Mystery Keepers
Geometric Continuity: Is the space that surrounds matter an extension of the matter?

J. "River" Kaster
Great Lakes WATER Institute
University of Wisconsin-Milwaukee
Milwaukee, Wisconsin 53208


The general relativity stress-energy tensor, Tuv is replaced with a relativistic stress-frequency tensor,
Tvuv, based on the assertion that relativistic frequency, hv = mc  /√(1 - v  /c  ), is basal to all other Lorentz transformations. This changes one of the main conceptual tenets of general relativity, that the curvature metric is determined by matter and energy, to that of frequency geometry being causal of space curvature. Is it possible that Einstein had the right mathematics emanating from the wrong concept? The relationship of frequency-curvature directly argues a geometric continuum of all matter-energy with space as opposed to Einstein’s concept of discrete matter-energy and space. Matter/space interplay can be thought of in a novel context that prescribes the idea of continuity: the space that surrounds matter is a continuity extension of matter-energy, demarcated by a sharp continuity transition.


general relativity, special relativity, gravity, gravitation, space curvature


The idea of ‘connectedness’ by a gravitational force particle that mediates the force of gravity has different implications than the idea of a ‘continuum’ by geometric continuity. The latter suggests a geometric force of gravity directly causal by a quantum wave distortion, ᴪd, of the homogeneity of space. Within the geometric framework of continuity, mediating force particles (e.g., the hypothetical graviton) are unnecessary and furthermore, their “connectedness” quality between particle bodies has resulted in a conundrum mired in conceptually erroneous ideas – spooky ideas. Albert Einstein’s ‘spooky connection at a distance’ does not derive from attributes within quantum mechanics.  Rather, it has precipitated from Einstein’s own conviction that matter and space are discrete, though influentially connected.  Einstein’s ‘spookiness’ originates from this conceptual shortfall that has been perpetuated for 100 years.

The conceptualization of Einstein’s suggestion that matter curves space, thereby delineating matter and space as separate, discrete entities is at the heart of this conundrum. Misner, et al. [1, p. 480] gave a very interesting variation of this conundrum: “If geometry tells matter how to move, and matter tells geometry how to curve, does one not have in one’s hands a Gordian knot? And how then can one ever untie it?” The concept of ‘continuum geometry’ cleanly unties the knot: space curvature is a physical continuity extension of matter, not a feature detached from matter.

Matter and space interplay can be thought of in a context that prescribes the idea of continuity: the space that surrounds matter is a continuity extension of matter, demarcated by a sharp continuity transition. This idea is disparate from Einstein’s conceptual interpretation of general relativity, but not from his field equations. Is it possible to have the correct mathematics based in general relativity, seed from a misdirected conceptual origin, i.e. the right math emanating from the wrong concept? This paper suggests that geometric continuity between matter and space is a conceptual idea that not only fulfills the mathematical requirements of general relativity, but can serve to obtusely redirect theoretical and empirical attention.

Relativistic Planck Quantum

Continuity is bolstered by special relativity given that frequency (sensu Planck hv) is basal to all other Lorentz transformations because these relationships can be derived from relativistic hv.  The relativistic relations, mass = mo/ √(1 - v  /c  ), length = lo√(1 - v  /c  ), time = to/ √(1 - v  /c  ), frequency = fo √(1 - v  /c  ), and so forth, can each be derived from the single generalized relativistic expression,
hv = mc  /√(1 - v  /c  ). Rather than using ‘matter’ or ‘energy’ to describe the “stuff” of the Universe, including gravitational space curvature, it is proposed that geometric continuity described by relativistic
hv is a basal measure of matter’s mass and energy. Geometric continuity also suggests that the Planck relationship, including its transitive form, m = hv /c  , is universal, relativistic, and parsimonious in this application.

Geometric continuity applies Planck’s relationship, hv, to Einstein’s field equation solution,

Guv = 8πTuv

which digests gravitational curvature (left side) as metric distortions of matter and energy (right side) and 8π being the correspondence factor between Newtonian and Einsteinian theories. When frequency ( relativistic hv) replaces matter and energy on the right side, then:

Guv = 8πTvuv

where Tv = stress-frequency tensor

This changes one of the main tenets of general relativity that the curvature metric is determined by matter and energy. If the physical form of the stress frequency metric (a basal aspect of mass and energy) originates from the homogeneity of space, i.e. a de novus quantum wave distortion of the homogeneity, then the geometric continuity relationship of frequency-wave distortion directly argues a continuum of all matter, energy, and the homogeneity of space.  The premise of this argument is that the de novus quantum wave is a distortion derivation of the homogeneity of space and thus continuous with the homogeneity.

The general background condition of the pre-Universe (denotes no energy, matter, or other variable, including time) is interpreted as a homogeneity (Ho). Ho is an isotropic pure linear and non-dispersive condition with no dimensional boundary restrictions and is an intact continuum of nothingness. Ho is non-descriptive with no curved spacetime; no curved space nor time (both being relative states that violate Ho). Asymptotically flat spacetime or  Minkowski space are not Ho. Flat spacetime holds energy and dimensionality, while Ho ‘holds’ neither. Ho does not equal G=0, (i.e. vanishing divergence, R=0). Ho is vacant of even electromagnetic energy and in this sense Ho is absolute nothingness from which everything, as we know it, stems from. 

The Ho can only support a frequency form composed of a disturbance of the Ho itself, i.e. a wave distortion, ᴪd, of the Ho is of the same essence as the Ho. Since Ho is the totality of a boundless empty set from which everything originates, any disturbance is a distortion of Ho; ᴪd is a ‘stable’ distortion. Here, this distortion is called a ‘wave distortion’ because our best model is electromagnetism that manifests as a wave (other names might include “string”, “wave-particle”, or quantum particle).  A wave distortion, ᴪd, for Ho has v = c0. Since ᴪd originates from Ho, there is continuity between the two, i.e., ᴪd is the Ho but in a dimensionally defined wave form (electromagnetics, e.g., photon).  If Ho has continuity with its own ᴪd (which it must because v = c0, and c0 derives from Ho, see ¶ below), then any energy or matter built from the ᴪd must also have a continuity with Ho and thus continuous with all other possibilities of energy and matter manifested within the Universe. From this, the Universe can be defined as a series of continuous distortions and curvatures within the Ho and demarcated at its outer edge by a boundary layer that expands deeper into Ho. Distortion of Ho is severe around ᴪd and is a less distorted curvature around composite matter (cf. general relativity). Severe, quantum distortion is continuous with the curved distortion at the surface of composite matter since composite matter is built from ᴪd. ᴪd is presumed to have a highly convoluted effect on local Ho. The convoluted effect is manifest as a sharp continuity transition at the quantum level homologous to curved space being the continuity expression around composite matter.

Ho must conserve its wave form. This suggests why c0 is constant and obeys the laws of conservation. ᴪd velocity is maximum at c0, and thus constant when moving through its own Ho, and is conserved because energy is not dissipated, e.g., light propagates across the Universe. A wave moving through its own composition travels at a maximum and constant speed. If one could remove all the extraneous makeup of a pure water wave including its boundary conditions, then it would have a non-violate constant and maximum speed. Perhaps a mid-ocean tsunami approaches its v0. Any ᴪd traveling through a Ho that differs from the wave itself, travels at less than maximum velocity, v < c0, and dissipates energy. This feature defines our Universe in that any different universe would have v0 ≠ c0.

Evolution’s Geometric Illusion

Biological evolution’s most illustrious illusion is differential perception of matter and space. This sentient perception is facilitated by the sharpness of the continuity transition between matter and curved space. General relativity describes this continuity transition. This is the hallmark of Einstein’s beautiful mathematics. Composite matter, e.g., you or a rock, is built of ᴪd. Einstein described concentrations of energy within a field as “bunched fields”. These conceptualized “bunched fields” are demarcated by a continuity transition producing an illusionary boundary that is biologically interpreted as “surface”. The continuity transition provides boundary coherence for ᴪd that could bestow a particle-like property to ᴪd. The quantum unit behaves like a discrete unit because of the steep continuity transition but is continuous from ᴪd to Ho. The space curvature of ‘you’ is continuous with the space curvature of a ‘rock’. This is the continuum of our Universe.

One would not declare that evolutionary genetic continuity does not exist because the recognizable particles that sustain the continuity, genes, appear as discrete units. What unleashed genetic evolutionary theory, was the arrival of genetic continuity.  In the same vein, general relativity is conceptually constrained by Einstein’s idea that matter-energy is discrete from space, a treatment that makes a similar naivety as pre-Mendalian genetics.

Einstein remarked that energy and matter’s mass are the “same thing”[2]. He did not say what that “same thing” was; however, relativistic hv would indicate that Einstein’s “same thing” is basal ᴪd, defined by vibration frequency. The internal motion of wave-particles within large objects is responsible for producing an objects aggregate mass and forms the basis of the universality of Einstein’s E =mc   and Planck’s E = hv. If Einstein was correct that “Physical objects are not in space, but these objects are spatially extended”, then the generalized hv relativistic expression must apply directly to describing physical reality as distortions of Ho. Einstein thought matter curved space and “The physical world is represented as a four-dimensional continuum.” [3] but he did not link it to the possibility of a geometric matter-space continuum.  A particle is no more than a snapshot of a flux process.

Gravitation as continuity, especially noted at continuity transitions, precludes a particle based force as causal. Rather gravitation is nature’s geometric force of curvature underpinned with continuity (“…Nature makes the geometry.”; “The coordinates are the work of man.”)[1]. Matter’s mass is a frequency measure of ᴪd and the de novo synthesis of composite matter that ultimately is a perceptual illusion of evolution. For sentient perception, the ‘particle’ does not exist except as an evolutionary adaptation to enhance ambient reception.

1. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman, New York (1970)
2. Einstein, A.: Relativity. Bonanza Books, New York (1961)
3. Einstein, A.: Essays in Science. Philosophical Library, New York (1934)
4. Heifetz, M., Bencze, W., Holmes, T., Silbergleit, A., Solomonik, V.: The Gravity Probe B Data Analysis Filtering Approach, pp. 57-66. In: Gravity Probe B Science Results—NASA Final Report, December  (2008). Accessed 25 July 2010