Obviously, sound (like every other causal phenomena) may not travel faster than the speed of light. I know that materials with a high bulk modulus and low density will typically have faster speeds of sound, but is there a theoretical limit due to either a condition relating the density and bulk modulus, or some relativistic condition beneath the propagation of sound?
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2$\begingroup$ the restriction is relativistic, you know in advance that the speed (or information) cannot be larger than the speed in vacuum. $\endgroup$– user83548Commented May 31, 2016 at 23:13
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3$\begingroup$ For an ideal ultrarelativistic gas it is $\frac{c}{\sqrt{3}}$ $\endgroup$– Count IblisCommented Jun 1, 2016 at 0:13
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1$\begingroup$ See also physics.stackexchange.com/q/54684 $\endgroup$– Martin BeckettCommented Jun 1, 2016 at 0:56
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1$\begingroup$ Related: What is the speed of sound in space?, which discusses another example where the speed of sound is comparable to the speed of light (baryonic acoustic oscillations). $\endgroup$– pelaCommented Jun 1, 2016 at 8:12
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2 Answers
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There is no restriction other than $c_s<c$. Relativistic plasmas and fluids explore this regime. A weakly coupled quark gluon plasma has $c_s=c/\sqrt{3}$. Even higher speeds are reached in neutron stars, see Is the speed of sound almost as high as the speed of light in neutron stars? .
The speed of sound is related to the adiabatic compressibility $$ c_s^2 = \left(\frac{\partial P}{\partial \rho}\right)_s \, . $$ This quantity also enters the neutron structure via the TOV equation. It constrains, in particular, the maximum mass and the mass-radius relation. The recent observation of a 2-solar mass neutron star implies that $c_s$ becomes quite large, certainly bigger than $0.5c$.
Adendum: See here for a more quantitative analysis based on the existence of a 2 M(solar) neutron star, and equ.(10) of this paper for a theoretical limit in which we can shows that $c_s\to c$.
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$\begingroup$ No equation from the link you provided indicates that $c_s > c/\sqrt3$. Is there a material where we'd expect $c_s > 0.6c$, or could one reasonably say that $c_s < c/\sqrt3$? $\endgroup$ Commented Jun 1, 2016 at 19:04
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Per the above link, the theoretical limit for the speed of sound in condensed matter is $$ v_{max} = \alpha c \sqrt{\frac{m_e}{2m_p}} \approx \frac{c}{8304} $$
where $\alpha$ is the fine structure constant, $c$ is the speed of light, $m_e$ is the mass of the electron and $m_p$ is the mass of the proton.
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1$\begingroup$ It is so satisfying that 4 years later the answer to this question changed because of new science. $\endgroup$ Commented Oct 15, 2020 at 17:24
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$\begingroup$ It's an amusing paper, but it's not rigorous (just dimensional analysis), and it only applies to solids for which the speed of sound is determined by the bulk modulus, and the scale for the bulk modulus is set by a typical chemical bonding energy. Certainly not relevant to relativistic fluids, such as neutron matter or the quark gluon plasma. $\endgroup$– ThomasCommented Oct 17, 2020 at 17:54
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$\begingroup$ Note that the equation refers to hydrogen; for nuclear matter the relevant mass-ratio would be something involving quark masses, and the fine structure constant replaced by the strong nuclear force counterpart. It is not a general answer to the question, despite the nice paper. $\endgroup$ Commented Oct 17, 2020 at 23:43
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$\begingroup$ Unfortunately, this paper is plain wrong since any ultrarelativistic gas already exceeds this limit by far. The only rigorous limit known to date is that sound (and any signal) must be subliminal. In fact, in neutron star studies it is only required that the speed of sound is less than "c", see e.g. the discussion in section 4.1 of arxiv.org/abs/2410.08008 ("Frozen and equilibrated f and p modes of cold neutron stars: nuclear metamodel predictions", 2024). $\endgroup$– QuilloCommented Oct 22 at 13:42