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Disintegration of Einsteinian general relativity,

towards a new ECE cosmology

M. W. Evans∗

Civil List and A.I.A.S.

and

H. Eckardt†

A.I.A.S. and UPITEC

(www.webarchive.org.uk, www.aias.us,www.atomicprecision.com, www.upitec.org)

Abstract

Keywords:

1 Introduction

2 Mathematcs in place of dogma

3 Numerical analysis with Maxima

We give some examples for the nature of the function m(r) in various contextsshowing that inconsistencies for this function arise. There is no transition fromEinsteinian theory to Newtonian mechanics.

The angular change or radius for an orbiting body is in Einsteinian as wellas ECE theory given by

dr

dθ= r2

(1

b2−m(r)

(1

a2+

1

r2

))1/2

(64)

where the functions m(r) are given by

m(r) = 1− r0r

(65)

in Einsteinian andm(r) = 2− exp

(2 exp(− r

R))

(66)

∗email: emyrone@aol.com†email: horsteck@aol.com

1

in ECE theory with constants of motion

a =L

m c, b =

L c

E, (67)

see Eqs. (11), (12), (17) and (24). The simplest transition to Newtonian theorywould be

m(r)→ 1. (68)

Then Eq. (65) becomes

dr

dθ→ r2

(1

b2− 1

a2− 1

r2

)1/2

. (69)

Because of (68) it isa

b=

E

m c2. (70)

Since E consists of the rest energy plus kinetic energy we have

E > mc2, (71)

hence

a > b and1

a2<

1

b2(72)

but the di�erence is small because of the huge rest energy of a celestial body.Consequently the condition from (69) for obtaining a real value of the squareroot,

1

b2≥ 1

a2+

1

r2, (73)

can barely be ful�lled for a ≈ b. The condition is much easier to ful�ll form(r) < 1, which is guaranteed for all values of r. This can be seen from thegraphs of Fig. 1 and 2. In Fig. 1 there are bound states for the m functions ofEinsteinian and ECE theory which are compatible with elliptic orbits. However,in order to obtain a non-negative square root argument for m = 1, one has tochoose a >> b. This is typical for the non-relativistic case. In other words,the function m = 1 can only be used in the non-relativistic case where ther-dependent m function is de�ned for total energies with exclusion of the restenergy. This is one reason why both types of theory do not pass into one another.In Fig. 2 there is no bound state for m(Einstein) and two distinct regions, onebound and one unbound, for m(ECE). Both cases do not describe ellipses.

If it is assumed that dr/dθ describes an orbits of a precessing ellipse, thegeneral form of orbit has to be equated by dr/dθ of elliptic orbits which has beendone in Eq. (21). As a result, a particular form of function m(r) is obtained,see (22). The radial part of this function is graphed in Fig. 3 for three valuesof eccentricity ε. It can be seen that these curves di�er only in the region verynear to the centre (note that we have set r0 = 1 throughout our calculations).However these curves di�er signi�cantly from the general form of m presentedin Eq. (66). This may be the reason why it was very di�cult to �nd boundelliptic orbits in Figs. 1 and 2.

The angular dependence of m is shown in Fig. 4 in a polar diagram. Forsmall radii, there is a signi�cant angular variation. For radii larger than r0this di�erences become indistinguishable as can also be seen from Fig. 3. This

2

Figure 1: Bound states of dr/dθ, for functions m(Einstein) and m(ECE). Pa-rameters were a = b = 10, R = 1, r0 = 1. For comparison: m = 1 witha = 10, b = 1.

Figure 2: Unbound states of dr/dθ, for functions m = 1, m(Einstein) andm(ECE). Parameters were a = 2.5, b = 2, R = 1, r0 = 1.

3

Figure 3: m(r) of Eq. (22) for a = b = x = 1, θ = π/2, in compar�son withm(r) from Eq. (24).

Figure 4: Angular dependence of m(r) from Eq. (22) for three radius values,ε = 0.2.

4

behaviour is to be expected for a spherical spacetime, there should be no angularvariance, otherwise the geometry would not be spherical. The behaviour nearto the center probably plays no practical role since this is inside a star whereadditional laws of physics are valid.

In section 2 an expression for the radial velocity of an orbiting body wasderived (Eq. (15)). In the non-relativistic approximation this expression shouldshade into the Newtonian expression given in Eq. (26). From equating bothvelocity expressions (Eq. (46)), one should obtain the function m which repre-sents the Newtonian theory. Due to the di�erent energy de�nitions we do notexpect a result of m = 1. Squaring Eq. (46) gives

c2b2m2(r)

(1

b2−m(r)

(1

a2+

1

r2

))=

2E

m+

2MG

r− L2

m2r2. (74)

This is a cubic equation in m(r) of the form

c1m3(r) + c2m

2(r) + c3 = 0 (75)

with constants

c1 = −c2b2(

1

a2+

1

r2

), (76)

c2 = c2, (77)

c3 = −2E

m− 2MG

r+

L2

m2r2. (78)

For consistency reasons, we write the constants L and E in terms of a and b asobtained from Eq. (67):

L = amc, E =a

bmc2. (79)

Eq. (75) can be solved, giving two complex and one real solution. The realsolution is

m(r) =

(√c3 (27 c21 c3 + 4 c32)

2 332 c21

− 27 c21 c3 + 2 c3254 c31

) 13

+c22

9 c21

(√c3 (27 c21 c3+4 c32)

2 332 c21

− 27 c21 c3+2 c3254 c31

) 13

− c23 c1

. (80)

Graphing this highly complicated expression yields the results shown in Fig. 5.All tested parameter combinations give qualitatively the same curve type. Thelimit of m for large r is negative and not +1 as it should be. In particularthis is not the function used by Einstein theory. This result makes evident thatNewtonian and Einstein theory are not compatible.

Finally we investigated the form of m(r) for a whirlpool galaxy. Equatingboth terms for dr/dθ as before, Eqs. (12) and (52), we obtain

r2(

1

b2−m(r)

(1

a2+

1

r2

))1/2

= αr (81)

5

Figure 5: Newtonian limit of m(r) for tow a,b parameters, other parametersbeing c = 1,m = 1,M = 100, G = 1.

or

m(r) =a2

r2 + b2

(r2

b2− α2

)(82)

which has been graphed in Fig. 6. The curve is similar to the general mfunction, Eq. (66), for values of α between 2.5 and 3. The far �eld limit is unityas required but convergence behaviour is slightly di�erent for all three curves.Compared to Fig. 3, this indicates that descibing spiral galaxies by ECE theorymay even be simpler than describing elliptical orbits, a surprising result whentaking into account that dark matter had to be assumed to bring Einstein theoryin agreement with the experimental velocity curve. In total we have shown bynumerical methods that Einstein theory is inconsistent and untenable.

6

Figure 6: m(r) for a spiral galaxy, with parameters a = 1, b = 1.

7

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