# Existence of any vacuum pressure

We know that there exists an underlying background energy in space throughout the entire Universe, called vacuum energy and this is a special case of zero-point energy that relates to the quantum vacuum. Wikipedia is saying about its current measured value:

Using the upper limit of the cosmological constant, the vacuum energy of free space has been estimated to be $$10^{−9}$$ Joules ($$10^{-2}$$ ergs), or $$\sim5$$ GeV per cubic meter.$$^{[3]}$$

I also know that there is some discrepancy about the measured value of this vacuum energy density and the current value of cosmological constant, known as Cosmological Constant Problem:

Depending on the Planck energy cutoff and other factors, the quantum vacuum energy contribution to the effective cosmological constant is calculated to be as little as $$50$$ and as much as $$120$$ orders of magnitude greater than observed,...

QUESTION:

• Associated with this energy density, does there exist any pressure (that may be thermodynamic or mechanical or whatever) in the vacuum [which may be termed as "(quantum) vacuum pressure"]?
• If yes, then what will be its expression for a general case?

My Attempt(s): In this paper on "Quantum vacuum pressure on a conducting slab", the author has found that the vacuum pressure on each surface of a conducting slab (in connection to the Casimir experiments) is, $$P = \dfrac{\hbar \omega_p^4}{24 \pi^2 c^3}$$ , where $$\omega_p = \sqrt{\dfrac{N_e e^2}{m_e \epsilon_0}}$$.
But this expression is not for a general case. I'm seeking for a comparatively general one. Is there anything so?

Based on Sakharov’s idea of a ‘metrical elasticity’ of space, i.e., of the emergence of a generalized force, preventing distortion of space, pressure of the vacuum is detected as per the geometry of the space around the local gravity system. The gravitational defect of mass is interpreted as the transfer of energy to the vacuum, which becomes apparent from its deformation. The gravitational impact of matter with dencity $$\rho$$ on the vacuum $$p=(1/3)c^2\rho$$ and opposite in the sign pressure of it $$p_v=-(1/3)c^2\rho$$ is determined in case of weakly gravitating static centrally symmetric distribution. It is assumed that it is the vacuum pressure $$p_v$$ that is the source of gravity. It is included in the energy-momentum tensor, which corresponds to the solution of the Einstein equations for a spherical static source of gravity with a constant density, described by the metric $$d{{s}^{2}}=\left( 1-\frac{8\pi G}{3c^2}\rho a^2 \right)d{{t}^{2}}-\frac{d{{r}^{2}}}{1-\frac{8\pi G}{3c^2}\rho r^2}-{{r}^{2}}(d{{\theta }^{2}}+{{\sin }^{2}}\theta \,d{{\phi }^{2}})$$ with radius of the sphere $$a$$.