Introducing Distance and Measurement in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale

Introducing Distance and Measurement in General Relativity: Changes

for the Standard Tests and the Cosmological Large-Scale

Stephen J. Crothers Sydney, Australia E-mail: thenarmis@yahoo.com

Relativistic motion in the gravitational field of a massive body is governed by the external metric of a spherically symmetric extended object. Consequently, any solution for the point-mass is inadequate for the treatment of such motions since it pertains to a fictitious object. I therefore develop herein the physics of the standard tests of General Relativity by means of the generalised solution for the field external to a sphere of incompressible homogeneous fluid.

1 Introduction

The orthodox treatment of physics in the vicinity of a massive body is based upon the Hilbert [1] solution for the point-mass, a solution which is neither correct nor due to Schwarz-schild [2], as the latter is almost universally claimed.

In previous papers [3, 4] I derived the correct general solution for the point-mass and the point-charge in all their standard configurations, and demonstrated that the Hilbert solution is invalid. The general solution for the point-mass is however, inadequate for any real physical situation since the material point (and also the material point-charge) is a fictitious object, and so quite meaningless. Therefore, I avail myself of the general solution for the external field of a sphere of incompressible homogeneous fluid, obtained in a particular case by K. Schwarzschild [5] and generalised by myself [6] to,

ds2=

(√Cn−α)

√

Cn

dt2− √

Cn

(√Cn−α)

C′

n

2 4Cn

dr2− −Cn(dθ2+ sin2θdϕ2),

(1)

Cn(r) =

r−r₀

n

+ǫn 2

n

,

α= s

3

κρ₀sin

3

χa−χ0 ,

Rca=

s 3

κρ0 sin

χa−χ0 ,

ǫ= s

3

κρ0 ₃

2sin 3

χa−χ0 −

− 9₄cosχa−χ0

χa−χ0

−

1 2sin 2

χa−χ0

13 ,

r0∈ ℜ, r∈ ℜ, n∈ ℜ+, χ0∈ ℜ, χa∈ ℜ,

arccos1

3<|χa−χ0|<

π

2 ,

|ra−r0|6|r−r0|<∞,

whereρ0is the constant density of the fluid,k2is Gauss’ gra-vitational constant, the signa denotes values at the surface of the sphere,|χ−χ0|parameterizes the radius of curvature of the interior of the sphere centred arbitrarily atχ0,|r−r0| is the coordinate radius in the spacetime manifold of Special Relativity which is a parameter space for the gravitational field external to the sphere centred arbitrarily atr0.

To eliminate the infinite number of coordinate systems admitted by (1), I rewrite the said metric in terms of the only measurable distance in the gravitational field, i .e. the circumferenceGof a great circle, thus

ds2=

1−2πα

G

dt2−

1−2πα

G

−1

dG2

4π2 − − G

2 4π2 dθ

2_{+ sin}2

θdϕ2

,

(2)

α= s

3

κρ0

sin3|χa−χ0|,

2π

s 3

κρ0

sin|χa−χ0|6G <∞,

arccos1

3<|χa−χ0|<

π

2 .

2 Distance and time

According to (1), iftis constant, a three-dimensional mani-fold results, having the line-element,

ds2= √

Cn

(√Cn−α)

C′

n

2 4Cn

Ifα= 0, (1) reduces to the line-element of flat spacetime,

ds2=dt2₋dr2_{− |}r₋r_0|2(dθ2+ sin2θdϕ2), (4) 06_|r−r0|<∞,

since thenra≡r0.

The introduction of matter makes ra6=r0, owing to the extended nature of a real body, and introduces distortions from the Euclidean in time and distance. The value of αis effectively a measure of this distortion and therefore fixes the spacetime.

Whenα= 0, the distanceD=|r−r0|is the radius of a sphere centred atr0. Ifr0= 0andr>0, then D≡rand is then both a radius and a coordinate, as is clear from (4).

Ifr is constant in (3), thenCn(r) =R2c is constant, and

so (3) becomes,

ds2=R2c(dθ2+ sin

2

θdϕ2), (5)

which describes a sphere of constant radius Rc embedded

in Euclidean space. The infinitesimal tangential distances on (5) are simply,

ds=Rc

q

dθ2_{+ sin}2

θdϕ2_.

Whenθandϕare constant, (3) yields the proper radius,

Rp=

Z s p_C

n(r)

p

Cn(r)−α

C′

n(r)

2p

Cn(r)

dr =

=

Z s p_C

n(r)

p

Cn(r)−α

dpCn(r),

(6)

from which it clearly follows that the parameter rdoes not measure radial distances in the gravitational field.

Integrating (6) gives,

Rp(r) =

q_√

Cn( √

Cn−α) +αln

p√

Cn+

p√

Cn−α

+K ,

K=const,

which must satisfy the condition,

r→r±a ⇒Rp→R+pa,

where ra is the parameter value at the surface of the body

andRpa the indeterminate proper radius of the sphere from

outside the sphere. Therefore,

Rp(r) =Rpa+

r p

Cn(r)

p

Cn(r)−α

−

− r

p

Cn(ra)

p

Cn(ra−α

+

+αln

q p

Cn(r) +

q p

Cn(r)−α

q p

Cn(ra) +

q p

Cn(ra)−α

,

(7)

which, by the use of (1) and (2), becomes

Rp(r) =Rpa+

r G

2π _G

2π−α

−

− s

r

3

κρ₀sin χa−χ0

r

3

κρ₀sin χa−χ0

−α

+

+αln

q

G 2π+

q

G 2π −α r

q ₃

κρ0

sinχa−χ0 +

r q ₃

κρ0

sinχa−χ0 −α

,

(8)

α= s

3

κρ₀sin

3

χa−χ0 .

According to (1), the proper time is related to the coord-inate time by,

dτ= q

g₀₀ dt= s

1−p α

Cn(r)

dt . (9)

Whenα= 0,dτ=dtso that proper time and coordinate time are one and the same in flat spacetime. With the intro-duction of matter, proper time and coordinate time are no longer the same. It is evident from (9) that bothτ andtare finite and non-zero, since according to (1),

1 9<1−

α

p

Cn(ra)

61−p α

Cn(r)

,

i .e.

1 9<cos

2

|χa−χ0|61−

α

p

Cn(r)

,

or

1

3dt6dτ 6dt , since In the far field, according to (9),

p

Cn(r)→ ∞ ⇒dτ→dt ,

recovering flat spacetime asymptotically.

Therefore, if a body falls from rest from a point distant from the gravitating mass, it will reach the surface of the mass in a finite coordinate time and a finite proper time. According to an external observer, time does not stop at the surface of the body, wheredt= 3dτ, contrary to the orthodox analysis based upon the fictitious point-mass.

3 Radar sounding

body along a radial line be(r2, θ0, ϕ0). Let the observer emit a radar pulse towards the small body. Then by (1),

1−p α

Cn(r)

!

dt2= 1−p α

Cn(r)

!−1

Cn′2(r)

4Cn(r)

dr2 =

= 1−p α

Cn(r)

!−1

dpCn(r)

2

,

so

dpCn(r)

dt = ± 1−

α

p

Cn(r)

!

,

or

dr dt= ±

2p

Cn(r)

C_n′_(r) 1− α

p

Cn(r)

!

.

The coordinate time for the pulse to travel to the small body and return to the observer is,

∆t = − √

Cn(r2)

Z √

Cn(r1)

d√Cn

1− α

√

Cn

+ √

Cn(r1)

Z √

Cn(r2)

d√Cn

1− α

√

Cn

=

= 2 √

Cn(r1)

Z √

Cn(r2)

d√Cn

1− α

√_C

n

.

The proper time lapse is, according to the observer, by formula (1),

∆τ =

r

1− α √

Cn dt = 2

r 1− α √ Cn √

Cn(r1)

Z

√

Cn(r2)

d√Cn

1−√α

Cn = = 2 r 1− α √ Cn p Cn(r1)−

p

Cn(r2)+αln

p

Cn(r1)−α p

Cn(r2)−α ! .

The proper distance between the observer and the small body is,

Rp=

√

Cn(r1)

Z √

Cn(r2)

s √

Cn

√

Cn−α

dpCn

= r

p

Cn(r1)

p

Cn(r1)−α

−

− r

p

Cn(r2)

p

Cn(r2)−α

+

+αln q p

Cn(r1) +

q p

Cn(r1)−α

q p

Cn(r2) +

q p

Cn(r2)−α

.

Then according to classical theory, the round trip time is

∆¯τ= 2Rp,

so∆τ6= ∆¯τ. If √ α

Cn(r)

is small for

p

Cn(r2)<

p

Cn(r)<

p

Cn(r1),

then

∆τ≈2

p

Cn(r1)−

p

Cn(r2)−

− α _p

Cn(r1)−

p

Cn(r2)

2pCn(r1)

+αln s_p

Cn(r1)

p

Cn(r2)

,

∆¯τ≈2

p

Cn(r1)−

p

Cn(r2) +

α

2 ln s_p

Cn(r1)

p

Cn(r2)

.

Therefore,

∆τ−∆¯τ≈α

ln

s_p

Cn(r1)

p

Cn(r2)

−

− _p

Cn(r1)−

p

Cn(r2)

p

Cn(r1)

i =

= α ln r

G1

G2 −

G1−G2

G1 ! = = s 3 κρ0 sin3

χa−χ0 ln

r

G1

G2 −

G1−G2

G1 !

,

(10)

G=G(r) = 2πpCn(D(r)).

Equation (10) gives the time delay for a radar signal in the gravitational field.

4 Spectral shift

Let an emitter of light have coordinates (t_E, r_E, θ_E, ϕ_E). Let a receiver have coordinates(t_R, r_R, θ_R, ϕ_R). Let u be an affine parameter along a null geodesic with the valuesu_E

andu_Rat emitter and receiver respectively. Then,

1−√α

Cn dt du 2 = 1−√α

Cn

−1

d√Cn

du

2 +

+Cn

dθ du

2

+ Cnsin2θ

dϕ du 2 , so dt du= "

1−√α

Cn

−1 ¯

g_ijdx

i

du dxj

du

whereg¯ij=−gij. Then,

tR−tE= uR

Z

uE

" 1−√α

Cn

−1 ¯

gij

dxi du

dxj du

#12 du ,

and so, for spatially fixed emitter and receiver,

t(1)R −t

(1)

E =t

(2)

R −t

(2)

E ,

and therefore,

∆tR=t

(2)

R −t

(1)

R =t

(2)

E −t

(1)

E = ∆tE. (11)

Now by (1), the proper time is,

∆τE =

v u u t1−

α

q

Cn(rE)

∆tE,

and

∆τR=

v u u t1−

α

q

Cn(rR)

∆tR.

Then by (11),

∆τ_R

∆τE

=  

1−_√ α Cn(r_R)

1−_√ α Cn(rE)

 

1 2

. (12)

If z regular pulses of light are emitted, the emitted and received frequencies are,

νE=

z

∆τE

, νR=

z

∆τR

,

so by (12),

∆ν_R

∆ν_E =

 

1−_√ α Cn(rE)

1−_√ α Cn(rR)

 

1 2

≈

≈ 1 +α 2

 

1 q

Cn(rR)

−q 1

Cn(rE)

 ,

whence,

∆ν νE

=νR−νE

νE

≈α₂  

1 q

Cn(rR)

−q 1

Cn(rE)

 =

= πα

₁

GR

− 1

GE

=

= π

s 3

κρ0 sin3

χa−χ0

1

GR

− 1

GE

.

5 Advance of the perihelia

Consider the Lagrangian,

L =1 2

"

1−√α

Cn

dt dτ

2# −

−1₂ "

1−√α

Cn

−1

d√Cn

dτ

2# −

−1₂ "

Cn

dθ dτ

2

+ sin2θ

dϕ dτ

2!#

,

(13)

where τ is the proper time. Restricting motion, without loss of generality, to the equatorial plane,θ=π

2, the Euler-Lagrange equations for (13) are,

1−√α

Cn

−1

d2√Cn

dτ2 +

α

2Cn

dt dτ

2 −

−

1−√α

Cn

−2

α

2Cn

d√Cn

dτ

2 −pCn

dϕ dτ

2 = 0,

(14)

1−√α

Cn

dt

dτ =const=K , (15)

Cn

dϕ

dτ =const=h , (16)

andds2=gµνdxµdxν becomes,

1−√α

Cn

dt dτ

2 −

−

1−√α

Cn

−1

d√Cn

dτ

2 −Cn

dϕ dτ

2 = 1.

(17)

Rearrange (17) for,

1−√α

Cn

_˙

t2 ˙

ϕ2−

1−√α

Cn

d√Cn

dϕ

2 −Cn=

1 ˙

ϕ2. (18) Substituting (15) and (16) into (18) gives,

d√Cn

dϕ

2 +Cn

1 + Cn

h2

1−√α

Cn

−K

2

h2 C 2

n= 0.

Settingu=_√1

Cn reduces (18) to,

du dϕ

2

+u2=E+ α

h2u+αu 3

, (19)

where E=(K

2 −1₎

h2 . The term αu

3 _{represents the} general-relativistic perturbation of the Newtonian orbit.

Aphelion and perihelion occur when du_dϕ= 0, so by (19),

αu3₋u2+ α

Let u=u1 at aphelion and u=u2 at perihelion, so

u16u6u2. One then finds in the usual way that the angle ∆ϕbetween aphelion and subsequent perihelion is,

∆ϕ=

1 +3α

4 u1+u2

π .

Therefore, the angular advance ψ between successive perihelia is,

ψ=3απ

2 u1+u2

=3απ 2

1 p

Cn(r1)

+p 1

Cn(r2)

! =

= 3απ2

₁

G1 + 1

G2

,

(21)

where G1 and G2 are the measurable circumferences of great circles at aphelion and at perihelion. Thus, to correctly determine the value of ψ, the values of the said circum-ferences must be ascertained by direct measurement. Only the circumferences are measurable in the gravitational field. The radii of curvature and the proper radii must be calculated from the circumference values.

If the field is weak, as in the case of the Sun, one may take

G_≈2πr, forr as an approximately “measurable” distance from the gravitating sphere to a spacetime event. In such a situation equation (21) becomes,

ψ≈3απ 2

₁

r1 + 1

r2

. (22)

In the case of the Sun, α≈3000 m, and for the planet Mercury, the usual value ofψ≈43 arcseconds per century is obtained from (22). I emphasize however, that this value is a Euclidean approximation for a weak field. In a strong field equation (22) is entirely inappropriate and equation (21) must be used. Unfortunately, this means that accurate solutions cannot be obtained since there is no obvious way of obtaining the required circumferences in practise. This aspect of Einstein’s theory seriously limits its utility. Since the relativists have not detected this limitation the issue has not previously arisen in general.

6 Deflection of light

In the case of a photon, equation (17) becomes,

1−√α

Cn

dt dτ

2 −

−

1−√α

Cn

−1

d√Cn

dτ

2 −Cn

dϕ dτ

2 = 1,

which leads to,

du dϕ

2

+u2=F+αu3. (23)

Let the radius of curvature of a great circle at closest approach bepCn(rc). Now when there is no mass present,

(23) becomes

du dϕ

2

+u2=F ,

and has solution,

u=ucsinϕ⇒

p

Cn(rc) =

p

Cn(r) sinϕ ,

and

u2c=

1 p

Cn(rc)

=F .

If p

Cn(r)≫α,

p

Cn(r)>

p

Cn(ra),

andu=uc> ua at closest approach, then

du

dϕ= 0 at u=uc,

soF=u2c(1−ucα), and (23) becomes,

du dϕ

2

+u2=u2c(1−ucα) +αu

3

. (24)

Equation (24) must have a solution close to flat space-time, so let

u=ucsinϕ+αw(ϕ).

Putting this into (24) and working to first order in α, gives

2

dw dϕ

cosϕ+ 2wsinϕ=u2c sin

3

ϕ−1

,

or

d

dϕ(wsecϕ) =

1 2u

2

c secϕtanϕ−sinϕ−sec2ϕ

,

and so,

w=1 2u

2

c 1 + cos2ϕ−sinϕ

+Acosϕ ,

whereAis an integration constant. If the photon originates at infinity in the directionϕ= 0, thenw(0) = 0, soA= −u2c,

and

u=uc

1−1₂αuc

sinϕ+1 2αu

2

c(1−cosϕ)

2

, (25)

to first order inα. Puttingu= 0andϕ=π+ ∆ϕinto (25), then to first order in∆ϕ,

0 = −uc∆ϕ+ 2αu2c,

so the angle of deflection is,

∆ϕ= 2αuc=

2α

p

Cn(rc)

= 2α

r_c−r₀

n

+ǫn

1

n

=4πα

Gc

Gc>Ga.

At a grazing trajectory to the surface of the body,

Gc=Ga= 2π

p

Cn(ra),

p

Cn(ra) =

s 3

κρ₀sin

χ_a−χ₀ ,

so then

∆ϕ= 2q_κρ3

0

sin3

χ_a−χ₀ q ₃

κρ0 sin

χ_a−χ₀

= 2 sin2

χ_a−χ₀ . (26)

For the Sun [5],

sin

χ_a−χ₀ ≈

1 500,

so the deflection of light grazing the limb of the Sun is,

∆ϕ≈₅₀₀22≈1.65′′.

Equation (26) is an interesting and quite surprising result, forsin

χ_a−χ₀

gives the ratio of the “naturally measured” fall velocity of a free test particle falling from rest at infinity down to the surface of the spherical body, to the speed of light in vacuo. Thus,

the deflection of light grazing the limb of a spherical gravitating body is twice the square of the ratio of the fall velocity of a free test particle falling from rest at infinity down to the surface, to the speed of light in vacuo, i .e .,

∆ϕ= 2 sin2

χa−χ0 = 2

va

c

2

=4GMg

c2_R

ca ,

where Rca is the radius of curvature of the body, Mg the

active mass, andGis the gravitational constant. The quantity

va is the escape velocity,

va=

s 2GMg

Rca

.

7 Practical constraints and general comment

Owing to their invalid assumptions about ther-parameter [7], the relativists have not recognised the practical limitations associated with the application of General Relativity. It is now clear that the fundamental element of distance in the gravitational field is the circumference of a great circle, centred at the heart of an extended spherical body and passing through a spacetime event external thereto. Heretofore the orthodox theorists have incorrectly taken the r-parameter,

not just as a radius in the gravitational field, but also as a

measurableradius in the field. This is not correct. The only measurable distance in the gravitational field is the aforesaid circumference of a great circle, from which the radius of curvature pCn(r) and the proper radius Rp(r) must be

calculated, thus,

p

Cn(r) =

G

2π,

Rp(r) =

Z q

−g11 dr .

Only in the weak field, where the spacetime curvature is very small, canpCn(r) be taken approximately as the

Euclidean value r, thereby making Rp(r)≡

p

Cn(r)≡r,

as in flat spacetime. In a strong field thiscannot be done. Consequently, the problem arises as to how to accurately measure the required great circumference? The correct de-termination, for example, of the circumferences of great circles at aphelion and perihelion seem to be beyond practical determination. Any method adopted for determining the re-quired circumference must be completely independent of any Euclidean quantity since, other than the great circumference itself, only non-Euclidean distances are valid in the gravita-tional field, being determined by it. Therefore, anything short of physically measuring the great circumference will fail. Consequently, General Relativity, whether right or wrong as theories go, suffers from a serious practical limitation.

The value of the r-parameter is coordinate dependent and is rightly determined from the coordinate independent value of the circumference of the great circle associated with a spacetime event. One cannot obtain a circumference for the great circle of a given spacetime event, and hence the related radius of curvature and associated proper radius, from the specification of a coordinate radius, because the latter is not unique, being conditioned by arbitrary constants. The coordinate radius is therefore superfluous. It is for this reason that I completely eliminated the coordinate radius from the metric for the gravitational field, to describe the metric in terms of the only quantity that is measurable in the gravitational field — the great circumference (see also [6]). The presence of the r-parameter has proved misleading to the relativists. Stavroulakis [8, 9, 10] has also completely eliminated the r-parameter from the equations, but does not make use of the great circumference. His approach is formally correct, but rather less illuminating, because his resulting line element is in terms of the a quantity which is not measurable in the gravitational field. One cannot obtain an explicit expression for the great circumference in terms of the proper radius.

the relativists. The lambda “solutions” and the expanding universe “solutions” are the result of such a muddleheaded-ness that it is difficult to apprehend the kind of thoughtless-ness that gave them birth. Since Special Relativity describes an empty world (no gravity) it cannot form a basis for any cosmology. This theoretical result is all the more interesting owing to its agreement with observation. Arp [12], for in-stance, has adduced considerable observational data which is consistent on the large-scale with a flat, infinite, non-expanding Universe in Heraclitian flux. Bearing in mind that both Special Relativity and General Relativity cannot

yield a spacetime on the cosmological “large-scale”, there is currently no theoretical replacement for Newton’s cos-mology, which accords with deep-space observations for a flat space, infinite in time and in extent. The all pervasive rolˆe given heretofore by the relativists to General Relativity, can be justified no longer. General Relativity is a theory of onlylocalphenonomea, as is Special Relativity.

Another serious shortcoming of General Relativity is its current inability to deal with the gravitational interaction of two comparable masses. It is not even known if Einstein’s theory admits of configurations involving two or more mass-es [13]. This shortcoming seems rather self evident, but app-rently not so for the relativists, who routinely talk of black hole binary systems and colliding black holes (e .g. [14]), aside of the fact that no theory predicts the existence of black holes to begin with, but to the contrary, precludes them.

Acknowledgement

I am indebted to Dr. Dmitri Rabounski for drawing my attention to a serious error in a preliminary draft of this paper, and for additional helpful suggestions.

Dedication

I dedicate this paper to the memory of Dr. Leonard S. Abrams: (27 Nov. 1924 — 28 Dec. 2001).

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