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### 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).