Making GR Weyl Invariant: Attempts and Possibilities

[kiki_danc]

General relativity is not invariant under Weyl transformations.
What would it take to make it Weyl invariant? And what are all the previous attempts to make it such (can you enumerate them)?

 

[Ambitwistor]

I would like to clarify that general relativity is a highly successful theory that explains the behavior of gravity and the structure of spacetime. However, like any scientific theory, it is not perfect and there are ongoing efforts to improve and refine it.

One of the limitations of general relativity is that it is not invariant under Weyl transformations. These transformations involve scaling the metric tensor (which describes the geometry of spacetime) by a positive function, and they do not preserve the physical distances between objects. This means that the theory is not fully consistent with the principle of relativity, which states that the laws of physics should be the same for all observers regardless of their relative motion.

To make general relativity Weyl invariant, one approach is to introduce an additional field, known as the Weyl field, which transforms under Weyl transformations in a specific way to cancel out the effects of the transformation on the metric. This approach has been explored in the 1920s by Paul Dirac and later by Hermann Weyl himself, but it has not been widely accepted due to various theoretical and conceptual issues.

Another approach is to modify the theory of gravity itself, such as by introducing additional terms in the gravitational action that are invariant under Weyl transformations. This has been attempted in the 1980s by Steven Weinberg and in the 1990s by Sergei Odintsov, but these attempts have not been fully successful in achieving Weyl invariance.

More recently, there has been renewed interest in making general relativity Weyl invariant in the context of quantum gravity, as it has been suggested that Weyl invariance may be a fundamental symmetry of the underlying theory. However, this is still an active area of research and no definitive solution has been found yet.

In summary, there have been various attempts to make general relativity Weyl invariant, including the introduction of a Weyl field and modifications to the gravitational action. However, none of these attempts have been fully successful, and the search for a Weyl invariant theory of gravity continues.