Showcasing LaTeX Rendering: The Einstein Field Equations

Posted at 2023-03-22 Edited at 2023-03-24 # LaTeX # Astro # Mathematics # Physics # Web Development

This post is primarily a demonstration of rendering LaTeX within this blog setup. I find it crucial for technical writing, especially when dealing with complex mathematical or scientific notation. In the future, I might detail how I integrated LaTeX rendering, perhaps specifically within the Astro framework.

For this test, I’ve chosen the Einstein Field Equations (EFE) from general relativity, presented in several forms to showcase different LaTeX features.

Einstein Field Equations: Compact Form

The most common and compact representation of the EFE is:

G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu} R = {8 \pi G \over c^4} T_{\mu\nu}

Here’s a brief explanation of the terms:

$R_{\mu\nu}$ : Ricci curvature tensor component, measuring curvature in spacetime caused by matter-energy.
$g_{\mu\nu}$ : Metric tensor, encoding distances and geometry in spacetime.
$R$ : Ricci scalar, representing overall curvature of spacetime.
$T_{\mu\nu}$ : Stress-energy tensor component, representing matter and energy distribution.
$G$ : Newton’s gravitational constant.
$c$ : Speed of light, a universal constant.

Expanded Form (Derivatives Explicit)

The compact form hides a lot of complexity. Expanding the Ricci tensor and scalar in terms of the metric tensor and its derivatives gives a much more involved expression. Here’s one representation (note: specific forms can vary based on conventions and intermediate steps):

\begin{gathered}\frac{1}{2} g^{\alpha \beta} \partial_\alpha \partial_\mu g_{\beta \nu}+\frac{1}{2} g^{\alpha \beta} \partial_\alpha \partial_\nu g_{\mu \beta}-\frac{1}{2} g^{\alpha \beta} \partial_\alpha \partial_\beta g_{\mu \nu}-\frac{3}{2} g^{\alpha \beta} \partial_\mu \partial_\nu g_{\alpha \beta}-\frac{1}{2} g^{\beta \lambda} g^{\alpha \rho} \partial_\alpha g_{\rho \lambda} \partial_\mu g_{\beta \nu} \\ -\frac{1}{2} g^{\beta \lambda} g^{\alpha \rho} \partial_\alpha g_{\rho \lambda} \partial_\nu g_{\mu \beta}+\frac{1}{4} g^{\beta \lambda} g^{\alpha \rho} \partial_\nu g_{\alpha \lambda} \partial_\mu g_{\rho \beta}+\frac{1}{4|g|} g^{\alpha \beta} \partial_\beta|g| \partial_\nu g_{\mu \alpha}-\frac{1}{4|g|} g^{\alpha \beta} \partial_\beta|g| \partial_\alpha g_{\mu \nu} \\ -\frac{1}{4|g|} g^{\alpha \beta} \partial_\beta|g| \partial_\mu g_{\alpha \nu}+\Lambda g_{\mu \nu}=\frac{8 \pi G}{c^4} T_{\mu \nu}\end{gathered}

This version explicitly includes partial derivatives ( $\partial$ ) of the metric tensor components ( $g_{\alpha \beta}$ , etc.) and introduces the cosmological constant $\Lambda$ . The term $|g|$ represents the determinant of the metric tensor.

Fully Expanded Form (Summations Explicit)

Using Einstein summation notation (where repeated indices imply summation) is standard practice. Making the summations explicit reveals the full extent of the calculations involved:

\begin{gathered} \frac{1}{2} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\alpha \partial_\mu g_{\beta \nu}+\frac{1}{2} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\alpha \partial_\nu g_{\mu \beta}-\frac{1}{2} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\alpha \partial_\beta g_{\mu \nu}-\frac{3}{2} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\mu \partial_\nu g_{\alpha \beta}-\frac{1}{2} \\ \sum_{\alpha=0}^3 \sum_{\beta=0}^3 \sum_{\rho=0}^3 \sum_{\lambda=0}^3 g^{\beta \lambda} g^{\alpha \rho} \partial_\alpha g_{\rho \lambda} \partial_\mu g_{\beta \nu}-\frac{1}{2} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 \sum_{\rho=0}^3 \sum_{\lambda=0}^3 g^{\beta \lambda} g^{\alpha \rho} \partial_\alpha g_{\rho \lambda} \partial_\nu g_{\mu \beta}+\frac{1}{4} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 \sum_{\rho=0}^3 \\ \sum_{\lambda=0}^3 g^{\beta \lambda} g^{\alpha \rho} \partial_\nu g_{\alpha \lambda} \partial_\mu g_{\rho \beta}+\frac{1}{4|g|} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\beta|g| \partial_\nu g_{\mu \alpha}-\frac{1}{4|g|} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\beta|g| \partial_\alpha g_{\mu \nu}-\frac{1}{4|g|} \sum_{\alpha=0}^3 \\ \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\beta|g| \partial_\mu g_{\alpha \nu}+\Lambda g_{\mu \nu}=\frac{8 \pi G}{c^4} T_{\mu \nu} \end{gathered}

Here, each $\sum_{\alpha=0}^3$ indicates summation over the spacetime indices (typically time and three spatial dimensions).

Example: Enumerating Terms

To further illustrate the complexity, let’s expand just the very first term containing summations:

\begin{aligned} & \frac{1}{2} \sum_{\alpha=0}^3 \sum_{\beta=0}^3 g^{\alpha \beta} \partial_\alpha \partial_\mu g_{\beta \nu}=\frac{1}{2} g^{00} \partial_0 \partial_\mu g_{0 \nu}+\frac{1}{2} g^{01} \partial_0 \partial_\mu g_{1 \nu}+\frac{1}{2} g^{02} \partial_0 \partial_\mu g_{2 \nu}+\frac{1}{2} g^{03} \partial_0 \partial_\mu g_{3 \nu} \\ + & \frac{1}{2} g^{10} \partial_1 \partial_\mu g_{0 \nu}+\frac{1}{2} g^{11} \partial_1 \partial_\mu g_{1 \nu}+\frac{1}{2} g^{12} \partial_1 \partial_\mu g_{2 \nu}+\frac{1}{2} g^{13} \partial_1 \partial_\mu g_{3 \nu}+\frac{1}{2} g^{20} \partial_2 \partial_\mu g_{0 \nu}+\frac{1}{2} g^{21} \partial_2 \partial_\mu g_{1 \nu} \\ + & \frac{1}{2} g^{22} \partial_2 \partial_\mu g_{2 \nu}+\frac{1}{2} g^{23} \partial_2 \partial_\mu g_{3 \nu}+\frac{1}{2} g^{30} \partial_3 \partial_\mu g_{0 \nu}+\frac{1}{2} g^{31} \partial_3 \partial_\mu g_{1 \nu}+\frac{1}{2} g^{32} \partial_3 \partial_\mu g_{2 \nu}+\frac{1}{2} g^{33} \partial_3 \partial_\mu g_{3 \nu} \end{aligned}

This shows the 16 individual terms generated just from the first double summation in the expanded equation. The full equation involves numerous such complex terms.

Conclusion

This demonstration shows that the current setup can handle complex, multi-line LaTeX expressions effectively. Displaying equations clearly is vital for technical accuracy, and I’m pleased with how these examples rendered. As mentioned, I plan to explore the implementation details in a future post.

Eigenigma

本徵矢無解