Zero-point energy stops space shrinking?

[johne1618]

According to John Baez,

Einstein's Field equations can be written in the following form:

V'' / V = - 1/2 (rho + P_x + P_y + P_z)

(http://math.ucr.edu/home/baez/einstein/node3.html

where

V is the volume of a small region of space,
rho is the energy density
P_x, P_y, P_z is the pressure in the x,y,z directions of space.
The second derivative is in terms of the proper time and the equation is valid at t=0.
and the units are such that 8 Pi G = 1 and c = 1.

Now the equation of state of isotropic electromagnetic radiation (a photon gas) is:

P = rho / 3

Consider a small region of space. If no zero-point energy is flowing out of the region then the zero point modes inside that region must be zero on the surface of the region. Thus only modes with wavelengths that fit inside the region are allowed. This implies that there is more zero point energy at points just outside the region than at points inside the region. Thus there is an inward electromagnetic pressure on the region - this is the Casimir effect.

Thus for zero-point electromagnetic energy the equation of state is:

P = - rho / 3

If you plug this into Einstein's field equations you find:

V'' / V = - 1/2 (rho - rho/3 - rho/3 - rho/3) = 0

Thus the zero-point energy has just the right qualities to stop space beginning to shrink!

Of course you can get the same effect if rho = 0 and P = 0. But this trivial solution is not consistent with quantum field theory's prediction of the existence of zero-point energy.

 

[Chalnoth]

Consider a small region of space. If no zero-point energy is flowing out of the region then the zero point modes inside that region must be zero on the surface of the region. Thus only modes with wavelengths that fit inside the region are allowed. This implies that there is more zero point energy at points just outside the region than at points inside the region. Thus there is an inward electromagnetic pressure on the region - this is the Casimir effect.

Since your region of space is only a hypothetical construct, you can't impose arbitrary boundary conditions upon it. In this case, you can only claim that the net flow of energy is zero, which doesn't place any limits on the wavelengths allowed.

Instead, the zero point energy of a vacuum must be Lorentz invariant, and the only way for that to be the case is if $p = -\rho$.

 

[Chronos]

That is a great answer, chalnoth.

 

[johne1618]

Instead, the zero point energy of a vacuum must be Lorentz invariant, and the only way for that to be the case is if $p = -\rho$.

Why must the zero point energy of a vacuum be Lorentz invariant?

 

[Chalnoth]

Why must the zero point energy of a vacuum be Lorentz invariant?

The energy of the vacuum is set by the laws of physics, not by the matter content of the universe. And the laws of physics are Lorentz covariant.