Negative energy and large-scale spacetime structure

Question

I was reading an essay from Stephen Hawking's on the Space and Time warps and I was trying to make sense on some statements referring to the Casimir effect such as:

The energy density of empty space far away from the plates, must be zero. Otherwise it would warp space-time, and the universe wouldn't be nearly flat. So the energy density in the region between the plates, must be negative.

Could anyone tell me what's the logic about energy from a far away place from plates being zero? If that was not the case how would the space-time warp? I might lack the knowledge, but I would like to understand the reasoning or have an intuitive explanation about those statements. Thank you very much!

Answer 1

The interaction between the geometry of spacetime (how precisely it is "warped"), and energy, is a fundamental notion in general relativity. Specifically, the Einstein field equations tell us that if there is energy or momentum near some spacetime point, then the geometry nearby will bend (warp, curve, whatever you'd like to call it) in a particular way. The equation itself is $$ \underbrace{G_{\mu\nu}}_{\text{geometric stuff}} = \underbrace{8\pi T_{\mu\nu}-g_{\mu\nu} \Lambda}_{\text{energy-momentum stuff}} $$ On the left hand side of these equations are quantities that tell you what the geometry of spacetime is, and on the right hand side are quantities that tell you about the energy-momentum content of spacetime (including, for example, the energy density that Hawking mentions). In particular, if the energy stuff on the right hand side is nonzero, then generically the equation tells us that there will be nontrivial geometric stuff on the left hand side, aka "warping."

So Hawking is basically alluding to the fact that general relativity tells us that the if there is a non-zero energy density in some region of spacetime, then it will not be flat, but since we know the universe to be essentially flat on large scales, the energy density in regions far from the plates must vanish, or else we would have a contradiction.

Physics Rocks.