Negative pressure, tension, and energy conditions

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Solution 1

Pressure is the (outwardly directed) force normal to any area. This definition most naturally fits hydrostatic pressure, e.g. in gases and liquids. In ideal media, this kind of pressure is never negative.

In real media, that is not necessarily true. The most obvious example occurs at the boundary of just about any liquid: There a negative pressure acts on the molecules at the surface. However, nobody uses the phrase "negative pressure" for it. The common way to call it is surface tension. Every other occurrence of negative pressures, created by attractive rather than repulsive forces in a medium, are treated likewise: They are tensions.

The example you gave, negative "pressure" in a solid is such an example: Engineers quantify the maximum of it that a material can take as ultimate tensile strength. However, pressure does not really describe the situation for solids very well, because forces acting at a surface need not necessarily be normal to that surface. A better concept than the (scalar) pressure is the stress tensor that can capture this force's direction and its variation depending on the orientation of the surface it acts on.

Solution 2

You could probably get a negative pressure in polymer physics, so you could view a big block of rubber as behaving this way.

Basically: negative pressures happen when an increase in volume causes a decrease in entropy. Polymers might be a good example because you have these molecules which "want" to be tangled up and kinked ("want" in the sense of "it is entropically favorable for..."). When you increase the volume of such a system by stretching it, it generally decreases the entropy, so you are opposing an entropic force which wants the system to return back to its "resting" size.

Solution 3

The Mie–Gruneisen equation of state for solids http://en.wikipedia.org/wiki/Mie%E2%80%93Gruneisen_equation_of_state is a model that combines the thermal pressure components and "cold" components of the pressure where the latter is derived thermodynamically from a model intermolecular potential. It has the form $p = p_T(\rho_0,T) + p_c(\rho_0,\chi)$, where $T$ is the temperature, $\chi=1-\rho_0/\rho$, $\rho_0$ is the initial density, and $\chi$ can be negative which corresponds to rarefaction.

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Updated on August 01, 2022

Comments

  • Timaeus
    Timaeus over 1 year

    We have lots of common everyday experience with positive pressure, the canonical example is a gas.

    But other examples of positive pressure are easy to imagine: for instance, a solid that gets compressed to be more compacted than its equilibrium density.

    To me it is straightforward that if a solid is instead pulled apart slightly so that it is still connected but at a lower density than its equilibrium density, that it can have a tension that is a negative pressure.

    But sometimes people object to negative pressure, so I think we could benefit for a comprehensive answer, that includes good definitions, justifications about why the definitions are good, and even includes comparisons to energy conditions (weak energy condition, strong energy condition, dominant energy condition, etcetera).

    An answer does not need to address cosmological constants or dark energy specifically, but I would like the answer to be comprehensive enough that people with questions about those issues can satisfy all their questions about negative pressure itself.

    What is negative pressure in general? How do we know that is the proper and fully general definition? Is it reasonable in light of known and acceptable physics? How/why do we know that? How, if at all, does it relate to tension? If different than tension, what is tension in general? How do we know that is the proper and fully general definition? Is it reasonable in light of known and acceptable physics? How does negative pressure relate to the classical energy conditions? Are any deviations or clashes with classical energy conditions justifiable or acceptable?

    • Jon Custer
      Jon Custer over 8 years
      I don't see a question - what, exactly, are you asking?
    • Timaeus
      Timaeus over 8 years
      @JonCuster Edited to put in question sentences.
    • Steeven
      Steeven over 8 years
      If positive pressure is easy to understand, then why is it difficult simply to think of negative pressure as pressure in the opposite direction? I mean, pressure has a direction - simply flipping it the other way will be the same as putting a minus in front of the magnitude.
    • HolgerFiedler
      HolgerFiedler over 8 years
      A tension can't be negative, it can be less then 1 but it can't be less than zero. Try to take the air out of a bottle. Let be the normal air pressure 1unit, you never will be able to reach less then zero. A negative pressure makes no sense
    • Void
      Void over 8 years
      This is something I am also pondering about at the moment, I guess that you are motivated by, to cite Einstein, the left-hand side of Einstein equations being of fine marble and the right-hand side of low-grade wood. BTW it is quite normal to have negative electromagnetic pressure, at least in the sense of the eigenvalues of the electromagnetic stress-energy tensor.
    • jabirali
      jabirali over 8 years
      @Steeven: I think it is easy to imagine negative relative pressure, as in less than the pressure of the atmosphere; but it is harder to think about negative absolute pressure, in the sense less pressure than vacuum, i.e. dark energy...
    • Timaeus
      Timaeus over 8 years
      @jabirali I actually think Steeven is correct that it is personally very easy to understand for instance a tension as pressure in the opposite direction. But I know people sometimes object to negative pressure and I want to address any issues that might arise by comparing it to accepted physics. I personally don't see the slightest issue, but a definition I make might not be general. For instance gravitationally bound objects act like they are under tension in that they resist being expanded, and a solid at equilibrium density resists changes in either direction.
    • Admin
      Admin over 8 years
      Is there anything wrong with taking negative pressure in a direction to be when the momentum flux along that direction has the opposite sign from the signed area orthogonal to that direction? (I guess this is a verbose way of saying "tension.") That is, define pressure as the spatial trace of the stress-energy tensor, as I did here and you yourself did here.
  • Timaeus
    Timaeus over 8 years
    There are already existing questions about negative pressure that confine themselves entirely to a thermodynamics context. I'm not sure that is general enough. For instance dark energy may or may not have an entropy. Also, for a continuous field that is already spread out over all space, there isn't a clear change in volume, even an electromagnetic field can have a pressure, but it is spread out so seeing a change in volume is hard. So I'm not sure thermodynamics is general enough.