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According to Visser M. in the book "Lorentzian Wormholes: from Einstein to Hawking", a realistic model for the stress-energy tensor of the Casimir effect is presented in equation (12.31): $$ T^{\mu \nu} = \sigma \hat{t}^{\mu} \hat{t}^{\nu} [ \delta (z) + \delta(z-a))] + \Theta (z) \Theta (a - z) \frac{\pi^2}{720} \frac{\hbar}{a^4} \left[ \eta^{\mu \nu} - 4 \hat{z}^{\mu} \hat{z}^{\nu} \right],\tag{12.31}$$ Where only the $z$ direction is considered, $\hat{t}$ is the unit vector in the time direction and $a$ is the distance between the plates. Also, $\sigma$ is the mass density of the plates. How can I derive that tensor?

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  • $\begingroup$ The second term, between the two plates, is more commonly written $\frac{\pi^2}{720}\frac{\hbar}{a^4}\text{diag}(-1, 1, 1, -3)$. (for example, see the page numbered 30 in this paper). Do you know how to derive that? $\endgroup$
    – Ghoster
    Jun 17, 2023 at 4:58
  • $\begingroup$ @Ghoster I do know how to derive that. But my question is more focused on the first term $\endgroup$ Jun 17, 2023 at 12:55
  • $\begingroup$ First term is just the energy density of the two plates, which are concentrated at z and z-a, therefore the Delta function terms $\endgroup$
    – KP99
    Jun 17, 2023 at 13:03
  • $\begingroup$ @KP99 I'm sorry but I still don't understand why the delta and $\Theta$ function appears $\endgroup$ Jun 18, 2023 at 3:39

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Delta functions are different from zero only in a point, while Heaviside $H(x)$ step functions are non-zero only at $x>=0$.

So the first term is the positive energy density of the two plates, since the plates are located at z=0 and z=a.

The second term is the negative energy density between the plates, since the product of step function is different from zero only when $0 <= z <= a$.

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