First, write down an frame Ansatz based upon assuming a combination of spacetime symmetry and what kind of motion you expect. For example, assume spherical symmetry and spherically symmetric collapse of spherical shells of test particles which our observers are riding on, a very simple frame Ansatz is:
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \partial_t - f \, \partial_r \\
\vec{e}_2 & = & \partial_r \\
\vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\
\vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi
\end{array}
[/tex]
where f is an undetermined function of r only, saying the rate of falling toward the origin depends only on r. Note that this Ansatz also says that differences in the t coordinate correspond to elapsed proper time measured by any of our infalling observers.
Second, demand that the Einstein tensor as measured by these observers have a given form, say that of an nonnull electrovacuum appropriate for a radial electrostatic field
[tex]
G^{ab} = \frac{q^2}{r^2} \, diag (1,-1,1,1)
[/tex]
where q is the charge. The great thing about using frames is that tensors like the EM stress-energy tensor have exactly the same form and behave the same way under boosts and rotations at an event as in flat spacetime.
In this example, we find an ODE for f which is easily solved giving
[tex]
f = \sqrt{ \frac{2m}{r} - \frac{q^2}{r^2} }
[/tex]
If we had demanded instead that the stress-energy tensor be the sum of the previous EM term plus a Lambda term
[tex]
G^{ab} = \frac{q^2}{r^2} \, diag (1,-1,1,1) + \frac{3}{R^2} \, diag \, (1,-1,-1,-1)
[/tex]
we would have found
[tex]
f = \sqrt{ \frac{2m}{r} - \frac{q^2}{r^2} + \frac{r^2}{R^2} }
[/tex]
The Einstein equation, also known as the Einstein field equation, is a set of ten non-linear partial differential equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. To use this equation to solve problems in general relativity, you need to have a good understanding of differential geometry, tensor calculus, and the principles of general relativity. You also need to have a specific problem that you want to solve, such as calculating the trajectory of a planet around a black hole or determining the gravitational waves emitted by a binary star system.
The Einstein equation is the cornerstone of general relativity, which is a theory that describes the fundamental principles of gravity. It has numerous applications in astrophysics, cosmology, and theoretical physics. Some examples include predicting the behavior of objects in strong gravitational fields, understanding the expansion of the universe, and exploring the nature of black holes and gravitational waves.
In most cases, the Einstein equation cannot be solved analytically, meaning that there is no exact mathematical solution. This is because the equations are highly non-linear and difficult to solve. However, there are some specific cases where the equations can be solved analytically, such as in the case of a perfectly spherical and non-rotating mass, known as the Schwarzschild solution.
One of the biggest challenges in using the Einstein equation is the complexity of the equations themselves. They involve higher-order mathematics and require a deep understanding of theoretical physics. Additionally, the equations are non-linear, meaning that small changes in the initial conditions can lead to vastly different outcomes. This makes it challenging to accurately model and predict the behavior of complex systems, such as black holes.
There are alternative theories of gravity, such as modified Newtonian dynamics and scalar-tensor theories, that have been proposed as alternatives to general relativity. However, these theories have not been able to fully explain all the phenomena that are predicted by general relativity. The Einstein equation remains the most widely accepted and successful theory of gravity, with many observational and experimental confirmations of its predictions.