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Home » General statements » Criticism of Ricci Flat Solutions (Stephen Crothers)

Criticism of Ricci Flat Solutions (Stephen Crothers)

Dear Scientists,

Ric = 0 violates Einstein's 'Principle of Equivalence' and so is inadmissible. Although one can find a formal solution for Ric = 0, the result is not a generalisation of Special Relativity, only of Minkowski space-time. Minkowski space-time and Special Relativity are not one and the same.

Ric = 0 excludes the presence matter by definition and so it cannot, even in principle, generalise Special Relativity (despite the usual Standard Model practice). Since the energy-momentum tensor is set to zero there is no source for a gravitational field, and no matter can be present in the associated space-time (so Einstein's equivalence of gravitational and inertial mass cannot manifest, his laws of Special Relativity cannot manifest, and his freely falling inertial frame – which is defined by the relative motion of two masses - cannot manifest). The Schwarzschild class of solutions and their extensions to Kerr space-time cannot contain any matter by definition.  The introduction of matter into the so-called "Schwarzschild solution" is a posteriori and ad hoc. There is no energy-momentum tensor accounting for the alleged matter so introduced (because Tμν = 0 by hypothesis).

When generalising from Minkowski space-time to the space-time of Ric = 0, matter does not appear in the Minkowski line-element. Matter is assumed to be able to be inserted into the space-time of Minkowski just as it is assumed to be able to be introduced into the 3-D Euclidean space of Newton. To generalise Minkowski space-time with matter (i.e. to generalise Special Relativity), matter must be introduced into the generalisation. This is done, according to Einstein, by means of a non-zero energy-momentum tensor that describes the configuration of the matter (Einstein does not banish matter as the cause of the gravitational field so that gravitational fields can exist in the complete absence of matter). But setting the energy-momentum tensor to zero excludes matter by definition, and hence the generalisation is of Minkowski space-time only, i.e. of a pure geometry (and hence a system of kinematics), not Special Relativity (a system of dynamics). In the process of generalisation, the resulting system of four simultaneous non-linear equations for Ric = 0 is a system in four unknowns – the four components of the metric tensor – and since Tμν = 0 the components of the metric tensor are thereby functions of one another and none of them contain any component for matter. Solving the system of equations results in a line-element in terms of functions of just one component of the metric tensor (the Schwarzschild class of solutions), thus:

goo = goo (√-g22),           g11 = g11 (√-g22),           g22,       g33 = g33 (g22),     

where g22 =  g22(r) is an a priori unknown analytic function of the variable 'r' of Minkowski space-time. All components of the metric tensor contain a constant and that constant is what modulates the curvature of the empty space-time (when this constant is set to zero Minkowski's geometry is recovered). As such the Schwarzschild solution is a pure geometry and one that cannot form a backdrop for Einstein's gravitational field (since it contains no matter). The entire (infinite) class of Schwarzschild solutions must be determined via the intrinsic geometry of the line-element and boundary conditions thereon, and is given by:

g22 = g22(r) = - (|r – ro| n  + αn)2/n

 ro is real,           n is positive real,

wherein the quantities ro and n are entirely arbitrary constants, and α is a constant (that which modulates the curvature).

The Standard Model a posteriori and ad hoc introduction of matter into the "Schwarzschild" solution by identifying the constant therein with matter, by a Newtonian "approximation", is therefore entirely spurious. This spurious association with Newton's potential function also involves a number of other misconceptions. First, Newton's potential is related to a two-body problem – his potential is the potential energy per unit mass of any mass that can be present a priori in the gravitational field of some other given mass, and his theory of gravitation is an interaction between two (or more) masses. But Ric = 0 contains no matter whatsoever. And even if the a posteriori and ad hoc introduction of mass into Hilbert's corruption of Schwarzschild's solution was allowed, for the sake of argument, that produces a one-body solution (in terms of its centre of mass) which can act on nothing (since there is no other matter present). There is surely no meaning to the notion of a gravitational field of one body which can act on nothing. Second, the asymptotic limit of the Schwarzschild solution is Minkowski space-time without matter (because Tμν = 0 from the outset, by hypothesis), not Special Relativity and not Newtonian gravitation, and matter cannot suddenly materialise asymptotically from a space-time that by definition contains no matter. Third, the quantity 'r' appearing in Hilbert's erroneous version of Droste's solution is not a distance of any kind in the Schwarzschild manifold, let alone a radial distance therein. It is the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section and as such it is not a distance, radial or otherwise, in the Schwarzschild manifold. The quantity 'r' is always treated by the Standard Modellers as a radial distance (although when cornered some of them, in futility, try to claim otherwise). They routinely call 'r' in Hilbert's corruption of the Schwarzschild/Droste solution, the radius or the radius of a sphere or the coordinate radius or the radial coordinate or the radial space coordinate or the reduced circumference or the radius of a 2-sphere or (according to G. 't Hooft) "a gauge choice; it defines the coordinate r". In their particular alleged case of r = 2m (using G = c = 1) it is always called the Schwarzschild radius or the gravitational radius. However, it is in fact none of these vague and various veils of ignorance, because it is easily proved that it is the radius of Gaussian curvature of the spherically symmetric geodesic surface in the spatial section, which is not a distance of any kind in the Schwarzschild manifold (recall that Gaussian curvature is an intrinsic property of a surface – Gauss' Theorema Egregium). But in Newton's theory 'r' is the radial distance between the centres of mass of two gravitationally coupled bodies (because his 'r' relates to a 3-D Euclidean space, not a Riemann space).

Steve Crothers.

Posted: 2008-05-28 


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