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The True Meaning of Einstein's Condition of "Emptiness"

Quoting Dirac (General Theory of Relativity p.25)


"Einstein made the assumption that in empty space

R_(ij) =0. (1)

It constitutes his law of gravitation. "Empty" here means that there is no matter present and no physical fields except the gravitational field. The gravitational field does not disturb the [sic] emptyness. Other fields do."


He then goes on to contradict himself in deriving the outer Schwarzchild solution. Dirac, following all other authors, writes the line element as


ds^2= b(r)dt^2-a(r)dr^2-r^2(d theta)^2-r^2sin^2 theta (d phi)^2,


where the unknowns u and v are functions of r only. "They must be chosen to satisfy the Einstein equations" (1):

-b'/abr+ (1/r^2)(1-1/a)=0

a'/a^2r + (1/r^2)(1-1/a)=0

(b'/b)'-(1/2)a'b'/ab +(1/2)(b'/b)^2 + b'/br -a'/ar =0

where the prime means differentiating with respect to r. From the first two equations there results

ab=1, say.

The third equation is then

b" +2b'/r =0. (2)


Now consider the displacement equation of equilibrium of linear elasticity:


(c_(ijkl) u_(k,l)),j=0, (3)


where c is the elasticity tensor, which is constant and strongly elliptic, u the generalized displacement, and the comma now stands for differentiation.


The displacement equation (2) applies to 2D, where the stress components


sigma_(ij) = c_(ijkl) u_(k,l)

are

sigma_(rr) = (lambda+2mu)u'+ lambda u/r


sigma_(theta theta) =sigma_(phi phi)= lambda u'+ (2mu+lambda)u/r


since v is a function only of r, and lambda and mu are the Lame' coefficients, which are assume constant in a homogeneous medium. The displacement equations of equilibrium (3)

reduce to a single equation

{(lambda+2 mu)u' + 2 lambda u/r}' + (4 mu/r) [u'-u/r] =0. (4)

In the homogeneous case, the Lame' coefficients are constant, and (4) reduces to

u" +2(u/r)' =0.

Integrating and setting the constant of integration equal to zero result in

u' +2 u/r =0. (5)

Setting u=b', it is seen that (5) is (2).


Thus, the Einstein conditions of "emptiness" (1) are actually, the compability conditions that ensure the tidal forces are derived from a scalar potential!








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