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Gravitational acceleration and the curvature distortion of spacetime.

1 Newtonian Introduction

When a test mass m' travels in the gravitational field of a point mass m' situated at r = 0, the Newtonian theory of gravity predicts that the acceleration experienced by the test mass

[d.sup.2]r/[dt.sup.2] = mG/[r.sup.2] (1)

is independent of the mass m'. In this theory the relative magnitudes of m' and m are arbitrary and lead to the following equation for the magnitude of the gravitational force between the two masses

m' mG/[r.sup.2] (m' [c.sup.2]/r)([mc.sup.2]/r)/[c.sup.4]/G

= (m'[c.sup.2]/r/[c.sup.4]/G)([mc.sup.2]/r/[c.sup.4]/G)[c.sup.4]/G (2)

when expressed in terms of the ratio [c.sup.4]/G.

In the PV theory [1] the force [mc.sup.2]/r represents the curvature distortion the mass m exerts on the PV state (and hence on spacetime), and the ratio

[c.sup.4]/G = [m.sub.*][c.sup.2]/[r.sub.*] (3)

represents the maximum such curvature force, where [m.sub.*] and [r.sub.*] are the mass and Compton radius of the Planck particles constituting the PV. The corresponding relative curvature force is represented by the n-ratio

[n.sub.r] [equivalent to] [mc.sup.2]/r/[c.sup.4]/G = [mc.sup.2]/r/m*[c.sup.2]/[r.sub.*] (4)

which is a direct measure of the curvature distortion exerted on spacetime and the PV by the point mass. Since the minimum distortion is 0 (m = 0 or r [right arrow] [infinity]) and the maximum is 1, the n-ratio is physically restricted to the range 0 [less than or equal to] [n.sub.r] [less than or equal to] 1 as are the equations of general relativity [2].

The important fiducial point at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Schwarzschild radius [r.sub.s] = 2[mr.sub.*]/[m.sub.*], where


The acceleration (1) can now be expressed exclusively in terms of the relative curvature distortion nr:

a([n.sub.r]) = - [d.sup.2]r/[dt.sup.2] = [mc.sup.4]/[r.sup.2][c.sup.4]/G = [c.sup.2]/r [mc.sup.2]/r/[m.sub.*][c.sup.2]/r

= [c.sup.2]/r [n.sub.r] = [c.sup.2]/[rn.sub.r] [n.sup.2.sub.r] 2[c.sup.2]/[r.sub.s][n.sup.2.sub.r] (6)

whose normalized graph a/(2[c.sup.2]/[r.sub.s]) is plotted in the first figure.

2 Affine Connection

The conundrum posed by equation (1), that the acceleration of the test particle is independent of its mass m', is the principle motivation behind the general theory of relativity [3, p. 4]; an important ramification of which is that, in a free-falling local reference frame, the acceleration vanishes as in equation (7). That result leads to the following development. Given the two coordinate systems [x.sup.[mu]] = [x.sup.[mu]]([[xi].sup.v]) and [[xi].sup.[mu]] = [[xi].sup.[mu]] ([x.sup.v]) and the differential equation

[d.sup.2] [[xi].sup.[mu]]/d[[tau].sup.2] = 0 (7)

applying the chain law to the differentials gives


Then using


to eliminate the coefficient of d2xv/dr2 in (8) leads to


Rearranging indices in(10) finally yields


where [u.sup.[mu]] d[x.sup.[mu]]/d[tau] is a typical component of the test-mass 4-velocity and


is the affine connection. The affine connection vanishes when there is no gravitational distortion; so for the point mass m, it should be solely a function of the curvature distortion [n.sub.r] given by (4).

The affine connection can be related to the the metric coefficients [g.sub.[alpha][beta]] via [3, p. 7]


which, for a metric with no cross terms ([g.sup.[alpha][beta]] = 0 for [alpha] [not equal to] [beta]), reduces to


with [mu] = 1 for example.

Since only radial effects are of interest in the present paper, only the [x.sup.0] and [x.sup.1] components of the spherical polar coordinate system ([x.sup.[mu]]) = ([x.sup.0], [x.sup.1], [x.sup.2], [x.sup.3]) = (ct, r,[theta],[phi]) are required. Then the affine connection in (11) for the [mu] = 1 component reduces to

[du.sup.1]/d[tau] = -[[GAMMA].sup.1.sub.v[rho]] [u.sup.v][u.sup.[rho]]

= - [[[GAMMA].sup.1.sub.00][([u.sup.0]).sup.2] + 2[[GAMMA].sup.1.sub.01][u.sup.0][u.sup.1] + [[GAMMA].sup.111][([u.sup.1]).sup.2]] (15)

which under static conditions ([u.sup.1] = dr/d[tau] = 0 for the test mass) produces

[du.sup.1]/d[tau] = - [[GAMMA].sup.1.sub.00][([u.sup.0]).sup.2]. (16)

In the spherical system with d[theta] = d[phi] = 0, the metric becomes

[ds.sup.2] = [c.sup.2]d[[tau].sup.2] = [g.sub.00] [c.sup.2][dt.sup.2] + [g.sub.11] [dr.sup.2] (17)

where [g.sub.00] and [g.sub.11] are functions of the coordinate radius [x.sup.1] = r. Under these conditions the only non-zero affine connections from (14) are:

[[GAMMA].sup.0.sub.10] [[GAMMA].sup.0.sub.01] [g.sup.00]/2 [partial derivative][g.sub.00]/[partial derivative][x.sup.1] (18)


Using (17), the velocity [u.sup.0] can be calculated from

cd[tau] = [g.sup.1/2.sub.00] 2[dx.sup.0] [[1 + ([g.sub.11]/[g.sub.00])[(dt/dt/c).sup.2]].sup.1/2] (20)

which for static conditions (dr/dt = 0) leads to

[u.sup.0] = [dx.sup.0]/d[tau] = c/[g.sup.1/2.sub.00]. (21)

Inserting (21) into (16) gives

[du.sup.1]/d[tau] = - [c.sup.2][[GAMMA].sup.1.sub.00]/[g.sub.00] = [c.sup.2]/[g.sub.00]([g.sup.11]/2 [partial derivative][g.sub.00]/[partial derivative]r) (22)

along with its covariant twin


Then combining (22) and (23) leads to the static acceleration


3 Static Acceleration

The metric coefficients [g.sub.00] and [g.sup.11] for a point mass m at r = 0 are given by (A6) and (A7) in the Appendix. After some straightforward manipulations, (24) leads to the (normalized) static gravitational acceleration (0 [less than or equal to] nr [less than or equal to] 1)


= [[n.sup.2.sub.r]/(1 - [r.sub.s] + [2.sup.n][n.sup.n.sub.r]).sup.1/n][(1 + [2.sup.n][n.sup.n.sub.r]).sup.3/2n] (26)

= [n.sup.2.sub.r]/[[[(1 + 2/[2.sup.n][n.sup.n.sub.r).sup.1/n] - 1].sup.1/2][(2[n.sub.r]).sup.1/2][(1 + [2.sup.n][n.sup.n.sub.r]).sup.3/2n] (27)

in terms of the relative curvature force [n.sub.r], all of which vanish for [n.sub.r] = 0. Formally, the acceleration in the denominator on the left of (25)

[DELTA][upsilon]/[DELTA]t = c/([r.sub.s] - [r.sub.s]/2)/c = 2[c.sup.2]/[r.sub.s] (28)

is the acceleration of a test mass starting from rest at r = [r.sub.s] ([n.sub.r] = 0.5) and accelerating to the speed of light c in its fall to [r.sub.s]/2 ([n.sub.r] = 1) in the time interval ([r.sub.s] - [r.sub.s]/2)/c.

The limits of (26) and (27) as n [right arrow] [infinity] are easily seen to be


where [n.sub.r] < 0.5 and [n.sub.r] > 0.5 are used in (26) and (27) respectively. Equations (26) and (27) are plotted in Figures 2 and 3 for various indices n, all plots of which are continuous in the entire range 0 [less than or equal to] [n.sub.r] [less than or equal to] 1. The curve that runs through the point (1,1) in Figure 2 is the Newtonian result from (6). It is clear from Figure 3 that the acceleration diverges in the range 0.5 [less than or equal to] [n.sub.r] [less than or equal to] 1 for the limit n [right arrow] [infinity]. In the range 0 [less than or equal to] [n.sub.r] [less than or equal to] 0.5 the acceleration is given by the upper equation in (29)--this result is identical with the static black-hole acceleration [3, p. 43].

4 Summary and Comments

The nature of the vacuum state provides a force constraint ([n.sub.r] [less than or equal to] 1) on any theory of gravity, whether it's the Newtonian theory or the general theory of relativity [2]. This effect manifests itself rather markedly in the equation for the Kerr-Newman black-hole area A for a charged spinning mass [4]:

A = 4[pi]G/[c.sup.4] x [2[m.sup.2]G-[Q.sup.2] + 2[([m.sup.4][G.sup.2]-[c.sup.2][J.sup.2]-[m.sup.2][Q.sup.2]G).sup.1/2]] (30)

where Q and J are the charge and angular momentum of the mass m. Using the relation in (3) and G = [e.sup.2.sub.*]/[m.sup.2.sub.*] [1], it is straightforward to transform (30) into the following equation


where all of the parameters ([e.sub.*], [m.sub.*], [r.sub.*], except c of course) in the denominators of the terms are PV parameters; and all of the terms are properly normalized to the PV state, the area A by the area 4[pi][r.sup.2], the angular momentum J by the angular momentum [r.sub.*][m.sub.*]c, and so forth.

The "dogleg" in Figure (4) at the Schwarzschild radius [r.sub.s] ([n.sub.r] = 0.5) and the pseudo-singularity in the black-hole metric at [r.sub.s] are features of the Einstein differential geometry approach to relativistic gravity--how realistic these features are remains to be seen. At this point in time, though, astrophysical measurements have not yet reached the [n.sub.r] = 0.5 point (see below) where the dogleg and the black-hole results can be experimentally checked, but that point appears to be rapidly approaching. Whatever future measurements might show, however, the present calculations indicate that the point-mass-PV interaction that leads to nr may point to the physical mechanism that underlies gravity phenomenology.

The evidence for black holes with all m/r ratios appears to be growing [3, Ch. 6]; so it is important to see if the present calculations can explain the experimental black-hole picture that is prevalent in today's astrophysics. The salient feature of a black hole is the event horizon [3, pp. 2, 152], that pseudosurface at r = [r.sub.s] at which strange things are supposed to happen. A white dwarf of mass 9 x [10.sup.32]gm and radius 3 x [10.sup.8]cm exerts a curvature force on the PV equal to 2.7 x [10.sup.45]dyne, while a neutron star of mass 3 x [10.sup.33]gm and radius 1 x [10.sup.6]cm exerts a force of 2.7 x [10.sup.48]dyne [2]. Dividing these forces by the 1.21 x [10.sup.49]dyne force in the denominator of (4) leads to the n-ratios [n.sub.r] = 0.0002 and [n.sub.r] = 0.2 at the surface of the white dwarf and neutron star respectively. The surfaces of these two objects are real physical surfaces--thus they cannot be black holes.

On the other hand, SgrA* [3, p. 156] is thought to be a supermassive black-hole with a mass of about 4.2 x [10.sup.6] solar masses and a radius confined to r < 22 x [10.sup.11] [cm], leading to the SgrA* n-ratio [n.sub.r] > 0.28. For an n-ratio of 0.28, however, the plots in Figures 2-4 show that the behavior of spacetime and the PV is smooth. To reach the [n.sub.r] = 0.5 value and the dogleg, the SgrA* radius would have to be about 12 x [10.sup.11] [cm], a result not significantly out of line with the measurements.

Finally, it should be noted that the black-hole formalism is the result of substituting [R.sub.n] = r in the metric (A1) of the Appendix. Unfortunately, since [R.sub.n]/r > 1 signifies a response of the vacuum to the perturbation nr at the coordinate radius r, the effect of this substitution is to eliminate that response. This is tantamount to setting [n.sub.r] = 0 in the second-to-last expression of (A3).

Appendix: Crothers Vacuum Metrics

The general solution to the Einstein vacuum field [5] [6] for a point mass m at r = 0 consists of the infinite collection (n = 1,2,3, ***) of Schwarzschild-like metrics that are nonsingular for all r > 0:

[ds.sup.2] = (1 - [r.sub.s]/[R.sub.n])[c.sup.2][dt.sup.2] - [(r/[R.sub.n]).sup.2n-2][dr.sup.2]/1 - [r.sub.s]/[R.sub.n] -

-[R.sup.2.sub.n](d[[theta].sup.2] + [sin.sup.2] [theta]d[[phi].sup.2]) (A1)


[r.sub.s] = mG/[c.sup.2] = 2 [mc.sup.2]/[m.sub.*][c.sup.2]/[r.sub.*] = 2[rn.sub.r] (A2)

[R.sub.n] = [([r.sup.n] + [r.sup.n.sub.s]).sup.1/n] = r[(1 + [2.sup.n][n.sup.n.sub.r]).sup.1/n] = [r.sub.s][(1 + [2.sup.n][n.sup.n.sub.r]).sup.1/n]/2[n.sub.r] (A3)

and where r is the coordinate radius from the point mass to the field point of interest and [r.sub.s] is the Schwarzschild radius. The ratio [R.sub.n]/r as a function of [n.sub.r] is plotted in Figure 4 for various indices n. The n-ratios 0, 0.5, and 1 correspond to the r values r [right arrow], [r.sub.s], and [r.sub.s]/2 respectively.

All the metrics in (A1) for n [greater than or equal to] 2 reduce to

[ds.sup.2] = (1 - 2[n.sub.r]) [c.sup.2][dt.sup.2] - [dr.sup.2]/1 - 2[n.sub.r] - [r.sup.2](d[[theta].sup.2] + [sin.sup.2] [theta]d[[phi].sup.2]) (A4)

for [n.sub.r] [much less than] 1.

It is clear from the expressions in (A3) that the requirement of asymptotic flatness [3, p.55] is fulfilled for all finite n. On the other hand, the proper radius [R.sub.n] from the point mass at r = 0 to the coordinate radius r is not entirely calculable:


due to the failure of the general theory in the region 0 < r < [r.sub.s]/2 [2].

The metric coefficients of interest in the text for d[theta] = d[phi] = 0 are

[g.sub.00] = (1 - [r.sub.s]/[R.sub.n]) (A6)

[g.sub.11] = [(r/[R.sub.n]).sup.2n-2]/1 - [r.sub.s]/[R.sub.n] = 1/[g.sub.11] (A7)

From (A3)

[partial derivative][R.sub.n]/[partial derivative] = 1/[(1 + [2.sup.n][n.sup.n.sub.r]).sup.(1-1/n)] (A8)

and from (A8)


Submitted on May 3, 2012 / Accepted on May 11, 2012

William C. Daywitt

National Institute for Standards and Technology (retired), Boulder, Colorado



[1.] Daywitt W.C. The Planck vacuum. Progress in Physics, 2009, v. 1, 2026.

[2.] Daywitt W.C. Limits to the validity of the Einstein field equations and general relativity from the viewpoint of the negative-energy Planck vacuum state. Progress in Physics, 2009, v. 3, 27-29.

[3.] Raine D., Thomas E. Black Holes: An Introduction. Second Edition, Imperial College Press, London, 2010. The reader should note that m is used in the present paper to denote physical mass--it is not the geometric mass m (= MG/[c.sup.2]) as used in this reference.

[4.] http ://www.phy. olemiss. edu/-luca/Topics/bh/laws.html

[5.] Crothers S.J. On the general solution to Einstein's vacuum field and its implications for relativistic degeneracy. Progress in Physics, 2005, v. 1, 68-73.

[6.] Daywitt W.C. The Planck vacuum and the Schwarzschild metrics. Progress in Physics, 2009, v. 3, 30-31.
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Author:Daywitt, William C.
Publication:Progress in Physics
Date:Jul 1, 2012
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