⚛️ general relativity

A Simple Pendulum in Schwarzschild Spacetime

This paper determines the small-oscillation period of a simple pendulum in Schwarzschild spacetime by modeling the rope as a spatial geodesic, deriving a formula expressed via the Schwarzschild lapse function that recovers the classical Newtonian result in the weak-field limit while differing from previous studies that assumed a coordinate-length constraint.

Original authors: Andrzej Czarnecki, Andrew Czezowski

Published 2026-08-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrzej Czarnecki, Andrew Czezowski

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery in a universe where space itself is a stretchy, heavy fabric. This is the world of General Relativity, Albert Einstein's theory that describes gravity not as an invisible force pulling things down, but as a curve in the very stage of the universe. In this curved world, "straight lines" are actually the paths objects take when they are just coasting, which physicists call "geodesics." If you drop a ball near a massive object like a black hole, it doesn't fall in a straight line; it follows the dip in the fabric. Now, picture a simple pendulum—the kind you might see in a grandfather clock or a playground swing. In our everyday world, we know exactly how long it takes to swing back and forth; it depends on how long the rope is and how strong gravity is. But what happens if you take that clock and hang it near a black hole, where space is stretched and time slows down? Does the clock still tick the same way? Does the swing still feel the same? This is the puzzle that physicists Andrzej Czarnecki and Andrew Czezowski decided to solve. They wanted to know the exact rhythm of a swinging weight in the extreme gravity of a black hole, a question that sounds simple but turns out to be a tricky dance between geometry and time.

The authors of this paper set out to calculate the precise "swing time" (or period) of a simple pendulum placed in the spacetime around a non-rotating black hole, known as Schwarzschild spacetime. Their main finding is a specific formula that tells us how long one full swing takes, as measured by an observer standing right next to the swinging weight. They discovered that the answer depends on a special function called the "Schwarzschild lapse," which is the same mathematical tool used to figure out how fast clocks tick and how light changes color (redshift) near a black hole.

To get this result, the team had to be very careful about how they defined the "length" of the rope. In the curved space near a black hole, the distance you measure with a tape measure (called "proper length") is different from the distance you would calculate just by looking at coordinates on a map. The authors insisted that the rope must have a fixed, unchanging physical length, like a real, inextensible string. They showed that when you pull the pendulum to the side, the rope doesn't just stay straight in a geometric sense; it naturally settles into a "geodesic," which is the shortest possible path through the curved space. This causes the bob (the weight at the end) to rise slightly higher than it would in normal space, a subtle effect that changes the swing time.

The paper explicitly argues against a different way of thinking about this problem that was proposed in an earlier study. That earlier study assumed the rope had a fixed "coordinate length," meaning the distance on the map stayed the same. The authors of this paper demonstrate that this is a different physical setup entirely. If you use the "fixed coordinate length" rule, the rope would actually have to shrink physically as it swings, which isn't how a real, stiff rope behaves. Because of this difference, the two models give different answers, especially when the pendulum is very close to the black hole's event horizon. The authors prove that their "fixed proper length" model is the correct way to describe a real, inextensible rope.

Their calculations show that as the pendulum gets closer to the black hole, the swing time changes in a very specific way. Near the horizon, the pendulum behaves as if it has a much shorter "effective length" than its actual rope, even if the rope is very long. This is because the gravity is so intense that it acts like a super-strong spring, making the swing incredibly fast. However, the authors also point out a limit to their own idea: they assume the rope is perfectly rigid and adjusts instantly. In reality, a signal (like a tension wave) takes time to travel down the rope. They show that for their "instant adjustment" model to work, the rope can't be too long, or the signal won't have time to travel back and forth before the pendulum swings again. But for most reasonable setups, especially near the horizon where time slows down so much, their model holds up.

In the end, the paper confirms that if you take a pendulum to a black hole, its rhythm is dictated by the same rules that govern time and light in that region. When they checked their math against the "weak field" limit (where gravity is weak, like on Earth), their complex formula perfectly matched the classic, simple formula we learned in school: $T = 2\pi \sqrt{L_0/g}$ . This gives them confidence that their new, exotic formula is the right one for the extreme world of black holes. They didn't just guess; they derived it step-by-step using the laws of physics, showing that even a simple playground swing has a deep, hidden connection to the curvature of the universe.

Technical Summary: A Simple Pendulum in Schwarzschild Spacetime

Problem Statement
The paper addresses the determination of the small-oscillation period of a simple pendulum in Schwarzschild spacetime. While the system appears elementary, the authors highlight that in curved spacetime, the definition of a "fixed length" constraint is non-trivial. Coordinate separations and invariant (proper) lengths do not coincide, particularly in strong gravitational fields. The study focuses on a pendulum with a fixed proper length $L_0$ , contrasting this with previous work that utilized a fixed coordinate length constraint. The goal is to derive the oscillation period as measured by a static observer located at the bob's equilibrium position.

Methodology
The authors model the system using a massless, inextensible, and taut rope connecting a fulcrum at radial coordinate $r_1$ to a bob at $r_2$ (where $r_1 > r_2 > r_s$ , and $r_s$ is the Schwarzschild radius). The analysis proceeds in several steps:

Geometric Derivation of Rope Shape:
By analyzing the stress-energy conservation of a massless rope ( $T^{\mu\nu}_{;\nu}=0$ ) in the static Schwarzschild metric, the authors prove that the rope, when displaced, follows a geodesic of the induced spatial metric $h_{ij}$ . This establishes that the shape of the displaced rope is determined by the spatial geometry rather than coordinate constraints.
Constraint Relations:
Using the fixed proper length condition, the authors relate the angular displacement $\vartheta$ to the radial lift $\delta r$ of the bob. They derive that for small amplitudes, the radial coordinate increases quadratically with the angle:
$\delta r \approx \frac{N_2 r_s}{4(N_1 - N_2)} \vartheta^2$
where $N(r) = \sqrt{1 - r_s/r}$ is the Schwarzschild lapse function, and $N_i = N(r_i)$ .
Lagrangian Formulation:
The dynamics are derived from the point-particle Lagrangian for the bob, subject to the holonomic constraint derived above. By substituting the constraint into the Lagrangian and expanding to quadratic order in the amplitude, the system is reduced to a harmonic oscillator form.
Period Calculation:
The coordinate-time period is calculated and then converted to the proper time period measured by a static observer at the bob's equilibrium position ( $r_2$ ).

Key Results
The primary result is the expression for the small-oscillation period $T^{(2)}$ measured by the observer at the bob:
$T^{(2)} = \frac{4\pi r_2^2}{c r_s} \sqrt{N_2 (N_1 - N_2)}$
where $N_1 = N(r_1)$ and $N_2 = N(r_2)$ .

The paper also provides the period in terms of the proper rope length $L_0$ by inverting the integral relation for $L_0$ , yielding an implicit function $T^{(2)}(L_0, r_2)$ .

Limiting Regimes

Newtonian Limit: In the weak-field limit ( $r \gg r_s$ ) and for short ropes ( $L_0 \ll r$ ), the result reduces to the classical formula $T = 2\pi \sqrt{L_0/g}$ , where $g = GM/r_2^2$ .
Close-to-Horizon Limit: As the bob approaches the horizon ( $r_2 \to r_s$ , so $N_2 \to 0$ ), the period scales as $T^{(2)} \propto \sqrt{N_2}$ . The authors interpret this behavior using an "effective length" $\ell_{\text{eff}} = 2N_1 r_s$ . Despite the rope potentially being long, the local observer perceives the pendulum as having a much shorter effective length due to the extreme spatial curvature and gravitational redshift. The local gravitational acceleration $g \propto 1/N_2$ becomes very large, explaining the rapid oscillation.

Comparison with Previous Work
The paper explicitly compares its findings with Ref. [1], which modeled the pendulum using a fixed coordinate length constraint ( $r' = L$ ). The authors argue that these are physically distinct problems:

In the fixed proper-length model (this work), the rope length is invariant, and the radial lift is a geometric consequence of the spatial geodesic.
In the fixed coordinate-length model, the proper length of the rope actually decreases as the bob is displaced sideways. Near the horizon, this shortening dominates the radial lift, leading to a different scaling law for the period ( $T \propto N_2$ vs. $T \propto \sqrt{N_2}$ ).
The paper notes that LaHaye and Poisson [7] analyzed a static string attached to a particle in Schwarzschild spacetime, considering massless and massive strings and gravitational backreaction, whereas the present work focuses on the dynamics implied by a proper-length constraint.

Consistency and Assumptions
The authors discuss the validity of the "fixed proper length" assumption, which implies the rope adjusts quasi-statically. They derive a condition requiring that a signal (e.g., a tension wave) must travel along the rope many times within one oscillation period.

For a fixed fulcrum, as the bob approaches the horizon, the oscillation period (as measured at the fulcrum) grows faster than the signal travel time due to redshift. Consequently, the fixed-length approximation becomes more consistent in this specific limit, as the signal delay becomes negligible relative to the period.
However, for very long ropes or distant fulcrums, the signal delay may become comparable to the period, necessitating a model of an elastic string with finite propagation speed.

Significance
The paper claims to provide the correct small-oscillation period for a pendulum defined by an invariant proper-length constraint in Schwarzschild spacetime. It demonstrates that the distinction between coordinate and proper constraints is crucial in strong-field regimes, leading to physically inequivalent predictions. The work serves as a pedagogical example of how spatial geometry and gravitational redshift jointly influence mechanical systems in general relativity, recovering classical results in the weak-field limit while offering a distinct interpretation of dynamics near the event horizon.

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