# Why is Brans-Dicke Theory considered as a failed attempt to incorporate Mach’s principle in a relativistic theory of gravity?

In Generalized Brans-Dicke theory: A dynamical systems analysis by Nandan Roy and Narayan Banerjee, Brans-Dicke theory is described as a failed attempt to incorporate Mach’s principle in a relativistic theory of gravity. How did it fail?

Roy and Banerjee seem to be trying to be intentionally provocative with the first sentence of their paper:

Brans-Dicke (BD) theory[1], a failed attempt to incorporate Mach’s principle in a relativistic theory of gravity,[...]

The following text doesn't clarify what they mean. There are several ways that this could be interpreted:

1. When Brans and Dicke originally proposed the theory, they said that there was no reason for the dimensionless constant $$\omega$$ to have any particular value, so the only reasonable thing to assume was that $$\omega\sim1$$. On the other hand, $$\omega\rightarrow\infty$$ reproduces GR. Since solar system tests force $$\omega$$ to be large, BD gravity could be said to be a failure on its own terms -- the terms stated by Brans and Dicke.

2. We could read this sentence as a claim that BD gravity doesn't fully incorporate Mach's principle. To me, this would seem off base, because Mach's principle was never a well-defined thing in Mach's formulation, and in fact BD gravity gave it a more clear-cut embodiment in which it could actually be tested.

3. They claim that there are problems with the $$\omega\rightarrow\infty$$ limit, so maybe this is why they call it a failure. They reference their own 1997 paper, which I couldn't find, and also this one: https://arxiv.org/abs/gr-qc/9902083

Brans-Dickie theory, can not be yet declared failed. Here is a very readable article on it. The evidence of variable G is low (it doesn't couple with matter).