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General relativity is the current model of gravity which has not yet been disproved. Are there still any anomalies such as the problem of Mercury's orbit during Newtonian gravity period that it failed to explain? If so are there other types of physics to be discovered?

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    $\begingroup$ I guess you could call dark matter and dark energy an anomaly. $\endgroup$ – celtschk Apr 27 '14 at 20:58
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    $\begingroup$ there is a very extensive answer to this question at math stack exchange: math.stackexchange.com/q/50521 $\endgroup$ – DavePhD Apr 27 '14 at 21:02
  • $\begingroup$ Related: physics.stackexchange.com/q/12093/2451 , physics.stackexchange.com/q/6980/2451 and links therein. $\endgroup$ – Qmechanic Apr 27 '14 at 21:07
  • $\begingroup$ Prof. David Tong's introductory notes to string theory feature an excellent summary of some of the issues with GR. In addition, see my answer to: physics.stackexchange.com/q/109924 $\endgroup$ – JamalS May 22 '14 at 10:10
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    $\begingroup$ how is that " problem of Mercury's orbit during Newtonian gravity period that GR fails to explain" ? On the contrary GR explains it and specifies it's causes in great detail, unlike any other theories $\endgroup$ – Mihai B. Feb 10 '17 at 19:19
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People usually focus on so-called strong field tests of GR, but I'm going to go completely the other direction to extremely weak fields, $a=GM/r^2\lesssim 10^{-10}\,{\rm m}\,{\rm s}^{-2}$.

One example that's been around for a few decades is the rotation curves of galaxies. Whereas GR (the fields are weak, so really it's just the Newtonian limit) would predict from the visible matter distribution that the rotation velocity rises steeply then falls off with radius, most galaxies seem to rise then stay approximately flat. One interpretation of this is that there is a lot of unseen 'dark matter', but an alternative is that in the weak field limit gravity isn't well described by GR. One point in favour of a modification to GR is the 'mass discrepancy-acceleration relation'. This relates the deficit of mass required to explain the observed rotation (i.e. the discrepancy w.r.t. the GR prediction) with the observed circular acceleration. These two quantities correlate very tightly, which could be an argument that it's something to do with the force law rather than the matter distribution. But others argue that the same relation can be explained within the dark matter paradigm (disclaimer: I'm an author of that paper). So it's currently an open problem. I think the only thing that nearly everyone agrees on is that there is some sort of new physics waiting to be explained here.

These issues, and some other weak-field anomalies, are discussed at a fairly accessible level in this recent paper. The author does a good job of pointing out the successes and failures of both the dark matter and modified gravity interpretations.

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  • $\begingroup$ Add a bounty and post an answer? That's pretty sly =P. I agree, though, this question has not received enough attention. $\endgroup$ – Emilio Pisanty Feb 8 '17 at 11:02
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    $\begingroup$ @EmilioPisanty Actually I posted the answer then added the bounty ;) But I'd rather give it to someone else who posts a nice answer! My example is a bit of an oddball in that it's weak field, I'd be happy to hear about any strong field anomalies that still exist. $\endgroup$ – Kyle Oman Feb 8 '17 at 17:07
  • $\begingroup$ The paper said disc galaxies in ΛCDM form at the centre of dark matter haloes. That's back to front ;) $\endgroup$ – John Duffield Feb 8 '17 at 17:09
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    $\begingroup$ @JohnDuffield you'd have the halo form in the disc galaxy? I don't think I follow. $\endgroup$ – Kyle Oman Feb 8 '17 at 17:27
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    $\begingroup$ @JohnDuffield I still don't follow, because you're speaking in riddles. Could you lay out plainly the physics you're alluding to? $\endgroup$ – Kyle Oman Feb 9 '17 at 17:24
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Perhaps the most important one is singularities: GR predicts its own downfall. The reason for this: General relativity is incompatible with quantum mechanics, as it doesn't follow the uncertainty principle. When the coordinate systems are very unstable, no reasonably sensible tensor analysis can be done, hence GR's very foundation is challenged. And this is exactly what happens in singularities, where quantum effects can't be ignored. And already, this approach to quantum gravity has shattered many "classical" assumptions; for example, Hawking radiation predicts emission of photons from a black hole, which classically didn't emit anything.

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  • $\begingroup$ -1: this answer is essentially wrong. General relativity is not 'incompatible' with quantum mechanics as it yields a non-renormalizable effective field theory; that it is not non-renormalizable does not mean it is 'incompatible.' Rather, it is simply we seek a UV completion. $\endgroup$ – JamalS Jan 3 '17 at 15:30
  • $\begingroup$ @JamalS you are right about the UV completion, but in case of some information paradoxes, there are situations where implementing both unitarity and equivalence principle does lead to an incompatibility. $\endgroup$ – Bruce Lee Feb 8 '17 at 7:57

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