In the following, the value of Einstein's constant will be calculated. To do so, at the beginning a field equation where the cosmological constant Λ is equal to zero is taken, with a steady state hypothesis. Then we use the Newtonian approximation with hypothesis of a weak field and low velocities with respect to the speed of light.
In this approximation, Poisson's equation appears as the approached form of the field equation (or the field equation appears as a generalization of Poisson's equation). The identification gives the expression of Einstein's constant related to quantities G and c.
The Einstein field equations in non-empty space
We have to obtain a suitable tensor to describe the geometry of space in the presence of an energy field. Einstein proposed this equation in 1917, written as:
(const) is what will become Einstein's constant. We will take the cosmological constant Λ equal to zero (one of the requirements of the properties of the gravitational equations is that they reduce to the free-space field equations when the density of energy in space Tαγ is zero, therefore that the cosmological constant λ appearing in this equation is zero) so the field equation becomes:
where Rαγ s the Ricci tensor, gαγ is the metric tensor, R the scalar curvature and κ is Einstein's constant we will calculate in the next section..
This equation can be written in another form, contracting indexes:
where T is the scalar Tαα which we shall refer to as the Laue scalar.
Using this result we can write the field equation as:
Classical limit of the gravitational equations
It will be shown that the field equations are a generalization of Poisson's classical field equation. The reduction to the classical limit, besides being a validity check on the field equations, gives as a by-product the value of the constant κ.
and respectively indicate and . Thus, means
Consider a field of matter with low proper density ρ, moving at low velocity v. The stress–energy tensor can be written:
If the terms of order and are neglected, it becomes:
One assumes the flow to be stationary and therefore expects the metric to be time-independent. We use the coordinates of special relativity ct, x, y, z that we write as x0, x1, x2, and x3. The first coordinate is time, and the three others are the space coordinates.
Applying a perturbation method, consider a metric appearing through a two-term summation. The first is the Lorentz metric, ημν which is that of the Minkowski space, locally flat. Formulating gives:
The second term corresponds to the small perturbation (due to the presence of a gravitating body) and is also time-independent:
Thus we write the metric:
Clarifying the length element:
If we neglect terms of order , the Laue scalar is:
And the right side of the field equations is to first order in all the small quantities , and is written:
Neglecting second-order terms in gives the following approximate form for the contracted Riemann tensor:
Thus the approximate field equations may be expressed as:
At first let us consider the case μ = ν = 0. As the metric is time-independent, the first term of the equation above is zero. What remains is:
Since the Lorentz metric is constant in space and time, this simplifies to:
Moreover, is time-independent, so [00,0] is zero. Neglecting second-order terms in the perturbation term , we get:
which is zero for β = 0 (which then corresponds to the derivative with respect to time). Substituting inside (*) we obtain the following approximate field equation for :
or, by virtue of time independence:
This notation is just a writing convention. The equation can be written:
which can be identified to Poisson's equation if we write:
Therefore, it is established that the classical theory (Poisson's equation) is the limiting case (weak field, low velocities with respect to the speed of light) of a relativistic theory where the metric is time-independent.
To be complete, gravity has to be demonstrated as a metric phenomenon. In the following, without detailing all calculation, the simplistic description of the complete calculation is given. Again, at first start from a perturbed Lorentz metric:
Suppose the velocity v to be low with respect to the speed of light c, with a small parameter . We have:
We can write:
Limiting to the first degree in β and ε gives:
Then one writes, as a classical calculation, the differential equation system giving the geodesics. Christoffel symbols are calculated. The geodesic equation becomes:
The approximate form of the Christoffel symbol is:
Introducing this result into the geodesic equation (**) gives:
This is a vector equation. Since the metric is time-independent, only space variables are concerned. Therefore, the second member of the equation is a gradient.
Coding the position-vector by the letter X and the gradient by the vector ∇, one can write:
This is no more than Newton's law of universal gravitation in classical theory, deriving from the gravitational potential φ if one makes the identification:
Conversely, if we set a gravitational potential φ, the movement of a particle will follow a space-time geodesic if the first term of the metric tensor is like:
That step is important. Newton's law appears as a particular aspect of the general relativity with the double approximation:
weak gravitational field
low velocity with respect to the speed of light
With the calculation above, we have made the following statements:
A metric g, solution of the Einstein field equation (with a cosmological constant Λ equal to zero).
This metric would be a weak perturbation in relation to a Lorentz metric η (relativistic and flat space).
The perturbation term would not depend on time. Since the Lorentz metric does not depend on time either, that metric g is also time-independent.
The expansion into a series gives a linearization of the Einstein field equations.
This linearized form is found to identify to Poisson's equation because a field is a curvature, linking the perturbation term to the metric and to the gravitational potential thanks to the relation:
And this rewards the value of the constant κ, called "Einstein's constant" (which is not the cosmological constant Λ or the speed of light c):
The Einstein field equation has zero divergence. The zero divergence of the stress–energy tensor is the geometrical expression of the conservation law. So it appears constants in the Einstein equation cannot vary, otherwise this postulate would be violated.
However, since Einstein's constant had been evaluated by a calculation based on a time-independent metric, this by no mean requires that G and cmust be unvarying constants themselves, the only postulate derived from conservation of energy is that the ratio must be constant. Depending on the choice of natural units, this ratio can be set to a defined constant value; subject to measurement is the dimensionlessgravitational coupling constant, variation in which would not necessarily amount to violation of the conservation of four-momentum.
^Ronald Adler; Maurice Bazin; Menahem Schiffer (1975). Introduction to General Relativity (2nd ed.). New York: McGraw-Hill. ISBN0-07-000423-4.
(see Chapter 10 "The Gravitational Field Equations or Nonempty Space", section 10.5 "Classical Limit of the Gravitational Equations" p. 345)