Many-body mass-centre for finite radiation-speed Homological Algebra approach to relativity theory of position-space and simultaneity

Many-body mass-centrefor finite radiation-speed

Homological Algebra approach to relativity theory

of position-space and simultaneity

Zbigniew OziewiczUniversidad Nacional Autonoma de Mexico

Facultad de Estudios Superiores

C.P. 54714 Cuautitlan Izcalli, Estado de Mexico

[email protected], [email protected]

December 7, 2013March 31, 2015 - improved notation

Presented at 6th European Congress of MathematicsSatellite International Conference 3Quantum:

Algebra Geometry InformationTallinn University of Technology, July 2012

Short version of 15 pages published in Journal of Physics:Conference Series 532 2014, 012021, pages 1-15.

doi:10.1088/1742-6596/532/1/012021

Abstract

The relativity of velocity is not completed without reduced mo-mentum relative to many-body mass-center. I present the intrinsiccenter-of-inertia of many-body interacting (bound) system for the caseof finite radiation-speed. The concept of a center-of-inertia is not pos-sible within theory of relativity postulating that each pair of reference

1

system must be related by Lorentz isometry-group transformation. Iam showing that center-of-mass is well defined concept within group-free homological algebra approach postulating energy-momentum con-servation.

Nicolaus Copernicus in 1543, Galileo Galilei in 1632, and GeorgeBerkeley in 1710, were aware of relativity of velocity and this impliesrelativity of position-space. During centuries the relativity of position-space was ignored. Each textbook on relativity has a Section Therelativity of simultaneity, however ‘The relativity of position-space’ isabsent. Does not exists in Nature universal absolute three-dimensionalposition-space. Material bodies with non-zero relative velocity possessown separate position-spaces. There are many position-spaces. Thisimplies that all motions, velocities, accelerations and rotations, mustbe relative, must be motions of one position-space relative to anotherposition-space. Must be motions of one material body relative toanother material body.

A position-space I see as the Grassmann factor-algebra of differen-tial forms where time-like material body with a positive mass is inter-preted as idempotent algebra epimorphism of the Grassmann algebraof a spacetime onto the Grassmann factor -algebra of correspondingposition-space of that material body. A material body as a referencesystem is a group-free split, and this allows us to express every mo-tion, velocities, accelerations and rotations, as relative with respect tothe choice of variable reference system.

Keywords: relative velocity, split-velocity, many-body time-like center-of-mass, many-body total mass (internal energy), reduced momentum, reducedmass

2000 Mathematics Subject Classification. 15A75 Grassmann algebra,18Gxx Homological algebra, 51B20 Minkowski geometries, 53A17 Kinema-tics, 53A35 Non-Euclidean geometry, 53B30 Lorentz metric, 83A05 Specialrelativity

Physics and Astronomy Classification Scheme (PACS) 2010. 03.30.+pSpecial relativity, 97.80.-d Binary stars

2

Zbigniew Oziewicz: Many-body center-of-mass for finite radiation-speed 3

Contents

1 Spacetime come to light because of absence of position-spacein the Nature 41.1 Relativity of position-space . . . . . . . . . . . . . . . . . . . . 6

2 Universal property of the differentialand other notations 9

3 From Keres’s position-spaceto algebra derivation 12

4 Leonhard Euler in 1757 15

5 Bad interpretation of the Riemannian tensor as a ‘metric’ 16

6 Group-free relativity postulate 17

7 Split-velocity for finite radiation-speed 18

8 Many-body centre-of-inertia 228.1 Maximal binding energy . . . . . . . . . . . . . . . . . . . . . 27

9 Reduced space-like momentumand reduced mass for finite radiation-speed 279.1 Two-body system . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 Three-body system modulo radiation . . . . . . . . . . . . . . 29

10 Universe as a bunch of Grassmann factor-algebras of relativeposition-spaces 3010.1 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11 Relativity theory versus gravity theory 35

12 Reference system as an iso-position process/monadwithin manifold-free and group-freehomological algebra approach 36


Wir mussen uberzeugt sein, daß das Wahre die Natur hat,durchzudringen, wenn seine Zeit gekommen, und daß es nur er-scheint, wenn diese gekommen, und deswegen nie zu fruh er-scheint noch ein unreifes Publikum findet; auch daß das Indi-viduum dieses Effekts bedarf, um das, was noch seine einsameSache ist, . . .

We must hold to the conviction that it is the nature of truthto prevail when its time has come, and that it appears only whenthis time has come, and therefore never appears prematurely, norfinds a public not ripe to receive it; also we must accept that theindividual needs that this should be so in order to verify what is asyet a matter for himself alone, and to experience the conviction,which in the first place belongs only to a particular individual, assomething universally held.

Musimy bowiem byc przeswiadczeni, iz w naturze prawdylezy to, ze toruje sobie droge wtedy, kiedy nadszed l jej czas, i zepojawia sie tylko wtedy, kiedy czas ow nadszed l, i dlatego nigdyani nie pojawia sie za wczesnie, ani nie staje wobec publicznoscinie dojrza lej; a takze i o tym, ze jednostce potrzebny jest tenefekt [dieser Effect] po to, by mog la utwierdzic sie w tym, co narazie jest jej przekonaniem samotnym, . . .

Georg Wilhelm Friedrich HEGEL (1770-1831)Phanomenologie des Geistes 1807

1 Spacetime come to light because of absence

of position-space in the Nature

Nicolaus Copernicus (1473–1543), Galileo Galilei (1564–1642) in [1632Second Day], and Irish Bishop George Berkeley (1685–1753) in 1710, ob-served and explained that the relativity of velocity implies immediately therelativity of position-space: are we changing a position or we stay in thesame position-location? - this is relative! Does not exists a concept of aposition in the Nature without an un-real choice of same material body atrest. A position (in some three-dimensional position-space/manifold) is a


mathematical convention and not a concept of physics.The relativity of velocity was fundamental observation by Copernicus in

1543:

For every apparent change in place occurs on account of themovement either of the thing seen or of the spectator, or on ac-count of the necessarily unequal movement of both. For no move-ment is perceptible relatively to things moved equally in the samedirections – I mean relatively to the thing seen and the spectator.

Nicolaus Copernicus 1543 Book One §5

For the fact that the wandering stars [planets] are seen tobe sometimes nearer the Earth and at other times farther awaynecessarily argues that the centre of the Earth is not the centre oftheir circles. It is not yet clear whether the Earth draws near tothem and moves away or they draw near to the Earth and moveaway. Nicolaus Copernicus 1543 Book One §5

If location is relative then motion . . . is meaningful only rela-tive to any other body.

Irish bishop George Berkeley (1685-1753) in 1710

There is no absolute [position]-space, and we only conceive ofrelative motion; and yet in most cases mechanical facts are enun-ciated as if there is an absolute [position]-space to which they canbe referred.

Henri Poincare (1854-1912), Science and HypothesisChapter 6: Classical Mechanics 1902

1.1 Postulate (Relativity of velocity). Each material body, say B, possesesinfinite many velocities with respect to many different material reference sys-tems, viz vRB denotes the velocity of B relative to variable reference body R;including always zero velocity relative to own reference system, 0B ≡ vBB,- that is the fundamental [Copernicus] axiom/postulate of the present pa-per. Notation for the relative velocity, vRB, means that the body B isowner of this relative velocity, however the reference body R is owner ofthe position-space where body B is moving; and R in own position-space isat rest, 0R ≡ vRR 6= 0B. Therefore the [involutive inverse] velocity of the Rrelative to the reference body B, viz vBR, does not possesses skew-symmetric


[reciprocal] meaning [for the finite radiation-speed], because here the ownerof the another position-space is the reference body B (where exactly B is atrest!), and R is owner of this [inverse] relative velocity, (vRB)−1 ≡ vBR. Pos-tulated here existence of many distinct three-dimensional position-spaces willdemands additional axioms of non-Euclidean geometry in four-dimensionalspace-time that will be introduced later on. Reader not interested in pos-tulates but in conclusions about necessarily non-reciprocity [reciprocity ofvelocity was crucial silent assumption made by Albert Einstein in 1905] cansee directly the Conclusion 7.6 below.

In reality what counts are events, four-dimensional spacetime manifoldof events: whereas ‘spacetime’ is not appropriate name for a manifold ofevents because do not possess neither position-space nor simultaneity relation(i.e. nor ‘time’). A position-location of an event in some position-space is anirrelevant extra convention [Oziewicz 2010, Oziewicz and Page 2012, 2014].

1.1 Relativity of position-space

In this subsection I will shortly review three publications by Harald Keres(1912-2010) [Keres 1972, 1973, 1976] that attracted 40 years ago my strongattention. In Section 3 I am presenting the Keres’s mathematical formal-ism. I am familiar with Harald Keres’s publications since 40 years. I wasimmediately strongly convinced by the Keres crystal clear concept of relati-vity of position-space, relativity of diachronic, relativity of ‘simul-localidad’(relativity to be in the same position-location). There are many position-spaces, and this holds irrespectively to the finite or infinite radiation-speed.

I like group-free and Lorentz-boost-free the Keres definition of arbitraryvelocity (i.e. not necessarily constant) of Keres’s x-position-space relative toy-position-space [Keres 1972 §2.6 page 68 expression (34)]. Moreover duringall these years I was permanently impressed by Keres’s insistence that everymotion must be relative motion, that the relativity theory must not endwith the relativity of arbitrary velocity, but must include also the relativityof acceleration and the relativity of rotation (the general relativity) [Keres1973]. The Keres group-free approach strongly motivated my understandingof isometry-group-free relativity for finite radiation-speed, i.e. the Lorentz-isometry-group-free relativity theory [Swierk 1988; Oziewicz since 2005].

Keres’s lecture in 1970 in Kiev’s Symposium on Philosophical Questionsof Relativistic Physics and Cosmology [Keres 1973] is of the fundamental


importance. In this lecture Keres states that the concept of inertial referencesystem is a convention, because every reference system can be considered asbeing inertial. Keres stressed that every motion, including acceleration androtation must be considered to be relative, to be dependent on the free choiceof the variable reference system, identified with material body of a positivemass.


There does not exist a universal [unique] absolute three-di-mensional [position]-space common for all phenomena, and there-fore does not exist universal absolute acceleration and rotation.Every motion is relative. Harald Keres 1973 page 142

The above opinion is contrasted with the long standing in history ofphysics opinion by Langevin in 1911 [Langevin 1921] that the accelerationand rotation must have an absolute meaning, reference-system-free, indepen-dent on the choice of the three-dimensional position-space.

Einstein in 1914-1916 defined acceleration as the Christoffel derivativeapplied to one-body system, acceleration =∇XX, is a geodesic vector field,see also [Weyl 1918, 1921 §14; Ehlers 1961-1962]. No reference body is in-volved, so it is an acceleration with respect to what? According to Coper-nicus, Berkeley, and Keres, one must specify another material body, anotherKeres’s x-position-space, with respect to which a relative velocity of one-body y-position-space are to be defined (velocity as a change of a position,but position-location of what reference material body?), and then relativeacceleration is expected to be given by a derivative of a relative velocity fortwo-body system [Keres 1972 §2.8, 1973, 1976].

The Christoffel derivative applied to one-body system was like throwingthe baby out with the bathwater: it made acceleration as defined by Ein-stein, Weyl, and later on by Ehlers, not relative but an absolute concept.Neither second reference material body nor centre-of-mass are involved. Un-fortunately, instead of generalising relativity of velocity (supposed to be doneby ‘special’ relativity in terms of an isometry-group and isometry-boost ge-nerated by Clifford bivectors), to relativity of acceleration, the Christoffelderivative introduced absolute concept.

Material body is identified with a congruence of time-like curves, like the‘Lagrangian description’ of the motion of a fluid material particle, in fact,due to Euler and not to Lagrange. Such congruence is called by Keres to bex-position-space, and for a pair of such congruences, for x-position-space andy-position-space, Keres defined arbitrary (not necessarily constant) relativevelocity between them (for infinite radiation-speed). Keres considered theGalilean relativity of three-dimensional position-space. Galilean relativity isdue to the relative velocity among two material bodies with positive masses.In Keres’s notation x denotes the conserved zero-grade scalar fields [Keres1972, 1976].

The Keres group-free concept of arbitrary relative velocity I am going to


present as Eulerian in terms of (Lie algebra of) derivations of commutativealgebra, F ⊗R F

m−−→ F , of scalar ‘fields’. An algebra derivation correspondsto geometric vector field, but I avoid geometric interpretation considering analgebra as a primordial concept within manifold-free approach. The idea torepresent material bodies in terms of (a Lie algebra of) derivations goes backto Leonhard Euler in 1757 (for velocity of a fluid), and in the case of thefinite radiation-speed to Hermann Minkowski in 1908.

Relativity of velocity needs at least two-body system, and each such sys-tem possesses two-body mass-centre, therefore the genuine relativity of ve-locity should include also the concept of velocity relative to mass-centre.Section 8 is devoted to a problem of a many-body mass-centre for the zero-mass radiation with finite radiation-speed.

2 Universal property of the differential

and other notations

In what follows, F ⊗R Fm−−−−→ F , denotes unital, associative, and com-

mutative R-algebra ‘of zero-grade scalar fields’, and derRm ≡ derR(m,m), isan abelian group (in an exact category) of Lie R-algebras. I need two Grass-mann algebras, the Grassmann algebra of differential multi-forms, (kerm)∧,and the Grassman algebra (derm)∧ of multi-vector ‘fields’,

(kerm)∧0 ≡ (derm)∧0 ≡ F , (derm)∧1 ≡ derm ≡ der(m,m). (2.1)

For X and Y ∈ (derm)∧, I use left Grassmann multiplication

∧X Y ≡ X ∧Y, (∧X) ◦ (∧Y) = ∧X∧Y; (2.2)

(∧X)2 = 0 for grade X = odd. (2.3)

2.1 Definition (Evaluation). Left evaluation, denoted by ev, is defined asan involutive pull-back of the left Grassmann wedge product

evX ≡ (∧X)∗ : (evX ω)Y ≡ ω(∧XY) = ω(X ∧Y), (2.4)

(∧X) ◦ (∧Y) = ∧X∧Y ⇐⇒ (evY) ◦ (evX) = evX∧Y . (2.5)

grade X = odd =⇒ (evX)2 = 0. (2.6)

2.2 Theorem (Cartan 1922). Elie Cartan (1869-1951) observed in 1922 thata derivation, X ∈ derm, extends to a graded derivation of the Grassmann


algebra of differential multiforms, evX ∈ der((kerm)∧). For differential mul-tiforms α and β we have the Cartan graded derivation with the followingCartan theorem,

derRm 3 X evX ∈ derF((kerm)∧), (2.7)

(evX)(α ∧ β) = (evX α) ∧ β + (−)αα ∧ (evX β), (2.8)

evX ◦∧α = ∧evX α + (−)α ∧α ◦ evX . (2.9)

Cartan denoted the above derivation by iX ≡ evX, and called inner pro-duct or inner derivation, however a name evaluation is more appropriate.

A pair {(kerm)∧, evX} with (2.6) is said to be the Grassmann complex,baptized by Bourbaki by the Koszul complex [Bourbaki 1980].

2.3 Clarification (Universal property of the differential). The differentiald ∈ der(F , (kerm)∧1) possesses universal property: each derivation X ∈der(m,m), is a composition of a differential with evaluation, X ≡ (evX) ◦ d.I suppose that differential d extends to graded nilpotent differential,

d ∈ der((kerm)∧, (kerm)∧) with an axiom, d ◦ d = 0, (2.10)

- this is not a Lemma. Thanks to the Grassmann condition (2.6) and becausethe evaluation by a derivation, evX, diminish grade by one for grade X = 1,we have

(evX)|F ≡ 0, (2.11)

X = (evX) ◦ d+ d ◦ (evX) = (d+ evX)2 ∈ der(m,m). (2.12)

2.4 Definition (Christoffel 1869). Ewin Bruno Christoffel (1829–1900) in1869 introduced a zero-grade derivation of all tensor fields. For X ∈ derm,and f ∈ F , by two conditions,

ev ◦∇X = ∇X ◦ ev, ∇X ◦ d 6= d ◦ ∇X and ∇fX = f ∇X. (2.13)

2.5 Definition (Slebodzinski 1931). W ladys law Slebodzinski (1884–1972)in his Ph. D. Thesis introduced a zero-grade derivation of all tensor fields bytwo conditions, for f ∈ F and X ∈ derm,

ev ◦LX = LX ◦ ev, LX ◦ d = d ◦ LX and LfX 6= fLX. (2.14)


D. van Dantzig in 1932 called the Slebodzinski derivation (2.14), the Liederivation, and introduced a notation L, however Sophus Lie passed awayin 1899 and has nothing to do with this concept of a derivation of a tensorfield. I keep the name the Slebodzinski derivation (or the Slebodzinski-Liederivation). From definition (2.5) one can prove the following

LX = (d+ evX)2 ∈ der((kerm)∧); ∀Y ∈ derm, LXY = [X,Y]. (2.15)

The Slebodzinski conditions (2.14) determine the unique derivation. TheChristoffel conditions (2.13) left a lot of freedom, and an extra condition puton by Christoffel, ∇ g = 0, for g being the metric tensor, do not determineChristoffel derivation ∇ uniquely as was observed by Cartan in 1923.

2.6 Notation (Algebra map versus derivation of an algebra). In the sequel Iuse the following notation for (1, 1)-tensors of grade Π = 0, and gradeD = 0,

Π ∈ alg((kerm)∧) ⇐⇒ Π (α ∧ β) = (Πα) ∧ (Πβ).

D ∈ der((kerm)∧) ⇐⇒ D(α ∧ β) = (Dα) ∧ β + α ∧ (Dβ).

Π∗ ∈ alg((derm)∧) ⇐⇒ Π∗ (X ∧Y) = (Π∗X) ∧ (Π∗Y).

D∗ ∈ der((derm)∧) ⇐⇒ D∗(X ∧Y) = (D∗X) ∧Y + X ∧ (D∗Y).

2.7 Definition (Grassmann algebra epimorphism). A splitting of (1, 1)-tensors Π and D, is a derivation, X ∈ derm, and a differential one-formα such that, αX 6= 0 ∈ F . Then Π is the Grassmann algebra epimorphism,and is also variably known as a product structure,

Πα ≡(evX) ◦ (∧α)

αX∈ alg

((kerm)∧,

(kerm)∧

(α)

)Dα ≡

(∧α) ◦ (evX)

αX∈ der((kerm)∧, (α)− principal ideal)

ΠX ≡(evα) ◦ (∧X)

αX∈ alg

((derm)∧,

(derm)∧

(X)

)DX ≡

(∧X) ◦ (evα)

αX∈ der((derm)∧, (X)− principal ideal)

(2.16)

Grassmann algebra derivation D is an involute permutation of an algebraepimorphism Π, and Definition 2.1 implies the following relations,

Πp = D, (Πα)∗ = ΠX, (Dα)∗ = DX. (2.17)


2.8 Exercise (Product structure). For the above (1, 1)-tensors the followingholds, and then these tensors are said to be the product structures,

Π ◦ Π = Π, D ◦D = D, D ◦ Π = 0 = Π ◦D,

D + Π =(evX +∧α)2

αX= id(kerm)∧ ,

(D + Π)∗ = D∗ + Π∗ =(evα +∧X)2

αX= id(derm)∧ .

(2.18)

2.9 Exercise (Ore). Let X ∈ derm, and α ∈ kerm. Then

(evX) ◦ ∧α ◦ (evX) = (αX) evX,

∧α ◦ (evX) ◦ ∧α = (αX) ∧α .(2.19)

3 From Keres’s position-space

to algebra derivation

Keres introduced three-dimensional position-space as a congruence oftime-like word-curves in four-dimensional spacetime manifold [Keres 1972§2.1, 1976 §3]. Each congruence corresponds to a material particle-body witha positive mass. Having explicitly a pair of two congruences, two position-spaces, called x-position-space and y-position-space, Keres defined conceptu-ally an arbitrary velocity (not necessarily constant) of one position-spacerelative to another position-space, i.e. an arbitrary velocity explicitly as therelative velocity, without invoking the group theory concept, i.e. group-freerelative velocity. This definition allows us to define also acceleration and rota-tion of Keres’s y-position-space relative to x-position-space, i.e. accelerationand rotation explicitly as relative acceleration and relative rotation [Keres1972 §2.8]. See also the Hypothesis 3.4 at the end of the present Section.

Each congruence of word-lines one can equivalently understand in Eule-rian way as integral curves of the corresponding vector field or synonymouslyas the integral curves of ordinary differential equation. This allows us to re-place congruence by an algebra derivation, and thus a material particle is thesame as a derivation of an algebra m ' F , where {x1, x2, x3} describe thecorresponding conserved scalar ‘integrals of motion’ of a body/particle. Withanalogous conserved scalar ‘integrals of motion’ of a second body/particle in‘adopted coordinate systems’ we have the following two-body free-system as


Figure 1: The reference system X is a material body as a permanent dia-chronic iso-position process-derivation, X ∈ derm. Process from own pastto own nearest future. Process X is going alongside of the crests of a three-wave of X-locations {dx1 ∧ dx2 ∧ dx3}, (evX)(dx) = Xx = 0. SynchronicX-simultaneity of a reference system X is a differential one-form, α, suchthat, αX ≡ 1.

dia

chro

nic

dx

synchronic

present of α

α = dt+ fidxi

X ≡(∂∂t

)x1,x2,x3

future of X

a pair of derivations of an algebra m ' F ,

t, x1, x2, x3, y1, y2, y3 ∈ F , (3.1)

X =

(∂

∂t

)x1,x2,x3

and Y =

(∂

∂t

)y1,y2,y3

∈ derRm, (3.2)

(evX)(dt) = X t ≡ 1 = (evY)dt, (evX)(dxi) ≡ 0, (evY)(dyi) ≡ 0. (3.3)

Each material body in (3.2) is basis-free because derivation of an algebra Fis an example of a basis-free tensor field.

In the present section for Keres’s two-body system (3.2), we set theGalilean simultaneity

(dt)X = 1 = (dt)Y. Then DX Y ≡ (∧X ◦ evdt)Y = X. (3.4)

3.1 Definition (Relative-velocity, Keres 1972 §2.6 formula (34)). An arbi-trary (not necessarily constant) velocity vXY of a material body Y (Keres’s


y-space) relative to a reference system ΠX (relative to Keres’s observer x-space) is defined as an ΠX-split of Y,

Definitions: ΠX ≡ evdt ◦∧X, DX ≡ ∧X ◦ evdt; (3.5)

vXY ≡ ΠX Y, Y = (DX + ΠX)Y = X + vXY, (3.6)

vXY = Y −X, (dt)vXY = 0. (3.7)

Above definition is valid for absolute simultaneity dt only (3.3)-(3.4), andonly in this particular case the relative velocity is reciprocal, vXY = −vYX.The Keres concept of a relative velocity is group-free: from group-centric toalgebra-centric philosophy.

The material body Y posses infinite many velocities with respect to manydifferent reference systems, posses many distinct splits (observation madeby Copernicus and well know to all drivers but not known to Universityprofessors)

Y = X + vXY = Z + vZY = . . . (3.8)

The Keres definition of relative split-velocity, Definition 3.1, expressed theobservation made by Nicolaus Copernicus in 1543, Galileo Galilei in 1632about the relativity of velocity, and by Irish bishop George Berkeley (1685-1753) cited above.

3.2 Corollary. The variable scalar components, vXY xi and vXY y

i, areKeres’s expressions for the velocity of y-position-space relative to x-position-space, and vice versa, [Keres 1972 §2.6; 1976 page 351 before formula (10)]

vXY xi = Y xi =

(∂xi

∂t

)y1,y2,y3

;

vXY yi = −X yi = −

(∂yi

∂t

)x1,x2,x3

.

(3.9)

3.3 Exercise. If vXYx = v = const, then, x = y+ vt, does the job, howeverthe Keres definition (3.7) is not restricted to constant velocities.

3.4 Hypothesis. We left intriguing questions/hypotheses about the mea-ning and a physical interpretation of a commutator [X,Y] as a relative acce-leration, and a bivector X ∧ Y as a relative rotation (for infinite radiationspeed, i.e. for absolute simultaneity).


4 Leonhard Euler in 1757

A relative velocity seen as a vector ‘field’ on four-dimensional space-timemanifold, must be an algebra derivation, v ∈ derRm. An acceleration is aderivation of a velocity vector field, a derivation of a derivation. Howeverbefore invention by Christoffel in 1869, a concept of a derivation of a vectorfield was not known, and therefore Newton in 1687, and Euler in 1757, definedan acceleration not as a derivation of a vector field, but as a second derivationof an algebra of a scalar fields. Newtonian definition of acceleration is stillin University Physics textbooks as a composition of two algebra derivations.Let Y ∈ derm and c∗ ∈ alg((kerm)∧, . . .) be an algebra epimorphism toalgebra on integral curve of Y. Then an acceleration is commonly defined asa composition of derivations,(

d

dt◦ ddt

)◦ c∗ = c∗ ◦ (Y ◦Y) 6∈ derRm. (4.1)

A derivation of an algebra (of scalar fields) is a basis-free vector ‘field’, butcomposition of derivations is not. Thus Newton’s and Euler’s acceleration isnot a tensor field. Einstein replaced the composition of derivations by theChristoffel derivation. The Christoffel derivation of a vector is a vector, andAlbert Einstein in 1914-1916 defined acceleration as the Christoffel derivativeof a one-body vector (giving the geodesic vector ‘field’).

The Keres relative velocity (3.7)-(3.8) is the same as the Euler definitionin 1757 of a fluid, and fluid’s relative velocity. The Euler derivative as a sumof vector fields, (4.2) below, is known incorrectly as substantial or material, orbarycentric, or hydrodynamic derivative. In the fluid context a material bodyis a vector field on space-time ∈ derm. The Euler derivative is interpretedin Fluid Mechanics textbooks in the following naive and wrong way: “thetotal rate of change is a sum of the local rate of change plus the convectiverate of change”

Y ≡ ∂

∂t+∑

vi∂

∂xi= X + vXY, (4.2)

∂

∂t≡(∂

∂t

)x1,x2,x3

,∂

∂xi≡(∂

∂xi

)t,xj 6=i

. (4.3)

In fact, the Euler derivative should be interpreted as a definition of a ve-locity v of a fluid Y relative to a reference system X in adopted coordinates


X = (∂t)x1,x2,x3 . Given two vector fields, X and v, then a fundamental the-orem of ordinary differential equations assure that for a sum, Y = X + v,there exists local adopted coordinates that Y takes a form as in (3.2).

For an observer X being the reference system, Xx = 0, if x is in Kerers’sx-space, cf. (3.2), vXX = 0X. Therefore velocity of Y relative to X is,vXY x = (Y −X)x = Y x. Euler defined an acceleration of a fluid followingNewton as a second derivative of scalar fields, i.e. not as a basis-free tensor,

aNewton = Y ◦Y 6∈ derRm,

aEuler = Y ◦ (Y −X) = (X + v) ◦ v 6∈ derRm,(4.4)

aEuler|xi = {(X + v) ◦ v}|xi = (X + v) vi

=

(∂vi

∂t

)xi

+∑

vj(∂vi

∂xj

)t,...

. (4.5)

5 Bad interpretation of the Riemannian ten-

sor as a ‘metric’

The Riemannian metric tensor g (g is short for ambient-gravity or forgeometry) is interpreted historically as playing a double role:

• Metric tensor is measuring distances in spacetime-manifold. This in-terpretation assumes a priori that spacetime-manifold exists in realityin Nature. That now all future events are known. During Einstein’slife nobody worried about the reality of spacetime-manifold in Nature.Moreover the distance between two events on a manifold is always givenin terms of an integral. Integral over closed curves is expected to givesthe zero distance, however it is hard to imagine closed curve in space-time of events. Metric tensor should not be interpreted as a distancebetween events.

• Metric tensor is a gravitational potential for the Christoffel derivative.This interpretation consider wrongly that dF = 0 is an analogy forthe Christoffel condition ∇ g = 0. Whereas the former equation is acondition on differential form F for the given (universal) differential dthat assure the existence of a potential due to an axiom, d ◦ d ≡ 0,- this is not a Lemma. Whereas the Christoffel latter condition is acondition on derivation ∇ for totally arbitrary Riemannian tensor g.


I consider that both above interpretations of the Riemannian tensor gare missing the most important and correct interpretation that tensor g isambient-dependent algebra isomorphism from the Grassmann algebra of mul-tivector fields to the Grassmann algebra of differential multiforms, and hisinverse g−1 is an algebra isomorphism from the Grassmann algebra of differ-ential forms to multivector fields. The genuine property of the ‘metric’-tensoris ambient-dependent particle-wave iso-duality.

6 Group-free relativity postulate

6.1 Definition (Riemannian ‘metric’ is algebra isomorphism). The genuinesignificance of the Riemannian ‘metric’ tensor ‘g’ is the Grassmann algebraisomorphism (as well as the tensor algebra isomorphism that I will not usein the present paper)

g ∈ alg((derm)∧, (kerm)∧), g(X ∧Y) = (gX) ∧ (gY), (6.1)

det(g| derRm) = −c2. (6.2)

6.2 Postulate (Minkowski factor). The following ‘reciprocity’ symmetryis postulated

(det g) γ ≡ (det g) γXY ≡ (gX)Y = (gY)X. (6.3)

6.3 Axiom (Absence of privileged reference material body). The relativitypostulate of the absence of a privileged reference body is expressed in thefollowing group-free way. Each time-like material body X (with a positivemass) as an iso-position/place process-derivation-traveller to his own nearfuture, possess own simultaneity differential Pfaff form, gX, that is ambient-dependent, i.e. metric-tensor g-dependent, such that for all time-like materialbodies X the following axiom holds, i.e. det g is the reference-body-free,

0 6= Xg−−−−→ gX, (6.4)

(gX)X =

det g, if X is time-like with mass > 0.

0 if X is light-like with mass = 0.

> 0 if X is space-like without mass.

(6.5)


7 Split-velocity for finite radiation-speed

7.1 Clarification (Contra coordinates). Many years ago professor ValeryDvoeglazov warn me that nobody is able to understand me if I do not usecoordinates. I must emphasise many times that I consider all coordinates(adopted and not-adopted) not only useless but detrimental because theyimpede wrong pre-conceptions that do not allow to understand neither math-ematics nor physics. In particular Cartesian coordinates (used by Einstein)drive me into despair. Descartes introduced such detrimental weed thatspread all over all textbooks. Nevertheless, against my conviction, in thissection I follow the Dvoeglazov’s advice and at the begining consider Exam-ple 7.2 in useless coordinates, and only after the same story is presented inthe correct coordinate-free way.

In this section I drop the Galillean condition of absolute simultaneity andI will consider two-body system for the case when, dt′ ∧ dt 6= 0,

X =

(∂

∂t

)x1,x2,x3

and Y =

(∂

∂t′

)y1,y2,y3

∈ derRm ≡ derR(m,m). (7.1)

7.2 Example. In adopted (irrelevant) coordinates in (7.1) we have,

gX = (det g)dt+ fidxi, and gY = (det g)dt′ + gidy

i. (7.2)

∂

∂xi≡(∂

∂xi

)t,...

,∂

∂yi≡(∂

∂yi

)t′,...

. (7.3)

=⇒ (gX)∂

∂xi= fi, (gY)

∂

∂yi= gi. (7.4)

The adopted dual bases of vector ‘fields’ associated to simultaneity differen-tial forms are as follows

Xi ≡∂

∂xi− fi

det gX, (gX)Xi ≡ 0, (dxi)Xj = δij,

Yi ≡∂

∂yi− gi

det gY, (gY)Yi ≡ 0, (dyi)Yj = δij.

(7.5)

id =X⊗ gX

det g+ Xi ⊗ dxi =

Y ⊗ gYdet g

+ Yi ⊗ dyi. (7.6)

Each material body as in (7.1) is coordinate-free and basis-free becausederivation of an algebra, F⊗RF

m−−−−→ F , is an example of a coordinate-freetensor field.


One can express a derivation X in a basis adopted for Y with gY andvice versa,

X =(gY)X

det gY + (X yi)Yi = γYX Y + (X yi)Yi,

Y =(gX)Y

det gX + (Y xi)Xi = γXY X + (Y xi)Xi.

(7.7)

The scalar components, X yi (and Y xi), are the Keres expressions for thevelocity of x-position-space relative to y-position-space [Keres 1976 page 351before formula (10)]

X yi =

(∂yi

∂t

)x1,x2,x3

and Y xi =

(∂xi

∂t′

)y1,y2,y3

(7.8)

7.3 Conclusion (Grassmann algebra epimorphism). Vide supra Definition2.7. Each time-like material body (as the reference system vector field, idest an observer) determine the Grassmann algebra epimorphism from theGrassmann algebra of the four-dimensional space-time, on the Grassmannalgebra of the three-dimensional position-space of this body. This Grassmannalgebra epimorphism (between algebras of multi vectors) is given by thefollowing idempotent

ΠY ≡evgY ◦∧Y

det g, ΠY(Z ∧W ) = (ΠYZ) ∧ (ΠYW ), (7.9)

evgY ◦ ∧Y + ∧Y ◦ evgY = (det g) id =⇒ ΠY ◦ ΠY = ΠY. (7.10)

ker ΠY = (Y)− principal ideal. (7.11)

7.4 Definition (Split-velocity for finite radiation-speed). The above expres-sions (7.7)-(7.8) lead to the following basis-free definition of the velocity ofx-position-space relative to the observer y-position-space in the case of thefinite speed of zero-mass radiation. This definition explicitly depends onisomorphism of Grassmann algebras g that can be identified with an envi-ronment (thanks to Albert Einstein idea), and with the simultaneity (thanksto Hermann Minkowski idea), where Minkowski factor γ is defined in Postu-late 6.2.

Velocity of X relative to epi-idempotent, ΠY ≡evgY ◦∧Y

det g, is

γ vYX ≡ ΠY X =1

det gevgY(Y ∧X) = X− γY. (7.12)


Velocity of Y relative to epi-idempotent, ΠX ≡evgX ◦∧X

det g, is

γ vXY ≡ ΠX Y =1

det gevgX(X ∧Y) = Y − γX. (7.13)

7.5 Proposition. Speed of relative velocities is the same, (vYX)2 = (vXY)2.

Proof.

γ2(vYX)2 ≡ (gγv)γv = (gX− γ gY)γv = (gX)γv

= (gX)(X− γY) = (det g)(1− γ2) = γ2(vXY)2,

=⇒ γ2

(1 +

v2

det g

)= 1, and v2 = −(det g)

(1− 1

γ2

). (7.14)

7.6 Conclusion. If simultaneity forms are not parallel, g(X∧Y) ≡ (gX)∧(gY) 6= 0, then the relative split-velocities cannot be reciprocal for finiteradiation-speed,

vYX ≡ (vXY)−1 =

−γ vXY −(γ − 1

γ

)X.

− 1γvXY −

(1− 1

γ2

)Y.

(7.15)

vYX ∧ vXY = −(

1− 1

γ2

)X ∧Y, (7.16)

(X ∧Y)2 = {(gY)Y}{(gX)X} − {(gY)X}{(gX)Y}= −(det g)2(γ2 − 1), (7.17)

vYX + vXY = −(

1− 1

γ

)(X + Y), (7.18)

(vYX + vXY)2 = 2(det g)(1 + γ)

(1− 1

γ

)2

. (7.19)

7.7 Contradiction with Einstein’s axiom. Albert Einstein in 1905 de-noted velocity of coordinate system k relative to coordinate system K byv ≡ vKk [§3]. On page 901, Einstein introduced third coordinate system K ′


that moves relative to k with velocity, −v ≡ vkK′ . Then on page 902 sup-posed that K ′ is identified with K, and thus follow the crucial reciprocitypostulate [not spelled explicitly],

vKk + vkK = 0. (7.20)

The motion of irrelevant Cartesian (or not Cartesian) coordinate systemsis funny, make laugh, but in spite of ridiculous (or maybe just because ofthis childish nonsense) was accepted by scientific community without mur-mur. Einstein himself later on in 1914-1916 was aware that all coordinatesare totally irrelevant, that mathematics and all physics phenomenas are co-ordinate free (are invariant with respect to change of irrelevant coordinates),i.e. must be expressed in terms of coordinate-free tensors, but WHY Einsteinnever come back to his-1905 ‘special/particular’ relativity with horrible coor-dinates and do not tried to reformulate relativity (of velocity) in coordinate-free way? Looks like I was among the pioneers presenting the Lorentz groupin coordinate-free way in terms of the Clifford bivectors [Oziewicz since 2006].

The Einstein reciprocity axiom pre-suppose wrongly the existence ofthe unique absolute position-space where both ‘coordinate-systems’ moves,and such wrong preconception dominate in all present University textbooks.More important is that the Einstein reciprocity postulate contradicts stronglyto his main observation about relativity of simultaneity. Namely, each rela-tive velocity is tangent to the present of observer, i.e. vKk must be tangentto simultaneity of K, whereas vkK must be tangent to simultaneity of ob-server k. Reciprocal relative velocities logically implies absolute simultaneityin sharp contradiction to the main conclusion in [Einstein 1905].

Preconceptions are difficult to overcome. Overcoming them, therefore,involves realising that the premises upon which they are based are wrong.Oftentimes disastrous event force us recognise preconception is wrong. Per-sonally for mi such disastrous event was discovery by Abraham Ungar in1988 that the addition/composition of Einstein’s reciprocal velocities, (7.20),is non-associative, it is the unital quasi-group, known as a loop. Non-associativity implies immediately that for the given pair of ‘coordinate sys-tems’, vKk cannot be unique, contrary to another Einstein supposition thatthe velocity vKk is unique [Oziewicz 2005, 2006, 2007]. I demonstrated in[Oziewicz 2005] that the composition of non-reciprocal relative velocities,(7.15)-(7.16), is evidently associative because they are unique by very defini-tion.


7.8 Clarification (Contra symmetry of Maxwell equations). This subjecthow history can drive in totally wrong direction deserve separate text/booklet,and not only shorter statement here. One can hear around many many timeswrong statement that symmetry group of Maxwell differential equations isthe Lorentz group, and therefore the genuine origin of the Einstein relativityis Maxwell theory, e.g. [Sommerfeld 1949, 1952 page 223]. This is false!Symmetry group of Maxwell theory is conformal group and this was knownfew years after Einstein 1905-paper [Cunningham 1909/1910; Bateman 1910].Evidently conformal group has nothing to do with relativity. Does this dis-astrous discovery of the conformal group changed a little wrong Einstein’s‘special/particular’ relativity (of velocity)? Not at all. I must say in fullvoice: the relativity theory (of velocity; of position-space observed by Coper-nicus before Maxwell was born) must be divorced from Maxwell theory. Theweed and illusion of the strong marriage of the Maxwell theory with relativitytheory spreader and invaded not only scientists deprived of intelligence, butinvaded also great philosopher of science Paul Feyerabend who described themeaning of relativity in the following words: “the relativity theory meansthat mechanics must be constructed taking into account determined aspectsof electromagnetism . . . ” [Feyerabend ‘Warum nicht Platon? 1980].

8 Many-body centre-of-inertia

The relativity of velocity is not complete without the concept of a mass-centre. Material body possesses many different velocities relative to choicesof different reference bodies, and this fact is crystal clear in Keres’s publica-tions. Many-body interacting system possesses the mass-centre and impor-tant are momenta and velocities relative to the mass-centre. Therefore thegenuine special relativity is complete only with the many-body mass-centreconcept. I consider that all phenomena, the Doppler shift, stellar aberrationdiscovered by Bradley, emissions, absorptions, scatterings, etc, should be an-alyzed in terms of momenta and velocities relatives to mass-centers. Thismeans that the important boost should be also the boost from mass-centre.In my opinion the concept of the mass-centre with own simultaneity is ofvital importance for special relativity theory.

However it is known, see e. g. [Pryce 1948; Møller C 1949, 1952, . . . ,1969] and any textbook on special or general relativity [Landau and LifshitzThe Classical Theory of Fields ], that within the Lorentz isometry-group,


the concept of intrinsic mass-centre does not exist for finite radiation-speed,and whenever many-body mass-centre is used in practice of cosmology, theNewtonian limit is accepted. I show in the present section that the group-free concept of a relative split-velocity, in the spirit of the Keres approach,extended to finite radiation-speed allows the perfect concept of mass-centre,split-momentum and split-velocities relative to mass-centre.

It is stressed in some publications that a real problem facing attemptto construct a relativistic many-particle theory (theory for finite radiation-speed) is the lack of a notion of simultaneity. Group-free approach, split-velocities (8.1) offer perfect solution of this problem: one must forget Lorentzisometry group, in a spirit of the group-free approach by Keres.

For two-body time-like system, as Keres x-space and y-space, a mass-center is in another Keres’s z-space with a total mass,

M =∑

mi − binding energy,

with nontrivial mass-dependence and variable velocity dependence in case ofthe finite radiation-speed.

One-body problem will stay always as one-body, however 2-body problemis in fact always 3-body, because it must involve a set of three Keres position-spaces: x- y- and mass-center z-position-space.

I consider a mass-centre with a total mass M, as a time-like derivation (avector ‘field’) Z with its own simultaneity differential one-form SZ, for a freetwo body system {X,Y} with corresponding masses {mx,my}.A mass-centerZ is a bound system and in order to be ionized - decoupled into free-systemthere must be external zero-mass gravity field G (or electromagnetic field),in terms of energy-momenta.

To write down the mathematical expression, we need the Minkowskisymmetric scalar factors γXY = γYX, (not Lorentz factor that enter to isom-etry group), in terms of simultaneity differential forms, see the MinkowskiPostulate 6.2. For two-body time-like system, X with a positive mass mx,and Y with a positive mass my, following Minkowski [1908] we set for group-free relative split-velocity vXY, see Proposition 7.5 and Conclusion 7.6,

(det g) γXY ≡ (gX)Y = (gY)X = (det g) γY X =det g√

1− (vXY)2

c2

. (8.1)

8.1 Notation. An internal self-energy of a k-body with a positive mass(relative to the same k-body) is denoted as Ekk = mkc

2. An internal energy


of a radiation is zero. If a G is a zero-mass radiation with (gG)G = 0, theninternal energy EGG ≡ 0. I denote by, Elk ≡ γlkEkk, an energy of a k-bodyrelative to l-body as a reference system (l-body is a reference system).

8.2 Definition (Schrodinger 1956). The energy of a zero-mass radiationG (gravity or electromagnetic radiation with (gG)G ≡ 0) relative to time-like material body X with a positive mass, (gX)X ≡ det g, was defined bySchrodinger as the following evaluation

(det g)EXG ≡ (gG) X = (gX) G. (8.2)

8.3 Definition (Many-body mass-center). With Definition 8.2 a mass-centerZ with own simultaneity differential one-form gZ, (gZ)Z ≡ det g, is definedby energy-momenta conservation law as follows. Many-body interacting ma-terial system, as a set of time-like vectors-derivations {Xi ∈ derRm}, possesstime-like mass-center, Z ∈ derRm, with a total internal self-energy (totalmass) EZZ = Mc2, according to the following law of conservation of energy-momenta,

G + EZZ Z =∑

Eii Xi, gZ =1

EZZ

(∑Eii gXi − gG

). (8.3)

On the right is a free decoupled many-body system, whereas on the left, Zis a mass-center, describing the bound system with radiation G 6= 0.

It is plausible to postulate the existence of the wave-front, gZ∧d(gZ) = 0,but I will not consider this postulate here.

The law of conservation (8.3) gives the two expressions for the total massfor finite radiation-speed. The first expression in terms of split-velocitiesand radiation energy relative to mass-center, whereas the second express thesame total mass in terms of split-velocities between constitutive materialparticles/bodies of a given many-body system.

8.4 Notation. I denote by γi ≡ γiZ the Minkowski factor relative to mass-center Z, EZi = Eiiγi.

8.5 Lemma.

EZZ =∑

EZi − EZG, (8.4)

M 2 =1

det g(M gZ)(M Z) =

∑mimj γij − 2

∑miEiG. (8.5)


8.6 Comment. Would be interesting to calculate Slebodzinski-Lie derivativeLZEZZ.

8.7 Lemma (Gravity ionization energy). The gravity (or alias electromag-netic) ionization energy EZG (resembles potential energy in four-dimension)in terms of Minkowski factors is as follows

EZG =√∑

mimj (γiγj − γij). (8.6)

Proof.

M EZG =1

det g(gG)

∑miXi =

1

det g

∑mi (gXi)G

=1

det g

∑mi(gXi)(

∑mj Xj −M Z)

=∑

mimj γij −M∑

mi γi. (8.7)

∑mimj γij = M

(∑mi γi + EZG

)=(∑

mj γj − EZG) (∑

mi γi + EZG

)(8.8)

In the Newtonian limit of the infinite radiation-speed we have∑mi γi

c→∞−−−−−−−→∑

mi,∑mimi γij

c→∞−−−−−−−→(∑

mi

)2

,

EZGc→∞−−−−−−−→ 0,

Mc→∞−−−−−−−→

∑mi.

(8.9)

Lemma 8.7 with ionization energy (8.6) contains mutual relative velocitiesbetween bodies of the system, γij, that one can express in terms of velocitiesrelative to mass-center γi. This will not reduce the number of independentvariables, because γij will be expressed in terms of γi and cosines of anglesvZi · vZj,

vZi · vZj = (det g)

(γijγiγj− 1

), γij = γi γj

(1− vZ i · vZ j

c2

)(8.10)


Above (8.10) is analogous (but not identical) to Sommerfeld identity [Som-merfeld 1909] for scalar speed of composed isometric-velocities (reciprocal)within Lorentz isometry group. In Sommerfeld identity in the last fac-tor in (8.10) there is plus sign, because isometric velocity is reciprocal,vZ i = −viZ, but this is not possible within group-free approach, whereby definition, (gZ) vZX ≡ 0 and (gX) vXZ ≡ 0, and simultaneity is relative,(gZ) ∧ (gX) 6= 0 [Oziewicz since 2005].

(EZG)2 =∑

mimj

√(γ2i − 1)(γ2

j − 1) cos(ij),√γ2i − 1 = γi

vic. (8.11)

For two body-system {mx,my} with mass-center {mx.my,M} there is thefollowing system of four scalar equations for the total mass M,

G +M Z = mx X +myY,

M EZG = mx EXG +my EY G,

EZG +M = mx γZX +my γZY ,

EXG +M γZX = mx +my γXY ,

EY G +M γZY = mx γXY +my.

(8.12)

The total mass of this bound system for finite radiation-speed is given interms of three Minkowski factors {γXY , γZX , γZY } (three Minkowski factorsfor two-body system), because in spacetime we are obliged to consider in factthree body system {mx,my,M},

M2 − 2M(mxγZX +myγZY ) +m2x + 2mxmyγXY +m2

y = 0, (8.13)

M = mxγZX +myγZY

−√m2x(γZX − 1) + 2mxmy(γZXγZY − γXY ) +m2

y(γZY − 1) (8.14)

8.8 Speculation. In fact all Minkowski factors are independent. Neverthe-less one can look for some speculative relationship.

∂M

∂γZX= 0 =

∂M

∂γZY=⇒ mx(2γZX − 1) = my(2γZY − 1) = . . . (8.15)


8.1 Maximal binding energy

What could be the maximal binding energy within group-free theory?Note that

γ − 1 ≤√γ2 − 1 < γ =⇒ 0 ≤

√γ2 − 1− (γ − 1) < 1. (8.16)

Binding energy =∑

mi −M = EZG −∑

mi(γi − 1). (8.17)

The maximal binding energy in (8.11) is for, cos(ij) = 1, and then

EmaxZG =

∑mi

√γ2i − 1,

√γ2i − 1− (γi − 1) = 1− γi

(1− vi

c

)=⇒ Maximal binding energy =

∑mi

(1− γi

(1− vi

c

))<

∑mi. (8.18)

9 Reduced space-like momentum

and reduced mass for finite radiation-speed

9.1 Notation (Relative split-velocity). Let vlk denotes a split-velocity ofk-body relative to a reference system l-body. By this definition the relativevelocity lay along the crest of a simultaneity differential form of a referencesystem, i.e. by definition (gXl)vlk ≡ 0. Therefore for finite radiation speedsplit-velocities cannot be reciprocal

vlk ∧ vkl =

(1− 1

γ2

)Xl ∧Xk 6= 0, (9.1)

vkl + vlk = −(

1− 1

γ

)(Xl + Xk)

c −→∞, γ −→ 1−−−−−−−−−−−−−−→ 0. (9.2)

9.2 Definition (Reduced momentum). Momentum of a k-body relative tomass-centre Z is said to be k-reduced momentum. Reduced momentum is


denoted by pZk, and it is given by ΠZ-split of k-momenta, cf Definition 2.7,

ΠZ ≡evgZ ◦∧Z

det g, (9.3)

pZk ≡ Ekkev(gZ)(Z ∧Xk)

det g= Ekk(Xk − γkZ) = EZkvZk. (9.4)

pZZ = 0 =⇒∑k

pZk − pZG = 0. (9.5)

The split-momenta (9.4) resembles the Planck expression deduced fromthe Lorentz boost [Planck 1906], but this visual resemblance must not hid-den a conceptual difference between group-free approach (9.2), and Lorentzisometry group where Albert Einstein in 1905 made crucial reciprocity pos-tulate, vkl + vlk = 0, i.e. the inverse of relative velocity was postulated adhoc to be, vlk = (vkl)

−1 = −vkl. I already emphasised before that reciprocityof velocity contradicts to relativity of simultaneity.

9.3 Clarification (Radiation in terms of relative momenta). Conservationof energy-momenta, Definition 8.3, and relative momenta of radiation leadsto the following expressions

G = EZG Z + pZG = EiG Xi + piG

=⇒ (piG)2 + (det g)(EiG)2 = 0, (9.6)

G

{EZZ + EZG

(1−

∑ EiiEiG

)}= EZZpZG − EZG

∑EiipiG. (9.7)

9.4 Proposition. The reduced momentum has the following expression interms of relative energies and relative velocities between material bodies whereactual forces and accelerations appears,

pZi =Eii

(EZZ)2

∑lk

EilElk(vli − vlk)

+∑k

Ekk (EiGXk − 2EkGXi)−

(EiG −

∑k

Eik

)G. (9.8)

9.5 Definition (Reduced mass of two-body subsystem). For each two-bodysubsystem the reduced mass for finite radiation-speed is defined as γ-dependent

µkl ≡ γklEkkEllEZZ

=EkkEklEZZ

=EllElkEZZ

. (9.9)


9.1 Two-body system

pZ1 = µ12 ·E22v21 − E11v12

EZZ

mod radiation

pZ2 = µ12 ·E11v12 − E22v21

EZZ

mod radiation

(9.10)

9.2 Three-body system modulo radiation

pZ1 = µ12(E22 + E23)v21 − E11v12

EZZ

+ µ31(E33 + E32)v31 − E11v13

EZZ

− E11

EZZ

· E12E23v23 + E13E32v32

EZZ

(9.11)

pZ2 = µ23(E33 + E31)v32 − E22v23

EZZ

+ µ12(E11 + E13)v12 − E22v21

EZZ

− E22

EZZ

· E21E13v13 + E23E31v31

EZZ

(9.12)

pZ3 = µ31(E11 + E12)v13 − E33v31

EZZ

+ µ23(E22 + E21)v23 − E33v32

EZZ

− E33

EZZ

· E32E21v21 + E31E12v12

EZZ

(9.13)

pZic −→∞−−−−−−−−→

∑µkivki (9.14)

About half the stars in the sky are double stars or binaries, with twostars orbiting a common center of mass. In cosmic-rays protons have beenobserved with γ ' 1011. Hernandez et all. [2012] observed that wide binarystars violate Newtonian gravity for higher relative velocities, i.e. for higherγ. Authors suggest breakdown of Kepler’s third law, keeping reduced massto be γ-independent.

Group-free and isometry-free relativity may offer an alternative explica-tion of experimental data of wide binary stars in terms of the γ-dependentreduced mass that modify Kepler’s laws.


10 Universe as a bunch of Grassmann factor-

algebras of relative position-spaces

The Grassmann complex implies the exact sequence of F -algebras

0 −→

(gX)-principal ideal.Simultaneity ideal.

Zero F -algebra

DX ≡∧gX ◦ evX

det g−−−−−−−−−−−−−→

GrassmannF -algebra

of differentialforms

ΠgX ≡

evX ◦∧gXdet g−−−−−−−−−−−−−→

{F -algebra

X-positions

}−−−→ 0 (10.1)

Grassmann factor-algebra of X-positions

=Grassmann F -algebra

(gX)-principal simultaneity ideal = ker ΠgX = imageDX

(10.2)

An algebra epimorphism (epi-morphism from one algebra to another factor-algebra) possesses kernel, and this kernel is two-sided ideal. Therefore factor-algebra is an image of algebra epimorphism Π – or equivalently, is a kernelof an algebra derivation D.

0 −→

F -algebra

of X-positionsdxi

ΠgX ≡evX ◦ ∧gX

det g−−−−−−−−−−−−−→

GrassmannF -algebra

of differentialforms

dxi ⊕ (dt ∧ dxi)

DX ≡

∧gX ◦ evXdet g−−−−−−−−−−−−−→

(gX)-principal ideal.Simultaneity ideal.

Zero F -algebra.dt ∧ . . .

−−−→ 0 (10.3)

For each time-like material body X of a positive mass, i.e. of a positiveinternal energy,

dimF(Grassmann algebra) = 24 = 16

dimF(X-factor-algebra) = 23 = 8.(10.4)


Figure 2: Idempotent ΠgX ' Πdt as the F -algebra mono-morphism is com-posed.

Simultaneityideal (dt)

F -algebraX-positions

dx1 ∧ dx2 ∧ dx3

Grassmannalgebra ofdifferential

formsdxi ⊕ (dt)

Anihilationev ∂

∂t

Creaction∧dt

Mono-morphism Πdt

Figure 3: Idempotent DX ' D ∂∂t

as the F -algebra epi-morphism is com-

posed.

F -algebraX-positions

dx1 ∧ dx2 ∧ dx3

Grassmannalgebra ofdifferential

formsdxi ⊕ (dt)

Simultaneityideal (dt)

Creation∧dt

Anihilationev ∂

∂t

Epi-morphism D ∂∂t

The Cartan theorem (2.9) implies that a kernel of an algebra derivation,ker(evX), is a Grassmann factor-algebra, or quotient-algebra, that we baptize


as the Grassmann factor-algebra of relative X-space, shortly ‘X-space’ is anF -algebra,

α ∈ ker(evX) and β ∈ ker(evX) =⇒ α ∧ β ∈ ker(evX). (10.5)

10.1 Terminology. A material body as a time-like vector field X is synonymof iso-position process in four-dimensional space-time, viz, (dxi)X ≡ Xxi ≡0 ∈ F .

Each time-like material body X is owner of the Grassmann factor-algebraof his own X-position-space. It is good even simply identify material bodywith Grassmann X-factor-algebra of his position-space. And when I say, X-factor-algebra (of positions), this means the Grassmann X-factor-algebra ofthe Grassmann F -algebra of the four-dimensional space-time.

Two-sided principal simultaneity ideal in the Grassmann algebra, i.e. anideal generated by a simultaneity Pfaffian differential form gX assigned byambient algebra-iso-morphism ‘g’ to iso-position process/vector X, is denotedby (gX).

10.2 Example. One can list some members of the Grassmann factor-algebraof X-position-space for Keres congruence in terms of a derivation X ∈ derm,given in (7.1). Let f ∈ F denotes any scalar of a zero-grade, then

If fdx is in ker(evX), then 0 = (evX)dx = {(evX) ◦ d}x = Xx

and x is conserved scalar ‘integral’ of X. (10.6)

{F , f, t, t′, x1, x2, x3, y1, fdx1, . . . , fdx1 ∧ dx2 ∧ dx3}∈ ker(evX) ≡ X-factor-algebra, (10.7)

(gX)-principal simultaneity ideal 3 {fdt, fdt ∧ dx1, . . .}6∈ ker(evX). (10.8)

In particular Grassmann X-factor-algebra contains all X-electric and X-magnetic relative fields.

10.3 Warning. We must stress very important two peculiarities. For amaterial body X (‘descendant’ of the Keres congruence as in (7.1)) being theGrassmann X-factor-algebra, the following two warnings holds:


• Every X-factor-algebra-of-positions is over the same entire algebra Fof zero-grade scalars, i.e. it is a different F -algebra for each body X,but always over the same algebra F of space-time scalars.

• Not each two-sided principal simultaneity-ideal (gX) possesses a factor-derivation dX ≡ d/(gX), and thus a factor-differential (dX)2 = 0.

ΠgX (gX) ≡ evX ◦∧gXdet g

(gX) = 0, (10.9)

dX ◦ ΠgX ≡ ΠgX ◦ d ⇐⇒ dX = d/(gX), (10.10)

=⇒ (dX)2 ◦ ΠgX = ΠgX ◦ d 2 = 0. (10.11)

The necessary and sufficient condition for existence of dX is

d (gX) ⊂ (gX), (10.12)

and then an ideal, (gX), is said to be closed. This fact is impor-tant for the concept of an inertial or non-inertial material bodies.If factor-differential does exists, then and only then the GrassmannX-factor-algebra-of-positions is invariant (stable) with respect to thefactor-differential:

(dX){Grassmann X-factor-algebra}⊂ {Grassmann X-factor-algebra}. (10.13)

A (gX)-simultaneity-principal-ideal in Grassmann algebra of the differentialmulti-forms is also known as an exterior differential system in the terminologyof Elie Cartan.

How one can see the Universe within manifold-free approach? Universeis a matter in an absolute motion from the own past to own future; and in arelative motions of one position-space relative to other position-spaces.

Just as there is no motion without matter, [so] there is nomatter without motion.

Georg Wilhelm Friedrich Hegel (1770–1831)

It is good to see the Universe as a collection of material bodies-processes-travelers to future (Universe is traveller to future) and each material body asthe Grassmann factor-algebra of the differential forms, always over the sameentire algebra m ' F of scalar fields. Universe is a bunch of Grassmannfactor-algebras.


10.1 Recapitulation

The Keres concept of an x-position-space as a congruence of word-lines ina space-time manifold [Keres 1976], I re-formulated in terms of a basis-freeand a manifold-free Grassmann X-factor-algebra over an algebra ‘m’ of scalarfields on space-time. I stress that Keres’s position-space must include entirealgebra m of zero-grade scalars. In this realm position-space is good to seeas a Grassmann factor-algebra of the Grassmann m-algebra - no change ofalgebra of scalars!

The Keres x-position-space include all coordinates,

t, x, y, . . . , t′, x′, y′, z′, · · · ∈ {x-space}.

However simultaneity one-form, dt ∈ (gX), is not in the Keres x-position-space if, gX = (det g)dt, and, (evX)(dt) = X t = 1.

DX ≡∧gX ◦ evX

det g∈ der((kerm)∧, (gX)− principal ideal), (10.14)

ΠgX ≡evX ◦∧gX

det g, ∀f ∈ F =⇒

{ΠgX f = f.

ΠgX dt = 0.(10.15)

{(dX) ◦ ΠgX}f = dXf = (ΠgX ◦ d)f = (id−DX)df = df −DXdf

= df − (X f)gX

det g∈ {x-position-space} ≡ ker(evX). (10.16)

dXt = dt− dt = 0 ∈ {x-space} (10.17)

Moreover I warn that existence of a factor-differential, dX, is not granteda priori.

I learn from Keres’s publications [Keres 1972, 1973, 1976] two importantlessons.

• Three-dimensional position-space is not a certain instant of a time asmajority of present-day textbooks claim. This wrong claim of whatit is ‘the physical position-space’ dominate in all present day scientificjournals and in scientific conferences on gravity and cosmology. Three-dimensional position-space is not a submanifold of four-dimensional


space-time manifold. Keres defined a position-space as a congruence ofword-lines. Therefore the Keres position-space is a quotient manifold:entire word-line is just one single location/position only.

No one instant of time, t = const, has a ‘conscience’ of a position/loca-tion. Instead, instance tell us about relative simultaneity. An event,as an element of a cross-section of a space-time for a given instant of atime, t = const, can belongs to infinite many different congruences ofworld-lines, and, as Keres was clearly aware, there is no way to assignto such single event some position/location in a ‘real physical position-space’. I explained on several illustrative examples in [Oziewicz 2010;Oziewicz and Page 2012, 2014], that position-space must be defined asa quotient-manifold and not as a sub-manifold,

Position-Space is a Grassmann factor-algebra

≡ Grassmann algebra

Simultaneity ideal(10.18)

• Keres is explaining that does not exists just one ‘physical position-space’ (as a factor-manifold; in the present paper re-interpreted as aGrassmann factor-algebra). His original enunciation is in terms of thedifferent congruences. The idea of many distinct ‘physical’ position-spaces can be traced and attributed to observations by Nicolaus Coper-nicus (1473–1543), Galileo Galilei in 1632, Irish Bishop George Berke-ley in 1710, Henri Poincare in 1902 and 1908, Ernst Mach in 1904, andby Minkowski 1908. However the relativity of simul-localidad was notelucidated so explicitly as in the Keres’s publications.

11 Relativity theory versus gravity theory

It is good, at least in the beginning for a clear pedagogical reasons, dis-tinguish conceptually theory of a gravitational interaction from theory ofrelativity. This is because the relativity theory (relativity of position-spaces,relativity of velocities, relativity of simultaneity [Einstein 1905; Minkowski1908; Ivezic 2012], etc) is fundamental for all kinds of interactions includingelectromagnetic interactions, weak and nuclear interactions. The relativitytheory for historical reasons is divided artificially into two parts, relativityof position/location, velocity and simultaneity – this is said to be special or


restricted relativity; and the second part deals with the relativity of accele-ration and rotation – this is how ‘general relativity’ should be understood.

The reference system is defined as a material body with a positive mass.Thus within the relativity theory, the velocity, acceleration, and rotation ofone material body depends on a free choice of another reference materialbody. To be relative means to be dependent on a choice of a materialreference system. There are still different non-equivalent understanding ofmathematical description of a reference system. Often ‘general relativity’ isunderstood as the theory of gravity only (excluding electromagnetic field),however I think it is more adequate do not mix ‘relativity’ with ‘gravity’, andinterpret ‘general relativity’ rather as the relativity of all motions, i.e. therelativity of position/location (to be in the same position is relative), relati-vity of velocity, relativity of acceleration and relativity of rotations, withoutselecting any specific kind of interaction. Thus consideration of the gravita-tional interaction would be under the name gravitation, avoiding the doubleterminology in some journals ‘general relativity and gravitation’.

12 Reference system as an iso-position pro-

cess/monad

within manifold-free and group-free

homological algebra approach

The idea to define a material body as a time-like vector field is due toHermann Minkowski in 1908. Einstein in 1905 adopted alternative and notequivalent presentation of a material body as a coordinate system alias a co-ordinate basis, tetrad and co-tetrad, and this is never accepted in my texts.I follow an approach by Minkowski. There are two advantages to define ma-terial body, and in this way every reference system, as a time-like derivation(alias a vector ‘field’), instead of Einstein’s coordinate frame=basis. Theoryof reference systems in terms of algebra derivations is known as a monadtheory.

1. Each algebra derivation, as in (7.1), is an example of a tensor field,thus it is coordinate-free and basis-free, and in this way we can forgetirrelevant coordinates, bases, (orthonormal or not orthonormal) frames,tetrads, co-tetrads and irrelevant moving frames of Elie Cartan.


2. The second advantage to define a material body as an algebra derivationis that this description is manifold-free and event-free in the spirit ofthe non-commutative point-free ‘geometry’. The commutative ‘geome-try’ deals with commutative algebra (of scalar ‘fields’) and for historicalreasons the name ‘geometry’ dominate over more adequate ‘algebra’.The recent birth (at the end of XX century) of non-commutative ‘ge-ometry’ open the way to manifold-free algebraic approach. The term‘geometry’ we are able to exchange for ‘algebra’, and for this ideologicalexchange non-commutativity or commutativity of the primary unitalalgebra of scalars is not so important: one can develop manifold-freeapproach also for commutative differential algebra as it was developedin the recent decades in the realm of manifold-free non-commutative‘geometry’.

What advantage I see in abandoning and forget, or never using, the space-time manifold of events, the world of events, so popular in the present-daytextbooks on relativity, on gravity, and on cosmology.

1. Franz Brentano during 1889–1891 published: past and future objectsare not real. Exists means exists now. Past and future are differentfrom NOW. See Doctoral Dissertation by Arianna Betti [Betti 2001].

2. Each manifold of higher then one-dimension must possesses by-definitionone-dimensional sub-manifolds, and in particular must possesses theclosed curves. Closed curve is natural in three-dimensional position-space, however it is hard to imagine the real closed curve in a manifoldof events.

3. In present paper a ‘space-time algebra’ is a set of two Grassmann alge-bras, of differential multi-forms and the Grassmann algebra of ‘multi-derivations’ (alias multi-vector fields). Such ‘space-time’ algebra withGrassmann functors, m 7→ {(kerm)∧, (derm)∧} allows to avoid to thinkabout universe as a four-dimensional manifold of events that existedsince eternity, exist now in this very moment, and implies that the fu-ture exist now. In the manifold-free approach universe is a collectionof material bodies and zero-mass radiation, a la Georg Hegel, and eachmaterial body is an iso-position process-derivation of a commutativealgebra as in (7.1) - reinstating the Keres idea of congruences in termsof algebra derivations.


The manifold-free algebraic approach assumes associative, unital andcommutative algebra, F ⊗R F

m−−−−→ F , as a primary concept. ‘Coordi-nates’ in (7.1) are elements of a commutative algebra without of necessityto use a concept of an event i.e. this is event-free and point-free approach torelativity theory. I avoid the name of a/the ‘vector field’ and replace it byadequate ‘time-like derivation of algebra of scalars’ that I use as a synonymof a material body with a positive mass, as it is clear in expressions (7.1),

derm 3 X −→ evX ∈ der((kerm)∧). (12.1)

Within Riemannian ‘geometry’, with non-singular ‘metric’ tensor ‘g’, itis important to consider also the Clifford algebras of differential forms andtensor algebra together with the Grassmann algebra, however for simplicitywe restricted ourself in the present paper to consider the Grassmann algebraalone.

Acknowledgments

I wish express my gratitude to friend Tomislav Ivezic (Ruder BoskovicInstitute, Zagreb, Croatia) for frequent inspiring Skype discussions. I bene-fited always from Tomislav’s comments during long Skype discussions, andbenefited from his papers pursuing philosophy of reality of four-dimensionalapproach, even more benefited if his papers are not accepted for publicationsby dogmatic referee system.

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