Complete Second-Order Relativistic Derivation of the Observed Pulsar Timing Modulations
Abstract
We present a fully nonlinear relativistic description of the observed pulsar timing modulations, defined through the ratio of the proper-time intervals between two successive emissions of radio pulses and their observations by a pulsar timing array. Working in the limit of a vanishing emission interval, we derive the observed pulsar timing modulation to second order in relativistic perturbation theory around a Minkowski background, without fixing a gauge and retaining the full scalar, vector, and tensor content of the metric perturbations. Although the derivation naturally introduces several gauge-dependent intermediate quantities, we explicitly demonstrate that they combine into a coordinate-independent expression. Moreover, we explicitly show that the second-order timing modulation from two neighboring radio pulses coincides with the observed redshift, defined as the fractional change in wavelength along a single geodesic. The agreement represents a nontrivial second-order realization of the exact nonlinear equivalence proven in Magi & Yoo 2026.
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BibTeX
@misc{magi2026complete,
title = {{Complete Second-Order Relativistic Derivation of the Observed Pulsar Timing Modulations}},
author = {Matteo Magi and Jaiyul Yoo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13364},
eprint = {2609.13364},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13364},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complete Second-Order Relativistic Derivation of the Observed Pulsar Timing Modulations
AU - Matteo Magi
AU - Jaiyul Yoo
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13364
UR - https://arxiv.org/abs/2609.13364
AB - We present a fully nonlinear relativistic description of the observed pulsar timing modulations, defined through the ratio of the proper-time intervals between two successive emissions of radio pulses and their observations by a pulsar timing array. Working in the limit of a vanishing emission interval, we derive the observed pulsar timing modulation to second order in relativistic perturbation theory around a Minkowski background, without fixing a gauge and retaining the full scalar, vector, and tensor content of the metric perturbations. Although the derivation naturally introduces several gauge-dependent intermediate quantities, we explicitly demonstrate that they combine into a coordinate-independent expression. Moreover, we explicitly show that the second-order timing modulation from two neighboring radio pulses coincides with the observed redshift, defined as the fractional change in wavelength along a single geodesic. The agreement represents a nontrivial second-order realization of the exact nonlinear equivalence proven in Magi & Yoo 2026.
ER -