General Relativity

Foreword

This is a personal-written article about General Relativity which is basically for myself.

Metric Tensor, Connection, and the Einstein Field Equations

The Relation Between Metric \(g_{ij}\) and Connection \(C^i_{kl}\)

In a general manifold, we can define the line element \(ds\) as:

\[ ds^2 = g_{ij} \, dx^i \, dx^j \]

When a tensor field \( A^i \) moves from one point \( x^i \) to a nearby point \( x^i + dx^i \), its change is not simply given by the ordinary differential \( dA^i \). It also involves a correction due to the connection, which reflects the curvature of the space. This correction is captured by the connection coefficients \( C^i_{kl} \), and the total change can be written as:

\[ DA^i = dA^i - \delta A^i = \frac{\partial A^i}{\partial x^l} \, dx^l - C^i_{kl} A^k \, dx^l \]

Now, we want to determine the exact form of \(C^i_{kl}\), which is related to the metric.

The metric can be used to lower indices: \( A_i = g_{ij} A^j \), and \( DA_i = g_{ij} DA^j \). Moreover, \( DA_i \) can also be written as \( D(g_{ij} A^j) \).

From this, we find that \( Dg_{ij} = 0 \).

The variation of the metric with respect to \( x^l \) should follow the relation:

\[ \frac{\partial g_{ij}}{\partial x^l} = C^k_{il} g_{kj} + C^k_{jl} g_{ik} \]

Following by a symmetrization operation,

\[ (\frac{\partial g_{ij}}{\partial x^l} + \frac{\partial g_{il}}{\partial x^j} - \frac{\partial g_{jl}}{\partial x^i}) = 2C^k_{il}g_{kj} \]

We can see that the connection coefficients \( C^k_{il} \) are related to the metric by the following equation:

\[ C^k_{il} = \frac{1}{2} g^{kj} \left( \frac{\partial g_{ij}}{\partial x^l} + \frac{\partial g_{jl}}{\partial x^i} - \frac{\partial g_{il}}{\partial x^j} \right) \]

The Einstein Field Equations

The Einstein Field Equations are a set of equations that describe the behavior of a tensor field in a general relativistic field. They are derived from the principle of least action, which states that the action of a tensor field on a stationary observer is independent of the path taken by the observer.

The Einstein Field Equations are:

\[ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 8 \pi T_{\mu\nu} \]

\[ \frac{1}{2} g_{\mu\nu} \left( \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j} - \frac{\partial}{\partial x^j} \frac{\partial}{\partial x^i} \right) R = \frac{8\pi}{c} \delta_{\mu\nu} \]

The first equation is the Einstein-Maxwell equation, which relates the radiation tensor \(R_{\mu\nu}\) to the stress-energy tensor \(T_{\mu\nu}\). The second equation is the Ricci curvature equation, which relates the Ricci tensor to the Riemann curvature tensor.

The Einstein Field Equations are the most important equations in general relativity, and they are used to determine the behavior of gravitational, electromagnetic, and strong-field forces.