Is there an experiment showing that general relativity and the standard model contradict each other? My friends and I are asking this question about the standard model and general relativity to put the discussion on a firmer footing:
Does some experiment show that the two theories contradict each other?
It is obvious that the two theories really do contradict each other - but only on paper. Their mathematical descriptions are not compatible and contradictory.
In reality, it seems that no experiment contradicts these theories, and no experiment
shows any contradiction between the two theories.
What is the exact status? Is there a discussion of this point somewhere?
ADDED:
(1) The incompleteness of either theory is clear and due to their different domains of application. But that does not show that they contradict each other. In nature, inertial and gravitational mass are equal; so letting something fall does not contradict the standard model. Just let the observer fall near the experiment, as a check.
(2) The mathematical formulations of the two theories do contradict each other, because general relativity is not probabilistic (e.g. the mass-energy tensor) whereas quantum theory is. But it seems that no experiment has observed the contradiction.
(3) Is there a thought experiment showing a contradiction?
(4) Any experiment (real or thought) confirming one theory and contradicting the other would qualify. In the meantime, a number of experts have confirmed to us that as of November 2022, no such experiment is known. Following the most recent reviews, general relativity and the standard model have no confirmed anomalies or exceptions. (Falling apples, quantum superpositions or double slit experiments do not qualify: they do not contradict the other theory.)
 A: From the standpoint of accelerator physics, the energy levels required to look at where gravity begins to muscle in on the standard model are unattainable now and will probably remains so forever (i.e., requiring an accelerator that was light-years in length), so there is no hope of ever performing an experiment that would, for example, sprinkle gravitons into a warm bowl of quark soup and let us taste the result.
Although the graviton/quark soup recipe was on the menu in the early stages of the big bang, there is no telescope that can reach earlier lookback times than the recombination era because the universe was opaque to electromagnetic radiation before then, so that path to understanding is blocked too.
This is why we have no direct experimental result on hand today that proves GR and QM incommensurable.
A: As pointed out in the comments, the Standard Model does not contain gravity. So the Standard Model cannot explain apples falling from trees.
However, there are ways to "glue together" classical gravity and the Standard Model, at least approximately. In modern language, this is the effective field theory of gravity. Essentially, since we expect quantum effects associated with the gravitational field itself to be small below the Planck scale, we can (to a very good approximation) treat the gravitational field as a classical background in which other quantum fields act.
On theoretical grounds, we can estimate the regime of validity of this approximation. Essentially we would only expect it to break down when the curvature scale becomes of order the Planck length; so, for instance, very close to the singularity of a black hole. There are no experiments that have been done, or conceivably could be done on any reasonable time scale, that would fall outside this regime of validity. So, there are no experimental results able to show this approximation breaks down.
There are theoretical possibilities that this approximation might break down in other ways, but these require some luck (they might just not be true) and the experiments have not been kind to these ideas. Some examples are:

*

*We could be living in a world with "large" (millimeter sized) extra dimensions, such that the true Planck length is actually much larger than the one we would naively guess confined to our four dimensional world. If this were true and the extra dimensions were large enough, we could detect deviations from the inverse square law at millimeter scales, or produce black holes in particle accelerators. Both experiments have been tried and found nothing.

*There are theories of quantum gravity where fundamental symmetries like Lorentz invariance are broken, and these broken symmetries could potentially manifest themselves observationally. Again, searches for these effects have been done, and not found.

*There are hints from the information loss paradox that the effective field theory of gravity should break down around the scale of the horizon of a black hole, rather than near the singularity. However, these effects imprint themselves in subtle ways on Hawking radiation, which is observationally very far out of reach. (In that sense this example is not like the other two; there are no searches for Hawking radiation of astrophysical black holes because it is just pointless).

To summarize: there is no direct experimental evidence contradicting the effective field theory of gravity coupled to the Standard Model. However, we also would not expect any experiment done to date (or in the conceivable future) to lie outside the regime of validity of this framework, barring a scenario like large extra dimensions. Nevertheless, the effective field theory of gravity coupled to matter does have a finite regime of validity, so on theoretical grounds we expect that it must be an approximation to something more fundamental.
A: Pretty much every single quantum mechanical experiment contradicts General Relativity. Take the double slit experiment. General Relativity predicts that Newtonian mechanics should hold in the double slit experiment, because General Relativity reduces to Newtonian mechanics in these approxomations.
So, the double slit experiment is one experiment that shows contradiction between the two theories.
Similarly, pretty much every observation in cosmology contradicts the Standard Model, because the Standard Model predicts no such thing as gravitation in its macroscopic limit.
EDIT I want to correct the last paragraph. The Standard Model can include an Effective Field Theory of Gravity, like explained here
This theory does reproduce General Relativity in the macroscopic limit. You could say that this theory accurately models all experiments until around the Planck scale.
