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
6)
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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