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Good evening everyone, I am new in the field of General Relativity and I have been reading and learning about the subject in recent months.

For example, I read several articles about experiments designed to test the Principle of Equivalence and other aspects of this theory.

So I asked myself:

Are there aspects of General Relativity that have yet to be tested? Are there any experiments or projects under development? How can I contribute as an engineering student?

My questions are not innocent. This year I have to work on a project of initiation to research in my engineering school and I wondered in what type of project I could start, and above all if it is wise to get into such a field at my level.

I am very curious and wish to deepen my knowledge in the field. Last year I studied the Three-body problem from a newtonian point of view and in particular the calculation of Lagrange points and the study of their stability in the Sun-Earth orbital plane. Would there be one aspect of this problem that needs to be explored from a relativistic point of view as much in mathematics, physics and computer science?

I hope my English is sufficiently well written and understandable.

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    $\begingroup$ Surprisingly little GR research has been done to calculate (analytically or numerically) its predictions in systems that are not spherically symmetric, except for a fairly modest number of specific examples, and doing those calculations would be a worthy contribution. But, finding a way to actually determine if observation confirms these predictions would be challenging, unless you could find some relevant systems that have been observed in astronomy with great precision. You couldn't build an experimental test of this with a grad student project's budget. (Also, your English is great.) $\endgroup$ – ohwilleke Aug 13 '18 at 23:04
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    $\begingroup$ I thought the [big-list] tag had pretty much vanished... (also I'm rather skeptical that it should be on this question, or any question that isn't too broad for this site) $\endgroup$ – David Z Aug 14 '18 at 0:19
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    $\begingroup$ Here is one current project for the biggest ever test of GR to determine the size of a black hole of an already known mass: eventhorizontelescope.org $\endgroup$ – safesphere Aug 14 '18 at 0:46
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    $\begingroup$ @ohwilleke: Surprisingly little GR research has been done to calculate (analytically or numerically) its predictions in systems that are not spherically symmetric, This doesn't sound accurate at all to me. Kerr (spinning) black holes are not spherically symmetric, and we have plenty of calculations of their properties, as well as observations of objects that act like them. Black hole mergers are not spherically symmetric. The gravitational waves emitted by such systems are not spherically symmetric. Solar system tests do not rely on spherical symmetry. The list goes on and on. $\endgroup$ – Ben Crowell Aug 14 '18 at 0:48
  • $\begingroup$ @BenCrowell I would agree that there are some special cases of that are not spherically symmetric including those you identify. But, I would disagree that the "list goes on and on", there is a huge wealth of asymmetric cases that are not well investigated. Solar system tests are largely based on two-point calculations (Sun and a rotating planet ignoring other masses, which may not necessarily be wrong in the Solar system where other masses are relatively small). $\endgroup$ – ohwilleke Aug 14 '18 at 1:02
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A good general resource on this kind of thing is the review article by Will, The Confrontation between General Relativity and Experiment, https://arxiv.org/abs/1403.7377 . He updates it every few years. The most recent version predates the direct detection of gravitational waves (although they were already verified observationally based on systems like the Hulse-Taylor pulsar [PSR B1913+16]).

If objects like Sag A* are correctly modeled as Kerr (i.e., spinning) black holes, then we have not yet observed whether they have event horizons, as predicted by GR. As safesphere noted in a comment, this may happen in the fairly near future.

GR incorporates the equivalence principle, and therefore it predicts that nothing special happens at an event horizon in terms of the local properties of space -- no firewalls or anything else crazy. This has not been tested.

There is quite a bit of work by theorists like Joshi suggesting that astrophysical collapse might not actually lead to a black hole but rather to some other object such as a timelike singularity. That is, the cosmic censorship conjecture is looking weaker and weaker. This has not been tested.

If it turns out that people like Joshi are right in their suspicions, then we could potentially observe singularities (because they wouldn't be hidden behind event horizons). This would allow us to test a prediction of GR, which is that singularities should be a generic thing that happens for most initial conditions. If we don't ever get such an opportunity, then the only singularity we'll ever have a chance to observe is the big bang.

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  • $\begingroup$ Which work of Joshi do you mean? I was under the impression that he considers very non generic and non stable cases, so they will have no relevance to any observation. $\endgroup$ – MBN Aug 14 '18 at 8:43
  • $\begingroup$ @MBN: They're now making some pretty strong claims about it being a stable result: arxiv.org/abs/1405.1146 $\endgroup$ – Ben Crowell Aug 14 '18 at 15:04
  • $\begingroup$ @BenCrowell : But they assume spherical symmetry and dust i.e. no pressure. They suggest that pressure can be allowed, I haven't looked at the references, but the stability and genericity they mention is only within the class of spherically symmetric space-times. It is interesting but far from realistic. $\endgroup$ – MBN Aug 14 '18 at 16:28
  • $\begingroup$ @BenCrowell Thank you for your detailed answer. I looked at the document written by Clifford M. Will that you shared and it seems to be a wealth of information ! As ohwilleke pointed out, it seems to me that these experiments require substantial financial means that are not allocated to students. I see you teach physics at college, I am 21 so what would you advise one of your students in my situation to work on ? For example, I said in the previous question that I studied the Three-body problem from a newtonian point of view, is there any work to be done on a relativistic point of view ? $\endgroup$ – Loïc Poncin Aug 14 '18 at 22:25
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    $\begingroup$ Loïc, I recommend Poisson and Will's textbook 'Gravity'. The authors have a more pragmatic approach to general relativity theory, which may be relevant to you since you are interested in experimental tests. You can also find online lectures by both authors on "post-Newtonian theory"; they are both excellent teachers. $\endgroup$ – Colin MacLaurin Aug 15 '18 at 7:47
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The Lense–Thirring precession, often referred to as "frame-dragging", has not been convincingly tested. The Gravity Probe B satellite experiment was supposed to do that but it was plagued by unexpectedly large torques acting on the gyroscopes, caused by the interaction between imperfections on the surface of the rotors and their housing. The Gravity Probe B team eventually managed to model that away and announce the detection of frame-dragging with a precision of 20%. But it seems to me that the dominant opinion in the scientific community is that a repeat of this experiment is needed.

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