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Can Gravity be Divorced from Matter?

Updated: Apr 18, 2022

Einstein would respond with an emphatic "yes". However, his condition for emptiness, i.e. the absence of matter, and his expression for "the 'energy components' of the gravitational field" mixes first and second order quantities. And since gravitational waves are linearized solutions of the Einstein equations, their source term is of second order, and, consequently, would never appear in either the near or far solutions, i.e., near or far from the sources of the gravitational field....And what is the "source" of the gravitational field if not the presence of matter? That's like saying there is a Coulomb law in the absence of charges.

The expression derived by Einstein was criticized by Levi-Civita. Einstein's expression consists of a bilinear form of Christoffel symbols of the second type. Now, the Christoffel symbols contain the metric coefficients as well as their first derivatives. Levi-Civita was quick to point out that "differential invariations of the first not exist." You need a second derivative to tell how the the metric curves space (and if, you will, time). Levi-Civita remarked about Einstein's perplexity of his result: Spontaneous gravitational waves would result in a "dispersion of energy through irradiation".

Moreover, Einstein's condition of emptiness mixes first and second order terms, i.e. linear terms in the derivatives of the Christoffel symbols, and quadratic terms in the products of the Christoffel symbols. That's equivalent to retaining quadratric terms in the gradients of the displacement in the strain tensor

E=1/2( grad u + u grad) + 1/2 (grad u)(grad u)^T, (A)

where T denotes the transpose. For a comparison with the Ricci tensor,

R_(ij)= {ij,l}_l -{il,l}_j +{lm,m}{ij,l} - {ik,l}{il,k} (B)

where repeated indices are to summed over, and {ij,k} is the Christoffel symbol of the second type leads to the inescapable conclusion that if the second term in (A) is to be neglected than the second term in (B) must also be neglected. For in elasticity theory, the Beltrami decomposition of a general, symmetric tensor is

T_(ij) = E_(ij) +(Curl(Curl g))_(ij), (C)

for a symmetric tensor, g, e.g. in the case of gravity, g would be the metric coefficients of the static metric. (C) is valid provided the non-linear terms in (A) and (B) are neglected, where the linear terms in the Ricci tensor (B) represent the solenoidal component in the break-down of T.

The decomposition (C) could be likened to the the sum the electric field and the curl of the magnetic field. The two are unrelated quantities if the fields are static since there would be no induction of the magnetic field by a time varying electric field. And it would also follow that there is no induction of an electric field by a time varying magnetic field. The lack of inductive terms in the circuit equations, to use Heaviside's terminology, means that there can be no traveling electromagnetic waves.

To see where Einstein was led to a mixing of first and second order terms, I consider his 1916 paper "The foundation of the general theory of relativity". In that paper, he wants to derive the energy of the gravitational field in the absence of matter. He uses his condition of emptiness: the Ricci tensor (B) must vanish in an empty universe. I take note of the following errors:

i) In equation (44), he wrongly contracts the Riemann tensor to get was is supposedly the symmetric Ricci tensor. The contaction should be with respect to the second index in the Ricci tensor, and not the third one since the Riemann tensor is antisymmetric in the first and last two indices.

ii) In that set of equations, he splits the Ricci tensor in terms which are independent of the determinant of the metric tensor,

S_(ij)=-((ln g^1/2)_(ij)+ {ij,k}((ln g^1/2)_k (C)

which are the second and third terms in (B), and

C_(ij)= {ij,k}_k -{ik,m}{jm,k}, (D)

which are the first and last terms in (B).

(iiI) Now he claims that he is free to choose coordinates in which g=1, i.e., a flat metric. Although the background metric in the derivation of gravitational waves is the Minkowski metric, there is no approximation here of considering a linear perturbation on a background metric. It is, therefore, somewhat surprising that his variational equation (48) is (D) but without any condition on g! That is, although he writes down the condition in (47a) is not used in the variation of his function H, in the expressions that follow.

(iv) There is also a problem with the variation of H,

H=g^(ij) {ik,m}{jm,k}. (E)

For he writes

dH= dg^(ij) {ik,m}{jm,k} + 2g^(ij){ik,m} d{jm,k},

where he takes advantage of the symmetry in the bottom two indices in the Christoffel symbols. Now he performs and integration by parts to get

dH= -{ik,m}{jm,k} dg^(ij) +2{ik,m}d(g^(ij){jm,k}).

But, he can't introduce the relation,


for a Christoffel symbol with a repeated indice since that would imply k=m, and convert his H-function (E) into


which is the second quadratic term in the Ricci tensor (B) because sum over repeated indices is implied. While it is also true that


because of the symmetry in the lower two indices of the Christoffel symbol, it is doesn't follow that


(v) This is revealed in his last expression in (49). He finds it necessary to introduce a Kronecker delta function before H in his expression his energy components of the gravitational field. The Kronecker delta function is between two 'mute' indices which should be summed over, leaving only the two indices on the metric coefficients, like that in the Ricci tensor (B).

(vi) In so doing, Einstein comes out with (47b), which is the vanishing of T in expression (D). Yet if everything went through without a hitch, there is no mention that (C) must also vanish in order to get the vanishing of the Ricci tensor, (B)!

(vi) Trying to make a bridge with the Poisson equation,

div grad V= 4 pi G rho, (F)

where V is the scalar potential and rho is the mass density, he sets (D) equal to zero and multiplies the right-hand side through by g^(km). Performing an integration by parts to get

(g^(km){im,n)_n= t^k_m, (G)

which is a mixed tensor in which the indices have nothing whatsoever to do with the original indices in (B). It's OK to use "different symbols for the summation indices", but not the original ones.

(vii) Einstein concludes that "the system of equations (51) [corresponding to (G) without the added condition g=1, which wasn't used in the derivation of his variational equation, C=0 in (D)] shows how this energy-tensor (corresponding to the density rho in Poisson's equation) is to be introduced into the field equations of gravitation....This will allow itself [ponderable energy together with gravitational energy] to be expressed by introducing into (51), in place of the energy-components of the gravitational field alone, the sum t_m^k+T^k_m of the energy components of matter and of gravitational field."

(viii) So on the right side of (G) we should add the mixed energy-stress tensor, T^k_m. Feeling rather uneasy about this, Einstein admits that "this introduction of the energy-tensor of matter is not justified by the relativity postulate alone." [Note the word "alone", and there is a typo on the left-hand side of (52) in the translation.]

Here, Einstein contradicts his whole approach. His field equation

R_(ij)= T_(ij)- (1/2)g_(ij) T,

supposedly represents a universal equivalence between geometry (left-hand side) and the energy-stresses that cause it (right-hand side). We already know this not to be true because there is a non-euclidean geometry even in the absence of T, e.g. the outer Schwarzschild solution. Yet, when Einstein tells us that by working backwards, he gets

{ij,k}_k -{il,k}{jk,l}= -[T_(ij)- (1/2)g_(ij)T], (H)

contradicts his whole theory. For the left-hand side is what is left of the Ricci tensor after he imposes his condition g=1. Yet, from his previous equation (52), he has

{ij,k}_k + 2{km,m}{ij,k}=-g^(ij)[t_(ij)+T_(ij)-1/2g_(ij)(t+T)]. (I)

Subtracting (H) from (I) results in

2{mk,m}{ij,k}+{im,k}{jk,m}=-g^(ij)[t_(ij)-(1/2)g_(ij)t]. (J)

But, because g=1, the first term in (J) vanishes, and we get his equation (50). But the first term can't vanish since otherwise

(1/g)g_k=g_(ij) g^(ij)_k =0, (K)

and his expression for the pseudo-tensor, t, vanishes. He should have notice that the equation before his (49), viz.,

[g^(ij)_k H_(g^(ij)_k]_m - H_k=0, (L)

is not (t_k^m)_k=0, because the differential in the first term in (L) is with respect to x_m, while the derivative in the second term in (K) is with respect to x_k. This explains the appearance of the Kronecker delta in the second term in the second equation of (49): the two indices must coincide. Moreover, if the first term in (J) is zero because g=1, then how does H become a function of g^(ij)_k? which is zero! The crucial equation is (K) which brings the curtain down on his whole approach.

Even if everything was hunky dory, there is still the problem of the meaning of setting the 'energy components' of the gravitational field equal to

t_(ij)= {ik,l}{jlk},

i.e., a bilinear product of Christoffel symbols of the second kind.

(viii) The relativity postulate doesn't enter at all, and neither does it in the identification of g^(km){im,n} as the analogy of the scalar potential V in Poisson's equation (F). A Christoffel coefficient is far from a scalar potential, which can be used to derive the tidal forces in Newtonian gravitation.

(ix) So Einstein has reached his desired conclusion that "we have here deduced it from the requirement that the energy of the gravitational field shall act gravitationally in the same way as any other kind of energy." Being written in by hand, provides no justification for the dichotomy of gravitation and matter. They are, in fact, one and the same since one cannot appear without the other, as Newton well knew.

Levi-Civita claims that ---in an empty universe--- the sum of the energy tensor and the gravitational energy should vanish because it corresponds to the vanishing of the Ricci tensor. In his words, "when the energy tensor vanishes the same occurrence must happend to the gravitatonal tensor [which is only a pseudo-tensor]. This fact entails the total lack of stresses, of energy flow, and also of a simply localization of energy." ["On the analytic expression that must be given to the gravitational tensor in Einstein's theory"]

As concluding remarks, there are parts of Levi-Civita's article that don't coincide with Einstein's interpretation or make sense. He considers the pseudo-stress tensor as the complete expression in the Ricci tensor (B), viz.,

G*=-g^(ik)[{ih,l}{kl,h} -{ik,l}{lh,h}].

He claims that since G* "is quadratic and homogeneous with respect to the Christoffel symbols, and hence also with respect to the g^(ik)_l, by virtue of Euler's theorem

G*_(g^(ik)_l) g^(ik)_l =2G*,

the invariant should reduce itself to G*." Although G* is bilinear in the Christoffel symbols, it is not homogeneously quadratic in the derivatives, g^(ik)_l, viz,

G*=2 (g^(ik)_l{ik,l}-(1/4)g^(ik) g^(mj)g^(ln) (g_(jn))_i (g_(lm))_k.

Levi-Civita's assertion would be true if he considered Einstein's function H, in (E).

One last point deserves mention. Landau & Lifshitz, in their first English edition of the "Classical Theory of Fields", write the Ricci tensor solely in terms of second derivatives of the metric tensor coefficients claiming that "at the point under consideration, all the {ij,k}=0". What they fail to appreciate is that all the g_(ij) are then constant. the Ricci tensor vanishes itself. Apart from the absurd expression for the pseudo tensor they find, they claim that conservation dictates

(T^(ik)-t^(ik))_k=0 (M)

is a condition for the conservation of energy and momentum.

However, t_(ij)=-{ik,m}{jm,k} is not the entire expression for the Ricci tensor in Einstein's view since the term {ij,k}_k is missing in the vanishing of the ordinary divergence in (M). And why is the ordinary divergence used when the whole idea is to take into consideration the non-euclidean geometry that gravity causes? In fact, in Landau & Lifshitz case, the pseudo-tensor is identically zero.

And the proof is in the pudding. After having written down the horrendous expression they find for the pseudo-tensor in their (82), no mention is made of it again. And if it is not an expression for the Ricci tensor on the previous page, then Einstein's equation cannot be fulfilled. In fact, in the fourth edition, the whole thing disappears. In place appears a new construct of "auxiliary quantities which are obtained upon going over from the exact equations of gravitation to the case of a weak field in the approximation we are considering....[they] contain along with terms obtained from the T^i_j also terms of second order from [the Einstein tensor]." This is totally incompatible with the linear equation they write down in (110.1). Moreover, they require it to be a conserved quantity in that the ordinary divergence should vanish which "replaces the general relation

div T^(ij)=(T^(ij))_j + {mj,i}T^(mj) =0,"

the vanishing of the covariant divergence.

Anything and everything goes. What is extremely amusing is how they bridge this with the expression they really want, the mass quadrupole expression. Again we are lead to spontaneous radiation, but now of two different polarizations. And this causes a loss of energy in a system which affirms its conservation laws!

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