Physics:Einstein's constant

Einstein's gravitational constant, denoted κ (kappa), is the constant appearing in the Einstein field equation which can be written:

[math]\displaystyle{ G^{\alpha \gamma} = \kappa T^{\alpha \gamma} , }[/math]

where G^αγ is the Einstein tensor and T^αγ is the stress–energy tensor.

This equation relates to the curvature of spacetime, indicating that stress–energy is what causes the curvature of spacetime, thus gravitation. The constant κ is directly related to Newton's gravitational constant G:

[math]\displaystyle{ \kappa = \frac{ 8 \pi G }{ c^2 } \approx 1.866 \times 10^{-26} \mathrm{\ m{\cdot}kg^{-1}} . }[/math]

The above is for the stress–energy tensor in units of mass density (i.e., mass per volume). The units of Einstein's constant depends on how the stress–energy tensor is defined, so an alternative choice for T with units of energy density (i.e., energy per volume) yields

[math]\displaystyle{ \kappa = \frac{ 8 \pi G }{ c^4 } \approx 2.076\,579 \times 10^{-43} \mathrm{\ s^2{\cdot}m^{-1}{\cdot}kg^{-1} \ (or \ m{\cdot}J^{-1} or \ N^{-1})} . }[/math]

Calculation

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.

The Newton law will arise and its corollary Poisson's equation.

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:

[math]\displaystyle{ G^{\alpha\gamma} + \Lambda g^{\alpha\gamma} = (\mathrm{const}) T^{\alpha\gamma} }[/math]

(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 T^αγ is zero, therefore that the cosmological constant Λ appearing in this equation is zero) so the field equation becomes:

[math]\displaystyle{ G^{\alpha\gamma} = \left( R^{\alpha\gamma} - \tfrac{1}{2} g^{\alpha\gamma} R \right) = \kappa T^{\alpha\gamma} }[/math]

where R^αγ is the Ricci tensor, g^αγ 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 indices:

[math]\displaystyle{ {R^\alpha}_\alpha - \tfrac{1}{2} {g^\alpha}_\alpha R = \kappa {T^\alpha}_\alpha~ }[/math]

Thus:

[math]\displaystyle{ R = -{\kappa T^\alpha}_\alpha = -\kappa T }[/math]

where T is the scalar T^α_α which we shall refer to as the Laue scalar.

Using this result we can write the field equation as:

[math]\displaystyle{ R^{\alpha\gamma} = \kappa \left(T^{\alpha\gamma} - \tfrac{1}{2} g^{\alpha\gamma} T \right) }[/math]

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 byproduct the value of the constant κ.

|i and |i|j respectively indicate ∂/∂xⁱ and ∂²/∂xⁱ ∂x^j. Thus, |i|i means ∂²/(∂xⁱ)².

Consider a field of matter with low proper density ρ, moving at low velocity v. The stress–energy tensor can be written:

[math]\displaystyle{ T_{\mu\nu}= \rho \begin{pmatrix} 1 & \dfrac{v_x}{c} & \dfrac{v_y}{c} & \dfrac{v_z}{c} \\ \dfrac{v_x}{c} & \dfrac{v^2_x}{c^2} & \dfrac{v_x v_y}{c^2} & \dfrac{v_x v_z}{c^2} \\ \dfrac{v_y}{c} & \dfrac{v_y v_x}{c^2} & \dfrac{v^2_y}{c^2} & \dfrac{v_y v_z}{c^2} \\ \dfrac{v_z}{c} & \dfrac{v_z v_x}{c^2} & \dfrac{v_z v_y}{c^2} & \dfrac{v^2_z}{c^2} \end{pmatrix} }[/math]

If the terms of order (v/c)² and ρ(v/c) are neglected, it becomes:

[math]\displaystyle{ T_{\mu\nu}= \begin{pmatrix} \rho_0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} }[/math]

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, 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, η_μν which is that of the Minkowski space, locally flat. Formulating gives:

[math]\displaystyle{ \mathrm{d}s^2 = \left(\mathrm{d}x^0\right)^2 - \left(\mathrm{d}x^1\right)^2 - \left(\mathrm{d}x^2\right)^2 - \left(\mathrm{d}x^3\right)^2 }[/math]

The second term corresponds to the small perturbation (due to the presence of a gravitating body) and is also time-independent:

[math]\displaystyle{ \varepsilon\gamma_{\mu\nu} }[/math]

Thus we write the metric:

[math]\displaystyle{ g_{\mu\nu} = \eta_{\mu\nu} + \varepsilon\gamma_{\mu\nu} }[/math]

Clarifying the length element:

[math]\displaystyle{ \mathrm{d}s^2 = \left(\mathrm{d}x^0\right)^2 - \left(\mathrm{d}x^1\right)^2 - \left(\mathrm{d}x^2\right)^2 - \left(\mathrm{d}x^3\right)^2 + \varepsilon\gamma_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu }[/math]

If we neglect terms of order ερ₀, the Laue scalar Tμμ is:

[math]\displaystyle{ T^\mu_\mu = \operatorname{Tr} \begin{pmatrix} \rho_0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} = \rho_0 }[/math]

And the right side of the field equations is to first order in all the small quantities ρ₀, v/c and εγ_μν is written:

[math]\displaystyle{ \begin{align} C\left(T_{\mu\nu}-\tfrac{1}{2} g_{\mu\nu}T \right)&\simeq C\left(T_{\mu\nu}-\tfrac{1}{2} g_{\mu\nu} T \right) \\ & \simeq C \left[ \begin{pmatrix} \rho_0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} - \frac{1}{2}\begin{pmatrix} \rho_0 & 0 & 0 & 0 \\ 0 & -\rho_0 & 0 & 0 \\ 0 & 0 & -\rho_0 & 0 \\ 0 & 0 & 0 & -\rho_0 \end{pmatrix} \right] \\ & \simeq \frac{C\rho_0}{2} \delta_{\mu\nu} \end{align} }[/math]

Neglecting second-order terms in εγ_μν gives the following approximate form for the contracted Riemann tensor:

[math]\displaystyle{ R_{\mu\nu} \cong \tfrac{1}{2}\left[\ln(-g)\right]_{|\mu|\nu|}-[\mu\nu,\beta]_{|\beta} }[/math]

Thus the approximate field equations may be expressed as:

[math]\displaystyle{ \tfrac{1}{2}\left[\ln(-g)\right]_{|\mu|\nu|}-[\mu\nu,\beta]_{|\beta} = \frac{\kappa \rho_0}{2} \delta_{\mu\nu} }[/math]

First consider the case μ = ν = 0. As the metric is time-independent, the first term of the equation above is zero. What remains is:

[math]\displaystyle{ [00,\beta]_{|\beta} = \left(g^{\alpha\beta}\left[00,\alpha\right]\right)_{|\beta} = -\frac{\kappa \rho_0}{2}\qquad (1) }[/math]

The Christoffel symbol of the first kind is defined by:

[math]\displaystyle{ \left[00,\alpha\right] = \tfrac{1}{2}\left(g_{0\alpha|0} + g_{\alpha 0|0} - g_{00|\alpha}\right) }[/math]

Since the Lorentz metric is constant in space and time, this simplifies to:

[math]\displaystyle{ \left[00,\alpha\right] = -\frac{\varepsilon}{2} \gamma_{00|\alpha} }[/math]

Moreover, γ_μν is time-independent, so [00,0] is zero. Neglecting second-order terms in the perturbation term εγ_μν, we get:

[math]\displaystyle{ g^{\beta\alpha} \left[00,\alpha\right] = \frac{\varepsilon}{2} \gamma_{00|\beta} }[/math]

which is zero for β = 0 (which then corresponds to the derivative with respect to time). Substituting into (1) we obtain the following approximate field equation for [math]\displaystyle{ \gamma_{00} }[/math]:

[math]\displaystyle{ \varepsilon\sum_{\beta = 0}^{3} \gamma_{00|\beta|\beta} = -\kappa \rho_0 }[/math]

or, by virtue of time independence:

[math]\displaystyle{ \varepsilon\sum_{\beta = 1}^{3} \gamma_{00|\beta|\beta} = -\kappa \rho_0 }[/math]

This notation is just a writing convention. The equation can be written:

[math]\displaystyle{ \begin{align} \sum_{\beta = 0}^{3} \gamma_{00|\beta|\beta} &= \sum_{i=1}^3 \frac{\partial^2 \gamma_{00}}{\partial {x_\beta}^2} \\ &= \frac{\partial^2 \gamma_{00}}{\partial {x_1}^2} + \frac{\partial^2 \gamma_{00}}{\partial {x_2}^2} + \frac{\partial^2 \gamma_{00}}{\partial {x_3}^2} = -\kappa \rho_0 \end{align} }[/math]

which can be identified with Poisson's equation if we write:

[math]\displaystyle{ -\frac{\varepsilon\gamma_{00}}{\kappa} = \frac{\varphi}{4\pi G} }[/math]

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:

[math]\displaystyle{ g_{\mu\nu} = \eta_{\mu\nu} + \varepsilon\gamma_{\mu\nu} }[/math]

made explicit:

[math]\displaystyle{ \mathrm{d}s^2 = \left(\mathrm{d}x^0\right)^2 - \left(\mathrm{d}x^1\right)^2 - \left(\mathrm{d}x^2\right)^2 - \left(\mathrm{d}x^3\right)^2 + \varepsilon\gamma_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu }[/math]

Suppose the velocity v to be low with respect to the speed of light c, with a small parameter β = v/c. We have:

[math]\displaystyle{ x^0 = ct }[/math]

We can write:

[math]\displaystyle{ \begin{align} \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^2 &= c^2 - v^2 + \varepsilon\gamma_{\mu\nu} \frac{\mathrm{d}x^\mu}{\mathrm{d}t} \frac{\mathrm{d}x^\nu}{\mathrm{d}t} \\ &= c^2 \left(1 - \beta^2 + \varepsilon\gamma_{\mu\nu} \frac{\mathrm{d}x^\mu}{\mathrm{d}x^0} \frac{\mathrm{d}x^\nu}{\mathrm{d}x^0}\right) \end{align} }[/math]

Limiting to the first degree in β and ε gives:

[math]\displaystyle{ \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^2 \cong c^2\left(1 + \varepsilon\gamma_{00}\right) }[/math]

Then one writes, as a classical calculation, the differential equation system giving the geodesics. Christoffel symbols are calculated. The geodesic equation becomes:

[math]\displaystyle{ \frac{\mathrm{d}^2 x^\alpha}{\mathrm{d}t^2} + [00,\alpha] c^2 = 0 \qquad (2) }[/math]

The approximate form of the Christoffel symbol is:

[math]\displaystyle{ [00,i] = \tfrac{1}{2}\varepsilon\gamma_{00|i} }[/math]

Introducing this result into the geodesic equation (2) gives:

[math]\displaystyle{ \displaystyle \frac{\mathrm{d}^2 x^i}{\mathrm{d}t^2} = -\frac{c^2}{2} \varepsilon \gamma_{00|i} }[/math]

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:

[math]\displaystyle{ \displaystyle \frac{\mathrm{d}^2 X}{\mathrm{d}t^2} = \frac{c^2}{2} \varepsilon \gamma_{00} }[/math]

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:

[math]\displaystyle{ \varphi = -\frac{c^2}{2} \varepsilon \nabla \gamma_{00} }[/math]

Conversely, if we set a gravitational potential φ, the movement of a particle will follow a spacetime geodesic if the first term of the metric tensor is like:

[math]\displaystyle{ g_{00} = 1 + \frac{2\varphi}{c^2} }[/math]

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:

[math]\displaystyle{ \varphi = \frac{c^2}{2} \varepsilon \gamma_{00} }[/math]

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):

[math]\displaystyle{ \kappa = \frac{ 8 \pi G }{ c^2 } }[/math]

One can then write the Einstein field equation:

[math]\displaystyle{ G^{\alpha\gamma} + \Lambda g^{\alpha\gamma} = \frac{8\pi G}{c^2} T^{\alpha\gamma} }[/math]

Dimensionally, each term on the left hand side of the equation has dimension 1/length². If the stress-energy tensor T on the right has units of energy/volume or mass/[length*time²], then the Einstein's constant must have units time²/[length*mass] to mediate between the two sides of the Einstein field equation. But see below for other unit choices in the tensor T.

About the possible forms for stress-energy tensor of each dimension

We have seen, neglecting the terms of order (v/c)² and ρ(v/c), that the Laue scalar could be written:

[math]\displaystyle{ T^\mu_\mu = \operatorname{Tr} \begin{pmatrix} \rho_0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} = \rho_0 }[/math]

which gives the corresponding Einstein's constant for a tensor with dimension of mass per volume:

[math]\displaystyle{ \kappa = \frac{ 8 \pi G }{ c^2 } }[/math]

But another valid choice for writing the form of the stress–energy tensor, is in units of energy per volume:

[math]\displaystyle{ T_{\mu\nu}= \rho \begin{pmatrix} c^2 & v_x c & v_y c & v_z c \\ v_x c & v^2_x & v_x v_y & v_x v_z \\ v_y c & v_y v_x & v^2_y & v_y v_z \\ v_z c & v_z v_x & v_z v_y & v^2_z \end{pmatrix} }[/math]

Neglecting the same term orders, the corresponding Laue scalar is:

[math]\displaystyle{ 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 }[/math]

which contains an additional term c² to convert units of mass density to units of energy density, so the corresponding Einstein's constant in the field equations is then:

[math]\displaystyle{ \kappa = \frac{ 8 \pi G }{ c^4 } }[/math]

This is just a question of choice, since for any chosen value unit, the Einstein field equations are the same.

Further reading

• Heller, Michael (1992). Theoretical Foundations of Cosmology: Introduction to the Global Structure of Space-time. World Scientific. p. 63. ISBN 981-02-0756-5. https://books.google.com/books?id=JJ1aNeTzmLQC&pg=PA63.