Constant of Einstein

In 1916, Albert Einstein proposes to describe the universe using an equation of field, tensorial. This one contains two constants, which do not have strictly anything to see one with the other. The first is the constant of Einstein , illustrated in the second member by the capital letter $C$ or (const) in the first equation written below. The second is the cosmological Constante $\ Lambda$ .

One proposes in what will follow to determine the value of the constant of Einstein. With this intention one will start from an equation of field to null cosmological constant and an assumption of stationnarity. Then one will pass to the Newtonian approximation by introducing the assumptions: weak field and low speeds in front of that of the light.

One will then see appearing the law of Newton and his corollary the Poisson's equation.

To some extent, in this approximation, the Poisson's equation appears as the approximate form of the equation of field (or the equation of field is presented in the form of a generalization of the Poisson's equation). The identification makes it possible to obtain the expression of the constant of Einstein according to the size $G$ (gravitational Constante) and $c$ (Speed of light).

The equation of field in a space not-vacuum

We must obtain a tensor suitable for describe the geometry of space in the presence of a field of energy. This equation was proposed by Einstein in 1917 and is written:

G^ {\ alpha \ gamma} + \ Lambda \ mathrm {G} ^ {\ alpha \ gamma} = (\ mathrm {const}) T^ {\ alpha \ gamma} ~

$\ Lambda$ is what is called the cosmological constant. We will place ourselves in a situation where this one is taken equalizes to zero. The equation of field becomes then:

G^ {\ alpha \ gamma} = \ left (R^ {\ alpha \ gamma} - \ frac {1} {2} \ mathrm {G} ^ {\ alpha \ gamma} R \ right) = CT^ {\ alpha \ gamma} ~

where $C$ is a constant. We will determine his value in the following section.

At the price of a certain handling we will write this equation in another form. Let us contract the indices in the equation above:

{R^ \ alpha} _ \ alpha - \ frac {1} {2} {\ mathrm {G} ^ \ alpha} _ \ alpha R = {CT^ \ alpha} _ \ alpha~

As follows:

R = - {CT^ \ alpha} _ \ alpha = - CT~

By using this result we can write the equation of field in the form:

Traditional limit of the equations of the gravitation

We wish that this equation of field of Einstein be a generalization of the Poisson's equation:

\sum_{i=1}^3 \varphi_-
\ begin {Bmatrix} & \ beta & \ \
\ \
\ driven & & \ naked
\end{Bmatrix}_-
\ begin {Bmatrix} & \ beta & \ \
\ \
\ driven & & \ naked
\end{Bmatrix}_

Notice 1

Certain readers will see in this result a difference with the value indicated in other pages of Wikipedia, for example that devoted to the cosmological constant, which also appears in the gate cosmology and which is:

C=- \ frac {8 \ pi G} {c^4}

It is about an error neither at the ones, nor at the others. All depends on the way in which one decides to write the tensor impulse-energy. If one considers speed of light

c

as an absolute constant then these two presentations are strictly equivalent. So in this calculation one had written the tensor impulse-energy in the form:

T^ {\ driven \ naked} = \ rho
\begin{pmatrix}
c^2 & \ nu_x C & \ nu_y C & \ nu_z C \ \
\ nu_x C & \ nu^2_x & \ nu_x \ nu_y & \ nu_x \ nu_z \ \
\ nu_y C & \ nu_y \ nu_x & \ nu^2_y & \ nu_y \ nu_z \ \
\ nu_z C & \ nu_z \ nu_x & \ nu_z \ nu_y & \ nu^2_z
\end{pmatrix}

and the corresponding trace:

T^\mu_\mu=\operatorname{Tr}
\begin{pmatrix}
\ rho_0 c^2 & 0 & 0 & 0 \ \
0 & 0 & 0 & 0 \ \
0 & 0 & 0 & 0 \ \
0 & 0 & 0 & 0
\ end {pmatrix} = \ rho_0 c^2

then one would have obtained the value of the constant of Einstein :

C=- \ frac {8 \ pi G} {c^4}

and the equation of field would be written:

G^ {\ alpha \ gamma} + \ Lambda g^ {\ alpha \ gamma} = \ frac {8 \ pi G} {c^4} T^ {\ alpha \ gamma}

Recently (subsequently to 2005) for example, Monique Signore, research director associated with the Observatory with Paris, published an article entitled “Basic principles of general relativity” where it uses the Greek letter $\ chi$ χpour to indicate the constant of Einstein , where it makes appear speed of light by her square.

Notice 2

The equation of Einstein is “with null divergence”. The null divergence of the tensor energy-impulse (energy-momentum tensor) is the geometrical expression of the law of conservation. This constraint implies that the constant of Einstein is an absolute constant, if not there would be violation of this postulate.

But, as this constant of Einstein was evaluated by melting calculation on metric independent of time this by no means does not imply the absolute constancy of $G$ and $c$ but only the absolute constancy of the report/ratio:

\ frac {G} {c^2}

References

Introduction to General Relativity (Ronald Adler, Maurice Bazin, Menahem Schiffer), Mcgraw-Hill, June 1975 - ISBN 0070004234

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