An Example of a Distill-Style Blog Post

Introduction

The general theory of relativity, published by Albert Einstein in November 1915, represents one of the most profound intellectual achievements in the history of science. Where Newton’s theory of gravity described a force acting instantaneously across space, Einstein’s framework reimagines gravity as the curvature of a four-dimensional spacetime continuum. Mass and energy warp the fabric of spacetime, and objects in free fall follow the straightest possible paths — called geodesics — through that curved geometry.

This post demonstrates the Distill-style layout for academic blog posts in the as-folio template. It showcases mathematical typesetting via KaTeX, code blocks, callout boxes, and structured sections that mirror the scholarly aesthetic of Distill.pub.

Equations

The Einstein Field Equations

At the heart of general relativity are the Einstein field equations:

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

where $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$ is the Einstein tensor, $\Lambda$ is the cosmological constant, $G$ is Newton’s gravitational constant, $c$ is the speed of light, and $T_{\mu\nu}$ is the stress-energy tensor encoding the distribution of matter and energy.

The left-hand side describes the geometry of spacetime; the right-hand side describes its content. This elegant equality encapsulates a deep physical truth: spacetime curvature and matter-energy are inseparably linked.

Special Relativistic Energy

From the special theory of relativity (1905), the rest energy of a body is given by:

$$E = mc^2$$

More generally, for a moving body with momentum $p$:

$$E^2 = (pc)^2 + (mc^2)^2$$

When $p = 0$ we recover the famous rest-mass relation. The full energy-momentum relation reveals that massless particles (such as photons) carry energy $E = pc$, consistent with observed light behavior in experiments such as the Compton effect.

Geodesic Equation

The trajectory of a freely-falling particle through curved spacetime satisfies the geodesic equation:

$$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$$

Here $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols — connection coefficients encoding how the coordinate basis vectors change from point to point — and $\tau$ is the proper time measured along the particle’s worldline.

Code Blocks

A common numerical task in general relativity is computing the Schwarzschild radius of a black hole:

import scipy.constants as const

def schwarzschild_radius(mass_kg: float) -> float:
    """
    Compute the Schwarzschild radius of a body.

    Parameters
    ----------
    mass_kg : float
        Mass in kilograms.

    Returns
    -------
    float
        Schwarzschild radius in metres.
    """
    G = const.G          # 6.674e-11 m^3 kg^-1 s^-2
    c = const.c          # 2.998e8 m s^-1
    return 2 * G * mass_kg / c**2


# Sun's Schwarzschild radius
M_sun = 1.989e30  # kg
r_s = schwarzschild_radius(M_sun)
print(f"Solar Schwarzschild radius: {r_s:.0f} m")   # ≈ 2954 m (~2.95 km)

The Sun’s Schwarzschild radius of approximately 3 km illustrates why ordinary stars are not black holes: the Sun’s actual radius ($\approx 696{,}000$ km) is far larger than this critical threshold.

Experimental Confirmations

General relativity has since been confirmed by a remarkable suite of experiments:

1. Perihelion precession of Mercury — The 43 arcseconds per century excess precession, unexplained by Newtonian mechanics, falls naturally out of GR.

2. Gravitational redshift — Photons climbing out of a gravitational well lose energy. The Pound–Rebka experiment (1959) measured this effect in a laboratory setting.

3. Gravitational waves — Predicted by Einstein in 1916, these ripples in spacetime were directly detected by LIGO on 14 September 2015 from the merger of two black holes roughly $1.3$ billion light-years away.

4. Black hole imaging — The Event Horizon Telescope collaboration produced the first direct image of a black hole shadow in M87 (2019) and Sgr A* (2022), matching GR predictions.

The Distill Layout

This post uses the Distill layout (distill: true in frontmatter), which renders with an academic-paper aesthetic:

• Authors are listed below the title with optional affiliations and links.
• The table of contents sidebar (desktop) provides quick navigation.
• Section headings use a bold bottom-border style reminiscent of journal articles.
• Block quotes are styled as aside panels with a colored left border.
• Custom .distill-note divs render as highlighted callout boxes.

The layout is defined in src/layouts/Distill.astro and activated automatically via src/pages/blog/[slug].astro when post.data.distill === true.

Conclusion

Einstein’s general theory of relativity remains one of the most accurately tested physical theories ever constructed. From GPS satellites (which must correct for both special and general relativistic time dilation to maintain centimetre-level accuracy) to the detection of gravitational waves from colliding neutron stars, GR continues to shape our deepest understanding of space, time, and gravity.

The mathematical machinery — Riemannian geometry, tensor calculus, differential topology — that Einstein marshalled to describe gravity has also found applications far beyond physics, influencing modern machine learning (differential geometry on manifolds), computer vision (gauge-equivariant networks), and theoretical computer science. The unreasonable effectiveness of mathematics, as Wigner noted, is on full display.