Curious thought i just had. The speed of sound is affected by a few factors, but the density plays a large role. As density increases, does this mean that sound could approach the speed of light? If so, does this mean sound exists in a black hole?
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2$\begingroup$ The "density" of a black hole decreases with mass (r goes proportional with m, not with r^1/3). So, no, it doesn't work that way. $\endgroup$– CuriousOneCommented Apr 26, 2015 at 22:42
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1$\begingroup$ I agree with CuriousOne, but at the same time, a black hole is more of a boundary around dense matter than matter itself. A black hole grows/forms around empty space, so if we ask specifically about the material inside a black hole, assuming it's not a point singularity, which it might be, that makes an answer to this question difficult but near light speed sound-waves could be possible. We're, ofcourse, not sure what matter inside a black hole is like. $\endgroup$– userLTKCommented Apr 26, 2015 at 22:52
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1$\begingroup$ It's also worth pointing out that infinitely dense suggests 0 volume. A wave doesn't make any sense across zero volume or distance. Infinitely dense could also mean infinite mass, in which case a wave would need infinite energy - and that also doesn't make any real sense. The question only works (I think) with super-ultra-high density but not infinite density. $\endgroup$– userLTKCommented Apr 26, 2015 at 22:57
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1$\begingroup$ @userLTK sure it makes sense! Infinite density means that the distance between any two particles of matter is zero, so pressure waves would propagate at infinite speed! Or, wait... $\endgroup$– AsherCommented Apr 26, 2015 at 23:29
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1$\begingroup$ I suppose I should have stated the question as, "as density approaches infinity, does the velocity of a sound wave approach the speed of light". Thanks all for the input. It was a curious thought. $\endgroup$– user38537Commented Apr 28, 2015 at 0:01
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1 Answer
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It is commonly believed that the speed of sound at high densities is bounded from above by $c/\sqrt{3}$, where $c$ is the speed of light. Calculations of this quantity in many theories, ranging from QCD to systems with scale invariance, have all shown it to either stay below or exactly saturate the bound. See the introduction of this paper for a recent overview of several models that were shown to behave this way. There is no proof for this conjecture, but if it holds true, then the answer to your question is that the speed of sound does not approach the speed of light, even at high densities.
Regarding black holes: the fact that in a classical description all mass is concentrated at a singularity is a sign that the theory breaks down at that point. In the context of quantum gravity, this problem should no longer arise.
answered Apr 27, 2015 at 0:20
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$\begingroup$ Might be relevant: physics.stackexchange.com/questions/259051/… $\endgroup$ Commented Dec 19, 2016 at 18:35
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$\begingroup$ +1. Minor remark: The upper limit is $c$ for interacting matter ($c/ \sqrt{3}$ is for perfect ultrarelativistic gases): the upper bound for the speed of sound in nuclear matter is $c$. Moreover, all possible sounds (i.e., all possible ways of propagating a signal) must be subliminal. See e.g. the discussion in section 4.1 of "Frozen and equilibrated f and p modes of cold neutron stars: nuclear metamodel predictions", doi.org/10.48550/arXiv.2410.08008 and references therein (especially Sec. V and App. A of doi.org/10.48550/arXiv.2204.11809 for a general rigorous discussion). $\endgroup$– QuilloCommented Oct 22 at 13:55