As recently mentioned in the news, before his passing, John Nash worked on general relativity. According to the linked article John Nash's work is available online from his webpage.

His work is summarized in this pdf of his lecture notes. One can see that he replaced Einstein's field equations in the vacuum with $$ \Box \, G^{ab} + G^{ps}\left(2R_{p}{}^a{}_s{}^{b} - \frac12 g^{ab}R_{ps}\right)=0. $$ Einstein's field equations for the vacuum read $$ G_{ab} + \lambda g_{ab}=0. $$

Nash's formula seems a lot more complicated, but my knowledge of general relativity ends here. Does anybody know whether similar modifications of Einstein's field equations have been considered in the literature? Is there any evidence that Nash's suggestion might be an interesting modification worth looking into more deeply?


1 Answer 1


Is Nash's equation interesting? That is a matter of taste, but objectively I can say that his equations have (independently) interested physicists in the recent past.

The equation of motion in your question originates from an action with higher-derivatives and without the usual Einstein-Hilbert action: $$ S = \int d^4 x \sqrt{-g}\left[2 R^{\mu\nu}R_{\mu\nu} - R^2\right] $$ This form of action interested Stelle (amongst others) in the 1970s. Stelle showed that such actions were re-normalizable. Their renormalizability stems from the fact that the equations of motion contain higher-derivatives. Those higher-derviatives are a double-edged sword, however, because they also introduce so-called "ghosts".

At the classical level, the Hamiltonian is unstable because of the Ostrogradsky instability. There are negative energy states that interact with positive energy states. This results in unstable runaway behaviour, where, for example, fields develop large positive and negative energies (that cancel).

In a quantum theory, higher-derivative interactions result (after a change in variables) in "ghost" fields - fields with negative norm. These negative norms result in negative probabilities and possibly a breakdown in unitarity, though there have been attempts to rehabilitate negative norm states and "live with ghosts."

Recently, Strumia and Salvio revived Stelle's work in their "agravity" theory. They were motivated to consider this action because of a principle of "classical scale invariance." With this principle, we write all dimension four operators (and nothing else). This omits the Einstein-Hilbert action, but keeps Stelle's higher derivative gravity. On the thorny issue of ghosts, they write that:

Sometimes in physics we have the right equations before having their right interpretation. In such cases the strategy that pays is: proceed with faith [11], explore where the computations lead [12], if the direction is right the problems will disappear [13].

Not everyone shares their viewpoint here. For example, Smolin remarks:

... it is just the old Kelly Stelle theory from 1977, which has been known since that time to be non-unitary ... the authors admit they have nothing to add to these issues. If this was the right answer, quantum gravity would have been solved long before string theory and LQG were even invented.

I'm not sure how the story of higher-derivative gravity will end - perhaps Strumia and Salvio will spark new work on this topic and the problems will dissappear, or perhaps the problems are truly intractable. Either way, Nash's notes are a curious footnote in the history of this idea.


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