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A generalization of Newton's law of gravitation to a Lorentz invariant version by using an analogy with electromagnetism is possible. Several slightly different ways to do it are described in wikipedia. In addition to predicting the wrong amount in the precession of Mercury's perihelion, it also fails to predict the curvature of light by a large gravitational mass. I assume somebody could have tried to introduce some additional equation or terms to the current ones that describe the interactions between the electromagnetic and gravitomagnetic fields. But I could not find any reference to attempts in this sense. Is there any obvious reason about why this is not even worth being tried?

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Gravitoelectromagnetism (GEM) is of some use for linearized gravity, i.e., a very weak field and slow speeds. That's because linearized gravity leads to similar equations as EM, and you can manipulate some correspondences. But really it is not of great value, linearized gravity has its own methodology and adds non-linear terms that can also be tested. That's the so called parametrized post Newtonian formalism, and is used in for instance some of the gravitational wave modeling and analysis. A 2010 summary of it for linearized gravity is at http://aforrester.bol.ucla.edu/educate/Articles/Derive_GravitoEM.pdf, but as you can see from googling GEM (spelled out) there is little new work on it.

Some alternatives to GR continue to be possible, but in almost all cases they've either been disproven as more experimental data is obtained, or they Have been theories with free parameters that at this point may be adjusted to match the same observations GR predictions match, but which have had no new predictions observed.

An example is f(R) gravity, where R is replaced by an arbitrary function of R, in essence introducing an infinite number of free parameters. Similarly antes sense and K-essence introduce more free parameters to try to explain the cosmological constant, and they can adjust them to match the fact measurements so far. One problem with other gravitational theories is that they need to match both the near earth and earth based experiments/observations, which apply to weak fields and small distances, and observations about stronger fields and longer distances such as in cosmology. So they do some contortions.

So far no strong competitor, but those remain as possible changes to GR if we find some anomalies.

The Pioneer anomaly had the Pioneer staellite being just a bit slower than all the gravitational effects predicted. But it has been explained and accepted since 2012 as having been due to recoil from thermal radiation emitted non-isotropically y from the satellite, and more in the direction to slow it down. See https://en.m.wikipedia.org/wiki/Pioneer_anomaly.

See also wiki, with all the alternatives considered for that anomaly, at https://en.m.wikipedia.org/wiki/Pioneer_anomaly.

Of course, continuing looks at any similar anomalies in other spacecraft continue here and there, but not as a 'mainstream' research activity.

A very important and serious set of results has been coming from cosmological measurements, and just started recently from LIGO (gravitational radiation). The latter, in its first detection of a gravitational wave also analyzed results to test numerical GR predictions, and they all came through well - for instance it confirmed that what was observed was consistent with the No Hair Theorem for Black Holes, a results by Hawking and others from GR. Black Holes will remain as mysteries as long as we don't have a good and accepted theory of quantum gravity to understand what really happens on the horizons and beyond.

Cosmological results have confirmed various GR effects that had not been verified before, such as the integrated Wolfe Sachs effect.

The main concern with GR and cosmological observations remain the origin of dark matter and dark energy. Both are also issues for the standard Model since it would require some new particles (for dark matter), and some undefined physics as the cause of dark energy, thought to be the vacuum energy.

See wiki's https://en.m.wikipedia.org/wiki/Alternatives_to_general_relativity for other alternatives it's the same rference as @Anna V's first reference. The article also talks about quantum gravity which are not really alternatives, but simply a quantum gravity that most of the time people try to reduce to GR in the low energy limit. We still don't have a good quantum gravity theory that has predicted anything new that was then observed.

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  • $\begingroup$ Thanks a lot. The only question that remains and could not find either in the article you link nor after searching an awful lot, is if there are any observational tests showing that the predictions of linearized gravity (the one that is a weak field approximation of GR) fail. What I mean is, I understand that they do not predict the same as full GR because they are a linear, weak field, approximation, but it is still not clear to me in what tests or observations this linear approximation breaks. $\endgroup$
    – user126422
    May 7 '17 at 19:01
  • $\begingroup$ They do not fail. They just don't apply and you have to take more terms, more and more nonlinear. The PPN approach is one way, it parametrizes the nonlinear terms. It's like any approximation, you have to take more terms into account when their impact would be measurable or relevant. In very strong gravity, like the latter stages of black hole mergers, linearization does not work, too many nonlinear terms and you have to take the full field equations which only can be handled numerically $\endgroup$
    – Bob Bee
    May 8 '17 at 17:19
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The introduction to this article in wikipedia explains why alternatives to general relativity fell by the way of mainstream research:

After general relativity (GR), attempts were made either to improve on theories developed before GR, or to improve GR itself. Many different strategies were attempted, for example the addition of spin to GR, combining a GR-like metric with a space-time that is static with respect to the expansion of the universe, getting extra freedom by adding another parameter. At least one theory was motivated by the desire to develop an alternative to GR that is completely free from singularities.

Experimental tests improved along with the theories. Many of the different strategies that were developed soon after GR were abandoned, and there was a push to develop more general forms of the theories that survived, so that a theory would be ready the moment any test showed a disagreement with GR.

By the 1980s, the increasing accuracy of experimental tests had all led to confirmation of GR, no competitors were left except for those that included GR as a special case. Further, shortly after that, theorists switched to string theory which was starting to look promising, but has since lost popularity. In the mid-1980s a few experiments were suggesting that gravity was being modified by the addition of a fifth force (or, in one case, of a fifth, sixth and seventh force) acting on the scale of meters. Subsequent experiments eliminated these.

Motivations for the more recent alternative theories are almost all cosmological, associated with or replacing such constructs as "inflation", "dark matter" and "dark energy". Investigation of the Pioneer anomaly has caused renewed public interest in alternatives to General Relativity.

There are researchers who were working on similar directions after 1980.It certainly is not a mainstream research path .

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