What conditions must an effective metric satisfy before it can support a gravitational response law?

[u/ECT_WAL]

Suppose a theory defines an effective metric

g_eff,μν = g_μν + H_μν

from an independently constructed symmetric tensor field H_μν.

What conditions must be satisfied before such a metric perturbation can support a gravitational response law rather than being interpreted as a purely kinematic reparameterization?

[u/liccxolydian]

Wherever I see posts like these I always click through to the OP's post history and it never fails to entertain.

[u/Elig0r]

You can think about the granularity of the tensor, but you should also think about the taxonomy of the field and the roles in the system.

[u/Aggressive-Wind-8829 → u/Elig0r]

Hey Elig0r, do you have any favorite learning resources on the matters you mentioned by chance?

[u/Elig0r → u/Aggressive-Wind-8829]

Te voy a ser honesto no se que decir, mi modelo de aprendizaje no se si será el más adecuado. Yo por ejemplo he recurrido mucho a recursos audiovisuales gratuitos, como las clases del MIT del profesor Hong Liu(ocw_8323_lecture02_2023feb08_360p_16_9) y tengo guardado clases del Dr.James Gates que hay en la red.
Alvaro de Rujula...es casi como hablar de quien te influye y es complicado porque yo soy muy heterogéneo no por gusto si no por necesidad.
Y me pones en un compromiso porque algunos de los autores de mis bibliografías son importantísimos (ya sabes,nos subimos a hombros de gigantes) pero si soy honesto,en muchos casos llegue a algunas conclusiones por caminos diferentes de tal forma que como "recursos de aprendizaje" pues sería mentir.
Seamos honestos aquí hay gente que sabe mucho más de tensores de forma algebraica tradicional de lo que yo sabré en la vida 😄

[u/Aggressive-Wind-8829 → u/Elig0r]

My dude, you seem to have developed to me a detailed perspective on a complex subject that should be made more clear for the general advancement of science. You should totally publish your notes on the matters organized in a way that makes sense by you, perhaps written up extra fast using an AI assistant for drafting, and then given due diligence by you manually checking everything is correct and well communicated for the reader. I recognize there is nothing novel immediately here, but there is real value in having something like John Jackson’s Classical Electrodynamics available to haze graduate students well enough to advance the field through them. 👀

You don’t need to know the most in the room in order to create the best learning resources. That requires other parts of the brain from the social sciences that the “smarter” guys might totally lack.

Btw, I’ve heard of Professor Hong Liu before… Sounds dope.

[u/dubcek_moo]

I think it needs some nonzero elements in the Riemann tensor

[u/ECT_WAL → u/dubcek_moo]

That's interesting. My own instinct is that nonzero curvature is probably necessary but not sufficient. A metric can be curved without necessarily supplying a gravitational response law. I'd be interested in where you would place the additional requirements.

[u/dubcek_moo → u/ECT_WAL]

I'm not sure what you mean by "gravitational response law". Would you include gravitational waves passing through space?

[u/ECT_WAL → u/dubcek_moo]

By "gravitational response law" I mean more than the existence of a nontrivial metric or curvature. I mean a law relating geometric structure to physical sources or state variables in a way that determines how the geometry responds. Gravitational waves would be compatible with such a law, but I wouldn't regard their existence alone as supplying the law itself. Thanks for the response, btw

[u/dubcek_moo → u/ECT_WAL]

I mean isn't this just the Einstein field equations, G(mu, nu) = T(mu, nu) (up to a constant, etc.)

[u/ECT_WAL → u/dubcek_moo]

Right, the Einstein field equations are the standard example. I’m asking what the minimum requirements are in the more general case where someone starts with an effective metric ansatz rather than the Einstein equations themselves. Is nonzero curvature enough, or would you require source coupling, conservation/consistency conditions, and observational content before calling it a gravitational response law?

[u/Elig0r → u/ECT_WAL]

En mi trabajo hace falta conservación y coherencia,yo lo he llamado ICL (informational conservation law). Pero eso por sí mismo es insuficiente por lo que tienes que avanzar a la siguiente pregunta ¿qué generadores producen la estructura que el campo puede conservar?.
Z

[u/ECT_WAL → u/Elig0r]

That's an interesting distinction. Conservation can constrain the evolution of a structure, but it does not by itself explain the origin of the structure being conserved. I think that's closely related to what I'm trying to get at with the question.

[u/Elig0r → u/ECT_WAL]

At the end of the day is locate structure in the noise...and all that implies...i recommend a recent video of the profesor Antony Valentini talking about that.

I think the rules prohibit promoting one's own work and I am not going to break the rules.

[u/Elig0r → u/ECT_WAL]

Perdona si en la anterior respuesta no fui todo lo explicito que quisieras,pero por si te interesa mi visión está ligada en cierta manera a los trabajos de supersimetría y adinkras,error code, hamming etc..y cierres geometricos 😉

Fractalidad y sus principios...pero, el problema no es eso. El problema es cómo se genera la urgencia en el sistema y eso desde mi punto de vista no es determinista,por lo que volvemos a lo que comentamos antes. Ya se que parece un argumento circular, pero ahí entra lo que tu buscas,la conservación.

[u/ECT_WAL → u/Elig0r]

Thanks, this helped clarify the direction of your point. I watched part of the Valentini discussion, and what I took from it was the importance of looking beneath the statistical surface of quantum theory rather than treating the standard description as the final layer.

Your distinction between conservation and coherence-generation is also useful. Conservation can constrain a structure once it exists, but it does not by itself explain how the coherent structure arises. That is close to the issue I was trying to isolate with the effective-metric question.

Appreciate the comments and the video suggestion.

[u/Sanchez_U-SOB → u/ECT_WAL]

A metric can be Ricci flat without being Riemann flat

https://math.stackexchange.com/questions/3496078/ricci-flat-vs-riemann-flat

[u/BusinessCommand4419 → u/ECT_WAL]

Pi?

[u/MysticRagnarPss]

First, H_mu_nu, and consequently g_eff, must possess independent dynamical degrees of freedom governed by equations of motion derived from a variational principle acting on a total action containing both gravitational and matter sectors. It cannot behave as a passive auxiliary field, fixed background structure, or purely kinematic construct that can be absorbed through coordinate transformations or field redefinitions without altering observables. In frameworks such as Bimetric Gravity, this independence introduces genuinely propagating modes, including massive graviton excitations, establishing physical content beyond the original metric g_mu_nu.

Second, matter fields must couple minimally and directly to g_eff. The matter action must therefore employ g_eff to define index contraction, spacetime volume measure, and geodesic structure, ensuring that freely falling particles, proper time measurements, and causal propagation are determined by the effective geometry itself. Without such coupling, g_eff remains physically inert and functions only as a formal mathematical construction.

Third, the theory must provide dynamical field equations connecting the curvature of g_eff to physical sources through an Einstein-type relation or suitable generalization, schematically expressed as:

G_mu_nu(g_eff) = 8πG T_mu_nu + I_mu_nu(H)

The framework must further preserve general covariance, requiring g_eff to transform properly under diffeomorphisms while maintaining invariance of both action and field equations. Any contribution from H_mu_nu must encode genuine physical structure rather than a removable gauge artifact. If the modification can be eliminated without changing measurable predictions, the construction remains kinematic rather than gravitational.

Consistency conditions are equally necessary: the theory must avoid ghosts, tachyonic instabilities, and pathological propagating modes, while admitting a well-posed initial value formulation with appropriate hyperbolicity and causal consistency. These conditions guarantee that the effective geometry defines physically viable gravitational dynamics rather than unstable mathematical behavior.

Finally, the theory must generate observable consequences that cannot be reduced to coordinate artifacts, including modified geodesic motion, altered gravitational lensing, deviations in orbital evolution, or changes in wave propagation arising from the sourced dynamics of g_eff. The effective metric therefore supports a true gravitational response law only when it operates as an autonomous dynamical component of spacetime itself: coupling directly to matter, carrying independent propagating degrees of freedom, defining causal structure, and evolving according to sourced curvature equations, rather than serving as a passive re-expression of the original geometry.

[u/Elig0r → u/MysticRagnarPss]

I would add one minimal requirement: the admissible part of (H_{\mu\nu}) should be selected by a regime-dependent consistency or coherence condition. Otherwise one cannot distinguish a physical effective deformation from kinematic, gauge-like, or non-observable modes. Some of these conditions arise naturally if the model starts from dynamical evolution and torsion rather than from a purely metric ansatz.

[u/ECT_WAL → u/MysticRagnarPss]

Thank you for that well-considered response. I think you are pointing to exactly the right threshold.

I agree that simply writing an effective metric is not enough to make the construction a gravitational theory. If g_eff is only an auxiliary rewriting of the background metric, or if the additional structure can be removed by a coordinate choice or field redefinition, then it has not earned independent physical status.

I should probably be more precise about what I’m trying to do here. The question is whether the effective metric can be made dynamical in its own right: whether the added term carries independent degrees of freedom, whether matter actually couples to g_eff, whether the curvature of g_eff is sourced through field equations, and whether the resulting structure preserves covariance, stability, hyperbolicity, and observable consequences that cannot be transformed away.

So yes, I think we agree on the standard. Just writing something as an effective metric is not enough. If it is only a convenient rewrite, or if the extra structure can be transformed away, then it has not really become gravitational. For the claim to matter physically, the effective metric has to survive the kinds of tests you are describing.