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2.1. Classical vacuum energy

Let us turn first to the issue of why the vacuum energy is smaller than we might expect. When Einstein proposed general relativity, his field equation was

Equation 1.1 (1.1)

where the left-hand side characterizes the geometry of spacetime and the right-hand side the energy sources; gµnu is the spacetime metric, Rµnu is the Ricci tensor, R is the curvature scalar, and Tµnu is the energy-momentum tensor. (I use conventions in which c = hbar = 1.) If the energy sources are a combination of matter and radiation, there are no solutions to (1.1) describing a static, homogeneous universe. Since astronomers at the time believed the universe was static, Einstein suggested modifying the left-hand side of his equation to obtain

Equation 1.2 (1.2)

where Lambda is a new free parameter, the cosmological constant. This new equation admits a static, homogeneous solution for which Lambda, the matter density, and the spatial curvature are all positive: the "Einstein static universe." The need for such a universe was soon swept away by improved astronomical observations, and the cosmological constant acquired a somewhat compromised reputation.

Later, particle physicists began to contemplate the possibility of an energy density inherent in the vacuum (defined as the state of lowest attainable energy). If the vacuum is to look Lorentz-invariant to a local observer, its energy-momentum tensor must take on the unique form

Equation 1.3 (1.3)

where rhovac is a constant vacuum energy density. Such an energy is associated with an isotropic pressure

Equation 1.4 (1.4)

Comparing this kind of energy-momentum tensor to the appearance of the cosmological constant in (1.2), we find that they are formally equivalent, as can be seen by moving the Lambda gµnu term in (1.2) to the right-hand side and setting

Equation 1.5 (1.5)

This equivalence is the origin of the identification of the cosmological constant with the energy of the vacuum.

From either side of Einstein's equation, the cosmological constant Lambda is a completely free parameter. It has dimensions of [length]-2 (while the energy density rhoLambda has units [energy/volume]), and hence defines a scale, while general relativity is otherwise scale-free. Indeed, from purely classical considerations, we can't even say whether a specific value of Lambda is "large" or "small"; it is simply a constant of nature we should go out and determine through experiment.

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