# Realistic model for the stress-energy tensor for the Casimir effect

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?

• 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? Jun 17, 2023 at 4:58
• @Ghoster I do know how to derive that. But my question is more focused on the first term Jun 17, 2023 at 12:55
• First term is just the energy density of the two plates, which are concentrated at z and z-a, therefore the Delta function terms
– KP99
Jun 17, 2023 at 13:03
• @KP99 I'm sorry but I still don't understand why the delta and $\Theta$ function appears Jun 18, 2023 at 3:39

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 $$0 <= z <= a$$.