October, 2005 PROGRESS IN PHYSICS Volume 3

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 1 π
arccos < |χa − χ0 | < ,
3 2
The orthodox treatment of physics in the vicinity of a massive
|ra − r0 | 6 |r − r0 | < ∞ ,
body is based upon the Hilbert [1] solution for the point-
mass, a solution which is neither correct nor due to Schwarz- where ρ0 is the constant density of the fluid, k2 is Gauss’ gra-
schild [2], as the latter is almost universally claimed. vitational constant, the sign a denotes values at the surface
In previous papers [3, 4] I derived the correct general of the sphere, |χ − χ0 | parameterizes the radius of curvature
solution for the point-mass and the point-charge in all their of the interior of the sphere centred arbitrarily at χ0 , |r − r0 |
standard configurations, and demonstrated that the Hilbert is the coordinate radius in the spacetime manifold of Special
solution is invalid. The general solution for the point-mass Relativity which is a parameter space for the gravitational
is however, inadequate for any real physical situation since field external to the sphere centred arbitrarily at r0 .
the material point (and also the material point-charge) is a To eliminate the infinite number of coordinate systems
fictitious object, and so quite meaningless. Therefore, I avail admitted by (1), I rewrite the said metric in terms of the
myself of the general solution for the external field of a only measurable distance in the gravitational field, i .e. the
sphere of incompressible homogeneous fluid, obtained in a circumference G of a great circle, thus
particular case by K. Schwarzschild [5] and generalised by    −1
myself [6] to, 2 2πα 2 2πα dG2
ds = 1 − dt − 1 − −
√   √  02 G G 4π 2
2 ( Cn −α) 2 Cn Cn (2)
ds = √ dt − √ dr2 − G2

Cn ( Cn −α) 4Cn (1) − 2 2
dθ + sin θdϕ , 2
4π 2
− Cn (dθ2 + sin2 θdϕ2 ) , s
3
 n  n2 α= sin3 |χa − χ0 | ,
Cn (r) =  r − r0  + n , κρ0
s s
3   3
α= sin3  χa − χ0  , 2π sin |χa − χ0 | 6 G < ∞ ,
κρ0 κρ0
s
  1 π
3 arccos < |χa − χ0 | < .
Rc a = sin  χa − χ0  , 3 2
κρ0
s 
33  
= sin3  χa − χ0  − 2 Distance and time
2
κρ0
  1 According to (1), if t is constant, a three-dimensional mani-
9     1   3
    
− cos χa − χ0 χa − χ0 − sin 2 χa − χ0  , fold results, having the line-element,
4 2
 √  02
Cn Cn
+
r0 ∈ < , r ∈ < , n ∈ < , χ 0 ∈ < , χ a ∈ < ,
2
ds = √ dr2 + Cn (dθ2 + sin2 θdϕ2 ) . (3)
( Cn −α) 4Cn

S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 41
Volume 3 PROGRESS IN PHYSICS October, 2005

If α = 0, (1) reduces to the line-element of flat spacetime, which, by the use of (1) and (2), becomes
r 
ds2 = dt2 − dr2 − |r − r0 |2 (dθ2 + sin2 θdϕ2 ) , (4) G G

Rp (r) = Rpa + −α −
2π 2π
0 6 |r − r0 | < ∞ ,
sr r 
since then ra ≡ r0 . 3   3  
The introduction of matter makes ra 6= r0 , owing to the − sin  χa − χ0  sin  χa − χ0  − α +
κρ0 κρ0
extended nature of a real body, and introduces distortions (8)
 
from the Euclidean in time and distance. The value of α is  q q 
 G
+ G
−α

effectively a measure of this distortion and therefore fixes  2π 2π 

+ α ln rq rq ,
the spacetime.      
 3
sin χa −χ0  + 3
sin χa −χ0  − α 
When α = 0, the distance D = |r − r0 | is the radius of a  κρ0 κρ0

sphere centred at r0 . If r0 = 0 and r > 0, then D ≡ r and is s

then both a radius and a coordinate, as is clear from (4). 3  
If r is constant in (3), then Cn (r) = Rc2 is constant, and α= sin3  χa − χ0  .
κρ0
so (3) becomes,
According to (1), the proper time is related to the coord-
ds2 = Rc2 (dθ2 + sin2 θdϕ2 ) , (5) inate time by,
which describes a sphere of constant radius Rc embedded q s
in Euclidean space. The infinitesimal tangential distances on α
dτ = g00 dt = 1 − p dt . (9)
(5) are simply, Cn (r)
q
ds = Rc dθ2 + sin2 θdϕ2 . When α = 0, dτ = dt so that proper time and coordinate
When θ and ϕ are constant, (3) yields the proper radius, time are one and the same in flat spacetime. With the intro-
duction of matter, proper time and coordinate time are no
Z s p longer the same. It is evident from (9) that both τ and t are
Cn (r) C 0 (r)
Rp = p pn dr = finite and non-zero, since according to (1),
Cn (r) − α 2 Cn (r)
(6)
Z s p p
1
<1 − p
α
61 − p
α
,
Cn (r) 9
= p d Cn (r) , Cn (ra ) Cn (r)
Cn (r) − α
i .e.
from which it clearly follows that the parameter r does not 1 α
measure radial distances in the gravitational field. < cos2 |χa − χ0 | 6 1 − p ,
9 Cn (r)
Integrating (6) gives,
q√ p √  or
√  p√  1
Rp (r) = Cn ( Cn − α) + α ln  Cn + Cn − α  + K , dt 6 dτ 6 dt ,
3
K = const , since In the far field, according to (9),
which must satisfy the condition, p
Cn (r) → ∞ ⇒ dτ → dt ,
r → ra± ⇒ Rp → Rp+a ,
recovering flat spacetime asymptotically.
where ra is the parameter value at the surface of the body Therefore, if a body falls from rest from a point distant
and Rpa the indeterminate proper radius of the sphere from from the gravitating mass, it will reach the surface of the
outside the sphere. Therefore, mass in a finite coordinate time and a finite proper time.
r p  According to an external observer, time does not stop at the
p
Rp (r) = Rpa + Cn (r) Cn (r) − α − surface of the body, where dt = 3dτ , contrary to the orthodox
analysis based upon the fictitious point-mass.
r p 
p
− Cn (ra ) Cn (ra − α +
(7) 3 Radar sounding
 qp qp 
 
 Cn (r) + Cn (r) − α  Consider an observer in the field of a massive body. Let the

+ α ln  qp qp ,
 observer have coordinates, (r1 , θ0 , ϕ0 ) . Let the coordinates
 Cn (ra ) + Cn (ra ) − α 
of a small body located between the observer and the massive

42 S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale
October, 2005 PROGRESS IN PHYSICS Volume 3

body along a radial line be (r2 , θ0 , ϕ0 ) . Let the observer emit Then according to classical theory, the round trip time is
a radar pulse towards the small body. Then by (1),
τ = 2Rp ,
Δˉ
! !−1 0
α 2 α Cn2 (r) 2 so Δτ 6= Δˉ
τ.
1− p dt = 1 − p dr =
Cn (r) Cn (r) 4Cn (r) If √ α is small for
Cn (r)
!−1
p p p p
α 2
Cn (r2 ) < Cn (r) < Cn (r1 ) ,
= 1− p d Cn (r) ,
Cn (r)
then
so 
p ! p p
d Cn (r) α Δτ ≈ 2 Cn (r1 ) − Cn (r2 ) −
=± 1− p ,
dt Cn (r) p 
p sp 
or ! α Cn (r1 ) − Cn (r2 ) Cn (r1 )
p − p + α ln p ,
dr 2 Cn (r) α 2 Cn (r1 ) Cn (r2 )
=± 1− p .
dt Cn0 (r) Cn (r) sp
 
p p α Cn (r1 )
The coordinate time for the pulse to travel to the small Δˉτ ≈2 Cn (r1 ) − Cn (r2 ) + ln p .
2 Cn (r2 )
body and return to the observer is,
√ √ Therefore,
Cn (r2 )
Z √ Cn (r1 )
Z √  sp
d Cn d Cn Cn (r1 )
Δt = − + = Δτ − Δˉ
τ ≈ α ln p −
√ 1 − √αC √ 1 − √αC Cn (r2 )
n n
Cn (r1 ) Cn (r2 )
p p 
√ Cn (r1 ) − Cn (r2 ) i
Cn (r1 ) √
Z − p =
d Cn Cn (r1 )
= 2 . (10)
√ 1 − √αC r !
n
Cn (r2 ) G1 G1 − G2
= α ln − =
G2 G1
The proper time lapse is, according to the observer,
s r !
by formula (1), 3   G G − G
1 1 2
√ = sin χa − χ0  ln
3
− ,
r r Cn (r1 )
Z √ κρ0 G2 G1
α α d Cn
Δτ = 1− √ dt = 2 1− √ =
Cn Cn 1 − √α p
√ C n
G = G(r) = 2π Cn (D(r)) .
Cn (r2 )

r p !
p p  C (r )−α  Equation (10) gives the time delay for a radar signal in
α  n 1 
= 2 1− √ Cn (r1 )− Cn (r2 )+α ln  p  . the gravitational field.
Cn  Cn (r2 )−α 

The proper distance between the observer and the small 4 Spectral shift
body is, √
Cn (r1 )s Let an emitter of light have coordinates (tE , rE , θE , ϕE ).
Z √
Cn p Let a receiver have coordinates (tR , rR , θR , ϕR ). Let u be
Rp = √ d Cn an affine parameter along a null geodesic with the values uE
√ Cn − α
Cn (r2 ) and uR at emitter and receiver respectively. Then,
r p     2  −1  √ 2
p α dt α d Cn
= Cn (r1 ) Cn (r1 ) − α − 1− √ = 1− √ +
Cn du Cn du
r p   2  2
p dθ 2 dϕ
− Cn (r2 ) Cn (r2 ) − α + + Cn + Cn sin θ ,
du du
 qp qp  so
  " # 12
 Cn (r1 ) + Cn (r1 ) − α  −1
+ α ln  qp qp . dt α dxi dxj
 = 1− √ gˉij ,
 Cn (r2 ) + Cn (r2 ) − α  du Cn du du

S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 43
Volume 3 PROGRESS IN PHYSICS October, 2005

where gˉij = − gij . Then, 5 Advance of the perihelia

ZuR" −1 # 12 Consider the Lagrangian,

α dxi dxj "
tR − tE = 1− √ gˉij du ,   2 #
Cn du du 1 α dt
uE L = 1− √ −
2 Cn dτ
" −1  √ 2 #
and so, for spatially fixed emitter and receiver, 1 α d Cn
− 1− √ − (13)
(1) (1) (2)
tR − tE = tR − tE ,
(2) 2 Cn dτ
"  2  2 !#
and therefore, 1 dθ 2 dϕ
− Cn + sin θ ,
2 dτ dτ
(2) (1) (2) (1)
ΔtR = tR − tR = tE − tE = ΔtE . (11)
where τ is the proper time. Restricting motion, without
Now by (1), the proper time is, loss of generality, to the equatorial plane, θ = π2 , the Euler-
v
u Lagrange equations for (13) are,
ΔτE = t u1 − q α ΔtE ,  −1 2 √  2
α d Cn α dt
Cn (rE ) 1− √ + −
Cn dτ 2 2Cn dτ
 −2  √ 2 p  2 (14)
and v α α d Cn dϕ
u − 1− √ − Cn =0 ,
u1 − q α
ΔτR = t ΔtR . Cn 2Cn dτ dτ
Cn (rR )  
α dt
1− √ = const = K , (15)
Then by (11), Cn dτ
  12 dϕ
α Cn = const = h , (16)
1− √ dτ
ΔτR Cn (rR )
=  . (12)
ΔτE 1− √ α and ds2 = gμν dxμ dxν becomes,
Cn (rE )
  2
If z regular pulses of light are emitted, the emitted and α dt
1− √ −
received frequencies are, Cn dτ
 −1 √ 2  2 (17)
z z α d Cn dϕ
νE = , νR = , − 1− √ − Cn = 1.
ΔτE ΔτR Cn dτ dτ

so by (12), Rearrange (17) for,

  12   2   √ 2
1− √ α α ṫ α d Cn 1
ΔνR Cn (rE ) 1− √ − 1− √ − Cn = 2 . (18)
=  ≈ Cn ϕ̇2 Cn dϕ ϕ̇
ΔνE 1− √ α
Cn (rR )
Substituting (15) and (16) into (18) gives,
 
α 1 1  √ 2   
≈ 1+ q −q , d Cn Cn α K2
2 + Cn 1 + 2 1− √ − 2 Cn2 = 0 .
Cn (rR ) Cn (rE ) dϕ h Cn h

whence, Setting u = √1C reduces (18) to,

n
   2
Δν νR − νE α 1 1 du α
= ≈ q −q = + u2 = E + u + αu3 , (19)
νE νE 2 dϕ h2
Cn (rR ) Cn (rE )
  (K 2 −1)
1 1 where E = h2 . The term αu3 represents the general-
= πα − =
GR GE relativistic perturbation of the Newtonian orbit.
du
s Aphelion and perihelion occur when dϕ = 0, so by (19),
 
3   1 1
= π sin3  χa − χ0  − . α
κρ0 GR GE αu3 − u2 + u + E =0, (20)
h2

44 S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale
October, 2005 PROGRESS IN PHYSICS Volume 3

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

p of curvature of a great circle at closest
u1 6 u 6 u2 . One then finds in the usual way that the angle approach be Cn (rc ). Now when there is no mass present,
Δϕ between aphelion and subsequent perihelion is, (23) becomes
 2
  du
3α
+ u2 = F ,
Δϕ = 1 + u1 + u2 π . dϕ
4
and has solution,
Therefore, the angular advance ψ between successive p p
perihelia is, u = uc sin ϕ ⇒ Cn (rc ) = Cn (r) sin ϕ ,
!
3απ
3απ 1 1 and
ψ= u1 + u 2 = p +p = 1
2 2 Cn (r1 ) Cn (r2 ) u2c = p =F .
Cn (rc )
  (21)
2 1 1 If
= 3απ + , p p p
G1 G2 Cn (r)  α, Cn (r) > Cn (ra ) ,
where G1 and G2 are the measurable circumferences of
and u = uc > ua at closest approach, then
great circles at aphelion and at perihelion. Thus, to correctly
determine the value of ψ, the values of the said circum- du
ferences must be ascertained by direct measurement. at u = uc ,
dϕ
the circumferences are measurable in the gravitational field.
The radii of curvature and the proper radii must be calculated so F = u2c (1 − uc α), and (23) becomes,
from the circumference values.  2
If the field is weak, as in the case of the Sun, one may take du
+ u2 = u2c (1 − uc α) + αu3 . (24)
G ≈ 2πr, for r as an approximately “measurable” distance dϕ
from the gravitating sphere to a spacetime event. In such a
Equation (24) must have a solution close to flat space-
situation equation (21) becomes,
time, so let
 
3απ 1 1 u = uc sin ϕ + αw(ϕ) .
ψ≈ + . (22)
2 r1 r2
Putting this into (24) and working to first order in α,
In the case of the Sun, α ≈ 3000 m, and for the planet gives
 
Mercury, the usual value of ψ ≈ 43 arcseconds per century dw

2 cos ϕ + 2w sin ϕ = u2c sin3 ϕ − 1 ,
is obtained from (22). I emphasize however, that this value dϕ
is a Euclidean approximation for a weak field. In a strong or
field equation (22) is entirely inappropriate and equation d 1

(21) must be used. Unfortunately, this means that accurate (w sec ϕ) = u2c sec ϕ tan ϕ − sin ϕ − sec2 ϕ ,
dϕ 2
solutions cannot be obtained since there is no obvious way
of obtaining the required circumferences in practise. This and so,
aspect of Einstein’s theory seriously limits its utility. Since 1

w = u2c 1 + cos2 ϕ − sin ϕ + A cos ϕ ,
the relativists have not detected this limitation the issue has 2
not previously arisen in general.
where A is an integration constant. If the photon originates at
infinity in the direction ϕ = 0, then w(0) = 0, so A = − u2c ,
6 Deflection of light and  
1 1 2
In the case of a photon, equation (17) becomes, u = uc 1 − αuc sin ϕ + αu2c (1 − cos ϕ) , (25)
2 2
  2
α dt to first order in α. Putting u = 0 and ϕ = π + Δϕ into (25),
1− √ −
Cn dτ then to first order in Δϕ,
 −1 √ 2  2
α d Cn dϕ
− 1− √ − Cn = 1, 0 = − uc Δϕ + 2αu2c ,
Cn dτ dτ
so the angle of deflection is,
which leads to,
 2 2α 2α 4πα
du Δϕ = 2αuc = p =  n1 = G ,
+ u2 = F + αu3 . (23) Cn (rc )  n
dϕ  r c − r0  +  n c

S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 45
Volume 3 PROGRESS IN PHYSICS October, 2005

Gc > G a . not just as a radius in the gravitational field, but also as a

measurable radius in the field. This is not correct. The only
At a grazing trajectory to the surface of the body,
measurable distance in the gravitational field is the aforesaid
p circumference
Gc = Ga = 2π Cn (ra ) , p of a great circle, from which the radius of
curvature Cn (r) and the proper radius Rp (r) must be
s
p 3   calculated, thus,
Cn (ra ) = sin  χa − χ0  , p G
κρ0 Cn (r) = ,
2π
so then Z q
q   Rp (r) = −g11 dr .
2 κρ3 sin3  χa − χ0   
= 2 sin2  χa − χ0  . (26)
0
Δϕ = q   Only in the weak p field, where the spacetime curvature
3
κρ
sin  χa − χ0  is very small, can Cn (r) be taken approximately as the
0 p
Euclidean value r, thereby making Rp (r) ≡ Cn (r) ≡ r,
For the Sun [5], as in flat spacetime. In a strong field this cannot be done.
Consequently, the problem arises as to how to accurately
  1 measure the required great circumference? The correct de-

sin χa − χ0 ≈  ,
500 termination, for example, of the circumferences of great
circles at aphelion and perihelion seem to be beyond practical
so the deflection of light grazing the limb of the Sun is,
determination. Any method adopted for determining the re-
2 00 quired circumference must be completely independent of any
Δϕ ≈ ≈ 1.65 .
5002 Euclidean quantity since, other than the great circumference
itself, only non-Euclidean distances are valid in the gravita-
Equation
 (26) is an interesting and quite surprising result, tional field, being determined by it. Therefore, anything short
for sin  χa − χ0  gives the ratio of the “naturally measured” of physically measuring the great circumference will fail.
fall velocity of a free test particle falling from rest at infinity Consequently, General Relativity, whether right or wrong as
down to the surface of the spherical body, to the speed of theories go, suffers from a serious practical limitation.
light in vacuo. Thus, The value of the r-parameter is coordinate dependent
the deflection of light grazing the limb of a and is rightly determined from the coordinate independent
spherical gravitating body is twice the square value of the circumference of the great circle associated
of the ratio of the fall velocity of a free test with a spacetime event. One cannot obtain a circumference
particle falling from rest at infinity down to the for the great circle of a given spacetime event, and hence
surface, to the speed of light in vacuo, i .e ., the related radius of curvature and associated proper radius,
 2 from the specification of a coordinate radius, because the
  va 4GMg
2
Δϕ = 2 sin χa − χ0 = 2  = 2 , latter is not unique, being conditioned by arbitrary constants.
c c R ca The coordinate radius is therefore superfluous. It is for this
where Rca is the radius of curvature of the body, Mg the reason that I completely eliminated the coordinate radius
active mass, and G is the gravitational constant. The quantity from the metric for the gravitational field, to describe the
va is the escape velocity, metric in terms of the only quantity that is measurable in the
s gravitational field — the great circumference (see also [6]).
2GMg The presence of the r-parameter has proved misleading to
va = .
R ca 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
7 Practical constraints and general comment formally correct, but rather less illuminating, because his
resulting line element is in terms of the a quantity which is
Owing to their invalid assumptions about the r-parameter [7], not measurable in the gravitational field. One cannot obtain
the relativists have not recognised the practical limitations an explicit expression for the great circumference in terms
associated with the application of General Relativity. It is of the proper radius.
now clear that the fundamental element of distance in the As to the cosmological large-scale, I have proved else-
gravitational field is the circumference of a great circle, where [11] that General Relativity adds nothing to Special
centred at the heart of an extended spherical body and passing Relativity. Einstein’s field equations do not admit of solutions
through a spacetime event external thereto. Heretofore the when the cosmological constant is not zero, and they do
orthodox theorists have incorrectly taken the r-parameter, not admit of the expanding universe solutions alleged by

46 S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale
October, 2005 PROGRESS IN PHYSICS Volume 3

the relativists. The lambda “solutions” and the expanding 5. Schwarzschild K. On the gravitational field of a sphere
universe “solutions” are the result of such a muddleheaded- of incompressible fluid according to Einstein’s theory.
ness that it is difficult to apprehend the kind of thoughtless- Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916,
ness that gave them birth. Since Special Relativity describes 424 (arXiv: physics/9912033, www.geocities.com/theometria/
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owing to its agreement with observation. Arp [12], for in- pressible fluid. Progress in Physics, 2005, v. 2, 43–47.
stance, has adduced considerable observational data which 7. Crothers S. J. On the geometry of the general solution for the
is consistent on the large-scale with a flat, infinite, non- vacuum field of the point-mass. Progress in Physics, 2005,
expanding Universe in Heraclitian flux. Bearing in mind v. 2, 3–14.
that both Special Relativity and General Relativity cannot 8. Stavroulakis N. A statical smooth extension of Schwarz-
yield a spacetime on the cosmological “large-scale”, there schild’s metric. Lettere al Nuovo Cimento, 1974, v. 11, 8
is currently no theoretical replacement for Newton’s cos- (www.geocities.com/theometria/Stavroulakis-3.pdf).
mology, which accords with deep-space observations for a 9. Stavroulakis N. On the Principles of General Relativity and
flat space, infinite in time and in extent. The all pervasive the SΘ(4)-invariant metrics. Proc. 3rd Panhellenic Congr.
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can be justified no longer. General Relativity is a theory of Stavroulakis-2.pdf).
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Another serious shortcoming of General Relativity is its Annales de la Fondation Louis de Broglie, 2002, v. 27, 3
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es [13]. This shortcoming seems rather self evident, but app- relativistic cosmology, Progress in Physics, 2005, v. 3, 7–18.
rently not so for the relativists, who routinely talk of black 12. Arp, H. Observational cosmology: from high redshift galaxies
hole binary systems and colliding black holes (e .g. [14]), to the blue pacific, Progress in Physics, 2005, v. 3, 3–6.
aside of the fact that no theory predicts the existence of black 13. McVittie G. C. Laplace’s alleged “black hole”. The Ob-
holes to begin with, but to the contrary, precludes them. servatory, 1978, v. 98, 272 (www.geocities.com/theometria/
McVittie.pdf).
14. Misner C. W., Thorne K. S., Wheeler J. A. Gravitation.
Acknowledgement W. H. Freeman and Company, New York, 1973.

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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S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 47