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GRAVITY (shown below) is a interferometric combiner of near infrared light from four very large telescopes called The Very Large Telescope in order to make careful astrometric measurements near the dusty galactic center where very exciting things happen near what we assume to be a supermassive black hole.

There have been publications of verification of general relativity by watching the star S2 pass through its periapsis around whatever the supermassive black hole-sized object is that it orbits. This was a major milestone in GRAVITY's contribution to science.

It may still be used for other things, but primarily...

Question: Was GRAVITY built to look at one star? Was the observation of S2 sufficient to justify this ESA effort? Or was it always also expected and required to do many more things?

From eso1825 — Science Release: First Successful Test of Einstein’s General Relativity Near Supermassive Black Hole; Culmination of 26 years of ESO observations of the heart of the Milky Way:

New infrared observations from the exquisitely sensitive GRAVITY [1], SINFONI and NACO instruments on ESO’s Very Large Telescope (VLT) have now allowed astronomers to follow one of these stars, called S2, as it passed very close to the black hole during May 2018. At the closest point this star was at a distance of less than 20 billion kilometres from the black hole and moving at a speed in excess of 25 million kilometres per hour — almost three percent of the speed of light [2].

The team compared the position and velocity measurements from GRAVITY and SINFONI respectively, along with previous observations of S2 using other instruments, with the predictions of Newtonian gravity, general relativity and other theories of gravity. The new results are inconsistent with Newtonian predictions and in excellent agreement with the predictions of general relativity.

[1] GRAVITY was developed by a collaboration consisting of the Max Planck Institute for Extraterrestrial Physics (Germany), LESIA of Paris Observatory–PSL / CNRS / Sorbonne Université / Univ. Paris Diderot and IPAG of Université Grenoble Alpes / CNRS (France), the Max Planck Institute for Astronomy (Germany), the University of Cologne (Germany), the CENTRA–Centro de Astrofisica e Gravitação (Portugal) and ESO.

[2] S2 orbits the black hole every 16 years in a highly eccentric orbit that brings it within twenty billion kilometres — 120 times the distance from Earth to the Sun, or about four times the distance from the Sun to Neptune — at its closest approach to the black hole. This distance corresponds to about 1500 times the Schwarzschild radius of the black hole itself.

GRAVITY is a second generation instrument for the VLT Interferometer and will allows the measurement of the positions and motions of astronomical objects on scales far smaller than (was) currently possible.

Source

A new instrument called GRAVITY has been shipped to Chile and successfully assembled and tested at the Paranal Observatory. GRAVITY is a second generation instrument for the VLT Interferometer and will allow(s) the measurement of the positions and motions of astronomical objects on scales far smaller than is was currently previously possible. The picture shows the instrument under test at the Paranal Observatory in July 2015.

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    $\begingroup$ If you have a good device that can look at stars, you can always use it to look at more stars. (Sumbit your research proposals! Limited time only!) The one scientific instrument to look (mainly) at one thing that comes to mind is the LHC: Namely, look at the Higgs Boson; and even then it's being repurposed for all matter of things, in particular: Look at lead ion collisions and maybe find events not described by the Standard Model but there has been no success there so far. $\endgroup$ – David Tonhofer Apr 4 at 11:39
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This link gives you all the papers that have cited the instrument description paper.

The list of papers shows that it has been used for studying: the centres of AGN, close binary systems, discs around young stars, the atmospheres of AGB stars and interferometric imaging of exoplanets at least.

Here is a paragraph from the instrument description paper

Inspired by the potential of phase-referenced interferometry to zoom in on the black hole in the Galactic center and to probe its physics down to the event horizon (Paumard et al. 2008), we proposed in 2005 a new instrument named GRAVITY as one of the second generation VLTI instruments (Eisenhauer et al. 2008). At its target accuracy and sensitivity, GRAVITY will also map with spectro-differential astrometry the broad line regions of active galactic nuclei (AGN), image circumstellar disks in young stellar objects and see their jets evolve in real time, and detect and characterize exo-planets especially around low mass stars and binaries – in short we will “Observe the Universe in motion” (Eisenhauer et al. 2011).

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    $\begingroup$ Could you summarize the conclusion to answer the specific question? $\endgroup$ – Mast Apr 3 at 20:48
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    $\begingroup$ @Mast The last bit acts as sort of a punch-line: "...in short we will “Observe the Universe in motion” (Eisenhauer et al. 2011) which was written after the proposal in 2005 but before first light in 2015 but I think it's pretty clear that "one star" doesn't fit at all with what's shown so we can see that the answer is "No, it wasn't." Is there anything i the answer that could be seen as saying "Yes, GRAVITY was built to look at one star?" $\endgroup$ – uhoh Apr 3 at 22:19

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