Not so ramparts. The claim that Ric = 0 describes the "field outside a body" is sophistic nonsense. You have given Einstein's field equations on your posting header. The left side of the equations describes the gravitational field, manifest in the geometry (and thereby curvature) of spacetime; the right side describes the configuration of matter that causes the gravitational field. So the field equations couple the gravitational field to the matter that causes it. Einstein's field equations:
"...couple the gravitational field (contained in the curvature of spacetime) with its sources."
Foster, J., Nightingale, J. D. A short course in General Relativity, Springer-Verlag, New York, Inc., 1995.
"Since gravitation is determined by the matter present, the same must then be postulated for geometry, too. The geometry of space is not given a priori, but is only determined by matter."
Pauli, W. The Theory of Relativity, Dover Publications, Inc., New York, 1981.
"In general relativity, the stress-energy or energy-momentum tensor T^a^b acts as the source of the gravitational
field. It is related to the Einstein tensor and hence to the curvature of spacetime via the Einstein equation".
McMahon, D. Relativity Demystified, A Self-teaching guide; McGraw-Hill, New York, 2006
"Thus the equations of the gravitational field also contain the equations for the matter (material particles
and electromagnetic fields) which produces this field."
Landau, L., ____ E. The Classical Theory of Fields, Addison-Wesley Publishing Company, Inc., Reading, Massachusettes, 1951.
"Again, just as the electric field, for its part, depends upon the charges and is instrumental in producing mechanical interaction between the charges, so we must assume here that the metrical field (or, in mathematical language, the tensor with components g_{ik} is related to the material filling the world. ... we have, in following the ideas set out just above, to discover the invariant law of gravitation, according to which matter determines the components Gamma^a_{bi} of the gravitational field, and which replaces the Newtonian law of attraction in Einstein’s Theory."
Weyl, H. Space Time Matter, Dover Publications Inc., New York, 1952.
"... the whole of the stress-energy tensor T" is "the source of the gravitational field."
Schutz, B. F. A first course in general relativity, Cambridge University Press, UK, 1990.
So, setting Ric = 0 eliminates all sources of a gravitational field by construction, yet the astrophysical scientists claim that this describes the "gravitational field outside a body"; their alleged "vacuum solutions". What is the source of the gravitational field "outside the body"? Well,e thereof) it must be described by a non-zero energy-momentum tensor in Einstein's field equations, as the astrop according to the very same astrophysical scientists, its the body! But if the body causes the gravitational field (i.e. a sourchysical scientists also assert (as cited above). As for the "Schwarzschild solution" in particular:[
"... the corresponding Newtonian potential is V = -MG/r, where M is the mass of the body producing the field, and G is the gravitational constant. ... we conclude that k = -2MG/c^2, and Schwarzschild's solution for the empty spacetime outside a spherical body of mass M is ..."
Foster, J., Nightingale, J. D. A short course in General Relativity, Springer-Verlag, New York, Inc., 1995.
"... the constant of integration m that has appeared ... is just the mass of the central body that is producing the gravitational field."
Dirac, P. A. M. General Theory of Relativity. Princeton Landmarks in Physics Series, Princeton University Press,
Princeton, New Jersey, 1996.
"... the quantity m_o introduced by the equation m = km_o occurs as the field-producing mass in it; we call m the gravitational radius of the matter causing the disturbance of the field."
Weyl, H. Space Time Matter, Dover Publications Inc., New York, 1952.
So the astrophysical scientists on the one hand remove all matter, including sources, by setting Ric = 0, and on the other hand, by a piece of sophistry, put it back in with the play on the words "outside a body". Not only that, when they get their "Schwarzschild solution" there is no matter present (its a solution for Ric = 0), so to satisfy their sophistic "outside a body" they put mass back in, post hoc, and to do so they use Newton's expression for escape velocity. Now Newton's expression for escape velocity is a two-body relation: one body escapes from another; and so it can't appear in an alleged one-body configuration (but which is in fact a no-body situation - a universe that contains no matter, as cited in my previous post as well). Furthermore, the alleged "Schwarzschild black hole" is alone in an otherwise empty Universe, according to the astrophysical scientists (see my previous post as well), and so there is nothing present, by construction, with which it can interact. Yet the astrophysical scientists also claim that these black holes exist in multitudes, in binary systems, colliding and merging with one another and interacting with other matter. So they are black holes that mutually persist in and mutually interact in a mutual spacetime that by construction contains no matter! All alleged black hole "solutions" pertain to one alleged mass in the entire Universe, so they can't interact with any other masses, since there are no other masses present by construction. There are no known solutions to Einstein's field equations for two or more masses, there is no existence theorem by which it can even be claimed that his field equations contain latent solutions for such configurations of matter, and the Principle of Superposition does not apply in General Relativity (so matter can't be simply piled up in any given spacetime - see the citations above). Einstein himself thoughtlessly talks about "the motion of a planet" in Schwarzschild spacetime, where "M denotes the sun's mass, centrally symmetrically placed about the origin of coordinates" . Einstein, A. The Meaning of Relativity, Science Paperbacks and Methuen & Co. Ltd., 56–57, 1967.
Also, there is not one shred of physical evidence to suggest that a gravitational field can be generated by anything other than matter.
Matter cannot be present in a spacetime that by construction contains no matter, and a gravitational field can't be described by Ric = 0 since Einstein's field equations couple the gravitational field to its sources. Moreover, Schwarzschild spacetime is not a generalisation of Special Relativity - it is only a generalisation of Minkowski spacetime. There is no mapping of matter from Special Relativity into Schwarzschild spacetime. Minkowski spacetime is not Special Relativity. Special Relativity is defined in terms of the presence of matter, Minkowski spacetime it not. And the speed of light that appears in the expression for Minkowski spacetime is not a photon - it's just a speed. The astrophysical scientists insert mass post hoc into Schwarzschild spacetime after they removed it in the first place from the mathematical relations - Schwarzschild spacetime is no generalisation of Special Relativity at all. And since Ric = 0 is thus inadmissible, Einstein's field equations violate the usual conservation of energy and momentum, placing them in conflict with the experimental evidence on a deeper level, and so General Relativity fails. To save his theory from this catastrophe Einstein invented, ad hoc, his pseudo-tensor, the components of which he says are
'the "energy components” of the gravitational field' and "expresses the law of conservation of momentum and of energy for the gravitational field."
Einstein, A. The Meaning of Relativity, Science Paperbacks and Methuen & Co. Ltd., 56–57, 1967.
But Einstein's ad hoc invention does not save his theory because his pseudo-tensor implies the existence of an invariant that depends solely upon the components of the metric tensor and their first derivatives - but the pure mathematicians G. Ricci-Curbastro and T. Levi-Civita, inventors of the tensor calculus, proved in 1900 that such invariants do not exist! Thus, reductio ad absurdum, Einstein's pseudo-tensor is a meaningless concoction of mathematical symbols and everything that relies upon it fallacious such as Einstein gravitational waves.
"It is not possible to obtain an expression for the energy of the gravitational field satisfying both the conditions: (i) when added to other forms of energy the total energy is conserved, and (ii) the energy within a definite (three dimensional) region at a certain time is independent of the coordinate system. Thus, in general, gravitational energy cannot be localized. The best we can do is to use the pseudotensor, which satisfies condition (i) but not condition (ii). It gives us approximate information about gravitational energy, which in some special cases can be accurate.
"Let us consider the energy of these waves. Owing to the pseudo-tensor not being a real tensor, we do not get, in general, a clear result independent of the coordinate system. But there is one special case in which we do get a clear result; namely, when the waves are all moving in the same direction."
Dirac, P. A. M. General Theory of Relativity. Princeton Landmarks in Physics Series, Princeton University Press,
Princeton, New Jersey, 1996.
It is not I, ramparts, who has abused science. The astrophysical scientists are the abusers of science.
Vacuum solutions (like one describing space outside the Sun) are still affected by matter from elsewhere - they just can't be examined where there actually is matter (like in the interior of the Sun).
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