## Abstract

Einstein’s theory of gravity—the general theory of relativity^{1}—is based on the universality of free fall, which specifies that all objects accelerate identically in an external gravitational field. In contrast to almost all alternative theories of gravity^{2}, the strong equivalence principle of general relativity requires universality of free fall to apply even to bodies with strong self-gravity. Direct tests of this principle using Solar System bodies^{3,4} are limited by the weak self-gravity of the bodies, and tests using pulsar–white-dwarf binaries^{5,6} have been limited by the weak gravitational pull of the Milky Way. PSR J0337+1715 is a hierarchical system of three stars (a stellar triple system) in which a binary consisting of a millisecond radio pulsar and a white dwarf in a 1.6-day orbit is itself in a 327-day orbit with another white dwarf. This system permits a test that compares how the gravitational pull of the outer white dwarf affects the pulsar, which has strong self-gravity, and the inner white dwarf. Here we report that the accelerations of the pulsar and its nearby white-dwarf companion differ fractionally by no more than 2.6 × 10^{−6}. For a rough comparison, our limit on the strong-field Nordtvedt parameter, which measures violation of the universality of free fall, is a factor of ten smaller than that obtained from (weak-field) Solar System tests^{3,4} and a factor of almost a thousand smaller than that obtained from other strong-field tests^{5,6}.

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## Acknowledgements

We thank P. Freire for pointing out how useful PSR J0337+1715 could be for testing the SEP, L. Shao for providing an independent cross-check on the signature of *Δ* and K. Nordvedt for explaining why the signature of *Δ* differs from that in lunar laser ranging. A.M.A. is supported by a Netherlands Foundation for Scientific Research (NWO) Veni grant. N.V.G. is supported by NOVA. J.W.T.H. acknowledges funding from an NWO Vidi fellowship and from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Starting Grant agreement number 337062 (‘DRAGNET’). A.T.D. is the recipient of an Australian Research Council Future Fellowship (FT150100415). D.R.L. also received support from NSF award OIA-1458952. S.M.R. and I.H.S. are Senior Fellows of the Canadian Institute for Advanced Research. I.H.S. is also supported by an NSERC Discovery Grant. The NANOGrav project (involving D.L.K., D.R.L., R.S.L., S.M.R. and I.H.S.) receives support from National Science Foundation (NSF) Physics Frontiers Center award number 1430284. The National Radio Astronomy Observatory is a facility of the NSF operated under cooperative agreement by Associated Universities. The Arecibo Observatory is operated by SRI International under a cooperative agreement with the NSF (AST-1100968), and in alliance with Ana G. Mendez-Universidad Metropolitana and the Universities Space Research Association. The Green Bank Observatory is a facility of the NSF operated under cooperative agreement by Associated Universities. The WSRT is operated by ASTRON with contributions from NWO.

*Nature* thanks P. Freire and C. Will for their contribution to the peer review of this work.

## Author information

### Affiliations

#### Anton Pannekoek Institute for Astronomy, University of Amsterdam, Amsterdam, The Netherlands

- Anne M. Archibald
- , Nina V. Gusinskaia
- & Jason W. T. Hessels

#### Netherlands Institute for Radio Astronomy (ASTRON), Dwingeloo, The Netherlands

- Anne M. Archibald
- & Jason W. T. Hessels

#### Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria, Australia

- Adam T. Deller

#### The Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), Hawthorn, Victoria, Australia

- Adam T. Deller

#### Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI, USA

- David L. Kaplan

#### Department of Physics and Astronomy, West Virginia University, Morgantown, WV, USA

- Duncan R. Lorimer

#### Center for Gravitational Waves and Cosmology, Morgantown, WV, USA

- Duncan R. Lorimer
- & Ryan S. Lynch

#### Green Bank Observatory, Green Bank, WV, USA

- Ryan S. Lynch

#### National Radio Astronomy Observatory, Charlottesville, VA, USA

- Scott M. Ransom

#### Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada

- Ingrid H. Stairs

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### Contributions

A.M.A. wrote the processing pipeline, orbital modelling, fitting and equation-of-state integration code, ran the data processing and fitting operations, wrote the manuscript, with substantial contributions from co-authors, and produced all figures and tables unless otherwise indicated. N.V.G. wrote the systematics analysis code, inspected observations for quality, carried out the systematics analysis, produced Table 1, Figs. 1, 2 and Extended Data Fig. 4, and wrote the section on orbital effects. J.W.T.H. carried out an intensive observing campaign with the WSRT. A.M.A., N.V.G., J.W.T.H., D.R.L., R.S.L., S.M.R. and I.H.S. carried out observations with Arecibo and the GBT. J.W.T.H., S.M.R. and I.H.S. carried out a preliminary version of the data processing. A.T.D. consulted on the astrometry. N.V.G. and D.L.K. carried out the tidal-effects analysis. All authors participated in discussions of the content of the paper.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Anne M. Archibald.

## Extended data figures and tables

### Extended Data Fig. 1 Template pulse profile used for timing.

This figure is based on the average 1,300–1,900-MHz profile from the GBT observation on MJD 56,412. The Stokes

*IQUV*data have been smoothed by a wavelet-based algorithm (*psrsmooth*, PSRCHIVE).**a**, Total intensity (*I*) and linear (*Q*,*U*) and circular (*V*) polarization after correcting for Faraday rotation. An offset*c*has been subtracted; see below.**b**, Polarization angle (P.A.) at the centre frequency of the observation. The linear polarization (red) at some phases is responsible for almost half the flux density and its profile has complicated polarization structure. Offsets have been added to*I*and to $\sqrt{{Q}^{2}+{U}^{2}}$ to ensure that*I*^{2}≥*Q*^{2}+*U*^{2}+*V*^{2}.### Extended Data Fig. 2 Timing-model truncation error.

The root-mean-square (RMS) arrival-time error caused by the finite time steps of the orbital integrator is shown as a function of the tolerance parameter. The vertical dotted line is the value used for all orbits in this work; the RMS error from truncation is below 0.1 ns. Blue triangles are calculations done in hardware 80-bit floating point; black stars are calculations done in software 128-bit floating point, which are much slower to compute. To estimate the errors in this plot, we compute a fiducial solution with 128-bit precision and a tolerance parameter of 10

^{−22}and compare all other solutions to this one.### Extended Data Fig. 3 Covariances between parameters that affect the orbit.

This plot does not include the parameters that are evaluated by linear least-squares fitting and marginalized out. Plots on the diagonal are single-parameter histograms; plots off the diagonal are pairwise two-dimensional histograms. See Extended Data Table 2 for parameter definitions.

### Extended Data Fig. 4 Distribution of residuals divided by uncertainty, for each telescope.

The standard deviation

*σ*represents the factor by which the scatter of the post-fit residuals exceeds the claimed uncertainties on pulse arrival times;*μ*is the mean of the distribution. Each colour represents a different telescope. Only observations in the 1,400-MHz frequency band are shown here. Here Δ*ν*and Δ*t*are the bandwidth and time, respectively, over which the data are averaged to produce each pulse arrival time.

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