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I am going to do a state of the art on Modified gravity models. I have found a talk that presents the problematic. In particular, it is said the following things :

Modifying General Relativity

  1. How to modify GR:
  • extra DoF(s): scalar, vector, tensor field(s);
  • going beyond the 2nd order differential equations;
  • diffeomorphism invariance breaking;
  • higher than 4 dimensions;
  1. Solar system constraints
  • screening mechanisms (Chameleon, Symmetron, k-mouflage, Vainshtein)
  1. In the following we will focus on theories with
  • an extra scalar and dynamical DoF;
  • higher order field equations (in spatial derivatives);
  • break diffeomorphism invariance;
  • 4 dimensions.

a) I woud like to know what means "going beyond the 2nd order differential equations" ? Is it related to the Einstein-Hilbert action with the Ricci scalar (this one contains second derivatives of the metric, that is to say, first derivatives on Christofell symbols ?).

b) Which diffeomorphism invariance is breaking ? I don't understand this sentence.

c) Finally, it is suggested "extra degrees of freedom" but theses extra degree of freedom are applied on the matter Lagrangian, or for example with the f(R) models, or also another models (in other words, are they applied only on the geometric component of Einstein-Hilbert's action ... ? )

Any suggestions/remarks to better understand are welcome.

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  • $\begingroup$ Btw, when you say 'I am going to do a state of the art on Modified Gravity', is there a word missing here? Some clarification on the sentences in the opening paragraph would be useful, thanks! $\endgroup$
    – Eletie
    Commented May 21, 2022 at 21:54
  • $\begingroup$ It's usual to give a reference (ot at least a link) to the source material that's being used in your question. In general being able to place the question you ask in proper context in the original material is often helpful to people answering. $\endgroup$
    – StephenG - Help Ukraine
    Commented May 21, 2022 at 23:05
  • $\begingroup$ Just be aware that non-metric alternatives are ruled out by the multi-messenger observation of the neutron star inspiral. $\endgroup$
    – rfl
    Commented May 22, 2022 at 7:22
  • $\begingroup$ @rfl . What do you mean by neutron star inspiral ? $\endgroup$
    – user87745
    Commented May 22, 2022 at 10:39
  • $\begingroup$ physics.aps.org/articles/v10/134 $\endgroup$
    – rfl
    Commented May 22, 2022 at 18:33

1 Answer 1

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  1. Going beyond second order means considering theories that contain terms that contain higher than second order derivatives of the metric in the equations of motions. E.g. a term like $\partial_{\mu} \partial_{\nu} R$, with $R$ is the Ricci scalar, has fourth-order derivatives of the metric $g_{\mu \nu}$
  2. Breaking diffeomorphism invariance means what it says: that the theory will no longer be invariant under diffeomorphisms (or general coordinate transformations). For example, theories with background fields or preferred fields break diffeomorphism invariance. This can be either spontaneous or dynamical, but the main point is that the theory is no longer fully covariant in the sense that coordinates would play some physical role.
  3. The extra degrees of freedom will originate in the gravitational sector, but can be equivalently expressed as something like a scalar field (e.g., scalar-tensor formulation of $f(R)$ gravity). In the vacuum case $S_m [g, \phi] =0$ there will be more than the two propagating degrees of freedom of usual GR. These will come from the modified gravitational action.

For much more detail about all these questions, I'd recommend reviews like the following https://arxiv.org/abs/1106.2476

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  • $\begingroup$ Thanks for your quick answer. a) How to jusitfy the using of derivatives of higher than 2-th order ? b) Is diffeomorphism invariance related to Bianchi's indenities when Einstein tensor covariant derivative equals to stress-energy tensor covariant derivative ? c) when you say "extra degrees of freedom will originate in the gravitational sector", you mean the Ricci scalar part (or f(R) or anyhting else provided it makes part of geometric part (not the matter Lagrangian), right ? . Regards $\endgroup$
    – user87745
    Commented May 21, 2022 at 22:57
  • $\begingroup$ There isn't really any natural justification, it's just that Lovelock's theorem says that you must break at least one of the assumptions (second-order, local, diff invariant, constructed solely from the metric, 4D) to deviate from standard GR. Indeed, higher order derivatives often come with issues relating to stability & Cauchy problems, but not always. $\endgroup$
    – Eletie
    Commented May 22, 2022 at 8:09
  • $\begingroup$ Yes, diff invariance is related to the Bianchi identities, but it's a little subtle and depends on the specific theory whether energy-momentum will be covariantly conserved. For c), yes I mean in the gravitational action $S_{\rm{grav}}$, which will be some modification of $S_{\rm{EH}}$ $\endgroup$
    – Eletie
    Commented May 22, 2022 at 8:11
  • $\begingroup$ when you say " In the vacuum case $S_m [g, \phi] =0$ there will be more than the two propagating degrees of freedom of usual GR. These will come from the modified gravitational action." : could you develop the expression of $S_m [g, \phi]$ which leads to zero. Excuse me, I don't want to be boring. Thanks in advance. Best regards $\endgroup$
    – user87745
    Commented May 23, 2022 at 19:55
  • $\begingroup$ By that I simply mean when working in the vacuum, where the matter action/Lagrangian is set to zero. $\endgroup$
    – Eletie
    Commented May 23, 2022 at 20:01

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