The standard definition of pressure (e.g. in wikipedia) is $p = \frac{F}{A}$, where $F$ is the magnitude of the normal force and $A$ is the area of the surface contact. In GR one sometimes also talks about pressure, e.g., in the perfect fluid model of matter, which is described by the energy-momentum tensor

$$T^{ab} = (\rho + p) u^a u^b + p g^{ab},$$ where $p$ is the pressure.

My questions are:

  1. Can one relate pressure in the sense of the perfect fluid model to the mentioned definition? If so, how to think about $F$? What kind of force is it?

  2. If not, does this model use a more general notion of pressure?

  3. Why are these particular components of the tensor regarded as representing pressure (and not the others)?


The expression $p = F/A$ is the mechanical pressure, and it applies to classical situations involving some material (i.e. matter).

You may also get "pressure" in an electromagnetic field acting on itself, while there isn't any matter at all! The magnetic field lines, for example, are applying a kind pressure on themselves, as if the lines were made of some effective elastic material: \begin{align}\tag{1} p_{\perp} &=\rho, & p_{\parallel} &= -\, \rho, \end{align} where $\rho = B^2 / 2 \mu_0$ is the field's energy density. In this case, $p_{\parallel} < 0$ is a tension along the lines, while $p_{\perp} > 0$ is a pressure between the lines (in the orthogonal plane). Of course, the formula $p = F/A$ doesn't make much sense in this case, since there is no matter. On average, electromagnetic radiation applies a pressure $p = \frac{1}{3} \, \rho > 0$.

You also have the thermodynamical pressure, which is a kind of statistical average of the formula $p = F / A$ (average mechanical pressure acting on the sides of some container): $$\tag{2} p = -\, \frac{\partial E}{\partial V}. $$

  • $\begingroup$ So, is there a general definition of pressure, whose special cases are the mechanical pressure, the pressure in an electromagnetic field, etc.? Or are they both called "pressures" only by analogy? $\endgroup$ – wiktoria Sep 15 '19 at 16:54
  • 1
    $\begingroup$ @wiktoria, they are all pressures because they have the same units (energy density) and are applying a net mechanical force on some boundary, if there is matter. I believe that formula (2) is the most general definition of pressure. $\endgroup$ – Cham Sep 15 '19 at 17:01

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