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Unformatted text preview: any Lagrangian invariant under general
coordinate diffeomorhpisms, the same group under which GR
is invariant. Thus general coordinate invariance cannot be the
deﬁning feature of GR.
Similarly, the principle of equivalence, which demands
that all mass and energy gravitate with the same strength,
is not unique to GR. It can be satisﬁed even in scalar ﬁeld
theories, if one chooses the interactions properly. For example, this can be achieved by iteratively coupling a canonical massless scalar to its own energy momentum tensor. Such
a theory in fact solves all the problems Einstein set out to
solve; it provides a universally attractive force which conforms to the principles of special relativity, reduces to
Newtonian gravity in the nonrelativistic limit, and satisﬁes
the equivalence principle.1 By introducing diffeomorphism
invariance via the Stuckelberg trick, it can even be made to
satisfy the principle of general coordinate invariance.
The real underlying principle of GR has nothing to do with
coordinate invariance or equivalence principles or geometry,
rather it is the statement: General relativity is the theory of a
nontrivially interacting massless helicity 2 particle. The other
properties are consequences of this statement, and the implication cannot be reversed.
As a quantum theory, GR is not UV complete. It must be
treated as an effective ﬁeld theory valid at energies up to a
cutoff at the Planck mass MP , beyond which unknown high
energy effects will correct the Einstein-Hilbert action. For a
given background such as the spherical solution around a
heavy source of mass M such as the Sun, GR has three
distinct regimes. There is a classical linear regime, where
both nonlinear effects and quantum effects can be ignored.
This is the regime in which r is greater than the
Schwarzschild radius r > rS $ M=MP . For M the mass of
the Sun, we have rS $ 1 km, so the classical linear approximation is good nearly everywhere in the Solar System. There
is the quantum regime r < 1=MP , very near the singularity of
the black hole, where the effective ﬁeld theory description
breaks down. Most importantly, there is a well-separated
1 This theory is sometimes known as the Einstein-Fokker theory,
ﬁrst introduced in 1913 by Nordstrom (1913a, 1913b), and later in a
different form (Freund and Nambu, 1968; Deser and Halpern,
1970). It was even studied by Einstein when he was searching for
a relativistic theory of gravity that embodied the equivalence
principle (Einstein and Fokker, 1914). Kurt Hinterbichler: Theoretical aspects of massive gravity middle ground, a classical nonlinear regime 1=MP < r < rS ,
where nonlinearities can be summed up without worrying
about quantum corrections, the regime which can be used to
make controlled statements about what is going on inside a
black hole. One of the challenges of adding a mass to the
graviton, or any modiﬁcation of gravity, is to retain calculable
yet interesting regimes such as this....
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