“generated” by the energy-momentum density of Mies
continuous matter [8], which later failed to replace point
masses of the pre-quantum Universe. This metric theory
of gravitational fields around still localized particles,
known today as General Relativity, can operate fluently
with curved spatial displacement d= dd
ij
N
Nij
lxx
of a
point mass
N
m

1
by accepting the Schwarzschild [9] or
Droste empty-space solutions [10] without specific restric-
tions on the space metric tensor
N
NN NN
ijoi ojooij
g
gg g

222
ddd=dd
N
.
GR solutions for dynamics of the considered probe parti-
cle N are related to its space-time interval,
N
NN
gxx l




s
, where the time element
12
2
1
dd d
oi
NN
NooNoo oi
gxggx

N
depends on the local
pseudo-Riemannian metric tensor
g
and, consequently,
on local gravitational fields. Hereinafter, 1, 2,3i
,
0,1,2,3
, and the speed of light c = 1 in the most of
equations.
The author intends to revisit time, d
N
, and space,
ddd
ij
N
Nij
lxx
, elements within the conventional GR
four-interval ddd
N
s
gxx

in order to prove that
the time element of the freely moving mass
N
m
dt de-
pends not only on the world time differential (with
dδddd >0
oo o
oo
txxx
di
) and gravitation, but also on
space differentials or matter displacements
x
in gra-
vitational fields. Then the ratio ddlv
NN
, called the
physical speed in Special Relativity (SR), should
non-linearly depend on spatial displacement
ddd
ij
N
Nij
lxx
, called the space interval in SR. Non-
linear field contributions to such an anisotropic (Finsler-
type) time element
d,d
N
x
x
within the four-interval
22 2
dd ,dd
s
xxl x
 of Einstein’s Relativity may
modify Schwarzschild-type metric solutions based on
curved three-space around non-physical point singulari-
ties for GR energy-sources [2]. Moreover, the calculated
ratio
dd =
NN
lvv
may differ from a real speed
ddl
N
O
measured by a motionless observer with local
proper-time

dd=0dlv

ON
. This metric-type
anisotropy of measured time rate was already confirmed
by observations of the gravitational Sagnac effect when
dd 0
i
gx
oi
. Rigorous consideration of anisotropic
physical time

d,d d
x
xv

dl
of each moving parti-
cle may preserve universal flatness of its 3-space element
. We shall start from the 1913 Entwurf metric formal-
ism for the geodesic motion of passive masses. Then, we
shall employ the tetrad approach and analyze non-linear
relations in the anisotropic relativistic time for a passive
mass under the geodesic motion. This will suggest to
keep for physical reality Euclidean 3D sub-intervals in
curved 4D intervals of moving probe particles.
The first attempt to interpret GR in parallel terms of
curved and flat spaces was made by Rosen [11], Ein-
stein’s co-author of the unpublished 1936 paper about the
non- existence of plane metric waves from line singulari-
ties of cylindrical sources. Later, Sommerfeld, Schwinger,
Brillouin and many other theorists tried to justify Eucli-
dean space for better modern physics. Moreover, the
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1467
original proposal of Grossmann (to use 4D Riemannian
geometry for geometrization of gravitational fields in the
1913 Entwurf version of GR) relied exclusively on 3D
Euclid- ean sub-space. Grossmann did not join further
GR metric developments with curved 3D intervals. In
1913 Einstein clearly underlined that space cannot exist
without matter in the Entwurf geometrization of fields.
However, at that pre-quantum time there were not many
options for geometrization of particles, because all (but
Mie) considered them localized entities for local events.
This might be the reason why in January 1916 Einstein
promptly accepted Schwarzschild’s warping of 3D space
around the point particle. Nonetheless, in 1939 Einstein
finally rejected Schwarzschild metric singularities for
physical reality. The well derived Schwarzschild’s solu-
tion has no mathematical errors in the empty-space para-
digm. However, we tend to use the non-empty-space
paradigm for the global superfluid overlap of self-co-
herent elementary particles, when each continuous parti-
cle is distributed over the entire Universe together with
the elementary field. This nonlocal approach to matter
can avoid difficulties of the Entwurf geometrization of
fields, proposed in 1913 without geometrization of parti-
cles, and, ultimately, can avoid non-physical warping of
the universal spatial ruler, which becomes the same for
all local observers in the flat Universe.
Contrary to non-metric approaches to gravitation with
spatial flatness, for example [12,13], we shall comply
with the Einstein-Grossmann extension of Special Rela-
tivity (SR) to gravitation through warped space-time with
non-Euclidean pseudo-geometry, founded by Lobachev-
sky, Bolyai and Riemann [11]. Inertia and gravitation
keep the same metric nature in our reiteration of the Ein-
stein-Grossmann approach. The proposed 4D geometri-
zation of matter together with fields will be made under
six metric bounds for
g
(called sometime intrinsic
metric symmetries) in the GR tensor formalism for every
physical object. In other words, the author is planning to
revise neither Einstein’s Principle of Relativity nor the
GR geometrization concept. On the contrary, I am plan-
ning further GR geometrization of continuous particles
together with the already available geometrization of
gravitational fields. Local nullification of the Einstein
tensor curvature for paired densities of the distributed
astroparticle and its field will be requested in their rest
frame of references. I intend to prove, for example, that
Schwarzschild’s solution for a central field is not “the
only rotationally invariant GR metric extension of the SR
interval”. One should admit non-empty (material) space
or Newtonian stresses of the material medium-aether
associated with continuous very low dense distributions
of non-local gravitation/inertial mass-ener- gies. Then
bound ensembles of elementary radial energies form so
called “macroscopic” bodies with sharp visual bounda-
ries (observed exclusively due to experimental restric-
tions to measure fine energy densities).
First, we discuss a local time element, dddlv

,
which should be considered as a chain function of speed
ddvl
or spatial displacement of a passive ma-
terial point in external gravitational field. Then, we dis-
cuss the electric Weber-type potential energy
dl

211
11
W
ooNoooo
UU vmUPUP
 for a point planet
with mass
N
mPmV and relativistic energy oNo
in the
Sun’s static field generated by the active energy-charge
M
E. Ultimately, this paper presents the self-contained
GR scheme with the energy-to-energy interaction poten-
tial oo M
UPGE r


for Machian mechanics of non-
local astroparticles with an analytical radial density
2
2
4π
o
nrrrr r
o
instead of the Dirac delta den-
sity
r
. One should see arguments for the singular-
ity-free gravitational contribution oo
UP to the smooth
metric tensor component

2
1gUP

P
ooo o. The main
challenge here was to keep the free fall universality and
the GR Principle of Equivalence for all carriers of probe
(passive, inertial) energies o in radial fields of the
Sun’s gravitational (active) energy E
M
.
In the speed-dependent time approach, the warped GR
four-interval
ddd,d
s
ll
cannot be approximated in
weak fields by pure time and pure space subintervals,
like in Schwarzschild-type solutions [9,10] with their
formal time and space metric split without chain relations.
In order to justify the indivisible non-linear involvement
of space displacements into physical time
ddl
N
of a
probe particle under the geodesic motion, one should
clarify how the already known gravitational tests of GR
can be explained quantitatively without departure from
spatial flatness. Then we discuss our energy-to-energy
attraction under the Einstein-Grossmann geodesic motion
in metric fields with flat 3-section (i.e. without Schwar-
zschild singularities). The author also accepts the Ein-
stein-Infeld-Hoffmann approach (but under flat 3-space)
to non-point slow-moving gyroscopes in order to de-
scribe the Gravity Probe B quantitatively.
In 1913, Einstein and Grossmann put weak Newtonian
field only into the temporal part of the Entwurf 4D inter-
val. Today, one tends to justify that strong-field GR met-
ric may also admit for reality six metric bounds ijij
which preserve universal 3D interval in specifically
curved space-time for any elementary particle N. Then
the metric tensor
g
for curved 4D with flat 3-section
depends on four gravitational potentials o
GUP

2
=1
oNoo
Pmg v
for the particle energy-charge .
This finding matches 6 metric bounds for spatial flatness
under any gravitational fields and their gauges. Since
2000, this post-Entwurf metric scheme with warped
space-time, but strictly flat three-space, became consis-
tent with the observed Universe’s large-scale flatness,
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV
1468
confirmed at first by balloon measurements of the Cos-
mic Microwave Background and then by all ongoing
Wilkinson Microwave Anisotropy Probe (WMAP) Ob-
servations of the flat Universe [17-20]. This new reading
of curved 4D geometry with non-linearly dilated anisot-
ropic time and flat non-empty space, explains quantita-
tively all GR tests, the known planet perihelion preces-
sion, the radar echo delay, and the gravitational light
bending, for example [21-23].
Speed-dependent time corrections to post-Newtonian
dynamics in Sun’s flat material space lead to computa-
tion results similar to numerical computations of other
authors who traditionally correct Newton in empty, but
curved 3-space. Observable dynamics of matter in mod-
erate and strong static fields provides, in principle, an
opportunity to distinguish our metric solutions with iso-
tropic flat space and speed-dependent time from Sch-
warzschild’s solutions, based on curved 3-space and di-
lated time. Alternative empty-space and non-empty space
paradigms can also be distinguished through different
probe body dynamics in stationary fields of rotating as-
trophysical objects.
2. Warped Four-Space with Intrinsic Metric
Symmetries for Flat Three-Space
To begin, we employ the GR tetrad formalism, for ex-
ample [24,25], in covariant expressions for an elementary
rest-mass
N
mi
in order to justify the mathematical op-
portunity to keep a flat 3D subspace
N
x
in curved
four-space
N
x
with a pseudo-Riemannian metric tensor
N
g
g

(for short). First, we rewrite the curved
four-interval,
 
dd
d d,
xx
x x


2
ddd
dd
N
NNN
sgxxg
ee xx

 



 



in plane coordinates

dd
x
ex

d and
 
d
x
ex

,
g1,1,


1, d with
1. One can findia

o
eg
 

bb
;
oo oo i
gg and 0, i
ee
from the equal-
ity

2
dd
i
ij
2
dd d
oi
oo ij
s
gxgxxx


, ioioo
g
gg .
irst glance, thad i
i
At f
e spatial tri
 
bb
N
ee(a, b = 1, 2, 3
and ,
= 0, 1, 2, 3) should apend essentially
on thvitational fields of other particles because this
triad is related to components of
lways de
e gra
N
g
. However, this
might not be the case when there are ernal metric rela-
tions or bounds in the general pseudo-Riemannian metric
with the warped tensor
int
N
g
. Shortly, a curved mathe-
matical 4D manifold does necessarily mean a curved
3D section for real matter (warped 2D paper in 3D trash,
for example, keeps parallel Euclidean lines due to steady
metric relations between neighboring points of paper).
It is not obvious that physical restrictions for four-v
not
e-
lo
rved pseud
cities of real matter, like 1gVV


, might require to
keep flat 3D sections of cuo-Riemannian 4D
manifolds. Therefore, let us look at three spatial compo-
nents i
V of the four-vector ddVgxs

by using
the conventional tetrad formalism,
12
1i
gg vvv

    
 

 

12
1.
oo i ii
ob
iii i
ob
bb
oo iibb
Ve
VeV eV
gg evvv
  
 
Here, we used

=
o
iooi
egg and
 

 
12 1
bb
b
Vv vv

 . Now one can
trace that the considered equalities

ii
VeV
12
1;
bb
v v

admit
trivial relations

ib
vvvv and
 
bb
ii i
bb
vevv
d velocities,

b
i

between the curve
ddd
joi
iij ooi
vxgxgxdd
j
ij
x

, and the plane
velocities,
 

dd
a
ab
b
vx
. All spatial triads for t
ations may be c
N
hese
“trivial” relonsidered as universal Kro-
necker delta symbols,
 
=
i
bb
i
e
, and, consequently, the
three-space metric tenrelevant to gravitation
fields, i.e. 1N
sor is ir
K
ijoi oj ooijijijij
ggg g

. All com-
ponents
g
, involved in these six relations
gra
N
, may de-
pend on vitations fields or system accelerations but
their combination should always keep spatial flatness
under admissible coordinate transformations. One could,
surely, ignore flat 3-space option within curved 4D ma-
nifold, as was suggested by the above tetrad analysis, by
trying curved 3D solutions in i
V when
 
i
bb
i
e
. But
we do not see much physical sense in suchtions
and, therefore, restrict GR geometrical constructions by a
partial case with six metric relations 1
oi oj ooijij
ggg g
complica
.
Applications of pseudo-Riemannian t
3-sections will quantitatively describe all known gravita-
tional experiments plus magnetic flux quantization. The
latter and the Aharonov-Bohm effect require only flat
3-space for satisfactory interpretations.
Again, we shall read

Kv
space-time with fla
g
ee
 
through

=;
oggg and

bb
oooo i
e
 
0,
b
i
e
for  all
physical cases we arested in describine interg. This
means for our consideration that
 
oo
ooo o
g
ee,
 
oo
oio i
g
ee, and
  
oo o
ij ijij
gee e
 
ab o
abij ij
ee e

.
And Euclidean spatial geometry,
e applied to pseudo-
Riemannian 4-intervals of
2
ddddd
ij ij
K
Kijij
lxxxx

, will b
all particles (due to intrinsic
metric relations
1
KK KK
oi ojooijij
gg gg
).
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1469
Contrary to univeements
variant four-interpe
rsal spatial acdl, in-
vals have differen ward metrics for
pa
displ
tly
rticles K and N, because N
K
g
g

and dd
K
N
s
s
in different external fields (for example, in the two-body
problem). The GR four-intervelected
ergy carrier,

al for a s mass-en-
2
22
dd 2d
d ddd,
i
oij
oi
oo ij
oo
gx
s
lgxxx
g





(1)
is defined for only one selectedobe mass


pr
N
m despite
notifications dd
N
s
s and dd
N
x
x are regularly
l s
m e
used for brevity. This geometrical 4-intervahould be
physically co mented in terms of tim
2
dd
x
and
space 2
ddddd
ij ij
ij ij
lxxxx

 elements, albeit 3-
space differentials di
x
contribute to particsical
time
le’s phy

dd
x
. We prove below that particles proper time
d
depends on dlen in constant gravitational fields
(wher is a first integral of motion o
Pconst
ev
e there
).
h an anisotropic time element
 
Suc

ddd d
oi
Noo i
x
gx xgx
 of the moving mss a
N
m always counts its spatial
ented gravitational field, des
imm e definition
r gravi
dis
pite
placement dl in a ori-
the fact that it is not
ediately obvious from the physical tim
for metrics with 0
oi
g. This post-Newtonian phe-
nomenon, related to the energy nature of anisotropic time,
appears in nonlineatational equations through the
energy (velocity)-dependent potentials. Our interpreta-
tion of the warped four-interval (1), based on warped
anisotropic time in isotropic non-empty flatspace rather
than in empty warped space, may be considered as a
prospective way for further developments of the 1913
metric gravitation through joint geometrization of dis-
tributed fields and distributed elementary particles.
Now we return to components of the four-vector
dd
NN
Vgxs
. Notice that

  
bo
b
V eV


   

 


1
o
bo oo
bo o
N
Ve eV
eVV eV
mU
 
 




with the
 
bo
bo
V V
four-velocity
 
eV




0 and
 
bb
i
e
22
;
11
NNNi
mmv
vv







22
1;
11
,
No
oNioo
NN
P
mg mg g
vv
KU





(2)
where

,
because b
eoi
. Flat t
to introduce
l pot
hree-space ge-
gauge invarianometry is a promising wayt
gravitatio entials,naoN
PG
 
GU
 with

oo
N
ooN
Ue mVUP
 

 , for the passive (probe)
mass
N
m, in close analoglectro-
al electric charge. The
point is that a four-momentum
y to four-c
tials for the classic
omponent e
magnetic poten
N
N
N
PmV
of the se-
lected scalar mass
N
m (without rotation) can be rigor-
ously decomposed into mechanical,
N
K
, and gravita-
tional,
N
U
, parts onunder strict spatial flatness, ly
j
iij
vv
, 2i
i
vvv, dd
ii
vx
,

12
dddsxx
,
dd
x
gx

, dd
N
x
x
,
g
ioioo
gg ;
j ooijij
ggg
ij iij
g

. Again, we use a time-lik
worldline witdd>
o
tx and
e
h 0
12
ddd>0
i
oo i
gxgx
 
onal energy
ofor the passive-inertial mN >
0. The gravitati-momentum part U
is de-
fined in (2) for a selected mass
N
m and its positively
defined passive energy 0
oNo
PmV, associated with
the global distribution of all other sses ma
K
m. This gra-
vitational part, UGP

o
, is not a full four-vector in
pseudo-Riemannian space-time, like
N
P
, r is the me-
chanical summanN
no
d Km
.
Because
 

0,
bbb
i
e
 and
 
dd
x
ex

, the
tetrad with the zeo
r (i.e. time) label takes the following
corom (2): mponents f


21 21
11 ;1
o
evUmvUm

 
21
1.
oN iN
oN
vUm


Ultimately, the tetrad

e
for the selected particle N
and the metric tensor

N
g
ee

, with gg


, depends in Cartesiaoordinates only on the gravi-
tational four-potential
n c
o
UPG
(ind for t e
e
troduceh
rlativistic energy-charge ooN
cP cP [26]),
 
 

 

 
  


1
11
22
21 1
21 21
1
22
2
2
1
1
11 11
11 1
1
1
oo oo
oo
N
oooooNo o
oo
N
oioio Ni N
N
ooi o
oo ab
N
ijijab ijijNij
N
ooij oij
oo ij
Noo i
m
UP UP
geevUm UP
geevUm vUm
gUP
gee eevUUm
gUUP
gUPUU








 

 

 21
,,,
oiij ij
jo NioN
Pg UPg
2
1
oN
evU



 


(3)
where we used

2
2
11
oo
ooo oo
gee vU and
22
1
ooo
Vg v to prove that
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV
1470

11
11
o oo o
1
oo oo
g
gU e, the paP UP

 . Therefors-
sive-inertial GR energy,

21
2
1
1
oo
v UP
mvU

2
11
,
ooo
o
Pmgv m
takes a linear superposition of kinetic and potential ener-
gies in all points of pseudo-Riemannian space-time
warped by strong external fields. Note that we did not
assign spin S
or internal angular mechanical momen-
tum to the Einstein-Grossmann “material point” or the
probe mass
N
m with the energy-momentum (2). The
affine connecions for the metric tensor (3) depend only
on four gravitational potentials
t
o
UP
in our space-
time geometr which is not relevant to warped mani-
folds with asymmetrical connections and torsion fields,
for example [27-29].
Every component of the metric tensor in (3) depends
on the gravitational part
y,
N
No
UmVm GP

 of
the probe carrier energy-momentum P
. At the same
time, all the components of the three-space metric tensor,
1
ijoi oj ooijij
ggg g
, aom
the gravitational potential
re always independent fr
o
from (3
the inhar
GUP

or its gauge.
Such inherent metric symmetries for 3D subspace may be
verified directly). In fact, our tetrad, and the metric
tensor, depends formally onmonic Weber-
type potentials,

211
11
No oo
UvmUP UP


 , as-
sociated with the particle speed 222
ddvl
. In 1848
Weber introduced [18] the non-Coulomb potential

2
2 212
12qq vr ac-
12 easurements of
ch q
perime
of
based on lab m
celerating forces between movin1 and 2
q
with the relative radial velocity 2
12 1v . This might was
the first extal finding that mechanical inertia and
acceleration depend on the kinetic energy or speed
interacting bodies.
By substituting the metric tensor (3) into the interval
222
ddddd
g arges
n
s
gxx l


, one can rewrite (1) and
submit the chain relation for the proper time dd
N
of the probe mass-energy carrier N in external gravita-


tional fields,


12
2
1
dd ddd
o
oi
NN
oooooi
N
lgxggxex
2
2
d
dd 1 .
dd
N
o
N
Ul
xx
ml

 



(4)
Notice that the proper-time differential,

1
1
oK
OoK
xUm
 , of the local observer K, with
e element (4)
ergy-charge
dd
i
d0x
of K and d0
K
l, differs from the tim
the moving mass m with the GR en
2
1
ooo
g v.
ing masits pro
Pm The proper interval ds of the mov-
s and per time element (4) depends, in
general, on all four components of U
. Therefore, the
observable three-speed dd
O
l
, of a moving particle
always differs in relativistic gravito-mechanics from the
non-linear ratio
dd dllv
, called the particle’s
physical speed (1). The chain relation
ddf
in
the physical time (4) of a moving particle changes the
GR interpretation of the geodesic motion and allows to
apply flat 3D space for gravitational tests.
The metric tensor (3), the interval (1), and the local
time element (4) are associated with warped space-time
specified by external fields for one selected mass
N
m or,
to be precise, for the passive energy-charge
N
o
P. We
may employ common three-space for all elementary par-
ticles (due to universal Euclidean geometry for their spa-
tial displacements), but we should specify arped
space-times with differently dilated times for the mutual
motion of gravitational partners. The particle’s time ele-
ment
w
ddd,
Nlv

in (4) may depend on the parti-
cles velocity or displacement. Ultimately, a nonlinear
time rate

dd
oo
exx
(hereinafter dd
f
tf
,
dd
o
txc) of moving material objects in (4) depends on
the ratio 22 2
lv
. This non-linear chain relation can
be simplified in several subsequent steps through the
following equalities to (4):
12
11
12
1
1d
dd 1
11
1
Ni
ioo oi
iN
i
vt
tUP PUv
vUm v
UP x



1
1
d.
1
o
io
oo
Um
tUP

(5)
Such anisotropic time dilatation in (5) by the ex
four-potential
ternal
N
NN
oo
GUP
results in the gravitational
Sagnac effect when an observer compares the dynamics
of different elementary energy-charges o
P in fields with
0
i
U
.
Now, one may conclude that the anisotropic time ele-
ment d
in the metric interval (1) and, consequently, in
the physical speed dd,vl
depends only on univer-
ur sal fopotentials G
for positive probe charges
0
o
P. The potential energy part

NNN
mUP mV


contributes to GR energy-momentum of the probe body
and, therefore, to its passive energy-charge,
N
oo
mV P
.
The universal ratio o
UP
should be tried
gravitation as a metric field four-potential (which is not a
covariant four-vector) of active gravitational charges for
passive energy-charges. Contrary to Newton’s gravita-
tion for masses, Einstein’s gravitation is the metric the-
ory for interacting energies. The static Sun, with the ac-
tive energy-charge 2
M
EMc, keeps the universal po-
tential
in Einstein’s
1;0
mM
UE GEr
 in the Sun’s frame of
reference for the passive, inertial energy content
2
om
cPEconst mc of the probe mass
N
m. Below,
we employ the unof the Sun’s potential, iversality
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1471
NN
ooMo
UP r all planets in our com-
putations for gravitational tests of General Re
(5) and flat material spce filled
everywhere by 2
r gravitational fields and the 4
r
GEr rr  , fo
lativity
with dilated time (4)-a
3. Flatspace for the Planetary Perihelion
Precession
Now we consider
extended masses.
the metric tensor (3) a central gra-
0
for
vitational field with a static four-potential, 1
io
UP
,
11
oo M
UPGE r

 , where 2
EMcrGcons 
active gravitaMo
of the “mo
ment with sp
t is
tionless” Sun
n’s cente
ricalatial flatn
th
of
e
sphe
tional energy
ymm
(in the moving Solar system). We use Euclidean geome-
try for the radial distance 1
ru
from the Sur
etry in agreeess
maintained by (3) for any gravitational four-potential
G
s
and its gauge
. Let us denote the energy con-
tent of a probe mass m in tc central field as a pas-
sive energy-charge

he stati
2
1
oNoNoo m
PmVmgv E .
n, the interval (1) for the passive energy carrier in a
central field with 0
i
U takes two equivalent presenta-
The
), tions due to (4) and (5

2
2
222
2
d
d1 1dd
dd
Mm
GE El
2
22
d1 d,
M
s
tl
l


 


(6)
rm
GE
tl
r




where iterations

 
2
22
1dd dd d
Mm
tGEErm lll





ion
22
d1
over the chain funct
2
ddl
in the Lorentz factor
resun lt i

2
d1 M
tGEr


for the Sun-Mercury po-
2
ten ergy tial enoMm
UGEEr . In other words, t e
-depend
iterations common for all p
h
specific, Weber velocityent potentials exhibit
after chainrobe particles local
time, 22
ddd d
N
K
sl
static metric solution (6
oes not coincide with the Schwarzschild me
herefore, the Schwarzschild exten-
si
 static fields. Spherical
coordinates can be equally used in (6) for the Euclidean
element 2222222
dd dsinddd
ij
ij
lrrrxx

  in
flat laboratory space.
, in
The) for probe elementary en-
ergy-charges in non-empty space of the radial energy-
charge dtric
[9] in empty space. T
on of the SR interval is not the only rotationally in-
variant solution which GR’s tensor formalism can pro-
pose for tests of space-time-energy self-organizations.
Ultrarelativistic velocities, dd1vl
and
2
10v, in the Weber-type energy-to-energy inter-
action in (6) revise the Schwarzschild singularity. The
latter is not expected at the finite radius in the energy-
charge formalism of Einstein. Einste
n’s gravitatioin, “the
reluctant father of black holes”, very strictly expressed
his final opinion regarding the Schwarzschild solution:
“The essential result of this investigation is a clear un-
derstanding as to why Schwarzschild singularities do not
exist in physical reality” [31]. In authors view, Schwarz-
schild’s metric solution, and all Birkhoff class solutions
for the empty space dogma, originates with ad hoc mod-
eling of matter in the 1915 Einstein equation in terms of
point particles. However, Einstein anticipated extended
sources for his equation and for physical reality. Below,
we prove that the static metric (6) corresponds to the 4
r
radial energy-charge or the extended source of gravity.
Therefore, our analysis denies the empty space paradigm.
Non-empty material space is in full agreement with Ein-
stein’s idea of continuous sources and Newton’s “
surd” interpretation of distant attractions through stresses
in an invisible material ether (called in 1686 as “God’s
sensorium”).
Our next task is to derive integrals of motion for the
passive (probe) mass-energy in a strong central field
from the geodesic equations
ab-
22 2
dd ddd
x
pxxp


 .
Nonzero affine connections
for the metric (6) take
the following components: rr

, 2
sin
rr

 ,
d2d,
r
ttoo
g
r ΓΓΓΓ1
rrrr
r

 
,
Γsin cos

 , ΓΓctg

 
 , and
d2d
tr rtoo oo
tt
g
gr, wh ere oo
g
is the function next
2
dt in the interval (6), 222
ddd
oo
to
s
gt l.
wing thpoach withBy folloe verified apr
π2const
for the ispic ceral field, for exam-
ple [25], and by substituting flatspace connections
otro nt
into GR’s geodesic eq
tric write the following grav
uations, one can define the para-
me differential dp andita-
tional relations,




22
2122
2
d ddconst
dd const,ddconst
d dconst
oooo m
m
oo m
psgtsEm
rpJrsJEmL
rpJr gmE


 
 

2
2
22
dd 1,d
ddd d1,
mo
o
gtp
rsrs Emg

(7)
with the first integrals ,,
m
Em and
J
of the relativis-
tic motion in strong static fields.
The last line in (7) is the interval equation
222
ddd
oo
s
gt l
with two integrals of motion
22222
dd
moo
Em gt s and π2
. Therefore, the
lly the n
in
sca-
lar invariant (6) is actuaequatio of motion for the
constant energy charge m
Econst a central field
with the static Weber-type potential
 

2
1
,
omo
MM
m vUEU
GErGE

 
W
oo
UU
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV
1472
which is inharmonic for t 20
W
o
U. Re-
call that Schwarzschild’s curved 3D
he Laplacian,
solution not only
differs from (6), but results in conceptual inconsistencies
[32,33] for the Einstein equation. We can use (6)-(7) for
relativistic motion in strong central fields in order to re-
inforce the ignored statement of Einstein that Schwarzs-
l reality. child singularities do not exist in physicaThere
are no grounds for metric singularities either in the in-
terval (6), or in the radial potential

W
o
Ur for 0r,
because

dd oo
tgrrGM
 is a smooth func-
tion. One can verify that the non-empty space metric
tensor (3), as well as 20
W
o
U, does correspond to the
continuous energy-source in the 1915 Einstein equa-
tion.
The strong field relations (6)-(7) can be used, for ex-
ample, for computations of planetary perihelion preces-
sion in the solar system. The planet’s gravitational en-
ergy for the GR energy-to-energy attraction,
1
oMmom
UGEEr rEu
, where 2const
o
rGMc
and 1ur, is small compared to the planet’s energy,
1const
om
urU E, that corresponds to the non-
relativistic motion of a planet N (with const1
m
Em,
mM
EE, and 222
dd 1vl
) in the Sun’s rest
frame, with 0U. The Gement for
i
s from (6) or (7) as
R time elt
read

the plane

1
12r

2
2
22 22
2
22
d
ddddd1 dd
d d,
m
o
oo
El
sllt ru
ml
utru l

 

 
(8)
where we set 1
o
ru  , 1
m
Em,
22
dddll
 ,
2
dt
.
with spatial displacement 2
d
o
rul
the right hand side of (8) belothe physical time
element withinvariant
and dd
The

22
dtl
field term on
ngs to
2
d
the in
s
. Th
u
is displacement
n-lire of anisotropiccorresponds to the near chain nat
time

no

ddlddlf
origiom the Weber , -
ty
nating fr
pe energy potential 2
1Uvm
in (3). Ths no
departure from Euclidean space geometry with the flat
metric
ere i

22 224222
d=π2d dddlrruuu


  in
the chain reading of geometrical intervals (6) or (8).
Againnear time with chain spatial dis-, a particle’s non-li
placement
ddl
differom the proper-time

rs in (8) f
12
d12 d
Oo
ru t
 of the local (motionless) observer.
Displacement corrections, 22
ddrult, for the non-rela-
tivistic limit are very small compared to the main gravi-
tational corrections,

2o
ru, to Newtonian time rate
o
222
12 dd
o
trurult
. However, the chain de-
pendence of a icle’s time element 2
d
o
part
from spatial
displacement dl accounts for the reverse value of this
time element,
2
22
dd
o
rul
, that is ultimately a way to
restore strict spatial flatness at all orders of Einstein’s
metric gravitation. Hee is some kind of analogy
re small contributions of
Maxwell’s displacement currents restore strict charge
conservation in Ampere’s quasi-stationary magnetic law.
Two integrals

re ther
with electrodynamics whe
e th
of motion12 dd
om
ru t sE m and
2ddrsL
result from (7) and (8) for weak fields in a
rosette motion of planets,

222222
1213=,
oo
ruLruuuE Lm

  (9)
where dduu
and 1ru . Indeed, (9) may be
o
fferentiated with respect todi the polar angle
,
2
2
93
oo
r
u uru2
3,
22
oo
ru uru
L
 
  (10)
by keeping only the largest gravitational terms. Thi
ay be solveo step
s equ-
ation md in tws when a non-corrected
Newtonian solution,

21cos
oo
urL

 , is substi-
tuted into the GR correction terms at the right hand side
of (10).
The most important correction (which is summed over
century rotations of the planets) is related to the “reso-
nance” (proportional to cos

) GR terms. Therefore,
one may ignore in apart from
224
2cosuL (10) all corrections

and 24
cos
o
uu rL

. Then the
approximate equation for the rosette motion,
u234
6cos
oo
urLrL

  , leads to the well known
perihelion precession
222
Δ6π6π1
oo
rLr a


 

,
which may also be derived through Schwarzschild’s met-
ric with w, as in [21-
25].
approximations arped three-space
It is important to empha
reces
size that the observed result
sion Δ
for a planet perihelion p
(in the Solar non
ated time by Sun’s energy densi
-
empty flatspace with dil-
ties) has been derived here from the invariant four-in-
terval (1) under flat three-space, ,
ij ij
rather than
under empty but curved three-space.
4. The Radar Echo Delay in Flatspace
The gravitational redshift of light frequency
can be
considered a direct confirmation that gravity couples to
the energy content of matter, includithe massless
photon’s energy Eng
, rather than to the scalar mass of the
2
c for
inverse
particle. Indeed, Einstein’s direct statement Em
all rest-mass particles is well proved, but the
reading, 2
mEc, does not work for electromagnetic
waves (with 0m
) and requires a new notion, the wave
energy-charge 20Emc

or the relativistic mass
0
m
.
In 1907, Einstein introduced the Principle of Eq
lence for a uniformly accelerated body and concluded
that its potential energy depends on the gravitationally
passive (“heavy”) mass associated with the inertial mass
[34]. This correct concluf Einstein was generalized
ron
uiva-
sion o
in a wg way that any energy, including light, has a
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1473
“relativistic mass” (the gravitational energy-charge in our
terminology) for Newtons mechanics. Proponents of this
generalization in question proposed that photon’s “rela-
tivistic mass” is attracted by the Sun’s mass
M
in
agreement with the measured redshift

12
S
EEmGMR mc
 

 . Nonetheless,
the coherent application (in the absence of the correct
EM wave equations in gravitational fields) of the “rela-
tivistic mass” to zero-mass waves promptly resulted in
the underestimated light deflection,
2
22
S
GMR crR
 , for the
al field [35]. In
1917, when Schwarzschild’s option [9] for spatial curva-
ture had been tried for all GR solutions, the new non-
Newtonian light deflection,
oS “mechanical free
fall” of photons in the Sun’s gravitation
4oS
rR
, had been pre-
dicted due to additional contributions
on. Later, all measurements
supported this curve-space modification for the “relativ-
istic mass” deflection by the Sun that provided false
“experimental evidences” of non-Euclidean three-space
in contemporary developmenravitation.
Below, we prove that Einstein’s GR for the Maxwell
wave equation firmly maintains the flatspace concept for
interpretation of light phenomena in gravitational fields if
one coherently couples the Sun’s rest energy to the pho-
ton’s wave energy E
from the supposed
spatial curvature in questi
ts of metric g
. We consider both the radar echo
delay and the gravitational deflection of light by coupling
its
imp
energy-charge with local gravitational potentials. Our
purpose is to verify that Euclidean space can match the
known measurements [21-23,36,37] of light phenomena
in the Solar system. Let us consider a static gravitational
field (0
i
g, for slicity), where the physical slow-
ness of photons, 1
nvc
, can be derived directly from
the covariant Maxwell equations [24], 1oo
ng


.
Recall that a motionless local observer associates oo
g
with the gravitational potential oo
UP at a given point.
The light velocity dd
O
vl
, measured by this observer,
as well as the observed light frequency dd
oO
t

,
is to be specified with respect to the observer’s time rate
dd
Ooo
g
t
. This consideration complies with Ein-
steins approach, where the light’s redshift is associa
with different clock rates (of local observers) in the Sun’s
gravitational potential [34].
Compared to the physical speed of light,
ted
1
dd
o
vl cn
, its coordinate speed
2
dd
2
1
oo oo
O
oo
tn
rr
cc
d
d=
O
lc
gcg
1
l
rr
 
 

(11)
 
is double-shiftedtial
 
 
by the gravitational potenoo
UP
o
rr, where 2
oS
rGMc oS
rrR
1.48km and
.
Notice that both the local physical slowness 1oo
ng
and the observer time dilation dd
O
t
sponsible for the double slowness of the coordinate ve-
locity (11), which is relevant to observations of light co-
ho reads
through relation (11) as
oo
g are re-
ordinates or rays under gravitational tests.
A world time delay of Mercury’s radar ec
22
o
2
44
ln 220μs,
oM
SES
S
2d
11 2
Δ2d
MM
EE
lx
lx
rx
tl
cc
l
x
y
and 7
5.7910 km
MS
r are
the Earth-Sun and Mercury-Sun distances, respectively.
Notice that in flat space we use the Euclidean metric for
spatial distance,

rr
r
cR

(12)





where 5
710km
S
yR is the radius of the Sun,
while 8
1.49510 km
ES
r
12
22
rxy , between th
ter (0,0) and any point (x, y) on the photonic ray. One can
measure in the Earth’s laboratory only the physical time
e Sun’s cen-
delay E
Eoo
g
t
, which practically coincides
with
the world time delay t
in the Earth’s weak field, i.e.
220
Ets

 . From here, the known experimental
results [24-25,37] the radar echo delay (12),
based on strictly flat three-space and dilated time as in
correspond to
A coordinate angu
1913 Entwu rf metric scheme.
5. Gravitational Light Bending in
Non-Empty Flatspace
lar deflection


 of a light
wave front in the Sun’s gravitational field can be
promptly derived in flat space geometry by using the
coordinate velocity (11) for observations,

00
2
2
2d2d o
r
l
lx
ycy 2
32
022
d4
x
41
.75.
So
oS
S
y
r

(13)
at principle to
light waves. This basic principle of physics should also
justify spatial flatness under suitable applications [38-40].
In agreement with Einstein’s original co
[34], one may relate the vector component o
Rx
rR
xR
 
The most rigorous classical procedure to derive the ray
deflection (13) is to apply the verified Ferm



 




nsideration
K
in the
scalar wave equation 0KK
to the measured
(physical) energy-frequency
of the photon
(dd ,
oooo
cK Etconst

 ). Recall that
o
P is also the measured particle’s energy in the similar
equation, 24
PP mc
, for a rest-mass particle. The
scalar wave equation 0
N
KKgKK


 has the
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV
1474
following solution for the electromagnetic wave,


22
22 222
dd
dd
dddd
ddd d,
oi
oo i
ij ooo
ooo
i
o
iioo iooo
Ktc gK
KKgKgK g
tcg
Ktxcgl
Kxlggtcg



oo
i
ijoo io
i
oo
gK
K



 

(14)
with

1
di
=, d
j
ij oo
K
EExg
lg c



.
The Fermat-type variationspect to with res
and
u
(1
ru
,
, and π2
aree spherical c
nates) f in a static gravitational field are
thoordi-
or photons

2
222
d
dd
d
dd1 0,
j
oij
ii
ioo
o
o
x
Kx x
cg l
uu ru




 

(15)
2
cu
(where

1
oo o
2
g
ru
 , 0
i
g
,ij ij
,
222ij
dddx
ij
lx rr
for o
ru
 ) resulting in a couple of
1,
22
const
oo
u

(16)

2
1 cos
light ray equations

2
2
(1 4)
2oo
ru uu
uuru





 
Solutions of o
ru
(16), 1
sin 2
oo
ur u

and 1
ooo S
rur R, may
n of

r
be used for the Sun’s weak
field. The propagatio light from
,

π
  to

r
 ,

corresponds  
to the angular deflection

1
arcsin41 cos4=1.75
oSo S
rRrR



 

fr l direction. This deflection coin
with (13) and
23-25].
We may that this no nee t
the “non-Ne
ne strictly follows Einstein’s
original approach to light in gravitational fields [34]. In
fact, the massless electromagnetic energy exhibits an
om the light’s initia-
cidesis in agreement with the known
measurements 1.660.18
 
 , for example [
concludeere do warp Eucli-
dean three-space for the explanation of wto-
nian” light deflections if o
inhomogeneous slowness of its physical velocity,
dd
Ooo
vl cg
, and, therefore, a double slowness of
the coordinate velocity, dd oo
ltcg. This coordinate
velocity slowness is related to the coordinate bending of
light measured by observers. In closing, the variational
Fermat’s principle supports Entwurf physics of Einstein
and Grossmann with dilated time and strict spatial flat-
6. Geodetic and Frame-Dragging Precessions
of Orbiting Gyroscopes
ness for light in the Solar system.
Precession of the
has been co
utions of masses. This
original GR approach practically coincides in the weak
ph
orbiting gyroscopes in the Gravity
Probe B Experiment [41] mpared only with
Schiffs formula [42,43] based on the Schwarzschild-type
metric for curved and empty 3D space. Here the author
plans to criticize the point spin model for GP-B compu-
tations in favor of the regular Einstein-Infeld-Hoffman
approach to slowly rotating distrib
Earths field with our flatspace reading of Einstein’s
ysics. Recall that our Entwurf-type space interval is
strictly flat due to the intrinsic metric bounds in the GR
four-interval (1) with the metric tensor (3). However, the
GR tensor formalism can be universally applied to any
warped space-time manifold with or without intrinsic
metric bounds.
By following Schiff and many other point particle
proponents in gravitation, one has to assume for a mo-
ment that the vector geodesic equation,
dd ddSp Sxp


 , in pseudo-Riemannian four-
space with only symmetrical connections,

 ,
may be applied to the point spin “four-vector” S
with
“invariant” bounds 0VS
or i
oi
SxS for or-
thonormal four-vectors,

ΓΓ ΓΓ .
oj okj jjk
ioikio ikj
dΓΓ
d
oj
i
SSx Sx
tio i j
x
xxx S 

(17)
Our flat-space for a strong static field with (3) and
0
oi
g




,

2
1
11
oo oo oo
g
UP g
 , and ij
gij
 d
formally maintain an inertial conservation,
, woul


22const,
ooij
ij i
ooooi ji
xg xg



SS
ij
oo
gg


SS SS
S
SSS
vS S
in agreement with Einstein’s teaching for a free-falling
bo
(curved space) tends to suggest [25,42,43] the n
dy. At the same time, Schwarzschild’s metric option d
on-com-
pensated Newtonian potential GMr

“free fall” equation,

2
const gSS

even in the
 
21 2
Sch

vS S.
Therefore, formal applications of the Einstein- Gross-
mann geodesic relations (derived for spatial translations
of material points) to localized spins S
(which are not
four-vectors in 4D manifolds with symmetrical affine
connections) contradict the spirit of GR inertial motion
and, ultimately, the Principle of Equivalence.
Our affine connections ΓΓ

, related to the met-
ric tensor (3), depend only on four field potentials
111
,GUP UPUP

 . This post-Entwurf metric
oooio

Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1475
tensor has been introduced for ergy-momen-
tum (2) without an
Moreover, neither the mechanical part,
the local en
y rotational or spin components.
K
, nor the gra-
vitational part, o
PG
, in (2) are separately covariant
four-vectors in warped space-time with the metric tensor
(3). Therefore, there are no optimistic grounds to believe
that four spin components S
might accidentally form a
covariant four vector in space-time with symmetrical
cof thennections for translatio energy-momentum
four-vector, PKPG
n o
o

l con
k

12
k
k o
oo
UPg


oo
UP
 . Nonetheless, we try by
nections for the point spin
avenue (17) in question in constant fields (when
0
og


, for simplicity),



 


2
2
111
122
11 1
211
11
1
2
21
2
1
j
ioj oijjoooij ooo
o
ioooioioo
oi j ioooijooo
j
ikjooo ijo
io ookjo
j
o oi
UPgUP gUP g
UPUP g
PU UPgUPg
UU PgU PgUP
UP gUP
UP UU


 


 

 



 

 

 
(18)
chance t
2
ik

One
h
i

U

could start
ese sym
i k
oo
ji
UP
P
UU
metrica
2

2
ij ooo
UUPg


11
M
GE r

 and
oo
o oo
oi
j o
P






11
2
ko ooiooo
i jook
ko
gUPg
U
P g


with
13
Ir

ss M rotating with l
2G
mio
UP
ai
rω f
o
or th
w a
e homogeneous sphe
ngular i.e. rrical
1
velocity,
,
21,
iio
UUP M
EM, and
2
25MR
nn
n
Im
keeping only
n for E14]. Then, by
linear terms to
E
xv Rr [
with respect io
UP, one
owly rotating gravitational field: can rewrite (17) for a sl


11
1
dln
d
.
2
ioo
kiooo
ooo
jjo j
SSx g
t
UPg
g
SU xxSg




vanish fo-rotating cewhen 0
1
2
ik
ooo
jk j
ikoooki
jk
oo
UP g
S
UP gUP
P


j
ji
i
g
r non
oo
(19)
The last three terms on the right-hand side of (19) are
responsible for frame rotation and frame dragging, which
nters
and
0UPecessions of magnitude vec-
tor

io . Pr

the constant
1
o
JSv UP
 , obtained for the weak-
field limit of
 

22
12
1
1
2const,
j
ooi ioj
iijij
jojii ji j
gSSUPUUPxS
UP SxSSSJJ









when
22
vS
1, 1
ii
oo
UP xx

, and iiioo
1
vUP
x

in (19),
 


11
11
1
2
,
jk
jik
o kio
jjoioo iojoo
JUP UP
JUP UP UPUP




11
1
d
d2
jji
iioo joo
J
JvUP vUP
t


 


(20)
mayf’s non-relativistic predic-
tion
be compared with Schif
geo fd
ddt

J
J = for Gravity Probe B.
The second summand at the right hand side of (20),

11
2
jk
jikokio fd
i
J
UP UPJ

, takes e
Schiff’s answer [42,43] for the frame-dragging precession,
xactly

332
12 .
2
GI
rr
r

 



r
r
3
fd GI 
r
(21)
The first and third precession terms in (20) depend on
the Earth’s radial field
1
ioo
UP
and they count to-
gether geodetic and frame phenomena. These terms pro-
vide
11 1
2.
gfoo o
P
P

 vU U Such a prec
for a point spin model, formally borrowed from the Ein-
stein-Grossmann theory for the probe mass without rota-
ession
tion, fails to reiterate the already well verified de Sitter
geodetic precession,

13
323 2
geoo o
UP GM
 vrvr ,
of the Earth-Moon gyroscope in the Sun’s field, where
122
,, 0UUU
U. Why does the Einstein-Grossmann
geodesic point mass fail for physics of spins and mass
rotations?
First of all, there is a clear mathematical reason to re-
je
roach t
ct point spins from the Einstein-Grossman metric for-
malism. The point spin appo GR matter cannot
justify that S
is a covariant four-vector in pseudo-
Riemannian space-timetric tensor is de-
fined exclusively for matter without self-rotations or for
the four-momentum of a probe particle without spin.
Therefore, one cannot place S
e where the m
into the Einstein-
Grossmann geodesic equation with symmetrical connec-
tions. Riemann-Cartan geometries with the affine torsion
and asymmetrical connection [27-29] are still under dis-
19 e
cussions for proper applications.
In 38 Einstin already answered the point spin ques-
tion by developing with Infeld and Hoffmann relativistic
dynamics of slowly moving distributions of active and
passive masses. It is well known (Weyl in 1923 and Ein-
stein-Infeld-Hoffmann in 1938 for example [24]) that the
inhomogeneous GR time dilation (or inhomogeneous
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV
1476

oo
g
r for mass elements rotating over a joint axis)
defines a relvistic Lagrangian for the classical non-
point gyroscope. Therefore, Einstein’s relativity quanti-
tatively explains the de Sitter precession through local
non-Newtonian time rates for distributed rotating sys-
tems. The non-Newtonian (three-times enhanced) pre-
cession originates exclusively from different GR time
rates in neighboring material points, rather than from a
local space curvature in question for the ill-defined GR
spin of a point mass. The autho
ati
r does not understand
Sc
all gra
Eu
metric scheme [2,26]. In order to achieve this main goal,
warping
around the localized gravitational source (including the
e contrary, our chain analysis of par-
allows us to infer that curved 4-in- hiffs reasons to ignore Einstein-Infeld-Hoffmann
physics and Weyl results for relativistic gyroscopes prior
to testing General Relativity through rotation of masses.
The Einstein-Hilbert tensor formalism for energy den-
sities of a gravitational source (rather than for a point
source) requires non-Schwarzschildian interpretation of
vitational tests, including Lunar-Laser-Ranging
and Gravity Probe B data. In authors view, the 1913 Ein-
stein-Grossmann geodesic motion in pseudo-Riemannian
space-time with flat space can provide a physical basis
for translational dynamics of only point particles, but not
for self-rotations of distributed relativistic matter. Point
spin models for geodetic and frame-dragging angular
drifts of free-falling gyroscopes cannot be reasonable for
GR physics even under formal success of point-spin ap-
proximations for the observable geodetic precession.
Possible speculations that the de Sitter geodetic preces-
sion of the Earth-Moon gyroscope or that the Mercury
perihelion precession have already confirmed non-
clidean space geometry are against proper applications
of the well-tested GR time dilation by gravitational fields,
and, therefore, against Einstein-Infeld-Hoffmann’s phys-
ics of slowly rotating systems having finite active/passive
masses at finite dimensions. In fact, the available GP-B
releases (einstein.stanford.edu) of the processed geodetic
precession data perfectly confirmed time dilatation for
Einstein-Infeld-Hoffmann rotating distributions of masses.
Lunar laser ranging of the Earth-Moon gyroscope and the
GP-B geodetic precession are irrelevant, in fact, to ex-
perimental proofs of space warping by the missing inch.
These tests are equally irrelevant to experimental proves
of black holes existence. On the contrary, all known pre-
cision measurements in gravitation confirms the strong-
field metric (3) with time dilation and continuous gravi-
tational masses in nonempty Euclidean 3-space.
7. Conclusions
There are a lot of disputes in modern gravitation and as-
troparticle physics. Our main goal was to reinforce spa-
tial flatness for real, non-point matter in a line of the
original En twur f geometrization of fields, rather than to
discuss other consequences of the selfcontained SR-GR
we derived quantitative geodesic predictions for Mer-
cury’s perihelion precession, Mercury’s radar echo delay,
and the gravitational light deflection by the Sun in strict-
ly flat three-space without references on the 1915 GR
equations at all. The numerical results are well known
from the Schwarzschild empty-space approximation of
reality. Recall that the conventional interpretation of
post-Newtonian corrections relies on space
“point” Sun). On th
ticles physical time
terval can keep strict spatial flatness and the Entwurf
metric scheme for strong-field gravitation. The GR dis-
placement dl may be referred as a space interval (like
in Special Relativity) in flatspace relativity of nonlocal
superfluid masses with mutual spatial penetrations. Con-
sequently, the integral dl
along a space curve does not
depend anymore on gravitational fields and takes a well-
defined meaning. Such a Machian-type nonlocality of
superfluid astroparticles reconciles 3D space properties
with the relativistic Sommerfield quantization along a
line contour. Indeed, these are no reasonable explana-
tions for quantized magnetic flux in laboratory SQUIDs,
unless one accepts 3D spatial flatness for any 2D surface
[3].
GR physics may attach all field corrections within the
GR invariant 2
d
s
to the time element
2
ddl
with
chain relations. Gravity indeed curves elementary space-
time intervals (therefore d
and d
s
are specific for
each moving particle), but their space sub-intervals dl
are always flat or universal for all particles and observers.
It is not surprising that our approach to relativistic cor-
rections, based on the strong-field Equation (7), resulted
in Schwarzschild-type estimations, which are based on
very close integrals of motion in the Sun’s weak field.
However, strong fields in (7) will not lead to further co-
incidences with empty-space Schwarzschild-type solu-
tions for dynamics of probe particles.
Both the Euclidean space interval ddd>0
i
i
lxx
and the Newtonian time interval
dtddd>0
oo
xxx
are independent from local
ery par
n
o
fields and prop parameters of elementarticles.
This absolute universality of world space and time rulers
is a mandatory requirement for these otions in their ap-
plications to different particles and their ensembles. Oth-
erwise, there would be no way to introduce for different
observers one universal ruler to measure three-intervals
and to compare dynamics of particles in common 3-space
under the common time parameter. For example, it is
impossible to measure or to compare differently warped
four-intervals

ddd
N
N
s
gxxx

of different parti-
cles. In other words, there is no universal, non-specific
Copyright © 2012 SciRes. JMP
I. E. BULYZHENKOV 1477
pseudo-Riemannian geometry for all world m
Th carriers can be ob-
when they maintain
cs.
Space-time-energy self-o
t 1015 m, then this ma
atter.
erefore, joint evolution of energy
served only in common sub-spaces
universal (for all matter) sub-metri
rganization of extended mat-
ter can be well described without 3D metric ripples,
which have no much sense in strictly flat material space.
Laboratory search of observable chiral phenomena for
paired vector interactions in flat material space is worth
to be performed before expansive projects to find 3D
metric ripples in cosmic space. Record measurements of
flat material space beyond the present limit 1018 m
might not be required for confirmation of the residual
EM nature of elementary masses under their Einstein-
type geometrization. Once chiral symmetry for hadrons
was violated ass-forming symme-
try was equally violated in the entire nonlocal structure
of the superfluid astroparticle [2] or in its infinite mate-
rial space. Non-empty Euclidean 3-space does match
curved 4D space-time in metric gravitation. Such a
matching allows the extended radial electron to move
(both in theory and in practice) without spatial splits of
mass and electric charge densities. Strict spatial flatness
is a real way for quantization of elementary fields and for
unified geometrization of extended gravitational and
electric charges.
8. Acknowledgements
I acknowledge useful discussions with Yu. S. Vladimirov
regarding relation nature for binary interactions of ele-
mentary matter and for the non-empty space concept.
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