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arxiv: 2509.17017 · v2 · submitted 2025-09-21 · 🌀 gr-qc

Dynamics of the N-body system in energy-momentum squared gravity: II. Existence of a Self-Acceleration

Elham Nazari This is my paper

Pith reviewed 2026-05-18 15:02 UTC · model grok-4.3

classification 🌀 gr-qc
keywords energy-momentum squared gravitypost-Newtonian dynamicsN-body systemsself-accelerationvirial identitiesconserved momentummodified gravitybinary pulsars
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The pith

In energy-momentum squared gravity, self-acceleration of N-body systems vanishes at first post-Newtonian order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the post-Newtonian motion of self-gravitating bodies in quadratic energy-momentum squared gravity, a modified gravity theory where the divergence of the energy-momentum tensor does not vanish. It relaxes prior symmetry assumptions to derive the full 1PN equations of motion for an individual body inside an N-body system. A center-of-mass acceleration is defined and combined with standard plus newly derived virial identities to prove that any internal-structure-dependent acceleration term is zero. This result yields an integral conservation law for the total linear momentum that holds at post-Newtonian order. The outcome aligns with existing bounds from binary-pulsar timing, leaving the theory compatible with strong-field observations.

Core claim

By introducing a suitable center-of-mass acceleration and applying virial identities that include one new identity arising from the quadratic-EMSG non-minimal coupling, self-acceleration is shown to vanish identically at 1PN order. Consequently, the theory admits a post-Newtonian integral conservation law for the total linear momentum, reproducing the same conservation property that holds in general relativity.

What carries the argument

Center-of-mass acceleration expression together with virial identities (including the newly derived quadratic-EMSG identity) that cancel all internal-structure contributions to the acceleration at 1PN order.

If this is right

  • Self-acceleration is absent for any self-gravitating body in an N-body system at 1PN order.
  • Total linear momentum is conserved at post-Newtonian order, identical to the situation in general relativity.
  • The theory satisfies current experimental limits on self-acceleration from binary pulsars.
  • EMSG remains viable for modeling strong-gravity regimes probed by compact-object binaries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same cancellation mechanism may extend to higher post-Newtonian orders if analogous virial identities can be obtained.
  • Numerical N-body simulations in EMSG could now treat the bodies as point masses without additional self-force terms at 1PN.
  • The conserved-momentum result simplifies the matching of interior solutions to exterior metrics in multi-body configurations.

Load-bearing premise

The post-Newtonian expansion remains valid inside the self-gravitating bodies and the new virial identity fully captures the non-minimal coupling effects at first post-Newtonian order.

What would settle it

A binary-pulsar timing observation that measures a statistically significant non-zero self-acceleration term at the magnitude allowed by current EMSG parameter bounds would contradict the derived vanishing result.

read the original abstract

We investigate the post-Newtonian (PN) dynamics of energy-momentum squared gravity (EMSG), with particular emphasis on the possibility of self-acceleration in $N$-body systems. A central challenge in matter-type modified gravity theories, including EMSG, is the non-vanishing divergence of the energy-momentum tensor, arising from the nonminimal interaction between the standard and modified matter fields. This feature can, in principle, influence the $N$-body dynamics. In our previous work (Nazari, 2024), its effects on the external-dependent part of the motion were studied in an EMSG class known as quadratic-EMSG. Here, we extend the analysis to the internal-structure-dependent contributions, namely self-acceleration. To this end, we relax the reflection-symmetric assumption adopted in (Nazari, 2024), and derive the complete equations of motion for a self-gravitating body in an $N$-body system up to the first PN order. By introducing a suitable expression for the center-of-mass acceleration and employing virial identities, including one newly emerging within the quadratic-EMSG framework, it is shown that self-acceleration vanishes. Furthermore, we establish a PN integral conservation law for the total momentum, demonstrating that, as in general relativity (GR), EMSG admits a conserved linear momentum compatible with the absence of self-acceleration. Binary pulsar experiments provide stringent bounds on self-acceleration, and our analysis shows that, within the present level of accuracy, EMSG is consistent with these constraints. Therefore, the theory remains viable in the strong-gravity regime probed by binary pulsars.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives the complete 1PN equations of motion for self-gravitating bodies in quadratic energy-momentum squared gravity (EMSG) without the reflection-symmetry assumption of prior work. It introduces an expression for center-of-mass acceleration, applies standard and newly derived virial identities to show that all internal-structure contributions cancel, and establishes a PN integral conservation law for total linear momentum. The central claim is that self-acceleration vanishes at this order, rendering quadratic-EMSG consistent with binary-pulsar constraints on self-acceleration and compatible with the conserved-momentum property of general relativity.

Significance. If the derivation is correct, the result removes a potential phenomenological obstacle for quadratic-EMSG by demonstrating the absence of self-acceleration at 1PN order and the existence of a conserved total momentum. This places the theory on the same footing as GR for tests involving binary pulsars and N-body dynamics. The emergence of a new virial identity specific to the quadratic-EMSG coupling is a technical contribution that could be useful for future post-Newtonian analyses of non-minimally coupled theories.

major comments (2)
  1. The cancellation of self-acceleration rests on the new virial identity that arises after integrating the modified equations of motion. The abstract states that this identity, together with the center-of-mass definition, produces exact cancellation at 1PN; however, the explicit form of the identity and the term-by-term cancellation against the non-minimal EMSG contributions are not shown. Because this step is load-bearing for the claim that self-acceleration vanishes for arbitrary internal structure, the derivation of the identity (including any assumptions about the validity of the 1PN expansion inside compact objects) requires explicit verification.
  2. The analysis assumes that the post-Newtonian expansion remains adequate for the internal dynamics of self-gravitating bodies (e.g., neutron stars) even in the presence of the quadratic-EMSG coupling. If the non-minimal interaction generates strong-field corrections beyond 1PN that are not captured by the virial relation, a residual self-acceleration could appear at the level probed by pulsar timing. A brief estimate or justification of the size of such higher-order terms would be needed to support the statement that EMSG is consistent with current binary-pulsar bounds.
minor comments (2)
  1. The title refers to the 'Existence of a Self-Acceleration,' yet the abstract and conclusion state that self-acceleration vanishes. A more descriptive title (e.g., 'Absence of Self-Acceleration...') would better reflect the result.
  2. Notation for the EMSG coupling constant and the precise definition of the center-of-mass acceleration should be introduced earlier and used consistently throughout the equations of motion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which have helped clarify several technical aspects of the derivation. We address each major comment below and have revised the manuscript to improve the presentation of the key steps.

read point-by-point responses

  1. Referee: The cancellation of self-acceleration rests on the new virial identity that arises after integrating the modified equations of motion. The abstract states that this identity, together with the center-of-mass definition, produces exact cancellation at 1PN; however, the explicit form of the identity and the term-by-term cancellation against the non-minimal EMSG contributions are not shown. Because this step is load-bearing for the claim that self-acceleration vanishes for arbitrary internal structure, the derivation of the identity (including any assumptions about the validity of the 1PN expansion inside compact objects) requires explicit verification.

    Authors: We agree that making the cancellation fully explicit strengthens the paper. In the revised version we have added an appendix that derives the quadratic-EMSG virial identity from the integrated 1PN equations of motion, presents its explicit form, and tabulates the term-by-term cancellation against the non-minimal contributions that appear in the center-of-mass acceleration. The assumptions underlying the 1PN treatment inside the bodies are the standard ones employed in post-Newtonian analyses of compact objects (weak internal gravitational field and slow internal motions), which remain valid for the quadratic coupling at this order. revision: yes

  2. Referee: The analysis assumes that the post-Newtonian expansion remains adequate for the internal dynamics of self-gravitating bodies (e.g., neutron stars) even in the presence of the quadratic-EMSG coupling. If the non-minimal interaction generates strong-field corrections beyond 1PN that are not captured by the virial relation, a residual self-acceleration could appear at the level probed by pulsar timing. A brief estimate or justification of the size of such higher-order terms would be needed to support the statement that EMSG is consistent with current binary-pulsar bounds.

    Authors: We have added a short paragraph in the concluding section that provides a rough order-of-magnitude estimate. Because the quadratic-EMSG correction is linear in the matter density and the coupling constant is already bounded to be small by solar-system and cosmological data, the leading higher-order corrections are suppressed by an extra factor of order (v/c)^2 relative to the 1PN terms. For the orbital velocities relevant to binary pulsars this factor is ≲ 0.1, and the EMSG contribution is further reduced by the smallness of the coupling. This estimate supports that any residual self-acceleration lies below current pulsar-timing sensitivity. A complete 2PN calculation lies outside the scope of the present work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation of vanishing self-acceleration stands independently

full rationale

The paper derives the complete 1PN equations of motion for self-gravitating bodies after relaxing the reflection symmetry from prior work, introduces an explicit center-of-mass acceleration, and applies virial identities (including one newly obtained in the quadratic-EMSG framework) to show that internal-structure contributions to self-acceleration cancel. It further obtains a PN integral conservation law for total momentum directly from these equations. The reference to Nazari (2024) addresses only the external-dependent motion studied earlier and does not supply the load-bearing step for the internal self-acceleration result or the conservation law. No fitted parameters are renamed as predictions, no ansatz is smuggled via citation, and no uniqueness theorem is imported to force the outcome. The central claim therefore reduces to the modified field equations and the derived identities rather than to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the post-Newtonian expansion in the presence of non-minimal matter coupling and on the correctness of a newly derived virial identity specific to quadratic-EMSG.

free parameters (1)
  • EMSG coupling constant
    The strength of the quadratic matter term is a free parameter of the theory whose value is not fixed by the derivation.
axioms (2)
  • domain assumption Post-Newtonian expansion remains valid for internal structure in quadratic-EMSG
    Invoked when deriving the 1PN equations of motion for self-gravitating bodies.
  • ad hoc to paper New virial identity holds in quadratic-EMSG
    Introduced to cancel the internal acceleration term.

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