Monogenic functions in 5-dimensional spacetime used as first principle: gravitational dynamics, electromagnetism and quantum mechanics

0.27 MB·English

by  Jose B. Almeida

Monogenic functions in 5-dimensional spacetime used as first principle: gravitational
dynamics, electromagnetism and quantum mechanics
Jos´e B. Almeida∗
Universidade do Minho, Physics Department, Campus de Gualtar, 4710-057 Braga, Portugal†
Monogenicfunctionsarefunctionsofnullvectorderivativeandarehereanalysedinthegeometric
algebraof5-dimensionalspacetime,G4,1,inordertoderiveseverallawsoffundamentalphysics. The
6 paperintroducestheworkingalgebraandthedefinitionofmonogenicfunctions,showingthatthese
0 generatetwo4-dimensionalspaces,onewithEuclideansignatureandtheotheronewithMinkowski
0 signature. Theequivalenceconditionsbetweenthetwospacesarestudiedandrelativisticdynamics,
2 notentirelycoincidentwithEinstein’sgeneraltheoryofrelativity,isdemonstrated. Themonogenic
condition is then shown to produce Maxwell’s equations and electrodynamics both classical and
n
quantized.
a
J
PACSnumbers: 02.40.Yy;03.65.Pm
1
1

I. INTRODUCTION D’Inverno [3, section 14.7]):
]
h
p r = χ m+ χ2 2mχ /2; (2)
- Our goal is to show how the important equations of − −
n physics, such as relativity equations and equations of (cid:16) p (cid:17)
and the new form of the metric is
e quantum mechanics, can be put under the umbrella of
g acommonmathematicalapproach[1,2]. We usegeomet- m 2
. 1
cs ric algebra as the framework but introduce monogenic dτ2 = − 2mr dt2 1+ m 4 (3)
i functions with their null derivatives in order to advance 1+  − 2r ∗
s theconcept. Furthermore,weclarifysomepreviouswork 2r (cid:16) (cid:17)
y in this direction and identify the steps to take in order dr2 r2 dθ2+sin2θdϕ2 .
h ∗ −
to complete this ambitious project.
p
From this equatio(cid:2)n we imm(cid:0) ediately define(cid:1)(cid:3)two coeffi-
[ Since A. Einstein formulated dynamics in 4-
cients, which are called refractive index coefficients,
dimensional spacetime, this space is recognized by the
1
vast majority of physicists as being the best for formu- m m 3
v 1+ 1+
8 lating the laws of physics. However, mathematical con- 2r 2r
n = , n = . (4)
7 siderationsleadtoseveralalternative4-Dspaces. Forex- 4 m r (cid:16) m(cid:17)
1 1
0 ample, the 4-dimensional space called 4-D optics (4DO) − 2r − 2r
1 is equivalent to the 4-D spacetime of the general theory
We devote the first part of this paper to deriving
0
of relativity (GTR) when the metric is static, and there-
them from a geometric algebra approach in a special
6
forethegeodesicsofonespacecanbemappedone-to-one
0 5D space with null geodesics, thereby establishing that
with those of the other. Then one can choose to work in
/ there is a 4DO Euclidean metric space equivalent to
s the space that is more suitable.
c the Schwarzschild metric space. We build upon previ-
i In the case of a centralmass, we can examine how the ous work by ourselves and by other authors about null
s
y SchwarzschildmetricinGTRcanbe transposedto4DO. geodesics, regarding the condition that all material par-
h The usual form of the metric is ticles must follow null geodesics of 5D space:
p
: 2m 2m −1 The implication of this for particles is
v dτ2 = 1 dt2 1 dχ2 clear: theyshouldtravelonnull5Dgeodesics.
i − χ − − χ −
X (cid:18) (cid:19) (cid:18) (cid:19) This idea has recently been taken up in the
r −χ2 dθ2+sin2θdϕ2 ; (1) literature, and has a considerable future. It
a means that what we perceive as massive par-
(cid:0) (cid:1)
where m is the spherical mass and χ is the radial coor- ticles in 4D are akin to photons in 5D.[4]
dinate, not the distance to the centre of the mass. This
Accordingly, particles moving on null
form is non-isotropic but a change of coordinates can be
paths in 5D (dS2 = 0) will appear as mas-
made that returns the expression to isotropic form (see
sive particles movingontimelike paths in 4D
(ds2 >0) ...[5]
We actually improve on these null displacement ideas
∗Electronicaddress: bda@fisica.uminho.pt
by introducing the more fundamental monogenic condi-
†TheauthorwishestothankFrankPotter,fromSciencegems.com,
tion,derivingtheformerfromthelatterandestablishing
for the enlightening discussions and corrections to the text and
equations. a common first principle.
2
II. SOME GEOMETRIC ALGEBRA follow the conventions set forth in Appendix A. We will
also assume this spacetime to be a metric space whose
Geometric algebra is not usually taught in university metric tensor is given by
courses and its presence in the literature is scarce; good
g =g g ; (8)
reference works are [6, 7, 8]. We will concentrate on αβ α· β
the algebra of 5-dimensional spacetime because this will
thedoubleindexisusedwithg todenotetheinnerprod-
be our main working space; this algebra incorporates as
uct of frame vectors and not their geometric product.
subalgebras those of the usual 3-dimensional Euclidean
The space signature is ( ++++), which amounts to
space, Euclidean 4-space and Minkowski spacetime. We −
saying that g < 0 and g > 0. A reciprocal frame is
00 ii
begin with the simpler 5D flat space and progress to a
defined by the condition
5D spacetime of general curvature (see Appendix C for
more details.) gα g =δα . (9)
β β
The geometric algebra G of the hyperbolic 5- ·
4,1
dimensional space we consider is generated by the co- Defining gαβ as the inverse of g , the matrix product
αβ
ordinateframeoforthonormalbasisvectorsσ suchthat of the two must be the identity matrix; using Einstein’s
α
summation convention this is
(σ )2 = 1,
0
(σ )2 =1−, (5) gαγgβγ =δαβ. (10)
i
σα σβ =0, α=β. Using the definition (8) we have
· 6
Note that the English characters i, j, k range from 1 to (gαγg ) g =δα ; (11)
γ β β
4 while the Greek characters α,β,γ range from 0 to 4. ·
SeetheAppendixAforthecompletenotationconvention comparing with Eq. (9) we determine gαwith
used.
Any two basis vectors can be multiplied, producing gα =gαγgγ. (12)
the new entity called a bivector. This bivector is the ge-
If the coordinate frame vectors can be expressed as a
ometric product or, quite simply, the product, and it is
linear combination of the orthonormed ones, we have
distributive. Similarly to the product of two basis vec-
tors,theproductofthreedifferentbasisvectorsproduces g =nβ σ , (13)
α α β
a trivectorandsoforthupto the fivevector,because five
is the dimension of space. where nβ is calledthe refractive index tensor or simply
α
We willsimplify the notationforbasisvectorproducts the refractive index; its 25 elements can vary from point
using multiple indices, i.e. σασβ σαβ. The algebra is to point as a function of the coordinates.[2] When the
≡
32-dimensional and is spanned by the basis refractive index is the identity, we have g = σ for the
α α
main or direct frame and g0 = σ , gi = σ for the
1 scalar, 1, − 0 i
• reciprocalframe, so that Eq. (9) is verified. In this work
5 vectors, σ , we willnotconsiderspaces ofgeneralcurvature but only
α
• those satisfying condition (13).
10 bivectors (area), σαβ, Thefirstusewewillmakeofthereciprocalframeisfor
•
the definition of two derivative operators. In flat space
10 trivectors (volume), σ ,
• αβγ we define the vector derivative
5 tetravectors (4-volume), iσ ,
• α =σα∂α. (14)
∇
1 pseudoscalar (5-volume), i σ .
01234
• ≡ Itwillbeconvenient,sometimes,tousevectorderivatives
Several elements of this basis square to unity: in subspaces of 5D space; these will be denoted by an
upper index before the and the particular index used
(σi)2 =(σ0i)2 =(σ0ij)2 =(iσ0)2 =1. (6) determines the subspace∇to which the derivative applies;
For instance m = σm∂ = σ1∂ + σ2∂ + σ3∂ . In
m 1 2 3
The remaining basis elements square to 1: ∇
5-dimensional space it will be useful to split the vector
−
derivative into its time and 4-dimensional parts
(σ )2 =(σ )2 =(σ )2 =(iσ )2 =i2 = 1. (7)
0 ij ijk i
−
= σ ∂ +σi∂ = σ ∂ + i . (15)
0 t i 0 t
Notethatthepseudoscalaricommuteswithalltheother ∇ − − ∇
basiselementswhilebeingasquarerootof 1;thismakes Thesecondderivativeoperatoristhecovariantderiva-
−
it a very special element which can play the role of the tive, sometimes called the Dirac operator, and it is de-
scalar imaginary in complex algebra. fined in the reciprocal frame gα
In 5-dimensional spacetime of general curvature,
spanned by 5 coordinate frame vectors g , the indices D=gα∂ . (16)
α α
3
Takingintoaccountthedefinitionofthereciprocalframe with the covariant derivative D. A generalized mono-
∇
(9), we see that the covariant derivative is also a vector. genicfunctionisthenafunctionthatverifiestheequation
In cases such as those we consider in this work, where
there is a refractive index, it will be possible to define Dψ =0. (23)
both derivatives in the same space.
Similarly to what happens in flat space, the covariant
Wedefine alsosecondorderdifferentialoperators,des-
Laplacianisascalarandamonogenicfunctionmustver-
ignated Laplacian and covariant Laplacian respectively,
ify the second order differential equation
resulting fromthe inner productof onederivative opera-
torbyitself. Thesquareofavectorisalwaysascalarand
D2ψ =0. (24)
the vector derivative is no exception, so the Laplacian is
a scalar operator, which consequently acts separately in
Itis possible to write a generalexpressionforthe covari-
each component of a multivector. For 4+1 flat space it
ant Laplacian in terms of the metric tensor components
is
(see[9,Section2.11])butwewillconsideronlysituations
∂2 where that complete general expression is not needed.
2 = + i 2. (17) When Eq. (23) is multiplied on the left by D, we are
∇ −∂t2 ∇
applying second derivatives to the function, but we are
Oneseesimmediatelythata4-dimensionalwaveequation simultaneously applying first order derivatives to the re-
is obtained by zeroing the Laplacian of some function ciprocalframevectorspresentinthedefinitionofDitself.
We can simplify the calculations if the variations of the
∂2 frame vectors are taken to be much slower than those of
2ψ = + i 2 ψ =0. (18)
∇ −∂t2 ∇ function ψ so that frame vector derivatives can be ne-
(cid:18) (cid:19)
glected. With this approximation, the covariant Lapla-
This procedure was used in Ref. [1] for the derivation cian becomes D2 =gαβ∂αβ and Eq. (24) can be written
of special relativity and will be extended here to general
curved spaces. gαβ∂αβψ =0. (25)
This equation can have a solution of the type given by
Eq.(21)ifagainthederivativesofp areneglected. This
III. THE MONOGENIC CONDITION α
approximationisusually ofthe sameorderasthe former
one and should not be seen as a second restriction. In-
Thereisaclassoffunctionsofgreatimportance,called
serting Eq. (21) one sees that it is a solution if
monogenicfunctions[6],characterizedbyhavingnullvec-
tor derivative; a function ψ is monogenic in flat space if gαβp p =0. (26)
α β
and only if
This equation means that the square of vector p=gαp
α
ψ =0. (19) iszero,thatis,pis avectorofzerolengthandis calleda
∇
nullvectorornilpotent. Vectorpisthemomentumvector
A monogenic function is not usually a scalar and has by
andshouldnotbeconfusedwith4-dimensionalconjugate
necessity null Laplacian, as can be seen by dotting Eq.
momentum vectors defined below.
(19) with on the left. We are then led to Eq. (18),
∇
which can also be written as
IV. EQUIVALENCE BETWEEN 4DO AND GTR
∂ ψ =∂ ψ. (20)
ii 00 SPACES
i
X
Thisrelationcanberecognizedasawaveequationinthe Bysetting the argumentofψ constantinEq.(21)and
4-dimensionalspace spanned by the σ which will accept differentiating we can get the differential equation
i
plane wave type solutions of the general form
p dxα =0. (27)
α
ψ =ψ ei(pαxα+δ), (21)
0 The lhs can equivalently be written as the inner product
of the two vectors p dx = 0, where dx = g dxβ is a
where ψ0 is an amplitude whose characteristics we shall · β
general 5D elementary displacement. In 5D hyperbolic
not discuss for now, δ is a phase angle and p are con-
α
space the inner product of two vectors can be null when
stants such that
the vectorsareperpendicularbut alsowhenthe twovec-
(p )2 (p )2 =0. (22) tors are null. Since we have established that p is a null
i 0
− vector, Eq. (27) can be satisfied either by dx normal to
i
X p or by (dx)2 = 0. In the former case the condition de-
When working in curved spaces the monogenic condi- scribesa3-volumecalledwavefrontandinthelattercase
tionisnaturallymodified,replacingthevectorderivative itdescribesthe wavemotion. Notice thatthe wavefronts
4
are not surfaces but volumes, because we are working We have now projected onto 4-dimensional space with
with 4-dimensional waves. signature (+ ), known as Minkowski signature. In
−−−
The condition describing 4D wave motion can be ex- order to check this consider again the special case with
panded as g =σ and the equation becomes
0 0
gαβdxαdxβ =0. (28) (dx4)2 = 1 (dx0)2 gmn dxmdxn; (35)
g − g
44 44
This condition effectively reduces the spatial dimension
to four but the resulting space is non-metric because all the diagonal elements g are necessarily positive, which
ii
displacements have zero length. We will remove this dif- allows a verification of Minkowski signature. Contrary
ficulty by considering two special cases. First let us as- to what happened in the previous case, we cannot now
sume that vectorg is normalto the other frame vectors obtain (dx4)2 by squaring a vector but we can do it by
0
so that allg factorsare zeroed;condition (28) becomes consideration of the bivector
0i
g00(dx0)2+gijdxidxj =0. (29) dx4ν = 1 gµg4dxµ. (36)
g g44
44
All the terms in this equation are scalars and we are
allowed to rewrite it with (dx0)2 in the lhs All the products g g4 apre bivectors because we imposed
µ
g tobenormaltotheotherframevectors. When(dx4)2
g 4
(dx0)2 = ij dxidxj. (30) is evaluated by an inner product we notice that g g4
−g 0
00 has positive square while the three g g4 have negative
m
We could have arrived at the same result by defining a square,ensuringthata Minkowskisignatureisobtained.
4-dimensional displacement vector Naturally we have to impose the condition that none of
the frame vectorsdepends onx4. Bivectorν issuchthat
dx0v = −1 g dxi; (31) ν2 = νν = 1 and it can be obtained by a Lorentz trans-
i
√g00 formation of bivector σ04.
and then squaring it to evaluate its length; v is a unit ν =T˜σ T, (37)
04
vector called velocity because its definition is similar to
the usualdefinitionof3-dimensionalvelocity;its compo- where T is of the form T = exp(B) and B is a bivector
nents are whose plane is normal to σ . Note that T is a pure ro-
4
tation when the bivector plane is normal to both σ and
dxi 0
vi = dx0. (32) σ4.
In special relativity it is usual to work in a space
Beingunitary,thevelocitycanbeobtainedbyarotation spanned by an orthonormed frame of vectors γµ such
of the σ4 frame vector that (γ0)2 = 1 and (γm)2 = −1, producing the desired
Minkowski signature [6]. The geometric algebra of this
v =R˜σ4R. (33) space is isomorphic to the even sub-algebra of G4,1 and
so the area element dx4ν (36) can be reformulated as a
The rotationangle is a measure of the 3-dimensionalve-
vector called relativistic 4-velocity.
locitycomponent. Anullanglecorrespondstov directed
Equations (30) and (34) define two alternative 4-
along σ and null 3D component, while a π/2 angle cor-
4 dimensionalspaces,thoseof4-dimensional optics(4DO),
responds to the maximum possible 3D component. The
with metric tensor g /g and general theory of rela-
ij 00
idea thatphysicalvelocitycanbe seenasthe 3Dcompo- −
tivity (GTR) with metric tensor g /g , respectively;
µν 44
nent of a unitary 4D vector has been explored in several −
in the former x0 is an affine parameter while in the lat-
papers but see [10].
ter it is x4 that takes such role. In fact Eq. (34) only
Equation(31)projectstheoriginal5-dimensionalspace
covers the spacelike part of GTR space, because (dx4)2
intoaspacewith4dimensions,withEuclideansignature,
is necessarily non-negative. Naturally there is the limi-
where an elementary displacement is given by the varia-
tationthatthe framevectorsareindependentofbothx0
tion of coordinate x0. In the particular case where g =
0 and x4, equivalent to imposing a static metric, and also
σ the displacement vector simplifies to dx0v = g dxi
0 i that g = g = 0. Provided the metric is static, the
0i µ4
and we can see clearly that the signature is Euclidean
geodesics of 4DO can be mapped one-to-one with space-
because the four g have positive norm. Although it has
i like geodesics of GTR and we can choose to work on the
not been mentioned, we have assumed that none of the
spacethatbestsuitsusforfreefalldynamics. Foraphys-
frame vectors is a function of coordinate x0.
icalinterpretationofgeometricrelationsitwillfrequently
ReturningtoEq.(28)wecannowimposethecondition
be convenient to assign new designations to the 5D co-
that g is normal to the other frame vectors in order to
4 ordinatesthat acquire the role of affine parameter in the
isolate (dx4)2 instead of (dx0)2, as we did before;
nullsubspace. Wewillthenmaketheassignmentsx0 t
g and x4 τ. Total derivatives with respect to these≡co-
(dx4)2 =−gµν dxµdxν. (34) ordinate≡s will also receive a special notation: df/dt=f˙
44
5
and df/dτ = fˇ. Special units conventions used in this to Schwarzschild metric, allowing 4DO to be used as an
paper are detailed in appendix B. alternative to GTR. Recalling that we derived trajecto-
Unless otherwise specified, we will assume that the riesfromsolutions(21)ofa4-dimensionalwaveequation
framevectorassociatedwithcoordinatex0 isunitaryand (28), it becomes clear that orbits can also be seen as 4-
normal to all the others, that is g = σ and g = 0. dimensionalguidedwavesby whatcould be describedas
0 0 0i
Recalling from Eq. (30), these conditions allow the def- a 4-dimensional optical fibre. Modes are to be expected
inition of 4DO space with metric tensor g . Although in these waveguides and we shall say something about
ij
we could try a more general approach, we would loose them later on.
the possibility of interpreting time as a line element and
this, as we shall see, provides very interesting and novel
interpretations of physics equations. In many cases it V. FERMAT’S PRINCIPLE IN 4 DIMENSIONS
is also true that g is normal to the other frame vec-
4
tors and we have seen that in those cases we can make Fermat’s principle applies to optics and states that
metric conversionsbetween GTR and4DO;it will be in- the path followed by a light ray is the one that makes
teresting,however,toexamineoneortwosituationswith the travel time an extremum; usually it is the path that
non-normal g4 and so we leave this possibility open. minimizes the time but in some cases a ray can follow a
Forthemomentwewillconcentrateonisotropicspace, pathofmaximumorstationarytime. Thesesolutionsare
characterized by orthogonal refractive index vectors gi usually unstable, so one takes the view that light must
whosenormcanchangewithcoordinatesbutisthesame follow the quickest path. In Eq. (30) we have defined a
for all vectors. Normally we relax this condition by ac- timeintervalassociatedwitha4-dimensionalelementary
cepting that the three gm must have equal norm but g4 displacement, which allows us to determine, by integra-
can be different. The reason for this relaxed isotropy is tion, a travel time associated with displacements of any
found in the parallel we make with physics by assign- size along a given 4-dimensional path. We can then ex-
ing dimensions 1 to 3 to physical space. Isotropy in a tend Fermat’s principle to 4D and impose an extremum
physicalsense needonly be concernedwith these dimen- requirement in order to select a privileged path between
sions and ignores what happens with dimension 4. We any two 4D points. Taking the square root to Eq. (30)
will therefore characterize an isotropic space by the re-
Ifrnadceteivdewinedceoxulfdraamlseoga0cc=eptσ0a, ngomn-=ortnhroσgmon,agl4g=wnit4hσi4n. dt= gij dxidxj. (41)
4 −g
therelaxedisotropyconceptbutwewillnotdosoforthe r 00
moment. Integrating between two points P and P
1 2
Equation (30) can now be written in terms of the
isotropic refractive indices as P2 g P2 g
t= ij dxidxj = ij x˙ix˙jdt. (42)
−g −g
dt2 =(n )2 (dxm)2+(n dτ)2. (38) ZP1 r 00 ZP1 r 00
r 4
Xm Inordertoevaluatethe previousintegralonemustknow
the particular path linking the points by defining func-
Spherically symmetric static metrics play a special role;
tions xi(t), allowing the replacementdxi =x˙idt. At this
this means that the refractive index can be expressed as
stage it is useful to define a Lagrangian
functions of r if we adopt spherical coordinates. The
previous equation then becomes g
L= ij x˙ix˙j. (43)
dt2 = (n )2 dr2+r2(dθ2+sin2θdϕ2) + −2g00
r
+(n4d(cid:2)τ)2. (cid:3) (39) The time integral can then be written
Since we have g4 normal to the other vectors we can ap- P2
ply metric conversionandwrite the equivalentquadratic t= √2Ldt. (44)
form for GTR ZP1
2 2 Time has to remain stationary against any small
dt n
dτ2 = r changeofpath;thereforeweenvisageaslightlydistorted
(cid:18)n4(cid:19) −(cid:18)n4(cid:19) ∗ path defined by functions xi(t)+εχi(t), where ε is arbi-
dr2+r2(dθ2+sin2θdϕ2) . (40) trarily small and χi(t) are functions that specify distor-
∗
tion. Since the distortionmustnotaffectthe endpoints,
As we stated in(cid:2)the introduction, the usu(cid:3)al form of thedistortionfunctionsmustvanishatthosepoints. The
Schwarzschild’s metric is given by Eq. (1) but a more time integral will now be a function of ε and we require
interesting, isotropic form is the one in Eq. (3). The that
latter can be compared to Eq. (40) allowing the deriva-
tion of the refractive indices in Eqs. (4). These refrac- dt(ε)
=0. (45)
tive indices provide a 4DO Euclidean space equivalent dε
(cid:12)ε=0
(cid:12)
(cid:12)
(cid:12)
6
Now,theLagrangian(43)isafunctionofxi,throughg and from Eq. (9)
αβ
and also an explicit function of x˙i. Allowing for a path
change, through ε makes t in Eq. (44) a function of ε √ g00v =givi =gigijx˙j =gjx˙j. (54)
−
P2 The conjugatemomentumandvelocityarethe same but
t(ε)= 2L(xi+εχi+x˙i+εχ˙i)dt. (46) their components are referred to the reciprocal and re-
ZP1 fractive index frames, respectively.[16] Notice also that
p
This can now be derived with respect to ε by virtue of Eq. (22) it is also
p
dt(ε) P2 1 v = i . (55)
i
= p
0
dε (cid:12)ε=0 "ZP1 √2L ∗
(cid:12) The Euler-Lagrange equations (51) can now be given
(cid:12) ∂L ∂L
(cid:12) χ˙i+ χi dt . (47) a simpler form
∗ ∂x˙i ∂xi
(cid:18) (cid:19) (cid:21)ε=0
v˙ =∂ L. (56)
Note that the first term on the rhs can be written i i
P2 1 ∂L P2 ∂(√2L) Thissetoffourequationsdefinestrajectoriesofminimum
χ˙idt= χ˙idt. (48) time in 4DO space as long as the frame vectors g are
ZP1 √2L∂x˙i ZP1 ∂x˙i known everywhere, independently of the fact thatαthey
may or may not be referred to the orthonormed frame
This can be integrated by parts
via a refractive index. By definition these trajectories
P2 ∂(√2L) ∂(√2L) P2 arethegeodesicsof4DOspace,spannedbyframevectors
χ˙idt = χi (49) g /√ g , with metric tensor g /g .
ZP1 ∂x˙i " ∂x˙i #P1 − iFol−low00inganexactlysimilar−proijced0u0rewecanfindtra-
jectories which extremize proper time, defined by taking
P2 d ∂(√2L)
χidt. the positive square root of Eq. (34). The Lagrangian is
−ZP1 dt ∂x˙i ! now defined by
The first term on the second member is zero because χi = 1gµνxˇµxˇν. (57)
vanishes for the end points; replacing in Eq. (47) L −2 g
44
dt(ε) 1 P2 d 1 ∂L Consequently the conjugate momenta are
= +
dε (cid:12)(cid:12)(cid:12)(cid:12)ε=0 +√21ZP1∂L(cid:20)dtχ(cid:18)id−t.√L∂x˙i(cid:19) (50) νµ = ∂∂xˇLµ = −gg4µ4νxˇν. (58)
√L∂xi
(cid:21) FromEq.(22)wehaveν =p /p ;theassociatedEuler-
µ µ 4
Therhsmustbezeroforarbitrarydistortionfunctionsχi, Lagrange equations are
soweconcludethatthefollowingsetoffoursimultaneous
equations must be verified νˇµ =∂µ . (59)
L
d 1 ∂L 1 ∂L ”These are, by definition, spacelike geodesics of GTR
= ; (51)
dt(cid:18)√L∂x˙i(cid:19) √L∂xi waimthetmhoedtrficortoennseo-tro-−ognµeνg/ego4d4easincdmwaepphianvgebtehtuwseednefi4nDeOd
these are called the Euler-Lagrangeequations. and spacelike GTR. Recalling the conditions for this
Consideration of Eqs. (31) and (34) allows us to con- mapping to be valid, all the frame vectors must be in-
clude that the Lagrangian defined by (43) can also be dependentofbothtandτ andg andg mustbenormal
0 4
written as L = v2/2 and must always equal 1/2. From tothe other 3framevectors. Intensorterms,allthe g
αβ
the Lagrangian one defines immediately the conjugate must be independent from t and τ and g =g =0.”
0i µ4
momenta
∂L g
v = = − ijx˙j. (52) VI. THE SOURCES OF REFRACTIVE INDEX
i ∂x˙i g
00
Notice the use of the lower index (v ) to represent mo- Thesetof4equations(56)definesthegeodesicsof4DO
i
menta while velocity components have an upper index space; particularly in cases where there is a refractive
(vi). The conjugate momenta are the components of the index, it defines trajectories of minimum time but does
conjugate momentum vector not tell us anything about what produces the refractive
index in the first place. Similarly the set of equations
givi (59) defines the geodesics of GTR space without telling
v = (53)
√ g us what shapes space. In order to analyse this question
00
−
7
we must return to the general case of a refractive frame We will now investigate spherically symmetric solu-
g without other impositions besides the existence of a tions in isotropic conditions defined by Eq. (39); this
α
refractive index. meansthattherefractiveindexcanbeexpressedasfunc-
Considering the momentum vector tions of r. The vector derivative in sphericalcoordinates
is of course
p=p gα =p n ασβ, (60)
α α β
1 1 1
D = σ ∂ + σ ∂ + σ ∂
with n γnβ = δβ, we will now take its time derivative. n r r r θ θ rsinθ ϕ ϕ −
α γ α r (cid:18) (cid:19)
Using Eq. (D4) 1
σ ∂ + σ ∂ . (67)
t t τ τ
− n
p˙ =x˙ (Dp)=x˙ G. (61) 4
· ·
The Laplacian is the inner product of D with itself but
By a suitable choice of coordinates we can always have
theframevectors’derivativesmustbeconsidered;allthe
g0 =σ0. Wecantheninvokethefactthatforanelemen-
derivatives with respect to r are zero and the others are
tary particle in flat space the momentum vector com-
ponents can be associated with the concepts of energy, ∂ σ =σ , ∂ σ =sinθσ ,
θ r θ ϕ r ϕ
3D momentum and rest mass as p = Eσ0 + p+ mσ4 ∂ σ = σ , ∂ σ =cosθσ , (68)
θ θ r ϕ θ ϕ
−
(see [1, 11] and Sec. VIII.) If this consequence is ex- ∂ σ =0, ∂ σ = sinθσ cosθσ .
θ ϕ ϕ ϕ r θ
− −
tended to curved space and to mass distributions, we
write p = Eσ0 +p+mg4, where now E is energy den- After evaluation the curved Laplacian becomes
sity, p=pmgm is 3D momentum density and m is mass 1 2 n′ 1
density. The previous equation then becomes D2 = (n )2 ∂rr+ r ∂r− nr ∂r+ r2 ∂θθ +
r (cid:18) r
E˙σ0+p˙ +mg˙4 =x˙ G. (62) cotθ csc2θ
· + ∂ + ∂ (69)
r2 θ r2 ϕϕ −
When the Laplacian is applied to the momentum vec- (cid:19)
1
tor the result is still necessarily a vector ∂ + ∂ .
− tt (n )2 ττ
4
D2p=S. (63)
The search for solutions of Eq. (63) must necessarily
VectorSiscalledthesourcesvector andcanbeexpanded startwith vanishing secondmember, a zerosourcessitu-
into 25 terms as ation, which one would implicitly assignto vacuum; this
is a wrong assumption as we will show. Zeroing the sec-
S =(D2nβα)σβpα =Sβασβpα; (64) ond member implies that the Laplacian of both nr and
n must be zero;consideringthatthey arefunctions ofr
4
where pα = gαβpβ. Tensor Sαβ contains the coefficients we get the following equation for nr
of the sources vector and we call it the sources tensor.
The sources tensor influences the shape of geodesics as n′′ + 2n′r (n′r)2 =0, (70)
weshallseeinoneparticularlyimportantsituation. One r r − n
r
importantconsequencethatwedon’tpursue hereis that
with general solution n =bexp(a/r). It is legitimate to
by zeroingthe sourcesvectoroneobtains the waveequa- r
make b = 1 because the refractive index must be unity
tionD2p=0,whichacceptsgravitationalwavesolutions.
at infinity. Using this solution in Eq. (69) the Laplacian
Ifσ0 isnormaltotheotherframevectorswecanwrite
becomes
p = E(σ0 + v) in the reciprocal frame, with v a unit
vector or p =E(−σ0+v) in the direct frame. Equation D2 = e−a/r ∂ + 2∂ + a ∂ + 1 ∂ + (71)
(61) can then be given the form rr r r r2 r r2 θθ
(cid:18)
E˙(σ0+v)+Ev˙ =σ0+v·G. (65) +cort2θ ∂θ+ csrc22θ ∂ϕϕ −∂tt+ (n1)2 ∂ττ;
(cid:19) 4
Since G can have scalar and bivector components, the
which produces the solution n = n . So space must
scalar part must be responsible for the energy change, 4 r
be truly isotropic and not relaxed isotropic as we had
whilethe bivectorpartrotatesthe velocityv. Thebivec-
allowed. The solution we have found for the refractive
tor part of G is generated by D p, which allows a sim-
∧ index components in isotropic space cancorrectly model
plification of the previous equation to
Newton dynamics, which led the author to adhere to it
v˙ =v (D v), (66) for some time [10]. However if inserted into Eq. (34)
· ∧ this solution produces a GTR metric which is verifiably
iftheframevectorsareindependentoft. Thisequationis in disagreement with observations; consequently it has
exactlyequivalenttothesetofEuler-Lagrangeequations purely geometric significance.
(56) but it was derived in a way which tells us when to The inadequacy of the isotropic solution found above
expect geodesic movement or free fall. forrelativisticpredictionsdeservessomethought,sothat
8
wecansearchforsolutionsguidedbytheresultsthatare Equation (73) can be interpreted in physical terms as
expected to have physical significance. In the physical containing the essence of gravitation. When solved for
worldwe are never in a situation of zerosources because sphericallysymmetricsolutions,aswehavedone,thefirst
the shape of space or the existence of a refractive index member provides the definition of a stationary gravita-
mustalwaysbetestedwithatestparticle. Atestparticle tional mass as the factor M appearing in the exponent
is an abstraction corresponding to a point mass consid- and the second member defines inertial mass as 2n .
4
∇
ered so small as to have no influence on the shape of Gravitationalmass is defined with recourseto some par-
space; in reality a point particle is a black hole in GTR, ticle which undergoes gravitational influence and is ani-
although this fact is always overlooked. A test particle matedwithvelocityvandinertialmasscannotbedefined
must be seen as source of refractive index itself and its without some field n acting upon it. Complete investi-
4
influence on the shape of space should not be neglected gation of the sources tensor elements and their relation
in any circumstances. If this is the case the solutions for to physical quantities is not yet done; it is believed that
vanishing sourcesvectormay haveonly geometricmean- 16 terms of this tensor have strong links with homolo-
ing, with no connection to physical reality. gous elements of stress tensor in GTR, while the others
The question is then what should go into the second are related to electromagnetic field.
member of Eq. (63) in order to find physically meaning-
ful solutions. If we are testing gravity we must assume
somemassdensitytosuffergravitationalinfluence;thisis VII. ELECTROMAGNETISM IN 5D
whatisusuallydesignatedasnon-interactingdust,mean- SPACETIME
ing that some continuous distribution of non-interacting
particles follows the geodesics of space. Mass density is Maxwell’s equations can easily be written in the form
expected to be associated with S4 ; on the other hand ofEq.(63)ifwedon’timposetheconditionthatg should
4 4
weareassumingthatthis massdensity is verysmalland remain normal the other frame vectors; as we have seen
so we use flat space Laplacian to evaluate it. We conse- in section IV this has the consequence that there will be
quently make an ad hoc proposal for the sources vector no GTR equivalent to the equations formulated in 4DO.
in the second member of Eq. (63) Wewillconsiderthenon-orthonormedreciprocalframe
defined by
S = 2n σ . (72)
−∇ 4 4 q
gµ =σµ, g4 = Aµσ +σ4; (77)
µ
Equation (63) becomes m
where q and m are charge and mass densities, respec-
D2x˙ = 2n σ ; (73)
−∇ 4 4 tively, and A = Aµσµ is the electromagnetic vector po-
tential,assumedtobeafunctionofcoordinatestandxm
as a result the equation for n remains unchanged but
r
but independent of τ. The associated direct frame has
the equation for n becomes
4
vectors
n′′ + 2n′4 n′rn′4 = n′′ + 2n′4. (74) g =σ q A σ , g =σ ; (78)
4 r − n − 4 r µ µ− m µ 4 4 4
r
When n is given the exponential form found above, and one can easily verify that Eq. (9) is obeyed. The
r
the solution is n = √n . This can now be entered into momentum vector in the reciprocal frame is p = Eσ0+
4 r
Eq. (34) and the coefficients can be expanded in series pmσm+qAµσµ+mσ4 andGinthesecondmemberofEq.
and compared to Schwarzschild’s for the determination (61) is G=qDA. We will assume DA to be zero,as one
·
of parametera. The final solution, for a stationarymass usuallydoesinelectromagnetism;alsoDcanbereplaced
M is by µ because the vector potential does not depend on
∇
τ. It is convenient to define the Faraday bivector F =
n =e2M/r, n =eM/r. (75) µ A, similarly to what is done in Ref. [6]; the dynamics
r 4
∇
equation then becomes
The equivalent GTR space is characterized by the
quadratic form p˙ +qA˙ =qx˙ F; (79)
·
and rearranging
dτ2 =e−2M/rdt2 e2M/r (dxm)2. (76)
−
m p˙ =qx˙ F qA˙. (80)
X · −
ExpandinginseriesofM/r thecoefficientsofthismetric The firstterminthe secondmemberis the Lorentzforce
one would find that the lower order terms are exactly and the second term is due to the radiation of an accel-
the same as for Schwarzschild’sand so the predictions of erated charge.
the metrics are indistinguishable for small values of the Recalling the wave displacement vector Eq. (D1) we
expansion variable. Montanus [12] arrives at the same have now
solutions with a different reasoning; the same metric is q
also due to Yilmaz [13, 14, 15]. dx=σαdxα Aµσ4dxµ. (81)
− m
9
This correspondsto a refractiveindex tensor whosenon- the other 3 square to minus unity. Those bivectors be-
zero terms are longto theevensub-algebraofG ,whichisisomorphic
4,1
the the algebra of Minkowski spacetime, as we have al-
q
nαα =1, n4µ = Aµ. (82) readystated. Itisperfectlylegitimatetoreplacethesaid
−m
bivector factors by Dirac matrices, as was demonstrated
According to Eq. (64) the sources tensor has all terms in the above cited references. We can then rewrite the
null except for the following monogenic condition as
q (γµ∂ +ip )ψ =0, (89)
S4 = D2A ; (83) µ 4
µ µ
−m
whichcanbeimmediatelyrecognizedasDirac’sequation
where D is the covariantderivative given by if p is assigned to the particle’s rest mass. The mono-
4
genic function given by Eq. (19) can then be given the
q
D=gα∂ =σµ∂ +(σ4+ A σµ)∂ . (84) usual physical interpretation of a Dirac spinor
α µ µ 4
m
ψ =ψ ei(Et+p·x+mτ); (90)
We can then define the current vector J verifying 0
where E is energy,p is 3-dimensionalmomentum andm
µ 2A= µ F =J, (85)
∇ ∇ is rest mass.
In order to separate left and right spinor components
where
we use a technique adapted from Ref. 6. We choose
J = mS4 σµ. (86) an arbitrary base element which squares to identity, for
µ
− q instance σ , with which we form the two idempotents
4
(1 + σ )/2 and (1 σ )/2. The name idempotents
4 4
Please refer to [6, Chap. 7] or to [8, Part 2] to see how means that they rep−roduce themselves when squared.
theseequationsgenerateclassicalelectromagnetism,par- Theseidempotentsabsorbanyσ factor;ascanbeeasily
4
ticularlyhowsettingthecurrentvectortozerogenerates checked(1+σ )σ =(1+σ )and(1 σ )σ = (1 σ ).
4 4 4 4 4 4
electromagnetic waves. Obviously we can decompose the wa−vefunction−ψ a−s
1+σ 1 σ
4 4
ψ =ψ +ψ − =ψ +ψ . (91)
+ −
VIII. MONOGENIC FUNCTIONS AND 2 2
QUANTUM MECHANICS
Thisapparentlytrivialdecompositionproducessomesur-
prising results due to the following relations
Dirac equation has been derived from the 5-
dimensional monogenic condition in previous works [1, eiθ(1+σ ) = (cosθ+isinθ)(1+σ )
4 4
11]; the motivation for returning to the subject here is
= (1cosθ+iσ sinθ)(1+σ ) (92)
4 4
the correctionofthe electro-dynamicsequation,which is
= eiσ4θ(1+σ ).
incorrectin the earlier paper [11] and absent in the later 4
one[1]. Becauseweareworkingingeometricalgebra,our
and similarly
quantummechanics equationswill inheritthatcharacter
but the isomorphism between the geometric algebra of eiθ(1 σ )=e−iσ4θ(1 σ ). (93)
4 4
− −
5D spacetime, G , and complex algebra of 4 4 matri-
4,1
ces, M(4,C), ensures that they can be trans∗lated into We could have chosen other idempotents, which would
the moreusualDiracmatrixformalism. The equivalence produce similar results. The available idempotents gen-
between the two formulations has been amply demon- erate an SU(4) group and it has been argued that they
strated in the two references above. may be related to different elementary particles.[11]
Recalling the monogenic condition (19), we will now Electrodynamics can now be implemented in the the
expand it into 3 terms same way used in Sec. VII to implement classical elec-
tromagnetism. The monogenic condition must now be
(σ0∂ +σm∂ +σ4∂ )ψ =0. (87) established with the covariant derivative given by Eq.
0 m 4
(84)
We have already established that this equation accepts
q
solutionsinthe formofEq.(21)andweusethattoeval- σµ∂ ψ+ σ4+ A σµ ∂ ψ =0. (94)
µ µ 4
uate the derivative with respect to x4 m
(cid:16) (cid:17)
Multiplying on the left by σ4 and taking ∂ ψ =imψ
(σ0∂ +σm∂ iσ4p )ψ =0. (88) 4
0 m 4
−
[γµ(∂ +iqA )+im]ψ =0. (95)
µ µ
If the equation is multiplied by σ4 on the left, the first 4
terms onthe firstmember acquirebivectorfactorsofthe This equation can be compared to what is found in any
form σ4µ, the first of which, σ40 squares to unity, while quantum mechanics textbook..
10
It is now adequate to saya few words about quantiza-
tion, which is inherent to 5D monogenic functions. We
have already seen that these functions are 4-dimensional
waves, that is, they have 3-dimensional wavefronts nor-
mal to the direction of propagation. Whenever the re-
fractive index distribution traps one of these waves a 4-
dimensional waveguide is produced, which has its own FIG. 1: Indices in the range {0,4} will be denoted with
allowed propagating modes. In the particular case of a Greek letters α,β,γ. Indices in the range {0,3} will also re-
central potential, be it an atom’s or a galaxy’s nucleus, ceive Greek letters but chosen from µ,ν,ξ. For indices in the
we expect spherical harmonic modes, which produce the range {1,4} we will use Latin letters i,j,k and finally for in-
well known electron orbitals in the atom and have un- dices in the range {1,3} we will use also Latin letters chosen
known manifestations in a galaxy. from m,n,o.
APPENDIX B: NON-DIMENSIONAL UNITS
IX. CONCLUSION
The interpretation of t and τ as time coordinates im-
Every physicist dreams of finding a unified formula- plies the use of a scaleparameterwhichis naturallycho-
tion for the fundamental laws of physics. It is usually sen as the vacuum speed of light c. We don’t need to
accepted that in order to achieve such objective a new include this constant in our equations because we can
paradigm is needed, meaning that one must surely step always recover time intervals, if needed, introducing the
back from accepted physics principles and start afresh speedoflightatalaterstage. Wecanevengoastepfur-
from new simpler ones. Ideally one should have a small ther and eliminate all units from our equations so that
setofprinciples,validforallareasofphysics,andallthe they become pure number equations; in this way we will
important relations should flow naturally from mathe- avoid cumbersome constants whenever coordinates have
matical reasoning. to appear as arguments of exponentials or trigonomet-
In this paper we extend a proposal previously made ric functions. We note that, at least for the macroscopic
in that direction, that one should accept 5-dimensional world, physical units can all be reduced to four funda-
spacetime as the adequate space to formulate the laws mental ones; we can, for instance, choose length, time,
of physics and introduce in this space the condition of mass and electric charge as fundamental, as we could
monogeneity. We had shown in another work that this just as well have chosen others. Measurements are then
condition is sufficient to arrive simultaneously at special made by comparison with standards; of course we need
relativity and the free particle Dirac equation; here we four standards, one for each fundamental unit. But now
show that by generalizing the monogenic condition to note that there are four fundamental constants: Planck
bentspacesoneisabletoobtainrelativisticdynamicsnot constant (h¯), gravitational constant (G), speed of light
entirely coincident with GTR but also electrodynamics, in vacuum(c) and protonelectric charge(e), with which
both classical and quantized. we can build four standards for the fundamental units.
Maxwell’s equations were also derivedfromthe mono- Table I lists the standards of this units’ system, fre-
genicconditionandwereunifiedtotheequationsrespon-
sibleforgravitationaldynamics. Theprocedureisnotyet
TABLE I: Standardsfor non-dimensional units’system
entirelysatisfactory,inthesensethatanad hoc proposal
hadtobemadeinrespecttoinertialmass;infuturework Length Time Mass Charge
we hope to find a suitable formulationfor the derivation
of curvature from first principles. G¯h G¯h ¯hc
e
c3 c5 G
r r r
APPENDIX A: INDEXING CONVENTIONS
quently called Planck units, which the authors prefer to
designate by non-dimensional units. In this system all
In this section we establish the indexing conventions the fundamental constants, h¯, G, c, e, become unity, a
used in the paper. We deal with 5-dimensional space particle’sComptonfrequency,definedbyν =mc2/¯h,be-
but we are also interested in two of its 4-dimensional comesequaltothe particle’smassandthe frequentterm
subspaces and one 3-dimensional subspace; ideally our GM/(c2r) is simplified to M/r. We can, in fact, take
choice of indices should clearly identify their ranges in all measures to be non-dimensional, since the standards
order to avoid the need to specify the latter in every aredefinedwithrecoursetouniversalconstants;thiswill
equation. ThediagraminFig.1showstheindexnaming be our posture. Geometry and physics become relations
conventionusedinthispaper;Einstein’ssummationcon- between pure numbers, vectors, bivectors, etc. and the
vention will be adopted as well as the compact notation geometricconceptofdistanceisneededonlyforgraphical
for partial derivatives ∂ =∂/∂xα. representation.
α