# Universal quantum modifications to general relativistic time dilation in delocalised clocks

Shishir Khandelwal1,2, Maximilian P.E. Lock3, and Mischa P. Woods1,4

1Institute for Theoretical Physics, ETH Zürich, Switzerland
2Group of Applied Physics, University of Geneva, Switzerland
3Institute for Quantum Optics and Quantum Information (IQOQI), Vienna, Austria
4Department of Computer Science, University College London, United Kingdom

### Abstract

The theory of relativity associates a proper time with each moving object via its world line. In quantum theory however, such well-defined trajectories are forbidden. After introducing a general characterisation of quantum clocks, we demonstrate that, in the weak-field, low-velocity limit, all good'' quantum clocks experience time dilation as dictated by general relativity when their state of motion is classical (i.e. Gaussian). For nonclassical states of motion, on the other hand, we find that quantum interference effects may give rise to a significant discrepancy between the proper time and the time measured by the clock. The universality of this discrepancy implies that it is not simply a systematic error, but rather a quantum modification to the proper time itself. We also show how the clock's delocalisation leads to a larger uncertainty in the time it measures – a consequence of the unavoidable entanglement between the clock time and its center-of-mass degrees of freedom. We demonstrate how this lost precision can be recovered by performing a measurement of the clock's state of motion alongside its time reading.

According to general relativity, the time read by an observer’s clock is
determined by their path through spacetime — an effect known as time
dilation. Quantum mechanics, on the other hand, says that such definite
paths are impossible, and that all objects will instead follow
superpositions of different paths. What time, then, does a quantum clock
measure?

In this paper, we give an answer to this question. We use the simple fact
that every clock must have some internal workings whose changes track the
passage of time. This internal structure is associated with some energy,
and general relativity tells us how this energy interacts with the energy
of the clock’s motion, in turn determining the time experienced by the
clock. We can separate this effect into a part which depends on the
particulars of the clock, and a part which doesn’t. Since the latter is the
same for any system regardless of its makeup, it is a universal time
dilation.

For the kinds of motion that most resemble the “classical” paths used in
general relativity, we find that a clock measures (on average) the
classical time dilation. For a superposition of such paths, however, we
predict a classical and a quantum time dilation effect, and the quantum
part is large enough to potentially be observed in the near future. We also
find that this relationship between the energy of the internal working of
the clock and its motion generates quantum entanglement between them,
reducing the accuracy of measurements of the clock time (unless the clock’s
motion is measured too).

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### Cited by

[1] Carolyn E. Wood and Magdalena Zych, "Composite particles with minimum uncertainty in spacetime", Physical Review Research 3 1, 013049 (2021).

[2] Emanuel Schwarzhans, Maximilian P. E. Lock, Paul Erker, Nicolai Friis, and Marcus Huber, "Autonomous Temporal Probability Concentration: Clockworks and the Second Law of Thermodynamics", Physical Review X 11 1, 011046 (2021).

[3] Mischa P. Woods, "Autonomous Ticking Clocks from Axiomatic Principles", Quantum 5, 381 (2021).

[4] Alexander R. H. Smith and Mehdi Ahmadi, "Quantum clocks observe classical and quantum time dilation", Nature Communications 11 1, 5360 (2020).

[5] Piotr T. Grochowski, Alexander R. H. Smith, Andrzej Dragan, and Kacper Dębski, "Quantum time dilation in atomic spectra", Physical Review Research 3 2, 023053 (2021).

[6] Esteban Castro-Ruiz, Flaminia Giacomini, Alessio Belenchia, and Časlav Brukner, "Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems", Nature Communications 11, 2672 (2020).

[7] Philipp A. Hoehn, Alexander R. H. Smith, and Maximilian P. E. Lock, "Equivalence of approaches to relational quantum dynamics in relativistic settings", arXiv:2007.00580.

[8] Maximilian P. E. Lock and Ivette Fuentes, "Quantum and classical effects in a light-clock falling in Schwarzschild geometry", Classical and Quantum Gravity 36 17, 175007 (2019).

[9] Yuxiang Yang and Renato Renner, "Ultimate limit on time signal generation", arXiv:2004.07857.

[10] Alexander R. H. Smith, "Quantum time dilation: A new test of relativistic quantum theory", arXiv:2004.10810.

The above citations are from Crossref's cited-by service (last updated successfully 2021-05-06 22:40:28) and SAO/NASA ADS (last updated successfully 2021-05-06 22:40:29). The list may be incomplete as not all publishers provide suitable and complete citation data.