General Relativity Primer

This page provides a high-level overview of some of the concepts in General Relativity. For more detailed treatments, see the references at the end of this page.

Einstein’s Field Equations

Einstein’s Field Equations (EFE) relate local spacetime curvature with local energy and momentum. In short, these equations determine the metric tensor of a spacetime, given the arrangement of stress-energy. The EFE are given as follows:

\[\boxed{R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}}.\]

Here, \(R_{\mu\nu}\) is the Ricci tensor, \(R\) is the scalar curvature (trace of Ricci tensor), \(g_{\mu\nu}\) is the metric tensor, \(\Lambda\) is the cosmological constant and lastly, \(T_{\mu\nu}\) denotes the stress-energy tensor. All other variables hold their usual meaning. If we introduce the Einstein tensor \(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}\), then the EFE can be written as:

\[\boxed{G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} - \Lambda g_{\mu\nu}}.\]

These equations form a ten-component tensor equation, which collectively denotes a system of coupled non-linear partial differential equations. These are usually intractable to approach analytically. However, under certain conditions, the EFE can be simplified and solved to yield exact solutions. For example, if the Einstein tensor is assumed to vanish, which implies that the stress-energy tensor also vanishes, then the EFE reduces to the vacuum Einstein equations, whose solutions are called vacuum solutions. The Minkowski, Schwarzschild and Kerr spacetimes are some examples of vacuum solutions.

Similarly, if an electromagnetic field is assumed to be present and also the only source of non-gravitational energy, then the EFE reduces to the source-free Maxwell-Einstein equations, whose exact solutions are termed electrovacuum solutions. The Kerr-Newman and Reissner-Nordström spacetimes are examples of electrovacuum solutions.

Metric Tensor

The metric tensor denotes a solution to the EFE. As such, it is a fundamental entity in general relativity, that captures the geometry of spacetime. It is a symmetric, indefinite rank-2 tensor, which can be represented by a matrix. It can be thought of as characterizing the differential line element for a given geometry:

\[\mathrm{d} s^2 = g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}\]

To get the matrix representation, the coefficients, \(g_{\mu\nu}\), in the above equation are substituted into the corresponding places designated by the indices, \(\mu\) and \(\nu\), in a matrix. For example, the metric tensor in the spherical-polar coordinate system can be written as:

\[\begin{split}g_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\theta \end{pmatrix}\end{split}\]

Note that the off-diagonal components are 0, since we are using an orthogonal basis. However, it is not always the case and in general, \(g_{\mu\nu} \ne 0\), for \(\mu \ne \nu\).

The metric tensor is also used to define the operations of lowering and raising indices. For this, we also need the inverse metric tensor, \(g^{\mu\nu}\) (or contravariant metric tensor to be precise), which is defined using the following identity:

\[g^{\mu\nu}g_{\nu\lambda} = \delta^{\mu}_{\lambda}\]

The inverse metric tensor can then be used to raise indices, while the metric tensor can be used to lower indices.

Notion of Curved Space

Imagine a bug traveling on a sheet of paper folded into a cone. The bug can’t see up and down. So, it lives in a 2D world. But it still experiences the curvature of the surface (space), as, after a long journey, it would return to the starting position.

Mathematically, the curvature of space is characterized by the rank-4 Riemann Curvature tensor, whose contraction gives the rank-2 Ricci tensor. Taking the trace of the Ricci tensor yields the rank-0 Ricci Scalar or Scalar Curvature.

Straight lines in Curved Space

Straight lines are used to describe the shortest path between two points in Euclidean space. Geodesics extend this notion of “shortest path” to curved spaces. In GR, geodesics represent the shortest path between two points in a possibly curved spacetime. They are completely characterized by the metric tensor and its derivatives, resulting in a set of coupled ordinary differential equations:

\[\boxed{\frac{\mathrm{d}^2x^{\mu}}{\mathrm{d}s^2} + \Gamma^{\mu}_{\alpha\beta}\frac{\mathrm{d}x^{\alpha}}{\mathrm{d}s}\frac{\mathrm{d}x^{\beta}}{\mathrm{d}s} = 0}\]

Here, \(x\) denotes the geodesic and the derivatives are taken with respect to \(s\), an affine parameter that uniquely parameterizes \(x\). \(\Gamma^{\mu}_{\alpha\beta}\) denotes Christoffel symbols of the second kind, which is essentially an array of partial derivatives of the metric tensor, computed with respect to the coordinate basis. It can be thought of as a “connection” between the derivates of nearby points along the geodesic and is given as:

\[\Gamma^{\mu}_{\alpha\beta} = \frac{1}{2}g^{\mu\gamma}\left(\frac{\partial g_{\alpha\beta}}{\partial x^{\gamma}} + \frac{\partial g_{\alpha\gamma}}{\partial x^{\beta}} - \frac{\partial g_{\beta\gamma}}{\partial x^{\alpha}}\right)\]

The Riemann Curvature tensor encapsulates the idea of curvature in GR. It can be written in a condensed notation using the Christoffel symbols and their derivaties:

\[\boxed{R^{\mu}_{\alpha\beta\gamma} = \partial_\gamma\Gamma^{\mu}_{\alpha\beta} - \partial_\beta\Gamma^{\mu}_{\alpha\gamma} + \Gamma^{\mu}_{\alpha\delta}\Gamma^{\delta}_{\beta\gamma} - \Gamma^{\mu}_{\beta\delta}\Gamma^{\delta}_{\alpha\gamma}}\]

A space with zero curvature implies that the Riemann tensor is zero and vice-versa. Since, we are dealing with tensors, i.e. objects that operate independently of basis choice, this statement holds for any coordinate system, i.e. \(R = 0\) in one coordinate system implies \(R = 0\) and by extension, zero curvature in all coordinate systems.

The Ricci tensor is another geometrical object in GR that is related to curvature. It can be obtained by contracting the first and third indices of the Riemann tensor:

\[R_{\mu\nu} = R^{\rho}_{\mu\rho\nu}\]

\(R_{\mu\nu}\) can be thought of as quantifying the deformation of a shape as it is translated along a given geodesic. The trace of the Ricci tensor gives the Scalar Curvature, \(R\):

\[R = g^{\mu\nu}R_{\mu\nu}\]

\(R\) relates the volume of infinitesimal geodesic balls in curved space to that in Euclidean space.

With this, we end our short and superficial look into some of the basic quantities that are used to characterize the structure of spacetime in General Relativity. Readers, who are interested in gaining a deeper understanding, are strongly recommended to peruse the resources listed below.