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An Information-Theoretic Analog of the Twin Paradox

Mladen Kovačević, Iosif Pinelis, Marios Kountouris

TL;DR

This work analyzes the information-theoretic twin paradox in a relativistic setting by modeling Doppler-modified AWGN channels under symmetric power and bandwidth constraints. The authors convert the problem into a probabilistic inequality and prove the conjecture in the constant-speed, closed-trajectory case, using extremal probability techniques to show that the static transmitter (Alice) can reliably send more bits per second to the traveler (Bob) than vice versa, with energy-per-bit correspondingly lower for Alice. A key result is an inequality relating the radial-velocity component and the Doppler factors that implies C̄_A > C̄_B, and the asymptotic behavior β → 1 yields unbounded growth for C̄_A and decay for C̄_B. The paper also discusses extensions to general trajectories and the infinite-bandwidth limit, highlighting fundamental limits on relativistic information transfer with potential implications for distributed systems and GPS within relativistic frameworks.

Abstract

We revisit the familiar scenario involving two parties in relative motion, in which Alice stays at rest while Bob goes on a journey at speed $βc$ along an arbitrary trajectory and reunites with Alice after a certain period of time. It is a well-known consequence of special relativity that the time that passes until they meet again is different for the two parties and is shorter in Bob's frame by a factor of $\sqrt{1-β^2}$. We investigate how this asymmetry manifests from an information-theoretic viewpoint. Assuming that Alice and Bob transmit signals of equal average power to each other during the whole journey, and that additive white Gaussian noise is present at both sides, we show that the maximum number of bits per second that Alice can transmit reliably to Bob is always higher than the one Bob can transmit to Alice. Equivalently, the energy per bit invested by Alice is lower than that invested by Bob, meaning that the traveler is less efficient from the communication perspective, as conjectured by Jarett and Cover.

An Information-Theoretic Analog of the Twin Paradox

TL;DR

This work analyzes the information-theoretic twin paradox in a relativistic setting by modeling Doppler-modified AWGN channels under symmetric power and bandwidth constraints. The authors convert the problem into a probabilistic inequality and prove the conjecture in the constant-speed, closed-trajectory case, using extremal probability techniques to show that the static transmitter (Alice) can reliably send more bits per second to the traveler (Bob) than vice versa, with energy-per-bit correspondingly lower for Alice. A key result is an inequality relating the radial-velocity component and the Doppler factors that implies C̄_A > C̄_B, and the asymptotic behavior β → 1 yields unbounded growth for C̄_A and decay for C̄_B. The paper also discusses extensions to general trajectories and the infinite-bandwidth limit, highlighting fundamental limits on relativistic information transfer with potential implications for distributed systems and GPS within relativistic frameworks.

Abstract

We revisit the familiar scenario involving two parties in relative motion, in which Alice stays at rest while Bob goes on a journey at speed along an arbitrary trajectory and reunites with Alice after a certain period of time. It is a well-known consequence of special relativity that the time that passes until they meet again is different for the two parties and is shorter in Bob's frame by a factor of . We investigate how this asymmetry manifests from an information-theoretic viewpoint. Assuming that Alice and Bob transmit signals of equal average power to each other during the whole journey, and that additive white Gaussian noise is present at both sides, we show that the maximum number of bits per second that Alice can transmit reliably to Bob is always higher than the one Bob can transmit to Alice. Equivalently, the energy per bit invested by Alice is lower than that invested by Bob, meaning that the traveler is less efficient from the communication perspective, as conjectured by Jarett and Cover.
Paper Structure (7 sections, 2 theorems, 39 equations)

This paper contains 7 sections, 2 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Model description and problem formulation
  3. The twin problem and the Doppler effect
  4. Communication model -- relativistic AWGN channel
  5. The Jarett--Cover conjecture
  6. Proof of the Jarett--Cover conjecture: Constant-speed case
  7. Concluding remarks

Key Result

Theorem 1

Let $f \colon [0,1] \to \mathbb{R}$ be a function satisfying $|f(x)| \leqslant b < 1$ and $\int_{0}^{1} f(x) dx = 0$. Then

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • proof : Proof of Lemma \ref{['thm:p']}
  • Remark 3
  • proof : Proof of Theorem \ref{['thm:inequality']}