### General relativity is not a field theory

There is a myth, perpetuated in physics textbooks and popular treatises, that states that general relativity is the theory of a gravitational field. The myth was started by Einstein, who often used the term gravitational field

during the development of the theory. The pioneers make a lot of mistakes because the territory is unknown, but there is no reason to continue perpetuating myths a century after Albert Einstein, Marcel Grossmann, and David Hilbert developed the theory of general relativity. Let me quote a recent tweet from cosmologist Will Kinney: in general relativity, the gravitational field doesn’t really exist

.

Understanding that general relativity is not a field theory is not only desirable for reasons of rigor and consistency, but has profound implications for research. For example, the sad status of quantum gravity research is partially because some physicists such as Feynman and Weinberg have tried to apply quantum field theoretic methods to a theory in which there is no gravitational field, and the result has been disastrous.

General relativity is a metric theory, where most gravitational interactions are described in geometrical terms as a consequence of the curvature of spacetime. I wrote most

, because self-gravitation defies a geometric description and has to be incorporated into general relativity by mean of the so-called self-gravitational forces

[1,2].

The popular form of the Hilbert-Einstein equations is

\[ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \]On the left-hand side, the Einstein tensor \( G_{\mu\nu} \) describes the curvature of spacetime in terms of the Ricci curvature tensor and the scalar curvature, \( g_{\mu\nu} \) is the metric tensor, and \( \Lambda \) is the cosmological constant. \( T_{\mu\nu} \) on the right-hand side of the equation is usually called the energy-momentum tensor. Those equations, one for each \( \mu\nu \) pair of indices, are sometimes called Einstein's field equations. A misleading name, and not just because they do not describe any gravitational field, but because Hilbert derived the equations first. The cosmological constant term can be moved to the right hand side of the equation and absorbed into the \( T_{\mu\nu} \)term. For the purposes of the current article, we will set the constant to zero.

Using the Hilbert gauge, the above equations can be rewritten as

\[ \square h_{\mu\nu}(\boldsymbol{x},t) = - \frac{16 \pi G}{c^4} \tau_{\mu\nu}(\boldsymbol{x},t) \]Here \( h_{\mu\nu} \) describes a deviation from a flat spacetime with metric \( \eta_{\mu\nu} \); that is, \( g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} \). This form of the equations of general relativity are often called the relaxed form.

The corresponding equation for a gravitational wave would be

\[ \square h_{\mu\nu}(\boldsymbol{x},t) = - \frac{16 \pi G}{c^4} \tau_{\mu\nu}(\boldsymbol{x},t) \]They appear to be the same equation, and this is the source of much confusion in the physics literature, but they are actually very different equations. Let us use a different notation for the general relativistic equation before discussing the differences

\[ \hat{\square} \hat{h}_{\mu\nu}(\hat{\boldsymbol{x}},\hat{t}) = - \frac{16 \pi G}{c^4} \hat{\tau}_{\mu\nu}(\hat{\boldsymbol{x}},\hat{t}) \]The term \( \square = \eta_{\mu\nu} \partial^\mu \partial^\nu \) is the wave operator (mathematicians call it the d'Alembert operator) and is a real derivative operator because it acts on a flat spacetime. \( \hat{\square} \) in general relativity is not a wave operator; it is not even a proper derivative operator, because only covariant derivatives \( \nabla^\mu \) are true spacetime derivatives in general relativity. Only when spacetime is flat \( \nabla^\mu = \partial^\mu \) and \( \hat{\square} \) is a true wave operator. This is the first difference between both equations.

The second difference between gravitational field theory and general relativity is that \( \hat{h}_{\mu\nu} \) is not the potential of a field. Not only does this quantity not satisfy the requirements associated with a field, but \( \hat{h}_{\mu\nu} \) is not even a physical quantity in general relativity. The only geometric element with physical meaning is the spacetime metric \( g_{\mu\nu} \). In the words of Norbert Straumann, the flat Minkowski spacetime becomes a kind of unobservable ether

[3] and, as a consequence, \( \hat{h}_{\mu\nu} \) is equally unobservable.

A third related difference is the pair \( (\hat{\boldsymbol{x}},\hat{t}) \). Those are coordinates associated with curved spacetime in general relativity, while \( (\boldsymbol{x},t) \) in the gravitational field theory are the coordinates associated with flat spacetime. General relativity describes motion in a curved spacetime, the field theory of gravity describes the same motion as being caused in flat spacetime by a force associated with the gravitational field.

*Gravity as warped spacetime versus gravity as force applied to the interaction between the Sun and an arbitrary body. The representation using the distorted rubber sheet is typical but misleading for two reasons: first, because general relativity assumes that spacetime is curved and not just space; second, because curvature is intrinsic, spacetime curves in on itself, whereas the rubber sheet analogy represents the curvature of a two-dimensional object in a third dimension.*

We can start analyzing the right hand side of the equations. \( \tau_{\mu\nu} \) is a true tensor, but \( \hat{\tau}_{\mu\nu} \) is not. It is called a pseudotensor in the literature. \( T_{\mu\nu} \) in the original Hilbert-Einstein equations is a proper tensor, defined in a curved spacetime. \( G_{\mu\nu} \) is also a tensor, but to obtain the relaxed form of the equations we must split this tensor into a linear term \( G_{\mu\nu}^{(L)} \) plus nonlinear corrections \( G_{\mu\nu}^{(NL)} \) to it. Those corrections have to be moved to the right hand side of the equations and absorbed into \( \hat{\tau}_{\mu\nu} \) which looses its tensor character.

There is one last difference: the energy momentum tensor in the field theory of gravitation includes the energy-momentum of the gravitational field \( \tau_{\mu\nu} = \tau_{\mu\nu}^{(matter)} + \tau_{\mu\nu}^{(gravit)} \), but there is no similar term in \( \hat{\tau}_{\mu\nu} \). Physicists have tried for decades to identify some component of \( \hat{\tau}_{\mu\nu} \) with a gravitational field, but even ignoring its pseudotensorial character, the chosen terms predict the wrong sign for the energy.

## REFERENCES AND NOTES

- A Rigorous Derivation of Gravitational Self-force 2008: Class. Quantum Grav. 25(20), 205009, 1–33. Gralla, Samuel E.; Wald, Robert M.
- A Rigorous Derivation of Gravitational Self-force (Corrigendum) 2011: Class. Quantum Grav. 28(15), 159501, 1. Gralla, Samuel E.; Wald, Robert M.
- General Relativity; Second Edition
**2013:***Springer Science+Business Media; Dordrecht*. Straumann, Norbert.