On the numerical evaluation of the `exact' Post-Newtonian parameters in Brans-Dicke and Entangled Relativity theories

This paper introduces two numerical methods to calculate exact Post-Newtonian parameters in Brans-Dicke and Entangled Relativity theories, revealing significant deviations from General Relativity for compact objects and demonstrating that the theory's testability critically depends on whether the matter Lagrangian is defined by energy density or the trace of the stress-energy tensor.

The Gist

Imagine gravity not just as a smooth, invisible sheet (like Einstein described), but as a sheet that has a hidden "volume knob" attached to it. This knob is a scalar field. In some theories of gravity, this knob can be turned up or down depending on how heavy and pressurized the object sitting on the sheet is.

This paper is a deep dive into two specific theories that use this "volume knob" idea: Brans-Dicke theory and a newer, stranger theory called Entangled Relativity.

Here is the breakdown of what the authors did, explained simply:

1. The Problem: The "Weak" vs. "Strong" View

For a long time, scientists have tested gravity using "Post-Newtonian parameters." Think of these as standardized ruler marks used to measure how much a theory deviates from Einstein's General Relativity.

• The Old Way: These ruler marks were calculated assuming gravity is weak (like on Earth or near the Sun). They are like measuring a rubber band while it's barely stretched.
• The New Reality: The authors realized that for super-dense objects like neutron stars (where gravity is crushing), the rubber band is stretched to its limit. The "ruler marks" change depending on the internal pressure and density of the star. You can't use the weak-field ruler anymore; you need an "Exact" ruler.

2. The Mission: Building the "Exact" Ruler

The authors developed two new computer methods to calculate these "Exact" parameters for stars.

• Method A: They simulated the inside of a star using complex math (Tolman-Oppenheimer-Volkoff equations) to see how the "volume knob" behaves under extreme pressure.
• Method B: They looked at the "outside" of the star and matched it to a known mathematical solution (Janis-Newman-Winicour) to double-check their work.

The Result: They found that for very dense stars, the difference between the "Old Ruler" and the "Exact Ruler" can be huge—sometimes over 80%. This means previous tests might have missed massive deviations because they were using the wrong tool for the job.

3. The Star of the Show: Entangled Relativity

The paper focuses heavily on a theory called Entangled Relativity. This theory is unique because it says matter and geometry are "entangled" (like two dancers who must move together). You can't have the dance floor (space) without the dancers (matter).

The authors asked: Does this theory look like Einstein's gravity for our Sun and Earth?

• The Good News: Yes, mostly. For the Sun and Earth, the deviations are tiny (invisible to current telescopes).
• The Catch: It depends on a specific assumption about how matter behaves.
  • Scenario A (The "Dust" Assumption): If matter acts like a simple cloud of dust, the theory predicts massive deviations for neutron stars. The "volume knob" turns way up.
  • Scenario B (The "Radiation" Assumption): If matter acts like light or radiation, the theory collapses back into Einstein's General Relativity. The "volume knob" stays off.

4. The Smoking Gun: Dipolar Gravitational Waves

How do we know which scenario is true? The authors looked at binary pulsars (two dead stars orbiting each other).

• In Einstein's theory, these stars lose energy by emitting gravitational waves in a specific, symmetrical pattern (like a lighthouse beam).
• In Entangled Relativity (under Scenario A), the "volume knob" difference between the two stars would cause them to emit dipolar waves (like a lighthouse beam that wobbles side-to-side). This would make them lose energy much faster.

The Verdict:
When the authors ran the numbers for a real binary system (PSR J1738+0333), they found that if Entangled Relativity works the way Scenario A suggests, the stars should be spiraling into each other 100 to 1,000 times faster than we actually observe.

The Bottom Line

• For Brans-Dicke Theory: The "Exact" parameters show that strong gravity behaves very differently than we thought, depending on the star's internal pressure.
• For Entangled Relativity: The theory is likely ruled out if it assumes matter acts like simple dust (Scenario A), because the stars would be moving too fast. However, if matter acts like radiation (Scenario B), the theory survives but becomes almost indistinguishable from Einstein's General Relativity, making it very hard to test.

In a nutshell: The authors built a better microscope to look at gravity in extreme conditions. They found that while some theories might still be hiding in the shadows, the "Entangled" theory is likely in trouble unless it behaves in a very specific, boring way that makes it look exactly like Einstein's theory.

Technical Summary

1. Problem Statement

Standard Post-Newtonian (PN) parameters ($\beta, \gamma, \delta$) are perturbative quantities derived in the weak-field limit of gravity theories. They assume that energy density ($\rho$) dominates over pressure ($P$) and are independent of the internal structure of the gravitating body. However, recent theoretical developments have shown that in scalar-tensor theories (like Brans-Dicke and Entangled Relativity), these parameters must be generalized into "exact" non-perturbative parameters ($\beta_{exact}, \gamma_{exact}, \delta_{exact}$).

These exact parameters depend explicitly on the internal structure (energy density and pressure profiles) of the celestial body. While analytical expressions for these parameters exist (derived by Chauvineau et al.), they had never been numerically evaluated. This gap prevents a rigorous comparison between strong-field predictions of alternative gravity theories and observational data from compact objects (e.g., neutron stars) where $P \sim \rho$.

2. Methodology

The authors developed two independent numerical methods to evaluate these exact parameters, ensuring robustness through cross-verification:

• Method A: Direct Numerical Integration (Tolman-Oppenheimer-Volkoff Framework)

  • The authors derived the modified Tolman-Oppenheimer-Volkoff (TOV) equations for both Brans-Dicke and Entangled Relativity theories.
  • They solved these differential equations numerically using a polytropic equation of state ($P = K\rho^\gamma$) to model the interior structure of stars (Sun, Earth, Neutron Stars).
  • They computed the integrated energy ($E^*$) and pressure ($P^*$) terms required by the analytical formulas for exact parameters.
  • A normalization technique was employed to handle boundary conditions at the star's center and infinity.

• Method B: External Solution Matching (Janis-Newman-Winicour)

  • The authors utilized the known vacuum exterior solution (Janis-Newman-Winicour metric) for scalar-tensor theories.
  • By matching the asymptotic behavior of the numerically integrated interior solution (from Method A) to the exterior analytical solution, they extracted the scalar charge ($d$ or $\alpha$).
  • This scalar charge was then substituted back into the analytical formulas to derive the exact PN parameters.

• Validation: The two methods were cross-checked, showing relative deviations of less than 0.07% for $\gamma_{exact}$ and 0.3% for $\delta_{exact}$, validating both the numerical implementation and the underlying analytical theory.

3. Key Contributions

• First Numerical Evaluation: This is the first study to numerically compute exact, non-perturbative PN parameters for realistic stellar models in scalar-tensor theories.
• Theory Extension: The framework was applied to Brans-Dicke theory and Entangled Relativity (a theory where matter and geometry are coupled non-linearly, satisfying Mach's principle).
• Connection to DEF Parameters: The authors established a direct analytical link between the exact PN parameter $\gamma_{exact}$ and the Damour-Esposito-Farèse (DEF) non-perturbative scalar charge ($\alpha_{DEF}$), bridging the gap between strong-field scalarization and PN formalism.
• Lagrangian Sensitivity Analysis: In Entangled Relativity, the authors rigorously tested two different assumptions for the matter Lagrangian ($L_m$):
  1. $L_m = -\rho$ (Total energy density).
  2. $L_m = T$ (Trace of the stress-energy tensor).

4. Key Results

A. Brans-Dicke Theory

• Deviation from Standard PN: For compact objects (neutron stars), the exact parameters deviate significantly from standard perturbative values.
• Magnitude: Relative deviations in $\gamma_{exact}$ and $\delta_{exact}$ can reach 44% to 83% depending on the coupling parameter $\omega$ and central density.
• Counter-Intuitive Behavior: Surprisingly, for ultra-compact objects where the speed of sound approaches $c/\sqrt{3}$ (implying $P \approx \rho/3$), the deviation from General Relativity (GR) decreases. This is because the source term for the scalar field ($\Box\phi \propto -\rho + 3P$) approaches zero, effectively suppressing the scalar field's influence in the most extreme limits.

B. Entangled Relativity

• Solar System: For the Sun and Earth, deviations are extremely small ($\gamma_{exact} - 1 \approx 10^{-6}$ for the Sun), consistent with current weak-field constraints but potentially detectable with future high-precision experiments.
• Neutron Stars ($L_m = -\rho$):
  • If the matter Lagrangian is $L_m = -\rho$, the scalar field is strongly sourced.
  • Deviations in $\gamma_{exact}$ range from 4.7% to 16.1%.
  • Deviations in $\delta_{exact}$ range from 11.2% to 40.1%.
  • These large deviations imply significant dipolar gravitational wave emission in binary systems (e.g., Pulsar-White Dwarf).
  • Constraint: The predicted orbital period decay ($\dot{P}_D$) for the system PSR J1738+0333 is 2 to 3 orders of magnitude larger than observational constraints. This suggests that Entangled Relativity with $L_m = -\rho$ is likely ruled out by current data.
• Neutron Stars ($L_m = T$):
  • If $L_m = T$ (often advocated in literature), the scalar field is not sourced by ordinary matter (dust/radiation).
  • The theory reduces effectively to General Relativity for standard compact objects.
  • Deviations would only occur for objects with extreme electromagnetic fields (e.g., Magnetars), where the source term is proportional to $B^2$. These effects are estimated to be $10^7$ times smaller than the $L_m = -\rho$ case, making them currently untestable.

5. Significance

• Strong-Field Testing: The paper demonstrates that weak-field PN parameters are insufficient for testing gravity in the strong-field regime. Exact parameters reveal that "spontaneous scalarization" and non-perturbative effects can drastically alter predictions for neutron stars.
• Theory Viability: The results provide a stringent test for Entangled Relativity. The assumption $L_m = -\rho$ appears incompatible with binary pulsar timing data, whereas $L_m = T$ renders the theory nearly indistinguishable from GR for most astrophysical objects, pushing the window for testing it to extreme magnetic environments.
• Methodological Benchmark: The development and cross-validation of the two numerical methods provide a robust toolkit for future investigations into scalar-tensor theories and the internal structure of compact objects.

In conclusion, the authors demonstrate that exact PN parameters are essential for a correct understanding of scalar-tensor gravity in strong fields. Their numerical evaluation reveals that while some theories (like Brans-Dicke) show complex density-dependent behaviors, others (like Entangled Relativity) face immediate constraints from binary pulsar observations depending on the specific form of the matter Lagrangian.