Einstein's Field Equations

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There is no single correct way to “derive” Einstein’s field equations of gravity. It has even been asserted that no rigorous derivation can be given. [1] However reasonable arguments can be presented which lead one to the field equations. The purpose of this page is to present one such “derivation.” There are several postulates on which this derivation will be based

1) Einstein’s Equivalence Principle
2) Correspondence Principle (Newtonian limit)
3) Require relativistic mass as the source of the gravitational field (gravitational charge)
4) Require that inertial (aka relativistic) mass be locally conserved
5) Require Poisson’s equation in Newtonian limit
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) Principle of general covariance 

The strong form of the equivalence principle states that the gravitational force is equivalent to an inertial force. This assumption is used to derive an expression for the gravitational force, which is expressible in terms of the components of metric tensor as

 

From this it follows that the components of the metric tensor form a set of 10 independent gravitational potentials. In the Newtonian limit the Einstein potential becomes the Newtonian potential. Recall the geodesic deviation equation 

 

where, Rmbsa, are the components of the Riemann tensor, R, which, in turn, is a function of the second derivatives of the gravitational potentials gab. The weak field limit of this equation is the Newtonian tidal acceleration, i.e.

 

 

Where tjk is the Newtonian tidal force tensor

Poisson’s equation is

These relations establish that the relativistic generalization of Poisson’s equation will be of the form

where g is the metric tensor and H is an operator which is a combination of gab and its first and second derivatives. M is a tensor which describes mass in all its aspects. It is expected from Eq. (5) that H is related to contractions of the Riemann tensor. The Ricci tensor, Rab, is one such contraction and is defined as

All other contractions vanish identically or equal, ± Rab. Such a contraction yields a second rank tensor. Also H must be a second order tensor since M is second order. The general expression that satisfies these conditions is 

 

a, s and l are constants to be determined. It is assumed that a ¹ 0. The relativistic generalization of the Newtonian mass is relativistic mass, the complete description of which requires a second rank symmetric tensor, T, the stress-energy-momentum tensor. It is to be noted that relativistic mass plays the role as the source of the gravitational field in the sense that it acts like gravitational charge. We require that mass is conserved. In a chosen Lorentz frame this requires that

g º rv is momentum density defined as momentum per unit volume, and r is the inertial energy density as measured by the observer. If Uobs is the 4-velocity of the observer then

 

The relativistic version of Eq. (9) is

Eq. (11) is a coordinate independent statement of the conservation of inertial energy. For Eq. (11) to valid for all choices of a metric then it follows that

Since it can be shown that the last term on the right hand side vanishes identically then it follows that

Comparing Eq. (12) to the Bianchi identity [2]

it follows that s/a = -1/2. Define the Einstein tensor, >G, as

The field equation then becomes

Consider the case where l = 0. Then

Contracting Eq. (17) we get

where T º Taa. Given this expression for R we can express Eq. (17) as

To find the constant a we note that Eq. (19) must hold in all cases and therefore it must hold in the Newtonian limit. For this purpose consider a distribution of matter in the zero momentum frame which is also an inertial frame of reference. Assume also that the contribution to T due to pressure is negligible compared to the inertial mass density. It follows that T00 @ rc2 and gmn @ hmn. The time-time component of Eq. (19) then becomes

 

It can be shown that R00 = 4pGr/c2. Then

Therefore we arrive at our final result

 

Multiply through by 2 to give

Let L º 2l/c2.. Substitution into Eq. 22 gives

This is known as Einstein’s field equation. L is known as the cosmological constant.


References: 

[1] On the “Derivation” of Einstein’s Field Equations, S. Chandrasekhar, Am. J. Phys. 40, Feb (1972).
[2] Gravitation and Spacetime – 2nd Ed. Ohanian and Ruffini, WW Norton & Co., (1994), page 327, Eq. (59).
[3] Ref 2, page 336, Eq. (102).
[4] Cosmological Physics, John A. Peacock, Cambridge University Press, (1999), page 24, Eq. (1.83).


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