Einstein Field Equations

The Einstein field equations or EFE are the 16 coupled hyperbolic-elliptic nonlinear partial differential equations that describe the gravitational effects produced by a specified mass in the theory of general relativity.

Essentially, they relate the geometry of spacetime to the distribution of matter within it. Because of the symmetry of G_μνand T_μν, the actual number of equations comes down to ten. However, G_μν satisfies an additional four differential identities known as the Bianchi identities, one for each coordinate.

Albert Einstein first published the Einstein field equations in 1915, in the form of a tensor equation that related the local spacetime curvature (represented by the Einstein tensor) with the stress, momentum, and local energy within that spacetime (represented by the stress-energy tensor). In other words, they determine the metric tensor of spacetime for a particular arrangement of stress-energy-momentum in spacetime. This is analogous to the relation of electromagnetic fields to the distribution of charges and currents via Maxwell’s equations.

By virtue of the relationship between the Einstein tensor and the metric tensor, we can write the EFE as a set of non-linear partial differential equations when used in the above manner. The metric tensor includes the solutions of the EFE as components. After that, we can calculate the inertial trajectories of particles and radiation (known as geodesics) in the resulting geometry using the geodesic equation.

Here, I will discuss the important aspects of the Einstein field equations in detail.

Mathematical form of the Einstein field equations

We can write the Einstein field equations in the following form:

G_μν + Λg_μν = κT_μν

Here, G_μν is the Einstein tensor, Λ is the cosmological constant, g_μν is the metric tensor, κ is the Einstein gravitational constant, T_μν and is the stress-energy tensor.

We can define the Einstein tensor in the following manner:

G_μν = R_μν - ½ Rg_μν

Here, R_μν is the Ricci curvature tensor and R is the scalar curvature. It is a symmetric second-degree tensor that only depends on its first and second derivatives and the metric tensor.

We can define the Einstein gravitational constant as follows:

κ = 8πG/c⁴ 2.077 × 10^-43 N^-1

Here, G is the Newtonian constant of gravitation and c is the speed of light in the vacuum.

Thus, we may also write the Einstein field equations in the following manner:

R_μν - ½Rg_μν + Λg_μν = κT_μν

Every term on the left-hand side has units of 1/length² in standard units.

The expression on the left hand side is a representation of the curvature as determined by the metric. On the other hand, the expression on the right-hand side stands for the stress-energy-momentum content of spacetime. We can subsequently interpret the EFE as a set of equations dictating how stress-energy-momentum determines the curvature of spacetime.

Along with the geodesic equation, the Einstein field equations form the core of the mathematical formulation of general relativity. Mathematically, they are tensor equations and each of them relates a set of symmetric 4×4 tensors. Each of these tensors possesses ten independent components. When we consider the four Bianchi identities, the number of independent comes down from ten to six. As a result, the metric is left with four gauge-fixing degrees of freedom.

Although the Einstein field equations were originally formulated in the context of a four-dimensional theory, some theorists have also considered their consequences in “n” dimensions. They’re actually much more complex than people generally make them out to be. When written out fully, they are seen to be a system of 10 coupled, nonlinear, hyperbolic-elliptic, partial differential equations.

Equivalent formulations

By taking the trace with respect to the metric of both sides of the Einstein field equations, we get the following result:

R - R + DΛ = κT (here, D is the spacetime dimension)

After solving for R and substituting it in the original EFE, we get the following equivalent “trace reversed” form:

R_μν - Λg_μν = κ(T_μν - T_μν)

In D-4 dimensions, the above equation reduces to the following:

R_μν - Λg_μν = κ(T_μν – ½Tg_μν)

By reversing the trace again, we can restore the original EFE. In some cases, the trace-reversed form may prove to be much more convenient.

Linearized form

Because of the nonlinearity of the Einstein field equations, it can be difficult to find exact solutions to them. One way of doing so is to perform an approximation because when we move so far from the source(s) of gravitating matter, the gravitational field becomes extremely weak and the spacetime approximates that of Minkowski space.

Subsequently, the metric is written as the sum of the Minkowski metric and a term that represents the deviation of the true metric from the Minkowski metric (ignoring higher-power terms). We can investigate the phenomena of gravitational radiation using this process of linearization.

Polynomial form

Although the EFE are written including the inverse of the metric tensor, we may arrange them in a form that contains the metric tensor in polynomial form, without its inverse. For that, we must first write the determinant of the metric in four dimensions, using the Levi-Civita symbol, as shown below:

det(g) = ε^αβγδε^κλμνg_ακg_βλg_γμg_δν

Similarly, we can write the inverse of the metric in four dimensions as shown below:

g^ακ =(ε^αβγδε^κλμνg_ακg_βλg_γμg_δν)/det(g)

By substituting this definition of the inverse of the metric into the equations and multiplying both sides by a suitable power of det(g) to eliminate it from the denominator, we can obtain polynomial equations in the metric tensor and its first and second derivatives.

We can also use appropriate redefinitions of the fields to write the action from which the equations are derived in polynomial form.

The cosmological constant

In the EFE (G_μν + Λg_μν = κT_μν), the term containing the cosmological constant Λ wasn’t present in the original version published by Einstein. Later, he introduced the term with the cosmological constant to provide scope for a universe that isn’t expanding or contracting. However, this attempt proved to be unfruitful due to the following reasons:

• Any required steady state described by this equation is found to be unstable
• Edwin Hubble’s observations proved that the universe is actually expanding

Thus, Einstein subsequently did away with the cosmological constant Λ and referred to it as the “biggest blunder” of his life. Nevertheless, its inclusion doesn’t create any inconsistencies as such.

For a long time, nearly all scientists and mathematicians assumed that the value of the cosmological constant was zero. Later astronomical observations demonstrated an accelerating expansion of the universe, the explanation of which required a positive value of Λ. The value of the constant is negligible at or less than the scale of a galaxy.

Solutions

The solutions of the EFE are metrics of spacetime that describe the structure of the spacetime along with the inertial motion of objects in the same. As mentioned previously, the nonlinearity of the EFE makes it difficult to completely solve them at all times without making approximations. For example, there’s no known solution for a spacetime containing two massive bodies (such as the theoretical model of a binary star system).

However, mathematicians usually make approximations (known as post-Newtonian approximations) in such instances. Nevertheless, there are many cases where complete solutions are available for the Einstein field equations; these are known as exact solutions. One of the most important activities of cosmologists is to study the exact solutions of the EFE. By doing so, we can predict black holes and develop different models of the evolution of the universe.

We can also use the method of orthonormal frames (put forth by Ellis and MacCallum) to arrive at new solutions of the EFE. When we take this approach, the equations are reduced to a set of nonlinear, coupled, ordinary differential equations.

Hsu and Wainwright have asserted that self-similar solutions to the EFE are fixed points of the dynamical system that is produced as a result. LeBlanc, Kohli, and Haslam have succeeded in discovering new solutions with the help of these methods.