# Can there be negative energy within the classical general relativity?

The positive energy theorem (Witten and Yau were awarded the Fields medal in mathematics in part for their work on this topic.) says that "the gravitational energy of an isolated system is nonnegative, and can only be zero when the system has no gravitating objects". I know that this is a classical result and in quantum field theory "virtual" particles (or more precisely particles that are off-shell) can have negative energy.

From this, it seems obvious that since the universe is isolated (neglecting fancy improbable ideas like wormholes connecting different universes) the gravitational energy of the universe is nonnegative.

But there are some hypotheses like the Zero-energy universe (on which papers were published decades after the positive energy theorem came) which clearly state that the gravitational energy is negative. Intuitively, I also feel like it should be negative. Even here it is stated to be negative.

I know that in General relativity the gravitational energy is ambiguous and there are several ways to incorporate it like the Landau–Lifshitz pseudotensor. But is it so ambiguous that even the sign is not fixed? Or am I misunderstanding?

• I also would like to know for what definition of energy the positive energy theorem holds. Commented Apr 7, 2022 at 18:50

"Gravitational energy is negative" is a statement from Newtonian gravity. It is true there when the preferred definition of the gravitational energy is assumed, which is zero when bodies are infinitely far from each other and decreases as function $$-\frac{1}{r}$$ when their separation $$r$$ gets smaller.

This preferred definition is not the only one. Can can use any of infinitely many definitions where energy comes out shifted by some amount $$C$$. For a given gravitationally interacting system, one can always change the definition in such a way that gravitational energy turns out positive.

In other words, sign of gravitational energy in Newtonian theory is arbitrary and does not matter.

In GR there is no single definition of gravitational energy either, but for a different reason; it is hard/impossible to find a formula that would always have the same properties as gravitational energy in classical theory. GR is just too different kind of theory.

There is a kind of widespread and quiet belief that value of energy in GR means something and cannot be shifted by $$C$$ willy-nilly. That is true for local energy carried by a massive body, where we require $$E=mc^2$$ and $$m$$ is real. It is not necessarily true for gravitational energy, which does not have a unique definition.

Maybe for ADM energy it can be proven that under some conditions, it is always positive. But that is one definition, there are others, such as the Landau-Lifshitz pseudotensor-defined energy, or the Einstein pseudotensor-defined energy, which are different ideas. It would be very surprising if these different formulae always gave the same number, or even the same sign.

• "There is a widespread ... cannot be shifted by C willy-nilly." I also thought it was true for gravitational energy also. Commented Apr 7, 2022 at 18:48
• Can you also explain for what definition of energy the positive energy theorem holds? Commented Apr 7, 2022 at 18:50
• Wikipedia and the original paper doi.org/10.1007/BF01208285 says the theorem is for ADM mass. Commented Apr 7, 2022 at 21:09
• Yeah and wiki also states that it was later extended to Bondi mass also. Commented Apr 8, 2022 at 3:44