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We say that the Standard Model of particle physics has been tested and validated up to energies of 10-13TeV. Similarly, have we been able to test the validity of General Relativity to very intense gravitational fields, and if so, can we quantify for how strong gravitational fields have we managed to do so? I am not asking about the quantization of GR.

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Quantifying how strong a gravitational field is, is notoriously tricky. The two main quantities that people use to qualify the strength of the gravitational field probed in an expertiment are:

  1. The depth of gravitational potential $\Phi$ (you can think of this as the gravitational binding energy of the system). If $M$ is the typical mass scale involved in the experiment and $L$ the typical length scale, then $\Phi$ scales as $M/L$.
  2. The curvature length scale $R$, you can roughly think of this as the size of the sphere that would have a similar curvature locally. This scales roughly as $\sqrt{M/L^3}$.

Experimental/observational tests of general relativity can put on a diagram plotting these two quantities against each other, see for example (note that this diagram includes both past tests, and projections for future tests using LISA):

Image from: B.S. Sathyaprakash et al., "Extreme Gravity and Fundamental Physics", 1903.09221.

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We see that the strongest gravitational fields (in both metrics) are currently probed by observations of gravitational waves from the merger of stellar mass black holes by ground-based gravitational wave observatories (like LIGO, Virgo, and KAGRA).

Observations of the event horizon by the event horizon telescope (EHT) probe similar potential depths $\Phi$, but because the imaged supermassive black hole is so much larger, the curvature scale $R$ is actually much smaller.

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