzig) 30, 1, 1909.
Dicke R, Gravitation and the Universe, Philadelphia: American Philosophical Society.
Ehrenfest P 1909 Physikalische Zeitschrift, 1970.
Einstein A, "ueber das Relativitatsprinzip und die aus demselben gezogenen Folgerungen",
Jahrb. Radioakt. Elektr. 4:411-462, trans, pp134-5, Relativity and Geometry, Torretti, R.,
New York: Pergamon Press, 1983 (1907).
Minkowski H , "Space and Time" in The Principle of Relativity, H.A. Lorentz, A. Einstein,
H. Minkowski, and H. Weyl, trans: W. Perrett and G.B. Jeffery, 1923 (1908).
Misner C, Thorne K, and Wheeler J, section 20.4, Gravitation, W. H. Freeman, 1973.
Taylor J, Fowler L, & McCulloch P, Nature, 277, 437-40, 1979.
... The gravity as the deformation of space-time is generally accepted, but the model seems to have open questions. (Arnold, 2013) Curved Space-Time Figure 4. The well-known and common picture of Einstein's curved space-time caused by mass or (a greater amount of) energy. ...
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![Tamas Lajtner]()
Thoughts and gravity have common roots. Gravity changes your thoughts, and your thoughts change gravity. How? According to current, widespread understanding, measurable thoughts are electromagnetic signals of the brain. We made a very simple experiment with force of thought using a paper wheel. We concluded that the energy carried by thoughts (expressed in frequency of energy wave) was eight orders of magnitude beyond the highest frequencies of the brain's electric waves. The brain's electromagnetic signal doesn't explain all effects of thought, it is just a part of measurable thought. Thought is a gravity-like force. According to the General theory of Relativity, gravity is the deformation of space-time. With this definition, however, we can only partially account for the peculiarities of the force of thought. For a complete understanding, we must redefine the concept of gravity; and for this, we must broaden our concept of the "space-time" conceptual system. This broader version can be described as "space-matter" model. Gravity can be regarded as changes in the frequencies of space waves. Thought manifests itself as a gravity-like force, as a new fundamental force. This new force can be given as the changes in the frequencies of space waves, too. Using thought force is possible in our daily practice, too. We can build devices and create methods that are run by thought force.
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![James Arnold]()
The evolution of the universe is described as an advancement of time, and only collaterally an expansion of space. An interpretation of time as proceeding at the equivalent of c across space, perpendicular to space, per a reconsideration of Minkowski's spacetime geometry, supports a description of the cosmos as a four-dimensional (hyper)spherical wavefront. By treating space as the surface of a four-dimensional sphere with a current radius of 13.82 billion years (equivalent to 13.82 billion light-years), a Hubble constant of 70.6 (km/s)/Mpc is derived from the measure of the expansion of a megaparsec arc on the surface, independent of empirical measurement or mathematical inversion. It is argued that a close correlation between the advancing temporal cosmic radius and the expansion of the arc subtending a Mpc suggests at least a remarkable coincidence, worthy of further investigation. The hypothesis also has the scientific virtue of economy of explanation, dispensing with the need for the (revived) cosmological constant, cosmic inflation, dark energy, and dark matter as a gravitational constraint on expansion, as well as questions of the shape of the universe and of the influence of gravitation on the rate of expansion. A reexamination of various cosmological parameters in terms of an advancing time hypothesis is expected to provide further confirmation and confer greater simplicity and general coherence to cosmology.
- M Born
- Die
Born M, Die Theorie des starren Elektrons in der Kinematik des Relativitäts-Prinzipes, Ann. Phys. (Leipzig) 30, 1, 1909.
- P Ehrenfest
Ehrenfest P 1909 Physikalische Zeitschrift, 1970.
Space and Time" in The Principle of
- H Minkowski
Minkowski H, "Space and Time" in The Principle of Relativity, H.A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, trans: W. Perrett and G.B. Jeffery, 1923 (1908).
- J Taylor
- L Fowler
- Mcculloch
Taylor J, Fowler L, & McCulloch P, Nature, 277, 437-40, 1979.
Born M, Die Theorie des starren Elektrons in der Kinematik des Relativitäts-Prinzipes
- J Arnold
Arnold J, "An Advancing Time Hypothesis", European Scientific Journal 9, 12, 2013. Born M, Die Theorie des starren Elektrons in der Kinematik des Relativitäts-Prinzipes, Ann. Phys. (Leipzig) 30, 1, 1909.
- C Misner
- K Thorne
Misner C, Thorne K, and Wheeler J, section 20.4, Gravitation, W. H. Freeman, 1973. Taylor J, Fowler L, & McCulloch P, Nature, 277, 437-40, 1979.
- R Dicke
Dicke R, Gravitation and the Universe, Philadelphia: American Philosophical Society. Ehrenfest P 1909 Physikalische Zeitschrift, 1970.
- A Einstein
- Relativity
- Geometry
- R Torretti
Einstein A, "ueber das Relativitatsprinzip und die aus demselben gezogenen Folgerungen", Jahrb. Radioakt. Elektr. 4:411-462, trans, pp134-5, Relativity and Geometry, Torretti, R., New York: Pergamon Press, 1983 (1907).
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